Low frequency oscillations in turbulent Rayleigh-Benard convection: laboratory experiments

FLUID DYNAMICS

RESEARCH ELSEVIER

Fluid Dynamics Research 16 (1995) 87-108

Low frequency oscillations in turbulent Rayleigh-Benard convection: laboratory experiments Ruby Krishnamurti Florida State University, Tallahassee, FL 32306-3017, USA Received 25 August 1992; revised version received 25 November 1994

Abstract This is a report on observations of low frequency oscillations in turbulent Rayleigh-Benard convection, in the Rayleigh number (R) range 107-108, with Prandtl number (Pr) equal to 7. It has been known that for convecting layers with large aspect ratio A, a steady large scale flow sets in at R = 2 x 106. Tilted transient plumes embedded in this flow, and maintaining it through Reynolds stresses, drift in one direction along the bottom of the layer, and in the opposite direction along the top. At a fixed point near the bottom or top boundary, there is a variability associated with the passage of these plumes. We call this the high frequency variability. A new kind of organization is observed for 107 < R < 108; clusters of transient tilted plumes travel in a horizontal direction as coherent units. These clusters are separated from each other by quiescent zones with almost no plumes. Now at a fixed point near the bottom boundary, there is a low frequency variability associated with the passage of clusters, as well as the high frequency variability from the passage of plumes within the cluster. Quantitative information on this low frequency oscillation derived from space-time portraits and from temperature time series is presented. Heat flux measurements show a Nusselt number (N)-Rayleigh number relationship for 106 < R < 108 which is hysteretic for A = 12, but when the convecting layer is partitioned into 144 cells each with A = 1, hysteresis is not present and the N - R relationship then agrees with earlier results for single cells with A = 1.

1. Introduction Large fluid systems such as the atmosphere of the Earth are no doubt turbulent but have in them considerable organization on many scales, a fact which is often used as an aid to prediction. Not all of these scales are directly forced; some arise as a result of internal rearrangements of energy and momentum, giving rise to such phenomena as the atmospheric jet streams. This paper is about an investigation of large scale organization that arises in turbulent Rayleigh-Benard convection. We know that in certain parameter ranges a steady large scale flow is generated in turbulent convection through the Reynolds stresses associated with tilted plumes. For Prandtl number Pr = 7, Rayleigh number R > 2 x 10 6, and large aspect ratio A = 10 to 25, (where A is the ratio of width to depth of the convecting fluid layer) a steady large scale circulation sets in, 0169-5983/95/$06.25 © 1995 The Japan Society of Fluid Mechanics and Elsevier Science B.V. All rights reserved SSDI 0 1 6 9 - 5 9 8 3 ( 9 5 ) 0 0 0 5 6 - 5

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R. Krishnamurti / Fluid Dynamics Research 16 (1995)87 108

embedded throughout with tilted transient plumes. This large scale flow is in one direction along the bottom and in the opposite direction along the top of the layer (Krishnamurti and Howard, 1981). All the plumes become organized with approximately the same tilt, the hot plumes from the bottom boundary drifting in one direction with the large scale flow, the cold plumes from the top drifting in the opposite direction. A measurement of the turbulent velocities and their correlations showed that (Krishnamurti and Howard, 1983) the Reynolds stress divergence balances the viscous force on the large scale (horizontally averaged) flow ti: 0 __

t~2tl

-~zUW = V Oz 2 .

(1)

The first integral of Eq. (1) is uw -

V~z = Mo,

(2)

where Mo is a constant independent of z and may be taken as the m o m e n t u m flux at the boundary. The tilt of the plumes is such that the m o m e n t u m transport by the Reynolds stress u w is counter to the down-gradient transport by the viscous stress on the mean flow. In a low order mathematical model (Howard and Krishnamurti, 1986) it was shown that tilted flows can result from a symmetry breaking bifurcation from upright cellular flow to one that tilts either to the right or to the left. This present report is on a new kind of organization in the form of travelling clusters of tilted plumes that results locally in long-period oscillations. The report consists of(i) space-time portraits of the flow, (ii) analyses of internal temperature time-series, and (iii) heat flux measurements, all in the Rayleigh number range 106-10 s, with Prandtl number of 7. The first two types of observations give quantitative information about the frequencies, length scales and speeds of the clusters. These observations will show that there are typical 5 to 10 transient tilted plumes in a cluster. Along a direction of travel of these clusters we see at any instant of time an alternating pattern of clusters of plumes, next to a quiescent region with almost no plumes, next to another cluster of plumes, etc. Along a 4 ft. line spanning the layer, there may be at any instant 3 to 4 clusters of plumes, each separated from the next by quiescent zones. At a fixed point near the bottom boundary, there is a high frequency (with periods on the order of 10 seconds) variation in temperature as the plumes within the cluster pass. The cluster passes in order 102 seconds. The quiescent epoch also lasts on the order of 102 seconds. Thus at a fixed point there is the high frequency variability associated with plume passage and the low frequency variability associated with cluster passage. The data will show how these periods vary with Rayleigh and Nusselt numbers. Tests will also be described to convince ourselves that this low frequency oscillation was not inadvertently forced. Finally comparison is made with previous high Rayleigh numbers and small aspect ratio studies. With water we are not able to have the tremendous range in Rayleigh number as in the low temperature gaseous helium experiments (Castaing et al., I989) but there is some overlap in R although Pr differs. We also compare our results with those of Tanaka and Miyata (1980) and of Zocchi et al. (1990) whose experiments were conducted in water, with aspect ratio unity, primarily at R = 1.2 x 10 9, but with some observations at R = 2.4 x 10 8.

