Buoyancy driven flow of two stratified liquids in a cavity

Pergamon Int. Comm. HeatMass Transfer, %/ol. 26, No. 7, pp. 955-964, 1999

Copyright © 1999 Elsevier Science Ltd Printed in the USA. All rights reserved

0735-1933/99IS-see front matter

PII S0735-1933(99)00085.8

BUOYANCY DRIVEN F L O W OF TWO STRATIFIED LIQUIDS IN A CAVITY

R. Tovar, F.E. Avila, J. Rojas and B. Vargas Centro de Investigaci6n en Energia y Posgrado en Ingenieria de la Universidad Nacional

Autrnoma de Mrxico. A.P., 34, Col. Centro, Temixco, Mor., 62580, Mrxico.

A B S T R A C T

(Communicated by J.P. Hartnett and W.J. Minkowycz)

A two dimensional study on the transient natural convection of two stratified liquids in a cavity is presented. We studied the motion of the fluids when one wall of the cavity is quickly heated up at a certain required temperature, the stratified liquids being initially at rest and in thermal equilibrium. The experiments were done at different inclination angles of the hot wall so that its influence on the dynamics of the process could be observed. Also, several initial concentrations for the denser liquid were tested for each inclination, and the global effect of the initial concentration investigated. It was possible to monitor the advance of the thermal intrusions by means of careful measurements of their thermal fronts. © 1999 Elsevier Science Ltd

Introduction

The motion of a single fluid within a cavity, originated by a temperature gradient applied across the

cavity has been extensively studied in analytical, experimental and numerical forms. The scale analysis

made by J.C. Patterson and J. Imberger in 1980 [1] for flows for which the Prandtl number is Pr > 1 and

the cavity aspect ratio A < 1 allowed these authors to classify the heat transfer into three general regimes:

conductive, transitional and convective; they further classified these regimes into a variety of different

modes for heat transfer according to the relative magnitudes of the dimensionless governing numbers.

This pioneer work gave birth to a number of investigations in which other researchers tried to verify or

overruled the validity of their results. A brief chronological account of the most relevant work for our own

research can be put as follows: In 1982 Yewell et al. [2] carried out experiments which were intended to

but did not show any evidence of oscillations. In 1984, Patterson proved that the experiments by Yewell et

al. did not meet the required conditions demanded by his theory. In 1984, G.N. Ivey [4] showed

955

956 R. Tovar et al. Vol. 26, No. 7

experimentally that a short period oscillatory approach to steady state exists for certain flow regimes but

that the nature of such behavior was a form of hydraulic jump. Later, in 1989, S.G. Schladow et al. [5]

carried out a numerical simulation of Ivey's experimental work and found no evidence of short period

oscillations. Instead, they clearly showed the presence of long period oscillations. In the same year,

Paolucei and Chenoweth [6] also reported numerical simulations based on Ivey's experiments. By

contrast, they observed only short-period oscillations and concurred with Ivey's hypothesis of a hydraulic

jump as their origen. In 1990, Schaldow [7] reconciled the differences between the experimental and

numerical observations by proposing that both the long and the short period oscillations exist as part of the

flow. They thought that the former are of the type suggested by Patterson and Imberger and the latter due

to instabilities in the vertical boundary layer rather than due to any sort of hydraulic jump. In 1990, J.C.

Patterson and S.W. Armfield [8] undertook a major research aimed at clarifying many of the

disagreements that had by then appeared. Their work contributed to deepen the understanding of the

transient features of the flow. The same authors further reinforced their conclusions in a second article

published in 1991 [9]. More recent works by M.R. Ravi et al. [10] in 1994 and C. De Saedeleer et al. [1 i]

in 1996, show that despite the large amount of efforts spent, the problem is not finished and other possible

effects such as Marangoni's may play an important role in the dynamics of the flow. Due to our particular

interest on the applications of stratified flows on energy storage systems, this work deals with two fluid

layers, pure water and solutions of common salt in water at different concentrations.

Experimental Setup and Procedure

The experiments were carried out in a cavity specially built to facilitate the making of the fluid

stratification and the changing of inclination for one of its walls. Figure 1 shows a schematic diagram of

the cavity and the coordinate system. The left wall of the cavity works as a heat exchanger and its bottom

part works both as an intermediate reservoir and as a diffuser for the denser fluid. This component is made

of two sheets of acrylic with a small space or reservoir left in between. The overall inner dimensions of

the experimental cavity when used in its parallel wall configuration are: 30 cm along x-coordinate (cavity

length, L); 20 cm along y-coordinate (cavity height), and 17 cm along z-coordinate (cavity depth). In an

actual experiment the two strata occupied only 10 cm in height. The heat exchanger is made out of acrylic

sheets (12 rnm thick) except for heat transfer surface made of stainless steel (0.8 mm thick), this surface is

the hot wall of the cavity. Two holes, 0.25 mm diameter, located on the hot wall at 25 and 75 mm height

were drilled so that one could inject dye directly inside the boundary layer. The tubing system for

injection was connected to the back of the hot wall through the heat exchanger. The heat exchanger was

Vol. 26, No. 7 FLOW OF TWO STRATIFIED LIQUIDS IN A CAVITY 957

attached to the base of the cavity by means of a leak-proof hinge so that it was easy to change its angle of

inclination with respect to the vertical.

