In this paper we present results from experiments with a system consisting of two immiscible fluid layers in rectangular and annular geometries, driven by a vertical temperature gradient. Time-dependent variations in the type of coupling observed between the two layers are described and characterized

Andereck, C. David

Colovas, Peter W.

Degen, Michael M.

English

NASA Technical Reports Server

/OBSERVATIONS OF TIME-DEPENDENT BEHAVIOR IN THE TWO-LAYER
RAYLEIGH-BI_NARD SYSTEM
C. David Andereck, Peter W. Colovas and Michael M. Degen
Department of Physics
The Ohio State University
Columbus, OH 43210
ABSTRACT
In this paper we present results from experiments with a system consisting of two immiscible fluid layers
in rectangular and annular geometries, driven by a vertical temperature gradient. Time-dependent variations
in the type of coupling observed between the two layers are described and characterized.
INTRODUCTION
Rayleigh-Bdnard convection is one of the oldest and most thoroughly studied of fluid dynamical systems.
The basic system consists of a single fluid confined between parallel horizontal plates held at a temperature
difference AT, usually with the higher temperature being on the lower plate. If AT is large enough, convection
begins in the form of parallel rolls, the result of the destabilizing effect of the buoyancy force overcoming
the stabilizing influences of viscous drag and thermal diffusion. The onset of convection in the one fluid
Rayleigh-B6nard problem is simply characterized by the Rayleigh number
gaATd s
Ra - --, (i)
t_v
the dimensionless temperature difference, and a ratio of the destabilizing force to the stabilizing forces. In
equation (1), g is the acceleration of gravity, a is the volume coefficient of expansion, d is the depth of the
fluid layer, _ is the thermal diffusivity, and u is the kinematic viscosity of the fluid. Another nondimensional
parameter, the Prandtl number P = u/s, does not enter into the determination of the onset of convection.
Thorough introductions to the stability analysis of the one fluid problem have been presented in the books
by Chandrasekhar[1], and Drazin and Reid[2]. One result of these analyses is the fact that the equations are
self-adjoint, and all the eigenvalues are real, leading to the "exchange of stability," i.e. the growth of small
disturbances leads to steady cellular convection, there being no oscillatory modes at criticality. Fluids from
water to liquid helium, and gases like air and C02 have been used in experiments which explore the one
fluid Rayleigh-Bdnard problem.
ff the upper surface is free, or bounded by another fluid, then surface tension may play a role in determin-
ing the onset of convection[3]. Various theoretical analyses have shown the possible existence of oscillatory
modes at the onset of convection, for cases of both deformable and nondeformable interfaces[4, 5, 6, 7, 8, 9,
10, 11, 12, 13, 14, 15, 16, 17]
Gershuni and Zhukovitskii [9] described a possible mechanism for oscillations which can appear in the
coupling between two convecting fluid layers. The basic types of coupling are thermal and mechanical.
Thermal coupling is the alignment of the hot rising fluid in both layers, causing rolls that are aligned one
above the other to turn with the same sense, either clockwise or counterclockwise. Mechanical coupling
is the alignment of the cold falling fluid in one layer with the hot rising fluid in the other layer, with the
result that the rolls turn with opposite senses, in a gear-like fashion. The oscillations were predicted to be
periodic variations between thermal and mechanical coupling. Experimental work on the two-layer problem
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has been reported by a number of groups [17, 10, 18, 19, 20], but none have found time-dependence in the
coupling at the onset of convection. In this paper we show the results of our experimental investigation of
a Rayleigh-B_nard system consisting of two immiscible fluids, concentrating on the search for oscillatory
states.
EXPERIMENTAL SYSTEM
The experimental system consists of two fluids confined between horizontal parallel plates and bounded
on the sides by transparent walls. The lower fluid, here designated fluid 1, occupies a depth fraction Ix = dl/d,
where dl is the actual depth of fluid 1 and d is the total thickness of the cell. Fluid 2, the upper fluid, similiarly