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2. Apparatus

A schematic diagram of the apparatus is shown in Fig. la. Starting at the b o t t o m is the heater which is a 122 cm by 122 cm mat of insulated resistance wire woven on a loom. Care was taken in its construction to assure fixed length of resistance wire per unit area of mat. This mat is attached to a 2.54 cm thick 122 cm by 122 cm aluminum tooling plate labeled 1 in Fig. la. Its temperature will be designated T1. All the aluminum tooling plates used in the apparatus are flat and of uniform thickness to a tolerance of 0.005 cm in 122 cm. Above this plate is an air layer which is so thin that it never convects for any temperature gradients imposed in our study. It was used as heat flux meter in other experiments but is not relevant to the experiments described here. Above the air layer is plate number 2 (at temperature T2) which is a 5 cm thick 122 cm by 122 cm aluminum tooling plate. Above plate 2 is the working fluid layer. It is a layer of water 122 cm by 122 cm in the horizontal, and of depth d which was set at 5.000 cm, 7.500 cm or 10.160 cm. All the data shown here will be for d = 10.160 cm. The side walls for containing this water were made of 5 cm thick Plexiglas bolted with O-ring seals to plate number 2. Above the working fluid layer is plate number 3 (at temperature T3). This is another 122 cm by 122 cm by 5 cm thick aluminum tooling plate.

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Fig. 1. Schematic diagram of apparatus, showing two arrangements of the boundary plates; (a) is the normal arrangement for most of the experiments, (b) is one of the test arrangements (see text).

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90

Above plate 3 is a layer of water of thickness 6, which acts as the heat flux meter. Its thickness was chosen sufficiently small so that it never convected for any temperature difference imposed in these experiments. Since this layer is in the conduction state the heat flux H is given by k H = -~(T3-T4), O

(3)

where k is the molecular thermal conductivity of water, T4 is the temperature of aluminum tooling plate 4 which sits above this water layer. Under externally steady conditions, i.e., T1, Tz, T3, T4 steady in time, H is also the heat flux through the convecting layer, neglecting losses out the side walls. This method of measuring heat flux was first devised by Malkus (1954) and has been used extensively (e.g., Krishnamurti, 1970a; b 1970b; 1973). The idea is that the aluminum plates, having high thermal conductivity (conductivity of alcoa aluminum tooling plate 7000 series is 0.33, of water is 1.4 x 10- 3 of Plexiglas is 5 × 10 -4 cal gm- 1 °C- 1 c m - 1), will have correspondingly small vertical temperature gradient. Thus it is impractical to measure heat fluxes by gradients in these plates. However, temperatures of the metal plates is easily measured accurately, as the placement of the probe in the plate is not crucial. By inserting a dummy layer of low conductivity between two high conductivity plates, Malkus was able to amplify the temperature gradient and thus measure the heat flux in terms of the known conductivity of this dummy layer. As explained earlier, (Krishnamurti, 1970a) the measurement becomes more accurate as the aspect ratio of the apparatus becomes larger. It is also important to operate experiments with the mean temperature at room temperature to minimize this lateral heat exchange. For example, operating at Nusselt number around 20, the lateral heat flux through the 2 inch Plexiglas is less than 1% of the vertical heat flux. To measure heat flux by the electrical power input to the heater has variable errors due to loss of heat downwards and out of the apparatus, in spite of our best attempts to insulate it. Losses out of the side walls become much less than this downwards loss when the aspect ratio is large. The top plate (number 4, at temperature T4) is made of 122 cm by 122 cm by 5 cm thick aluminum tooling plate. Long channels of semicircular cross-section were cut into the top of this plate and 1.91 cm (inside diameter) copper pipes were embedded in these channels. The pipes are spaced as in a counter-current heat exchanger; those pipes carrying the incoming coolant water are placed as close as possible (0.476 cm separation) to those carrying the returning coolant water so that heat conduction between the cold incoming and warmed returning flows removes much of the horizontal temperature gradient that might otherwise result. Such pairs of pipes then lie parallel to the next pair with a 2.54 cm separation between pairs. Fig. lb shows a different arrangement of the plates. This was used to test the importance or lack of importance of the periodic arrangement of the cooling pipes. This will be discussed in the observations. The apparatus sits upon a table which is supported by three stainless steel pillars of 10.16 cm diameter. These pillars contain a threaded shaft which could be turned with a long cross bar, as in a capstan screw. By turning these, the apparatus could be leveled to __+2 x 10 -5 radians.