A constant temperature bath, _+0.1 °C, connected to the heat exchanger was used to provide the hot

water needed to keep the hot wall at that fixed temperature required, such that the driving temperature

difference were AT = 5°C (initial temperature in all experiments of 25°C).

25.0

i.

I - ~ 335.0 ~ 1

. , . 0

189.0

/

FIG. 1 Experimental cavity (measured in mm). 1 Heat exchanger, 2 cavity for strata,

3 reservoir-diffuser (bottom part), 4 inclination angle, 5 holes for dye injection.

As it can be seen from Figure 1 the bottom of the cavity has many small holes (about 500) through

which liquid may come in into it. This component is connected through a flexible hose to a container

where the denser fluid is kept. The opposite wall of the cavity is made of acrylic (12 mm thick), and its

fore and back walls are made of glass (4 mm thick). The cavity was not covered leaving the surface of the

upper fluid layer open to the ambient conditions. Two salt solutions and three geometrical configurations

were used: 2.5 and 0.5 % concentrations by weight and 90, 60 and 45 ° inclination angles, respectively. A

video camera was set up in order to record the flow when flow visualization was needed. These conditions

implied: Ra=O(107), A=0.166 (h/L), and Pr=6, as defined in [1] and therefore classifying the global

behavior of this flow in Region IV of the same reference.

The strata were prepared by doing the following routine: from the container, brine was brought into

the reservoir-diffuser just enough to fill up the small holes; then, distilled water was slowly poured inside

the cavity building up a layer 5 cm thick. Once the layer of water was still, brine was let inside the cavity

through the reservoir-diffuser until a 5 em thick layer of brine was formed. This process had to be done


very slowly so that the incoming brine uniformly pushed the layer of water up causing the least possible

mixing of the fluids. Inevitable, the inter space between the two liquids constituted a third layer with a

steep concentration profile. The strata were let still for some minutes before starting an experimental run.

Since it was desirable to assess the amount of mixing between the strata during the period of

strongest convection effects, the initial concentration profile was measured for each experiment to be used

later as a reference. In order to do this, a Conductivity Meter was used to measure locally the initial

concentration profile across the y-coordinate (MSCI model 125). The probe of the instrument (5 mm

diameter) was mounted in such a way that it could be slowly moved downward along y-coordinate

without disturbing the strata. Although such profile represents the concentration distribution along the y-

axis in only one point of the x-z plane, we took it as being representative of the concentration distribution

everywhere. Preliminary tests for initial concentration profile had repeatedly shown the same

concentration profiles in different points of the x-z plane for the same batch of fluids. Figure 2 shows an

example of an initial concentration profile. One can see that the initial concentration gradient spanned for

a distance of approximately 10 mm which was the typical separation between pure water and brine at the

beginning of all of our experiments. In the same plot, a concentration profile taken at the end of the same

experimental run (1800 seconds duration in this example) is also plotted. Anticipating the discussion in

the next section, one can say that in spite of the strong convection in the first minutes of the flow (1800

sees.), the concentration profile is well preserved.

11

10

9

8

7

6

g o

3

2

1

0

-1

T

i 0 0 0.5 1.0 1.5 2 0 2.5

% SALT CONCENTRATION BY WEIGHT

FIG. 2 Concentration profile before and after 1800 seconds of an experimental run.


Ten thermocouples (T type gage 36 Copper-Constantan, Omega, Co.) were placed inside the cavity,

their positions are given in Figure 3. They were used for two purposes: to monitor the temperature

evolution toward steady state and to estimate the average velocity of the thermal horizontal intrusions in

their way to the cold wall.

Hot Wall ~ _ 8 ,12 Number x (mm) y(mm)

1 0 25 2 0 75 4 20 98 6 20 98 8 120 98 12 200 98 3 20 45 5 60 45 7 120 45 11 200 45

FIG. 3 Position of the thermocouples.

Experimental Results and Discussion.

At the sudden increase of the temperature in the hot wall, a double (thermal and viscous) vertical

boundary layer is formed, one in each layer. Due to buoyancy forces each of these will go up until it turns

towards the opposite wall. The hot fluid coming up from the vertical thermal boundary layer forms a

lingulate structure which advances horizontally towards the opposite wall. This structure is called a

thermal intrusion. There will be a general re-circulation in the cavity until the temperature gradients died

out and the liquids remain motionless (all boundaries are considered to be adiabatic except for the hot

wall).

