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Int. Comm Hcor Mass Tnwufec Vol. 28, No. 2, 221-231, 2001 pp. Comrinht 0 2001 Elsevier Science Ltd Ri’iGin the USA. All rights reserved
07351933/01/S-see front matter
PII: !SO735-1933(01)00229-9
BUOYANCY DFUVEN FLOW OF TWO STRATIFIED LIQUIDS NUMEIUCAL VS. EXPERIMENTAL BESULTS
R. Tovar, F. E. Avila’ and J. Rojas Centro de Investigaci6n en Energia y Posgrado en Ingenieria de la Universidad National
Aut6noma de Mexico, A.P. # 34, Col. Centro, Temixco, Mor., 62580, Mexico.
(Communicated by J.P. Hartnett and W.J. Minlcowycz)
ABSTRACT A study on a 2-D transient natural convection flow of two stratified Liquids in a cavity is presented. The motion was analyzed numerically and compared with previous experimental results. The numerical initial concentration profile, across the two layers, was fed with real values measured in the experimental array. During the measurements of the concentration profile, special attention was given to the points across the diffuse interface. Such strategy brought immediate agreement between experiments and numerics allowing undertaking with great detail the analysis of certain fluctuations of the flow. Certain time and length scales were estimated in our experiments as well as in our numerics. These values were compared. whenever possible, against the theoretical time and length scales reported in the literature. Q 2001 Ebevier Sdenee Ltd
In a recent communication [l] we reported on the experimental set up, the techniques and the obtained
results for a natural convective flow of a two-layer fluid in a side-heated cavity, the hot wall of which could
be moved to an inclined position. In there, it was also established that the most relevant works for our own
work were those by J.C. Patterson and J. Imberger [2], Patterson [3], G.N. Ivey [4], S.G. Schladow et al. [S],
S.G. Schladow [6], J.C. Patterson and SW. Armfield [7], and SW. Armfield and J.C. Patterson [S]. Our
experimental results in the previous communication seemed to fall, for the interval of the time the
experiments were monitored, in Region IV given by Patterson and Imberger [2].
’ deceased.
221
222 R. Tovar, F.E. Avila and J. Rojas Vol. 28, No. 2
A clear conclusion from the experimental results was that the global features of the motion did not
change with the angle of inclination of the hot wall neither did they with the initial concentration for the
lower stratum. However, for each experiment performed, the behavior of the upper stratum was found to be
very different than that of the lower one. For instance, the average velocity of the thermals, as estimated
through their temperature signals, and the wide variety of thermal and hydrodynamic oscillations found,
were observed to be different weather the point of observation was located in the upper or the lower stratum.
In what follows we intend to give an account of these observations and provide a numerical approach that
describes them reasonably well.
Summaw of the Ex_primental Work
As a way to advance toward the objective of this paper we begin by recalling the type of flow we are
considering. The temperature of a vertical wall of a two-layer fluid cavity was suddenly increased, then a
boundary layer was formed in each of the strata. They went up due to buoyancy effects until they reached
their top boundary. At that moment, they both turned toward the opposite wall, each one becoming then what
is known as a thermal horizontal intrusion. This problem, for a straight wall’s cavity, is graphically defined
in Figure 1, and a set of typical temperatures signals for both strata is shown in Figure 2. These signals kept
track of the motion of the thermal intrusions towards the cavity’s opposite wail. The main results of that
experimental work can be summarize as: 1) the angle of inclination (45 to 90 degrees) and the initial
concentration (0.5 to 2.5 % by weight) did not have an appreciable effect on the global motion of the thermal
intrusions in their way towards the opposite wall; 2) as a result of a less active interaction between the liquid-
air interface, a faster motion of the thermal intrusion in the upper stratum with respect to that in the bottom
layer was evident for all angles of inclination (90,60 and 45 degrees) and for the two concentrations (2.5 and
0.5 % by weight); and, 3) no much transversal mixing or entrainment of fluids occurred during the most
intense stage of the transient motion.