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91

Finally this entire apparatus was surrounded by a 5 cm thick layer of Styrofoam and then a 8.9 cm thick batting of fiberglass insulation, except for holes for laser beams to enter and a slit through which the laser beam was viewed and photographed. Heating was provided by a constant voltage power supply. Maximum usage was approximately 1kilowatt. Cooling was accomplished by circulating cold water at a set temperature through the pipes in plate 4. The flow rate was 189 litres per minute so that at the highest power input, the outflow was at most 0.07 °C warmer than the inflow. At this highest power input, three refrigeration units worked constantly to maintain the cooling water temperature. There was also a heater with a fine control of 0.05°C in the cooling system to maintain the desired temperature. The temperatures of the aluminum blocks were measured using copper-constantan thermocouples embedded in the blocks. The placement of these thermocouples in the blocks is not crucial since, even at the highest heat flux, the vertical temperature gradient in the aluminum was approximately 10-2 °C cm-1, while 10-2 °C is comparable to the precision attainable from the amplified thermal emf. Thus from Tz-T3 we can determine the Rayleigh number, and from T3-T4 the heat flux. There is another thermocouple Tib which measures internal temperature at a point in the working fluid, 1 to 2 m m above the bottom boundary. A data acquisition system was used to record Tib once per second for durations of 4 hours, for the purpose of studying power spectra of these temperature time series. The experiments were later repeated with a thermocouple placed at mid-depth in the fluid layer, giving Tim once per second for 4 hours for the purpose of studying probability distributions as well as power spectra. The space-time portraits of the flow were obtained with the following equipment. Two 3 milliwatt He/Ne lasers with their beams anti-parallel and overlapping illuminated tracer particles in a line of fluid from two opposite directions. (We call this the x-direction.) This line was usually chosen to lie parallel to the bottom boundary and 2 to 4 mm above it. (In fact, the thermocouple measuring Tib was usually just below this beam.) With the beam so fixed in space, a camera with its optic axis perpendicular to the beam (call this the y-direction) recorded the flow in time; 70 mm photographic film was slowly wound onto its take-up reel while constantly exposing the film, thus producing an (x, t) record where t is the time. Exposures usually lasted 4 hours, coinciding with the recording of Tib. Time markers were placed on the film at fixed intervals. These were useful since the reel was wound at constant angular speed so that distance along the film does not remain linearly proportional to the time. Even without these, however, time could be deduced from the location on the roll of film. The process could be repeated with the beam at different depths and at different y. For (y, t) records, the lasers were moved, and a second 35 m m camera with its optic axis parallel to x recorded the y-time evolution of the flow. Another type of boundary could be introduced into the apparatus. These were vertical walls of 0.635 cm Plexiglas which would partition the fluid layer into many small cells. Equally spaced parallel strips of Plexiglas were interlocked with another set of equally spaced parallel strips at right angles to the first set to make a mesh resembling an eggcrate. For example, one set divided the working fluid, originally 122 cm by 122 cm by 10.16 cm deep, into 16 cells each 30.48 cm by 30.48 cm by 10.16 cm deep for an aspect ratio A = 3. Another set of walls divided the working layer into 144 cubical cells each 10.16 cm on a side making A = 1. The former we call "ostrich eggcrate" and the latter "goose eggcrate".

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R. Krishnamurti / Fluid Dynamics Research 16 (1995) 87-108

Finally, as each of the 5 cm thick aluminum plates weighed 211 kg, chain-hoists and moveable A-frames played an important role in assembly, cleaning and re-assembly.

3. Experimental Procedure The heater voltage control was put at its lowest setting and the cooling system adjusted so that the mean temperature of the working fluid remained at 21°C. The temperature of the four aluminum blocks reached nearly steady values in 4 to 6 hours, but usually more time elapsed before their temperature changes became no more than the noise level of the amplifiers; usually their values were recorded after one day at the fixed setting of heater and cooler. After this external steady state was reached, an x, t photograph and/or a y, t photograph and simultaneous recording of the temperature time series Tib o r Tim were made. The heater setting was then increased by a small fraction of its total range, the cooler adjusted to maintain mean temperature, and the entire procedure above repeated. Thus at each setting of the heater, we have a determination of heat flux and Rayleigh number, as well as time series of internal temperature, and an x, t portrait of the flow. The experiment was never shut down until the end of the range (the highest heating rate attainable with the present apparatus) was reached. In some studies of hysteresis, the experiment was not shut down until after going in small steps to the top of the range, then back down in small steps to the bottom of the range. A power failure in this months-long process meant dismay. Fig. 2 illustrates how x, t portraits were made and what information they contain. Fig. 2a show streamlines in a vertical plane intersecting steady convecting rolls. The laser beam near the bottom boundary illuminates tracers in a band parallel to the x-direction. Tracers enter this beam and move from left to right in one cell, and from right to left in the adjacent cell. They exit the illuminated region as they rise upwards in their cells. A camera photographs this illuminated region of fluid as it evolves in time by placing the image on a continuously moving film. The x, t photograph that would result from this steady cellular flow is shown in Fig. 2b. Tracer particles in adjacent cells make a herringbone pattern. From the slopes of these particle tracks, the x-component of the speed can be determined. In steady flow, the cell boundaries, where there is no x-component of flow, remain fixed, leading to the straight, vertical line on the x, t portrait. Fig. 2c shows an actual x, t photograph for R -~ 104. The tracer used was Rheoscopic Fluid AQ 1000 from Kalliroscope Corporation, used at a dilution of tens ofml in the 150 1 of water, x, t photographs are most useful in detecting time dependence. Fig. 2d shows an example for R = 10 6, Pr = 860. Here the flow is still cellular, although cell boundaries oscillate and drift with time. The main feature in this regime is the nearly periodically occurring bright regions (i.e., regions of differential shear) that move horizontally from one cell boundary to the next in the direction of the cellular flow. These were identified as thermal plumes (Krishnamurti, 1970b; 1973) that originate in the thermal boundary layer. Similar behavior has been observed for Prandtl number from 7 to 103. The clearest and most persistent examples were found for Pr -~ 10 2 t o 10 3. Since these plumes take several minutes to move from one cell boundary to the next, their motion is hard to recognize without the x, t record. The speed of film motion was picked to best display the phenomenon of interest. As the tracers would settle out, about 5 ml of tracer was added every few days. This was done very carefully so as not to disturb the convection. Usually the tracer would slowly settle on the

R. Krishnamurti / Fluid Dynamics Research 16 (1995) 87 108

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d Fig. 2. x, t portraits of cellular flows. The total extent in x, Ax and the total time duration At are given with each photograph; (a) schematic diagram of streamlines and laser beam, (b) schematic (x, t) portrait, (c) steady cellular flow, R = 2 × 1 0 4 , P r = 5 7 , A x = 1 8 c m , A t = 10.7minutes, (d)cells with plumes, R = 1 0 6 , P r = 8 6 0 , A x = 4 0 c m , At = 10.7 minutes.