The experimental results are organized in two ways: global trends and a more detailed analysis of

the characteristics of the flow, which are apparent only when closely looking into its evolution. To

estimate the average velocity of each thermal intrusion (its leading edge), the horizontal distance from the

hot wall to the position occupied by the thermocouples was divided by the elapsed time between the

beginning of the experiment and the particular time at which the jump in the thermocouple temperature

signals appeared.


~ ~ T U M

~ ' 5 1 ! 1 1 ! 1 1 1 1 1 1 !

~ . o i . . . . . . . . i . . . . . . . . .

26,o i , i , i i ~ ; ~ ; i ! x ~

= , o i i ! i i : i i ; i ~ ® :

~ . o i i : i 7 i i z 7 ! : ~ -

~ . 5 : . i i i ~ ! i i : ! .

24.5

(a)

29.5

29 .0

28.5

28.0

275

27.0

26.5

26.0

265

250

24.5

L ~ ~ T ~ I I I I I I I I I I I I

: i

i i [ ~ 2 0

. ~ _

i i , , , , , i , i i i

t ( s )

LIPPER STRATUM 29.5 I I I I I I I i I I I

2 9 . 0 . ~ 7

28.5

28.0

27.5

27,0

: - 26.5

26 .0

25.5

25.0

24 .5

. ~ i . ! ; ; ~ : x = 2 6

i , i , l , i . i , i , i , i , l , i , i , i ,

t ( 5

27.5

27.0

26.5

26.0

25.5

25.0

24 .5

UPPER ~ M ~ ' 5 1 1 1 1 1 1 1 1 1 1 1 1

28 .5 . . . . ; . . . . . . . . . . . . . . . . .

2 8 . 0 - i ~ : [ i i • i -

. i i ; i : : x = ~

. i i [ i i l i i l ; ~ l i _

i i i i i i i i i i i i

100 150 ~ 0 260 ~ 0 ~ 0 4 0 4 0 ~ 0 550 ~ 0

t ( s )

(b)

i - -

(c)

29.5 LOWER STRATUM i = i ! ! ! , , ! ! ! i

29,0 : :

• i 28.5

28.o : : . . . . . .

27.5 - : : : :

2 7 , 0 - i . i =

~ . 5 - i i i x = 8

26.0 . i ;

26,5

0 50 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0 4 0 0 4 5 0 5 0 0 5 5 0 6 0 0

t (s)

29.5

29.0

28.5

280

275

27.0

265

260

25.5

250

245

LOWER STRATUM I I I I I I I I I I I I

X=l ."

• : x = 2 "

0 50 100 150 260 250 300 350 400 450 500 550 6~0

t (s)

FIG. 4 Temperature signals for the thermocouples positioned in the upper and lower strata for

configurations at an angle of: (a) 90 °, (b) 60 ° and (c) 45 °. Notice the distance x to the hot wall (in mm).


Figure 4 shows the Temperature vs. Time plots for the actuai experiments for 90, 60 and 45 degrees

configurations and 2.5 % of salt concentration. Similar signals were obtained for the 0.25 % concentration.

Notice that the abrupt temperature change in the pure water stratum and the smoother change in the brine

stratum may be taken as a consequence of the fact that, in the former ease, thermals move up (and change

direction) against a sharp inter phase (water-air) with minimum interaction, whereas, for the latter, they

move against a medium with a strong density gradient where mixing and a more active transport of heat

may be possible. This fact may also be responsible for the faster motion of the thermal intrusions in the

upper stratum as compared with the their corresponding motion in the lower one.

Tables 1 and 2 show the mean velocity for both 2.5% and 0.25% concentrations. The value in each

entry represents the average value of three experiments _+ a dispersion value. The average value for the

thermal intrusion velocity does no depend on the wall inclination, at least within the range of our

parameters regardless the 45 ° difference in angle between extremes. Consistently, and independently on

angle and concentration, the thermals in the upper stratum move faster than those in the lower layer. For

the 2.5 % experiments, this difference is of about 30, 60 and 40 % for the corresponding inclination angles

of 45, 60 and 90 °, respectively.

TABLE 1 Salt Concentration of 2.5% by Weight.

ANGLE AVG. VELOCITY OF INTRUSION (mm/s) UPPER LOWER

INTRUSION INTRUSION 45 ° 1.56 + 0.07 1.13 + 0.04 60 ° 1.63 + 0.06 0.93 + 0.06 90 ° 1.57 + 0.07 1.02 + 0.04

TABLE 2 Salt Concentration of 0.25% by Weight.