__.--.---. Horizontal intrusion
WATER
FIG. 1 Schematic representation of the phenomena.
Opposite Wdl
Vol. 28, No. 2 BUOYANCY DRIVEN FLOW OF TWO STRATIFIED LIQUIDS 223
Temperature evolution on the points localized in the upper and lower thermal intrusions.
In addition, short period oscillations of the motion as well as long oscillations were identified in the
temperature signals. They vary according to position, the strongest ones being detected near the point of
turning from a vertical thermal boundary layer to a horizontal intrusion. Large hydrodynamic fluctuations
were also observed and recorded. As it was said in the previous communication, a numerical counterpart was
being simultaneously carried out, the results of which are the central part of the present work.
Numerical ADDLE@
The conservation equations along with the proper boundary and initial conditions were solved
numerically by means of a commercial program 191 which makes use of finite volumes to discretize and
convert them in a set of algebraic ones.
Due to the different thermal and concentration regions found in the experiments, the numerical grid
was divided in 16 parts, each with its own mesh size according of the expected temperature and/or
concentration gradient. A region with a larger gradient in one of the properties was given a finer grid mesh.
The distribution by regions and the individual characteristics of the grid cells can be seen in Table 1. A total
number of 37200 cells were used; 300 along the horizontal coordinate, x; and 124 along the vertical
coordinate, y. The inclination of the wall. for the cases of 60 degrees, was simulated by making used of a
special command furbished by the program, which assigns a value between 0 and 1 according the degree of
obstruction of a cell to any property of the flow. Full obstruction within any side of a cell means no-transfer
of any property across it. Figure 3 shows the numerical grid used for the parallel wall case; this was modified
to simulate the inclined wall cases.
224 R. Tovar, F.E. Avila and J. Rojas Vol. 28, No. 2
region Xl
regions Y6, Y7andY8 +
4 ,
region X2
1
i region Y5 -_j I
regions Y2 __i Y7 mld Y4 I
i mgionY1 -*
-1=3oomm m-
FIG. 3 Numerical grid used to run the calculations.
TABLE 1 Distribution of cells along the x and y coordinates.
t b=loomm
I
Regioa Xl x2
Number of cells 193 107
Cell length 0.3ooo 2.2626
Total length
Total length of the region 57.90 242.10 300.00 mm
The boundary conditions fed into the program for the rectangular cavity are the following (1 is the
length and h is the height of the cavity):
x=0, OSySh.
x=l, osym y=o, osxsl, y=h, OSxSl,
u=o. v=o, T= 2.6*tanh(t/4-1.5)+27.4 aciax=o u=o, v=o$ n/ax=0 acbo WO, v=o, niay=o acfaFo
way=o. ~0, a77aFo acfay=o
Vol. 28, No. 2 BUOYANCY DRIVEN FLOW OF TWO STRATIFIED LIQUIDS 225
Since, in the experiments, it was not possible to achieve a truly instantaneous temperature increase in
the hot wall, and neither it was possible for miscible liquids to have a perfect step function for the initial
concentration profile. It was necessary to devise a strategy to reproduce the experimental conditions in the
numerical solution. So, the increase with time of the wall temperature was simulated by means of a
hyperbolic tangent iimction which, when compared with experimental tests. closely followed the behavior of
the experimental temperature signal at the hot wall. For the initial concentration distribution to be fed into
the numerical program, a real measured concentration profile was used. This distribution had been
previously measured in situ. By proceeding in this way, the numerical results were quickly brought to agree
with the experimental results.
Results and Discussion
Most of the experimental behavior was described in the previous communication. Here we complete
the description of the motion by saying that the global behavior of the flow was described satisfactorily by
our numerical scheme. The following discussion about the characteristics of the stratified flow will pertain to
the straight wall’s cavity with a 2.5 7%~ of salt concentrations in the lower strata. Then, some remarks will be
made regarding the flow in the inclined wall cavities.
