VOLUME 56, NUMBER 17 P H Y S I C A L R E V I E W L E T T E R S 28 APRIL 1986

Nonsteady End Effects in Hele-Shaw Cells

J. N. Koster,(a) P. Ehrhard, and U. Muller Institutjur Reaktorbauelemente, Kernforschungszentrum Karlsruhe, 7500 Karlsruhe 1, West Germany

(Received 7 June 1985) Two new results are presented on the development of convective flow in a finite Hele-Shaw cell.

First, it is shown that within a certain range of Rayleigh numbers there exist three regions in the fluid layer: the center region having a steady flow and the two end regions exhibiting time-dependent flow. Second, in the two end regions each flow oscillation is always found to be mono-periodic and the two are not correlated. Transition to a more complicated oscillation is not ob­served, and the flow never becomes chaotic. The time-dependent flow regime is bounded at low and high Rayleigh number by transitions to steady-state flow.

PACS numbers: 47.25.Qv, 02.50.-fs

Time-dependent convection is a timely subject for studies of the transition to turbulence.*~5 In all these experiments, when the temperature difference across the fluid layer is increased, the developing unsteady flow is generally characterized by more than one fre­quency in the power spectrum of a signal.

In contrast to these experiments, we observe a flow pattern that is time-dependent in end regions only, whereas the central flow remains steady. This type of end effect has not been reported before; other types have been described by Elder.6 He observed a flow structure in a Hele-Shaw cell that is controlled by hor­izontal thermal boundary conditions. Observation of confined localization of sources for flow oscillations in fluid layers has been reported by Dubois and Berge2

and by Walden et al? Apart from problem-oriented modification, the

Hele-Shaw cell investigated here (Fig. 1) is similar to those described previously.5 The vertical walls of the cell are made of 4-mm-thick Plexiglas which categor­izes the cell as a low-conductivity cell. To improve the linearity of the vertical temperature profile in the sidewalls, the Plexiglas cell is sandwiched between two BK7-glass windows in thermal contact with the upper and lower isothermal boundaries. The dimensions of the Hele-Shaw cell are gap width rf = 4.1 mm, height /?=40.3 mm, and length 6 = 111.2 mm. The aspect ratios are A/6 = 0.36 and /?/</ = 9.83, or hi bid = 0.36/1/0.037. The fluid used is silicone oil of viscosity i = 20 cS (centistokes) at F = 2 0 ° C and Prandtl number NPT^P/K == 235, with kinematic viscosity v and thermal diffusivity K. The Hele-Shaw cell is built into a low-conductivity frame (Novotex) with windows for optical access. In addition, the framed cell is insulated with 30-cm-thick, partly re­movable, heat-insulation material to reduce the heat transfer to the surroundings.

The fluid layer is heated from below and cooled from above with a symmetric heating rate of 0.02 K/min. Between the heating periods, equilibration periods of between 30 min and 24 h are provided be­fore measurements at constant temperature difference

are taken. The temperature difference A T and the upper temperature Tx are measured in order to calcu­late the Rayleigh number which is defined as NR&

= = ^ / 3 A 7 1 / ? 3 ( ^ K ) _ 1 with acceleration of gravity g, thermal expansion coefficient /3, and vertical tempera­ture difference A T across the fluid layer of height h The Rayleigh number is normalized as N^—N^/ (iVRa)osc, where (AfRa)0SC defines the onset of oscilla­tions.

At the upper and at the lower side there are four 0.25-mm-diam thermocouples. They protrude approx­imately 2 mm into the fluid. These thermocouples provide measurements of temperature fluctuations in the horizontal thermal boundary layers of individual convective roll cells.

Time-dependent signals from the horizontal thermal boundary layers are amplified and filtered at a frequen-

(T3-TJ | ice JT2 IT, 1——1 E?z

1 v| j v| lyl |

3 0

m i i n ' i V i i i i i i i

-•

2ZzS

[A]

[yl

1 1 i

h

i L

AT) He IT? ice t (T5-T6)

FIG. 1. Sketch of the area h x b of the Hele-Shaw cell with thermocouple locations drawn to scale. Shaded area represents the visualized area.

1802 © 1986 The American Physical Society

VOLUME56, NUMBER 17 PHYSICAL REVIEW LETTERS 28 APRIL 1986

L

FIG. 2. One period of oscillation at /VRa= 1.58x 106 with period Af = 47.6 s. Interferograms are taken at Af = 10-s in­tervals.

cy of / = 10 Hz, and their dc offset is removed before storage on tape. Segments of 2048 values are digitized with a time step of 2 s < A t < 5 s. The auto­correlation and cross-correlation coefficients and the auto- and cross-power-density spectra of different sig­nal combinations are calculated. A detailed error analysis of these experimental procedures has been given previously.5

The threshold of the onset of convection is mea­sured in several runs and found to be (NRaL)c

= 3.9xl0 5 ( A r - 1 . 3 5 K). A four-roll flow pattern develops in each run. With increasing Rayleigh number, more vortices grow into the layer. At the on­set of oscillations ten vortices are established.