occupies 12 = d2/d. Two different pairs of fluids were selected for use in our experiments. The first pair
consists of a silicone oil, Rhone-Poulenc Rhodorsil 47v10, over a perfluorinated hydrocarbon, Fluorinert
FC-70, from 3M. They were chosen because theey are nearly immiscible, and because their properties are
similiar enough that convection in both layers should start at the same overall AT for nearly equal depths.
The other pair of fluids is 2 cs silicone oil, Rhodorsil 47v2, over water.
[n the rectangular geometry, the fluids are confined within 9.5 mm thick glass walls. The interior
horizontal dimensions of the cell are 78 mmx 21 ram, and the total height of the cell d = 12 ram. The
dimensions of the cell were chosen so that the onset of convection leads to a nearly one dimensional set of rolls
in each layer. These rolls form parallel to the short dimension of the cell, as in single layer convection. The
top and bottom of the cell are aluminum blocks. The blocks are thick to assure uniformity of temperature
across the surfaces which contact the fluids. The top and bottom blocks are cooled and heated, respectively,
by water which is conditioned by temperature controlled baths. In each circulation loop is a thermal low
pass filter, which removes the short term fluctuations present in the bath. The system is shown in Figure 1.
The almular cellhas innerradiusr_ --44 ram and outer radiusro = 64 ram. Ithas the same heightas the
rectangularcell,and isheated and cooledin the same manner.
The temperature differencebetween the platesismeasured by thermistorsembedded in the centerof
each block,approximately 1.2cm from the contactsurface.The long term stabilityAT, is±.02 C, while
the horizontalvariationinplatetemperatureislessthan +.01 C.
The thermal patternsin the rectangularsystem were observed using Schlierenand shadowgraph tech-
niques,whileonly shadowgraph couldbe used forthe annularcell.The measurement, whilelocalalong the
lengthof the cell,isan integraloverthe width ofthe fluidlayer.A representativeSchlierenpatternisshown
in Figure 2. The dark regionsindicaterollsturningin one direction,while the lightregionsindicaterolls
turningin the other direction.To analyze the space-timebehaviorof the system,a singlehorizontalline
through the centerof each layerwas chosen,and the intensityrecordedovera periodof time encompassing
severaloscillations.A representativeplotisshown forthe rectangularcellin Figure3, and forthe annular
cellin Figure 4. The resultingdata can then be analyzed to findperiods,wavelengths,wave speeds,etc.
Our analysiscomes primarilyfrom thisdata.
PROCEDURE
While the parameter space of the two-layer Rayleigh-Bdnard problem is large[21], practical consideratic_ns
simplify the experimental process considerably. Upon choosing the fluids, the only experimental variables
are the depths of the layers, 11 and lz, and the temperature difference, AT, between the top and bottom
plates. We proceed by filling the cell with the appropriate amount of each fluid to achieve the desired I1
and 12 The depth of the lower fluid is measured precisely by an optical telescope, accurate to ±.02ram.
The full system is left to equilibrate for a period of 12 hours before the temperature ramping begins. The
temperature is then ramped approximately 0.1°C at a time, starting from AT -- 0, and the system is left
for at least three hours before data taking begins. As the period of oscillations is on the order of one hour,
314
a data run consists of one "flame" of a single data line in each layer, taken every 30 seconds over a period
of 6 to 8 hours. This gives us an effective ramping rate of _. 0.2°C/day.
Wavelengths and frequencies are computed from power spectra in space and time, respectively. A time
average at each spatial point is first subtracted from the data, to remove non-uniformities in lighting due to
the system optics. A one-dimensional FFT is then performed on the resulting data to compute the desired
spectrum. We report our results in terms of AT, the temperature difference across the total height d.
RESULTS AND DISCUSSION
We began our experiments with 47v10 over FC-70, at a depth fraction It = 0.50, where calculations
predicted the onset of convection to occur at nearly the same AT in both layers. At this fraction, we found
only mechanical coupling, at AT values up to 2.0 C. From here, we systematically decreased li until we