94

R. Krishnamurti / Fluid Dynamics Research 16 (1995) 87 108

bottom boundary near a side wall where it was introduced. The convective flow would then carry the tracer throughout. Different ways of introducing the tracer were tried (e.g. slowly, as described above or quickly, introducing 5 ml at once) but no difference in x, t portraits could be seen to result. Internal temperature times series were sometimes obtained in distilled water without tracers as a check by comparison with those obtained with tracers. This was usually done when the tank was cleaned and fresh water added.

4. Observations 4.1. The x, t portraits

In Fig. 3 we show x, t portraits of non-cellular flows. These occur for Pr = 7 and R > 10 -6. Although only a few hours duration out of the hundreds of hours of x, t photography is shown here, the features described below are typical of that Rayleigh number regime. The others not shown differ in that the large scale flow may be from left to right or from right to left. Some were x, t portraits at various values of y, some were y, t portraits at various x. They showed the same general features but were useful in determining parameter dependence of the low frequency oscillations, described below. 106 < R < 2 x 106: Random plumes, no large scale flow. Fig. 3a shows an x, t portrait typical of this narrow Rayleigh number range. There are no cell boundaries, plumes appear now and then, here and there, and they show no organized drift. 2 x 106 < R < 107: Steady large scale flow with randomly occurring plumes throughout. Fig. 3b shows a typical x, t portrait. Each randomly occurring plume drifts in the same direction across the tank. This is the regime of the steady large scale flow described by Krishnamurti and Howard (1981; 1983). Fig. 3b shows x, t photographs of this flow at various depths. Plumes move steadily from left to right over most of the lower half layer, and from right to left over most of the upper half layer. The point emphasized here is that at mid-depth the right-going and left-going plumes meet in a shearing "collision". As R is increased the efficiency of these "collisions" appears to increase so that x, t photographs at mid-depth show little structure. However, this efficient collision picture will be seen to be in agreement with the narrowness of the probability density distributions of the mid-depth temperature (Fig. 6 below). 107 < R < 2 x 108: Low frequency oscillations. Throughout this range, the x, t photographs, of which Fig. 3c is typical, show a periodic repetition in time with a narrow dark region that moves steadily at speed Uo in the x direction, followed by several bright plumes also travelling steadily in x at the same speed, followed again by a narrow dark region, and so on. At a fixed time, 5 to 10 plumes can be seen across the bright strip. These are the plumes of a cluster. At a fixed point Xo there is an alternation between the dark region (the quiescent epoch) and the bright region with many plumes (the active epoch). The period varies weakly with AT, definitely depends upon the layer depth d, and is of the order of 3 to 6 minutes in these experiments. The oscillation persists for many periods before there is any appreciable phase shift. As R is lowered to 107 from above, occasional oscillations are seen but these are less persistent and are coherent for fewer cycles. The disappearance of the periodic behavior is gradual with R decreased towards 107 .

R. Krishnamurti / Fluid Dynamics Research 16 (1995) 87 108

95

As with the steady large scale flow, the direction of travel of the dark region and of the cluster of bright plumes may be, in a given experimental realization, from left to right or right to left (in the x-direction) at the bottom of the layer (and in the opposite direction at the top), or it may be from front to back or back to front (in the y-direction). In some x, t photographs, travel was in the x-direction for part of the 50 cm width and in the y-direction for the remainder of this width. That is,

a

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b

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t Fig. 3. x, t portraits of non-cellular flows ( n o t a t i o n as in Fig. 2). (a) Showing only plumes, no cell boundaries, a n d n o large scale flow. R = 1.5 × 106, P r = 7, Ax = 48 cm, At = 61.6 minutes; laser b e a m at z/d = 0.03. (b) Showing steady large scale flow with drifting plumes. R = 7.2 x 106, P r = 7, Ax = 48 cm, At = 61.6 minutes, z/d = 0.03. (bb) Showing the vertical variation of steady large scale flow. R = 7.2 x 106, P r = 7, Ax = 12 cm, At = 9.0 minutes; from left to right, z/d = 0.05, 0.2, 0.5, 0.7, 0.97, the p h o t o g r a p h s were t a k e n sequentially, not simultaneously. (c) Showing the periodic arrival of clusters of plumes at station Xo. The alternating bright a n d dark b a n d s are indicated by arrows. The bright b a n d s consist of m a n y plumes; at a fixed time 5 to 10 plumes can be seen across the bright b a n d which is the steadily m o v i n g cluster (see text). R = 6.2 x l0 T, P r = 7, Ax = 48 cm, At = 61.6 minutes, z/d = 0.03. O n the right is a n enlargement of a p o r t i o n of this p h o t o g r a p h .