The onset of time-dependent convection occurs at (i¥Ra)osc==1.58xl06 (ARa = l ) with a dimensionless period of oscillation of T = K / / / I 2 = 1.02X 10~2. After the onset of oscillatory flow the total number of con­vection rolls does not change further.

The new phenomenon in this experiment is that only the two end rolls on each side become time dependent. Interferograms of one period of oscillation and related qualitative stream-line sketches are shown in Fig. 2. End rolls of this type have not yet been seen in containers with h — d The periodic growth of the

Delay time f / s

FIG. 3. Cross-correlation coefficient and cross-power-density spectrum of the signals F3 - T4 and T8 at one side of the Hele-Shaw cell at WRa = 4.8x 106. The signals are well correlated.

corner vortex can be observed in these interferograms. At one time the growing upper vortex is sheared off by the lower upflow and a cold, descending thermal plume forms. The optically measured period of 40-50 s is confirmed by the frequency analysis of the thermal-boundary-layer signals. The oscillations are periodic at both ends throughout the range of time-dependent flow. The center rolls remain stable.

The autocorrelation coefficients of thermocouple signals show that the signals have great coherence. The frequency analysis, made only at a few Rayleigh numbers, shows that, in the range investigated, all the oscillations at both ends of the Hele-Shaw cell are monoperiodic; i.e., they have one basic frequency, fu

and higher harmonics, mf\ (m = l - 4 ) . As suggested by the temperature traces, this monoperiodic behavior is also anticipated at Rayleigh numbers between the analyzed measuring points.

The two time-dependent flow regions are separated by an area of steady flow, a pattern that has not been observed before. Cross correlation (Fig. 3) of the sig­nals measured at one end of the Hele-Shaw cell (either the left-hand side or the right-hand side) reveals that in the whole range of Rayleigh numbers these signals are well correlated. Cross correlation of any sensor signals at the left end with any sensor signals at the right end of the Hele-Shaw cell (Fig. 4) reveals that the two oscillations, separated by the region of steady convection, are not correlated. However, the frequen­cies of oscillation are very close.

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VOLUME 56, NUMBER 17 P H Y S I C A L R E V I E W L E T T E R S 28 APRIL 1986

Frequency f /Hz

0-k v/X/V/X/N/^^

J -1 -U

0 Delay time t / s 550

FIG. 4. Cross-correlation coefficient and cross-power-density spectrum of the signals 73 — T4 and T\ which moni­tor signals at the two ends of the Hele-Shaw cell at Ar

Ra = 4.8x 106. Note the absence of correlation.

As in previous experiments, a reverse transition from time-dependent flow to steady flow is observed at high Rayleigh number. Note that the oscillations cease at different Rayleigh numbers at the different sides. On the right-hand side they cease at ;VRa = 6.54x 106 {N^ =4.14) and at the left-hand side at A ^ ^ . O S x l O 6 (jVRa=4.48). Time-dependent motion is no longer detectable either in the interfero-grams or in the thermocouple signals. There is

currently no physical explanation for this transition. When the Rayleigh number is reduced again, time-

dependent convection begins at the right-hand side of NRa = 5.95xl06 (^Viu-3.77), and the left-hand side at iVRa = 6.21 x 106 (iVRa =3.93). This indicates that a pronounced hysteresis exists at this reverse transition, in conformance with previous results.5

Note that the central high-wave-number cells do not develop oscillations at any Rayleigh number ap­plied.

The results reported here show that three regions of different flow structure may coexist at one Rayleigh number in a fluid layer. One central region is steady and the two end regions are time dependent. The two individual oscillations are always monoperiodic and uncorrected at any Rayleigh number in the time-dependent regime.

We wish to thank J. P. Gollub for suggestions. We appreciate the English editing by D. Gutzler.

(a)Present address: Department Mechanical Engineering, University of Colorado, Campus Box 427, Boulder Colo 80309.

1G. Ahlers and R. P. Behringer, Phys. Rev. Lett. 40 712 (1978).

2M. Dubois and P. Berge, Phys. Lett. 76A, 53 (1980). 3J. P. Gollub and S. V. Benson, J. Fluid Mech. 100, 449

(1980). 4A. Libchaber and J. Maurer, J. Phys. (Paris), Colloq. 41

C3-51 (1980). 5J. N. Koster and U. Muller, J. Fluid Mech. 139, 363

(1984). 6J. W. Elder, J. Fluid Mech. 27, 29, 609 (1967). 7R. W. Walden, P. Kolodner, A. Passner, and C. M. Sur-

ko, Phys. Rev. Lett. 53, 242 (1984).

1804

i * FIG. 2. One period of oscillation at /VRa = 1.58 x 106 with

period A/= 47.6 s. Interferograms are taken at A/= 10-s in­tervals.