'ANGLE AVG. VELOCITY OF INTRUSION (ram/s) UPPER LOWER

INTRUSION INTRUSION 45 ° 1.68 ::k 0.03 1.16 + 0.03 60 ° 1.58 + 0.05 1.25 + 0.04 90 ° 1.65 :i: 0.03 1.07 + 0.04

Temperature depends naturally on the position in which the measurement is taken. A thermocouple

like T4, 20 nun away from the hot wail, shows a signal with small fluctuations but a thermocouple like T2,

0.5 nun away from it, shows large oscillations whose amplitudes vary from plus to minus 0.5 °C and


more, with periods about 8 seconds, see Figure 5. One can see in the same figure that the small

temperature oscillations seem to be riding a long period-large amplitude secondary oscillation whose

period is about 180 seconds. It seems that these long period oscillations scale with the length of the cavity:

a rough estimate based on the average velocity of the thermal intrusion at 240 mm from the hot wall,

would give us a time for such intrusion to reach the opposite wall around 190 seconds. In this elapsed

time, one parcel o f fluid would have completed one full cycle around the cavity.

o~

( . .

2 9 . 5

2 9 . 0

2 8 . 5

2 8 . 0

2 7 . 5

2 7 . 0

2 6 . 5

2 6 . 0

2 5 . 5

2 5 . 0

- 5 0

i i : ! i i

z • i , i • i , • , i , i • .

0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0 4 0 0 4 5 0 5 0 0

T I M E (SEC)

FIG. 5 Evolution of temperature in a region next to the hot wall. This plot corresponds to a

45 ° of wall inclination. Similar traces were obtained for the other angles of inclination.

y=lO cm

y = 4 c m mr"- - -

I x=0

] > x x = 1 0 c m

FIG. 6

Digital version of the hydrodynamical oscillations observed for all experiments.


It is important to remark that not only were thermal oscillations detected but also were

hydrodynamic waves observed, the images were recorded in videotape. Indeed, by feeding dye direetly

inside the boundary layer through the injecting holes, it soon became clear for all experiments that the

streaks of dye took on a wave-like configuration in the region were the hot fluid coming from the

boundary layer had to change direction. These waves resembled those reported by Ivey [4]. Figure 6

shows a digitalization of one of the images recorded on magnetic tape.

Conclusions

The angle of inclination and the initial concentration did not have a significant influence on the

global motion of the thermal intrusions in their way towards the cold wall. However, a faster motion of the

intrusion in the upper layer with respect to the corresponding one in the bottom layer was evident, for all

angles, for all concentrations. This was explained on the basis of a more active interaction between liquid-

liquid strata than between liquid-air boundary. Also, no much transversal mixing of fluids did occur

during the period of time in which the motion was at its most intense stage. Several experiments with

higher initial concentrations were performed but their results were in all similar to those corresponding to

the highest concentration presented in this paper.

Short period oscillations as well as long period oscillations were identified in the temperature

signals. They were strongly dependent on the position of the thermocouples. Large hydrodynamical waves

were also observed and recorded. At the moment we are carrying out a numerical counterpart for a more

comprehensive analysis of the experimental results presented here.

For the interval of time the experiments were monitored, the flow in both strata seems to be

correctly described by the classification given in [1], Region IV.

Acknowled2ments

The authors wish to thank the Programa Universitario de Energia as well as the DGAPA of the

Universidad National Aut6noma de M6xico (UNAM) for their financial support. R. Tovar holds a

scholarship from the Direcci6n General de Estudios de Posgrado, UNAM.


Nomenclature

h Height of one stratum L Length of the cavity (parallel wall configuration) A Aspect ratio h/L Ra Rayleigh number based on h Pr Prandtl number

References

1. J.C. Patterson and J. Imberger, J Fluid Mech. 100, 65 (1980).

2. R. Yewell, D. Poulikakos and A. Bejan, d~ Heat Transfer 104, 533 (1984).

3. J.C. Patterson, Transactions of ASME 106, 104 (1984).

4. G.N. Ivey, J. FluidMech. 144, 389 (1984).

5. S.G. Schladow, J.C. Patterson and R.L. Street, J. FluidMech. 200, 121 (1989).

6. S. Paolucci and D.R. Chenoweth, J FluidMech. 201,379 (1989).

7. S.G. Schladow, J. FluidMech. 213, 589 (1990).

8. J.C. Patterson and S.W. Armfield, J. FluidMech. 219, 469. (1990).

9. S.W. Armfield and J.C. Patterson, IJHMT 34, 929 (1991).

10. M.R. Ravi, R.A.W.M Henkes and C.J. Hoogendoorn, ~ FluidMech. 262, 325 (1994).

11. C. Saedeeler, A. Garcimartin and J.K. Platten, Phys. Fluids 8, 670 (1996).

Received June 10, 1999

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Buoyancy driven flow of two stratified liquids in a cavity