R. Krishnamurti / Fluid Dynamics Research 16 (1995) 87 108

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R. Krishnamurti / Fluid Dynamics Research 16 (1995) 87 108

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the direction of travel was not necessarily parallel to the side walls. The direction of travel seems to depend on the initial conditions at the time of onset of the large scale flow. Because of the way the experiments were performed, starting at one R and changing R in small steps, never turning off the experiment until the completion of the range, the direction of travel was the same for the whole range of R. In another realization it was in another direction. Also, as with the steady large scale flow, the travelling plumes are tilted away from the vertical, with the bases of the plumes leading and the tops trailing in their horizontal motion, so that the Reynolds stress uw is non-zero and has the same sign for all the plumes. Thus as with the steady large scale flow, the vertical transport of horizontal momentum would be up the gradient of mean horizontal velocity. The balance must be different from that of Eq. (1) since there may be now a local time derivative ~u/t. When the periodic behavior was first noticed on the x, t photographs, an explanation was sought from any periodic external forcing. Heating and cooling systems were either on constantly or were switching on and off with a period of tens of seconds, not the 3 to 6 minutes of the observed flow. Nevertheless, refrigeration units were moved away from the convection tank and care was taken to direct the hot exhaust out of the area to reduce any mean gradients in room temperature. However, could a large scale flow past periodically placed cooling pipes be the source of this periodic behavior? It seemed unlikely since both x-periodic and y-periodic flows were observed with the same orientation of the cooling pipes. However, to test this idea the apparatus was arranged as shown in Fig. lb. The lower water layer of depth 5 cm was bounded above as well as below by inert 5 cm thick aluminum plates. The upper water layer, also of depth 5 cm, was bounded below by the inert 5 cm aluminum plate, but its upper boundary was the 5 cm aluminum cooling plate with the embedded cooling pipes. Time-periodic x, t portraits were obtained in each layer, but the directions of travel of the clusters of plumes were not in any obvious relationship in the two layers. The bottom layer had x-directed flow, where the upper layer had approximately y-directed flow. It was concluded that the periodic arrangement of the cooling pipes was not related to the periodic flows observed. In summary, for 107 < R < 108, Pr -- 7, A -- 12, the x, t portraits show that at a fixed point x, there is a low frequency time-periodic variability but travelling in the x direction at a constant speed Uo, there is no variance with time. 4.2. Temperature time series 4.2.1. Power spectra Time series of the temperature Tib at a point in the fluid near the bottom boundary were analyzed and some typical power spectra are shown in Fig. 4. In each plot the frequency range shown along the abscissa is from 0.00 to 0.08 s- 1, but the plot of the spectrum is from 0.002 to 0.08 s- 1 in each case. (Other frequency ranges are not shown here because of scale differences.) However, the power plotted along the ordinate has a different range in each plot. It is to be noted that in the first plot at R = 4.11 x 10 6, in the range of steady large scale flow, there are narrow peaks in the spectrum throughout the range. (The four hours long records are necessary in order to resolve these peaks.) The x, t photographs show plumes passing a fixed point with periods of approximately 20 to 30 seconds. In good agreement, this power spectrum has its largest peaks at 18 and 32 seconds. In the remaining spectra shown in Fig. 4 for R > 107 large

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Fig. 4. Typical power spectra of the time series of temperature Tib. From top to bottom, R = 0.624 x 107, 1.06 x 107, 2.50 × 107. The scale on the abscissa is from 0.00 to 0.08 s- 1 with the plot starting at 0.02 s- 1 for each spectrum, but the scale on the ordinate varies. At R = 0.624 x 107 the largest peaks are at 18 s and 32 s. For R > I × 107 large peaks appear at periods of order 300 s. a m p l i t u d e p e a k s are seen in the low f r e q u e n c y range, c o r r e s p o n d i n g to periods of several minutes. T h e higher frequencies (tens of seconds) are still significant, b u t the largest p o w e r is in the r a n g e of 102 seconds. These spectra clearly illustrate the g e n e r a t i o n of low f r e q u e n c y m o d e s in a g r e e m e n t

R. Krishnamurti/ Fluid Dynamics Research 16 (1995)87 108

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Fig. 5. A plot of oscillation period z versus q, where q = (T2-T3)/(T3-T4) and is inverselyproportional to the Nusselt number. * period derived from x, t photographs; ® period derived from power spectra. with the x, t photographs. Spectra made in water without tracers were not noticeably different fromspectra made in water with tracers. Direct comparison of the temperature trace of Tib when plume passage was over the thermocouple as seen from the x, t photo was attempted with limited success. Correspondence between plume passage and increase in temperature could be followed for several oscillations but the general turbulent nature of the flow and corresponding irregularity of Tib made interpretation difficult. In Fig. 5 we have taken the half-periods z/2 of oscillation as deduced from the x, t portraits as well as from the lowest frequency of the significant peaks in the power spectra, and plotted them against r / = ( T 2 - T 3 ) / ( T 3 - T 4 ) which is inversely proportional to the Nusselt number. The best linear fit is z = 0.801r/+ 1.374. 4.2.2. Frequency distribution From the time series of the mid-depth temperature T i m a few representative frequency distributions are shown by the histograms in Fig. 6a. For each value of R, the 1.44 × 104 measurements of temperatures, taken at one second intervals, were subtracted from the time mean of the series. Then each observation was normalized by dividing by AT appropriate for this value of R. The number of occurrences in a certain fixed temperature interval is shown plotted along the ordinate, against temperature expressed as a percent of AT along the abscissa. The total range is + 10% of AT. Obviously, if hot plumes often reached the mid-depth sensor the number density distribution would be broadened to positive temperatures. Similarly if cold plumes reached the sensor frequently, the distribution would be broadened to negative temperatures. But if hot and cold plumes were efficiently mixed, the distribution would be narrowed around zero temperature. Narrowing is clearly evident in Fig. 6a, as the Rayleigh number is increased. At R = 7 x 107, occurrences of temperatures __ 2% of A T are negligible. The standard deviation in this case is + 0.89% of A T. In general, the standard deviation, expressed in percent of AT, decreased somewhat irregularly from 10% to 0.5% as R was increased from 1 0 6 t o 108. This supports the concept of efficient mixing of