reached the value It = 0.39, where we discovered oscillations slightly above the onset of convection. A plot of
our results is shown in Figure 6. Oscillations near onset continue at lower fractions, until the value li = .345 is
reached, where no oscillations are observed, and the coupling is predominantly thermal. We have calculated
wavenumbers and frequencies for the oscillations in both the top and bottom layers from the power spectra.
The results are close to those predicted by the numerical analysis of Y. Renardy (unpublished). Figure 6
shows that in order for time-dependent behavior to develop with this combination of fluids, the coupling at
the onset of convection must be mechanical.
Our experiments with the combination of fluids 47v2 over water show different behavior at the onset of
convection. Calculations predict simultaneous onset of convection for ll = 0.69 in th.is system. Our work
near this depth fraction shows that in the range 0.71 _> ll _> 0.60 the critical mode at onset is time dependent.
A space-time plot of tiffs behavior is shown in Figure 5. This is consistent with numerical work done by Y.
Renardy, who describes a parameter
r = ---- (2)
/¢2 _2p 2 '
which should be far from one for the possibility of oscillations to be large. The 47v10/FC-70 system has
Y = .776, while the 47v2/water system has Y = 0.374, implying that the range of depths where oscillations
can be expected is much larger.
CONCLUSIONS
Numerical studies done on the two-layer Rayleigh-B6nard problem predict that the coupling between the
two layers at onset should be predominantly thermal for values of ll just below the range where the onset
should have oscillatory behavior, and predominantly mechanical for ll, just above the oscillatory range. The
size of the window of oscillations is determined by the parameter Y. Our work in the 47v10/FC-70 shows
thermal coupling at low li and mechanical coupling at high li. In addition, this system shows oscillatory
behavior above onset for a range 0.39 _> li _}20.345. Further work with the 47v2/water system shows the
occurrence of oscillations at the onset of convection for 0.71 > li _-20.60, supporting the usefulness of Y as
a qualitative predictor for the range of ll where oscillatory behavior may be expected.
ACKNOWLEDGEMENTS
We would liketo thank Y. Renardy for her discussionson the nature of the system and her numerical
predictions. We would also like to acknowledge discussions with P. Kolodner and M. Schatz, and D. A.
Campbell & Co. for supplying the siliconeoils.This work was supported by NASA grant NAG3-1612.
315
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316
. Thermal
E Low Pass
Filter AI Block
Themuston _ " Fluad2
AI Block
C_u_Uag
Temperai_m Th_
Bath Low Pass
Cvculalmg
Ternpecar_re
Bath
Figure 1 Schematic of the convection cell and temperature
control, including thermal low pass filters and in-line
active heaters
e-.
(a)
I i
10 20 30 40
x (mm)
E
1
100
70
Figure 2: Schlieren image of the 47v 10/FC-70 system.
Light regions are rolls turning in one direction, Dark rolls
turn in the opposite sense. AT=I.6 C, 1,=0.38. (b)
, I I
10 20 30 40
x (mm)
Figure 3: Space-time plot of the (a) top and (b) bottom
layer in the rectangular ceil, for the state shown in Fig. 2.
Time advances up the vertical axis, and a time-averaged
background has been subtracted.
(a)
Sprier I')
370
360
350
340
330
320
310
300
15 25 35 45 55
x (nun)
N
|
_e, e. u_ e, •
_pece I'1
Figure 4: Background subtracted spac_-time plot of the (a)
top and (b) bottom layer m the annular cell, for AT=2.0 C,
1,=0.37. The horizontal axis traits are degrees around the
azimuthal dimension.
Figure 5: Background subtracted space-time plot of the
upper layer m the 47v2/water system near the onset of
convection, showing the time-dependent behavior of the
rolls. AT--0.74C, 1]=0.6.
2.2 ¸
2.0
18,
16
1.4-
I-,
12
I0
08
06
i • l • | • i • w • v - ! • | - | • ! •
032 034 036 0.311 040 042 044 046 0411 0_0
I n
Figure 6: Plot of AT vs. I, showing the range of coupling
and the onset of oscillations in the 47v 10/FC-70 system
(X) and (+) indicate the onset of mechamcally coupled
convection m the top and bottom layers, respectively. (o)
and (ra) indicate the onset of thermally coupled convection
in the top and bottom layers. (e) indicates the onset of
oscillations.
318


Observations of Time-Dependent Behavior in the Two-Layer Rayleigh-Benard System

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