R. Krishnamurti / Fluid Dynamics Research 16 (1995) 87 108

100

hot and cold plumes by shearing collisions as implied by the mid-depth x, t portraits. However, the exponential probability distribution (described below) suggests higher than Gaussian occurrences of hot and cold temperature extremes. Fig. 6b shows some typical probability density functions obtained from the same data. The histograms were normalized by dividing the height of each rectangular bar by n c where N is the total number of observations, c is the bin width, thus making the area of the histogram equal unity. The abscissa has been normalized by dividing by the standard deviation. The nearly linear shapes on the semi-log plot at R = 7 × 1 0 7 shows that the distribution function is reasonably approximtated as a double exponential.

a)

HISTOGRAMOF

HISTOGRAM OF

FREQUENCY-LINEAR P L O T

'6r 0

FREQUENCY-LINEAR P L O T

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26°r

FREQUENCY-LINEAR PLOT

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24°r

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-9.50-6.50-5.50-0.50 2.50 5.50 9 . 5 0 TEMPERATURE e HISTOGRAM OF FREQUENCY-LINEAR P L O T Z3

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TEMPERATURE e

Fig. 6. (a) Frequency distribution of temperature Tim at the mid-depth sensor, showing the decrease in deviation from the mean as R is increased. The abscissa is in percent of AT. Experiment G6, R = 8.8 × 106; G5, R = 2.1 × 107; G4, R = 3.5 × 107; Z3, R = 5.1 x 107; G3, R = 5.3 x 107; Z4, R = 7.2 x 107. (b) L o g a r i t h m o f p r o b a b i l i t y d e n s i t y d i s t r i b u t i o n

plotted against percent of AT divided by the standard deviation. At R = 7 x 107, the approximately linear profiles imply a double exponential probability density function.

R. Krishnamurti / Fluid Dynamics Research 16 (1995)87-108

(b)

d

HISTOGRAM o~.o~o~.~oo~.~

101

HISTOGRAM OF.OOOFF.~O0~.C~

~8

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HISTOGRAM OF LOG OF FREQUENCY 3.4-

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e/oFig. 6. (Continued).

2.3. Heat flux

The Nusselt n u m b e r N, being the ratio of total heat flux to conductive heat flux, is easily determined as N - d(T3-T4) ~(T2-T3)'

where d is the convecting layer depth, 5 the depth of the d u m m y layer between plates 3 and 4. Since water fills both layers, only the depths and not the conductivity enter into the expression for N. The measurement error varies with the range but typically it is on the order of 1%. F o r calculating the Rayleigh number, 0~, K, and v at the m e a n temperature of the layer was used. Although almost all the data was obtained for a layer mean temperature of 21 °C there were a few points with means o f 4- 1 °C difference. For these, the appropriate values of ct, K and v were used.

R. Krishnamurti / Fluid Dynamics Research 16 (1995) 87-108

102

Fig. 7 shows plots of N R versus R. Each curve is a best fit to a power law through the data points. The arrow indicates whether the data are for a sequence of increased R or decreased R. Table 1 summarizes the power "laws" obtained where N = CR ~. The 90% confidence limits on these powers (~) vary but are of the order of ~ _+ 0.007. This was obtained by assuming a Gaussian distribution of the data about the fitted curve. Since the total

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Fig. 7. H e a t flux NR versus Rayleigh n u m b e r R. (a) A s p e c t ratio A -- 12, R decreased, N = 0.259 R 0"246. (b) S h o w i n g hysteresis (A = 12); I R decreased, N = 0.259 R °'2'.6, II R increased, N = 0.428 R °'2°5, III R decreased, N = 2.566 R °'1°9, I V R increased, N = 0.205 R °'253. (c) (i), (ii), and (iii) s h o w i n g the effect of different values of A; AI A = 12, R decreased, N = 0.259 R °'246, AII A = 3, R decreased, N = 0.223 R °'253, AIII A = 3, R increased, N = 0.223 R °'25a, A I V A = 1, R decreased, N - 0.148 R °'279, A V A = 1, R increased, N = 0.176 R °'271

R. Krishnamurti / Fluid Dynamics Research 16 (1995) 87-108

c(i)

210-

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(Continued).

range of R is small, ~ may well be different for a wider range of R, at A = 12. Nevertheless, within this range hysteresis and the effects of changing A are clearly seen. One other interesting feature of the heat flux data might be mentioned. Some of the data points in Fig. 7a are enclosed in rectangular boxes. Each box contains data at a single setting of the heater and cooler. At R "-~ 8.4 x 10 7, the spread is much larger than at R - 6.5 x 10 7. These points were obtained when an external steady state appeared to be attained; temperatures were practically unchanged for a day, more or less. But in a time on the order of a few hours, R along with H would

R. Krishnamurti / Fluid Dynamics Research 16 (1995)87-108

104

A2

A~.A~-/

c(iii) 210-

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R* I0.*-7 Fig. 7. (Continued).

Table 1 Nusselt number (N) Rayleigh number (R) relationship: N = CR ~ Range of R

Sequence of R

C

~

Comments

0.9 x 10 s 106 106-0.9 x 108

decreased increased

106 0.9 × 108

decreased decreased or increased

0.259 0.428 0.206 0.223

0.246 0.205 0.253 0.253

106-0.9 x 108

decreased

0.148

0.279

full tank, A = 12 full tank, A = 12 full tank, A = 12 16 cells (ostrich egg crate), A = 3 144 cells (goose

increased

0.176

0.271

eggcrate), A = 1

change and appear to settle down at another "fixed" R - N R point. Here it would remain for typically 1/2 to 2 days, but then drift in a few hours to another point. N o obvious external cause for this drift could be found, nor could any obvious pattern in the sequence of points be discerned. At first, the hope was that the system would settle into a fixed point, as it did at lower R, but after following it for some weeks it was concluded that the wandering was "random" and not ending.

R. Krishnamurti / Fluid Dynamics Research 16 (1995)87-108

105

Although the spread of data points is somewhat smaller than the separation due to hysteresis, it is supposed that the two are related. Both this wandering and the hysteresis disappeared when the "ostrich eggcrate" partition was in place, and is very small with the "goose eggcrate" partition.

5. Discussion

Clusters of travelling tilted plumes have been observed moving past a fixed point is nearly periodic repetition. Although such regularity seemed strange in a turbulent flow, a "scavenging plume model" will be presented in a following paper which has such a periodic behavior. In this model, prior passage of a plume or cluster of plumes has depleted the thermal boundary layer in a history-dependent way: the boundary layer is very thin where the plume has just passed, but has had time to thicken where the plume passed some time ago. Thus at any m o m e n t there is a thermal boundary layer of varying thickness, and the pressure gradient in it drives a flow towards the thicker regions. Taking this to be slow viscous flow, there is then a shear and a down-gradient m o m e n t u m flux at the wall. This boundary layer is supposed to erupt when and where its thickness reaches some critical value. Then, assuming that the interior m o m e n t u m flux is entirely by Reynolds stresses, we can, by matching interior and boundary layer heat and m o m e n t u m fluxes, show that plumes move away from regions of thick boundary layer at just such a rate that the ever-thickening boundary layer keeps the plume "fed" with buoyant fluid. The periodic recurrence at a fixed position is from the successive passage of clusters of plumes. The predicted period will be shown to be in reasonable agreement with that obtained from the x, t photographs and the power spectra shown in this paper. These clusters of travelling tilted plumes, seen at A = 12, R = 107-108 may not be related to the waves and spirals that Zocchi et al. (1990) observed with water, A -- 1, R = 1.2 × 10 9. In that case the waves seemed to be excited by plumes near a corner of the cell striking from the opposite horizontal boundary. We, however, do not see plumes from one horizontal boundary reach the opposite boundary, at least not in the interior of the tank. There has been in recent years much discussion of "soft" and "hard" turbulence (e.g., Casting et al., 1989). For cells with aspect ratio A = 1.0, and Pr - 1, there is a transition at R ~- 107. Above this Rayleigh number it is found that: (i) there is a quasi-steady large scale flow; (ii) the probability density function of the mid-cell temperature is a double exponential; (iii) the scaling law for N ~ R ~ has ~ 2 These constitute hard turbulence. Siggia (1994), however, points out that for other cells these three criteria do not onset together but occur singly, and that furthermore "the hard" regime is unlikely to be asymptotic". To add to the confusion, we are reporting here for A = 12, Pr = 7, that: --

7"

(i) steady large scale flow sets in at R =- 2 × 106; (ii) the probability density function approaches a double exponential at R -- 7 × 107; (iii) ~ is never 2 for R < 108 (unless the A = 12 tank is partitioned into 144 cells each with A = 1). The N - R relationship with 144 cells of A = 1, Pr = 7 is N = 0.148 R °'28.

106

R. Krishnamurti / Fluid Dynamics Research 16 (1995) 87-108

This is in good agreement with Tanaka and Miyata (1980) who found, for Pr = 6.8 and A = 1.4 to 3.5, that N = 0.145 R °'29. However, many other values of ~ have been reported in the past. In their summary, Goldstein, Chiang and See (1990) cite values of ~ ranging from 0.23 to 0.33 for values of A between 1.5 and 2.5. Since N is the ratio of the total heat flux H to the conductive heat flux kAT/d, N would be proportional to d, and hence to R a/3, if H were independent of d. H independent of d implies, for large d, that the interior does not influence the heat flux; this then would be controlled entirely by processes in the boundary layer. Conversely, if N --~ R ", ~ ~ ½, then the interior does influence the heat flux. Now, the side walls of a small aspect ratio tank would surely have negligible effect on the thin boundary layers of high R (and finite Prandtl number) convection, but the interior flow would certainly be affected by the side walls. These walls would still be unimportant to the heat transfer if = ½, but not otherwise. In the A = 1 cell of water ofZocchi et al. (1990), hot plumes from the bottom boundary reach the top boundary at one corner of the cell, while cold plumes at the opposite corner reach the bottom boundary (perhaps as part of the large scale flow). It seems reasonable to assume that these plumes contribute to the total heat transfer. If A were made larger and larger, such corner plumes would contribute to a smaller and smaller fraction of the total heat flux. For A = 12 and 2 × 106 < R < 108, the following model is suggested for the remainder of the heat flux (i.e., that carried by the flow not near corners or other vertical walls): instead of hot plumes carrying heat to the cold wall, they carry heat only to mid-depth where they are "annihilated" by shearing the cold plumes, which sink to mid-depth, but travel horizontally in the opposite direction to the hot plumes. This is suggested by the mid-depth x, t photographs and by the absence at mid-depth of temperature signals from the hot or cold boundaries. But, of course, much more study would be needed to test the validity or invalidity of such a model. We do know however, that at a given R, there is a relationship between the upper bounds on heat and on m o m e n t u m fluxes (Howard, 1990). The largest heat flux allowed implies zero m o m e n t u m flux, (hence no large scale flows which are maintained by m o m e n t u m fluxes). Perhaps it is this relative heat to m o m e n t u m transfer that is affected by the aspect ratio. It is certainly true that the A = 12 tanks of water have a smaller heat flux than the 144 A = 1 cells of water through out the range measured. The wandering of the heat flux and Rayleigh number at high R might be due to the existence of several possible states of the system, together with the finite heat capacity of the boundaries. If one such state of turbulent flow were organized so as to transport a great amount of heat from bottom to top boundary, the Rayleigh number would drop if the heat capacity of the boundary was not large. If there were a (metastable) state of high heat flux at this lower Rayleigh number, the system may remain unchanged for some time. The wandering behavior is reminiscent of a phase point near a heteroclinic orbit, moving very slowly near an unstable critical point, which it leaves, then moving quickly until it approaches another (unstable) critical point. One kind of ideal boundary would be of infinite heat conductivity and infinite capacity. The present apparatus has a high conductivity ratio (of boundary to fluid) of 0.7 × 103. On the other

R. Krishnamurti / Fluid Dynamics Research 16 (1995) 87 108

107

hand, the heat capacity of the two aluminum plates of the top b o u n d a r y is approximately 0.56 of the heat capacity of the water, when this last layer is 10 cm deep. However, something more than the finite heat capacities must be involved, since in the eggcrate experiments there was no wandering and very little hysteresis.

6. Conclusions Our main new result is the observation of clusters of tilted transient plumes travelling horizontally across the tank as coherent units. These clusters, typically containing 5 to 10 plumes, are separated from one another by quiescent regions with almost no plume activity. Thus at a fixed point in the fluid near the bottom boundary, the temperature signal shows variability with periods of tens of seconds as the plumes within a cluster pass by. This epoch lasts for some hundreds of seconds, then is followed by a quiescent epoch also lasting some hundreds of seconds, after which the next cluster passes by. Thus there is both a high frequency (10 second periods) and a low f r e q u e n c y (102 second periods) variability. These results were deduced from the x, t portraits and the power spectra of the internal temperature time series as well as from direct visualization. The Nusselt number-Rayleigh number relationships for Prandtl number Pr = 7, aspect ratio A = 12 were hysteretic and had lower heat fluxes than when the tank was partitioned into 144 cells each with A = 1. In the former case, with A = 12, N -,~ R 0"25 ifR is always decreased in small steps; N ,-~ R ° ' 2 ° if R is increased in small steps, but any other method of approach can produce another power. It may then be questionable to speak of a power "law" or a unique relationship between N and R at least for 106 < R < 108. In the latter case, with A = 1, there is no hysteresis and good agreement was found with previous studies.

Acknowledgements This research was supported by the Office of Naval Research, Marine Meteorology Division, under contract no. N00014-89-J-1662 and by the Air Force Office of Scientific Research under contract no. A F O S R 49620-92-J-0498. This support is gratefully acknowledged. It is a pleasure to acknowledge the highly skilled assistance of Mr. M a r k Richards of the FSU Physics Departr~ent in the construction of the optical system of the apparatus. Also I am grateful to T.N. Krishnamurti of the FSU Meterology Department whose staff assisted with the data analysis. This is contribution number 362 of the Geophysical Fluid Dynamics Institute.

References Castaing, B., G. Gunarantne, F. Heslot, L. Kadanoff, A. Libchaber, S. Thomae, X. -Z. Wu, S. Zaleski and G. Zanetti (1989) Scaling of hard thermal turbulence in Rayleigh-Benard convection, J. Fluid Mech., 204, 1 30. Goldstein, R.J., J.D. Chaing and D.L. See (1990) High-Rayleigh-Number convection in horizontal enclosure, J. Fluid Mech., 213, 111-126. Howard, L.N. (1990) Limits on the transport of heat and momentum by turbulent convection with large-scale flow, Studies in Appl. Math., 83, 273-285.

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Howard, L.N. and R. Krishnamurti (1986) Large-scale flow in turbulent convection: A mathematical model, J. Fluid Mech., 170, 385 410. Krishnamurti, R. (1970a) On the transition to turbulent convection. Part I: The transition from two- to threedimensional flow, J. Fluid Mech., 42, 295 307. Krishnamurti, R. (1970b) On the transition to turbulent convection. Part II: The transition to time dependent flow, J. Fluid Mech., 42, 309-320. Krishnamurti, R. (1973) Some further studies on the transition to turbulent convection, J. Fluid Mech., 60, 285-304. Krishnamurti, R. and L.N. Howard (1981) Large-scale flow generation in turbulent convection, Proc. Nat. Acad. Sci., 78, 1981 1985. Krishnamurti, R. and L.N. Howard (1983) Large scale flow in turbulent convection: Laboratory experiments and a mathematical model, Papers in Meteorological Research, J. Meterological Soc. Rep. China, 6 (2), 143 159. Malkus, W.V.R. (1954) Discrete transitions in turbulent convection, Proc. Roy. Soc. A, 225, 185-195. Siggia, E.D. (1994) High Rayleigh number convection, Annu. Rev. Fluid Mech., 26, 137 168. Tanaka, H. and H. Miyata (1980) Turbulent natural convection in a horizontal water layer heated from below, Int. J. Heat Mass Transfer, 23, 1273 1281. Zocchi, G., E. Moses and A. Libchaber (1990) Coherent structures in turbulent convection, an experimental study, Physics A, 166, 387 407.