Figure 4a shows how the upper thermal intrusion advances faster than the lower one, the time of
observation ls 150 set; the upper thermal intrusion seems to touch the opposite wall in about this time. It was
pointed out that this effect was caused by the more active transfer of heat and momentum between the layers
of brine and pure water than the transfer occurring between the layers of pure water and air. The numerical
scheme seems to reproduce this fact. Figure 4b shows the velocity distribution for the same interval of time.
It can be seen a persistent wave near the turning point. We recall from our flow visualization the persistent
wavy formation near the hot wall for both strata, much in the same way that it is depicted in the numerics.
Figure 4c shows the concentration distribution. One can see that the original concentration profile was well
preserved in the numerical scheme, this agrees with the experhnental fact that the initial concentration profile
was well preserved for the first 1800 set of the experiments, the more intense stage of the motion [ 11.
The development of the boundary layers in the numerical scheme followed closely the description
given in reference [2]. Figure 5 shows for the upper layer the growing of both the thermal and viscous
boundary layer, taken at y=80 mm. The time it took for the thermal boundary layer to reach a steady state
was of about 12 s and its thickness then was about 3 mm; for the viscous boundary layer it took about 15 s
226 R. TOW, F.E. Avila and 3. Rojas Vol. 28, No. 2
and its thickness was 6 mm. The small negative values for y components of the velocity, for times greater
than 9 s and beyond x=4 mm from the wall, were due to fluid entering the boundary layer. Figure G shows
the double boundary layer at y=28 mm, i.e., within the bottom stratum; its characteristics were in all similar
with the ones just described corresponding to the upper layer. This is expected since at that height there was
yet no change in the flow direction due to the encounter with the diffusive interface.
(4
(b)
(cl
30
P _2__..______._____44 ____ ____^ ________ _ ____ -. ._ -7 _ _ _ _.. , .---___ _ _ - - _ _ _ _ _ - - _ _ _ - - - - - - - - _ - - - - - - - - - - - - _ - . . . .
. ~
(. ., -.- . . . . . . . . . . . . . . . (
I !. ..............................................
\. ............................................
.............................................
-: .02 m/a.
0
-.-~--
2.5
FIG. 4 (a) Thermal intrusions in a 90 degrees cavity, t= 150 sec. (b) Velocity distribution. It can be seen a persistent
wave near the turning point. (c) Concentration distribution. One can see that the original concentration profile is well preserved.
Vol. 28, No. 2 BUOYANCY DRIVEN FLOW OF TWO STRATIFIED LIQUIDS 227
(a)
FIG. 5 The growing of both the thermal and viscous boundary layer for the upper stratum. at y=80 mm
FIG. 6 The growing of both the thermal and viscous boundary layer for the lower stratum, at y=28 mm
A very different picture was obtained when one calculates these proflles in a zone located where the
flow change of direction. within the dlffisive interface near the hot wall. In this zone, horizontal thermal
intrusions were being generated from a vertical thermal boundary layer. Figure 7 shows the events occurring
at y=&l mm ‘Ihe first thing one can see is that. due to the strong sahnlty gradients over that height which
cause a ceiling effect, the temperature variations were effected along the x direction forming the lower
thermal intrusion Figure 7 also revels in great detail many interesting aspects of the thermal intrusions.
Between x=0 and x=2.5 mm, the temperature variation were similar than in the cases described earlier (Y=28
mm and ~80 mm). In the intraval x (2.5, 3.75 mm) and between 18 and 33 sec. it was possible to
distinguish small temperature oscillations; though negligible, they indicate, to a much lesser scale due to the
strong stratification, the same type of oscillations found experimentally for the upper stratum. Figure 7 also
228 R. TOW, F.E. Avila and J. Rojas Vol. 28, No. 2
shows how a thermal front of a constant temperature moves towards the cold wall. This can be seen by
drawing a horizontal line at the desired temperature. Each intersection with a temperature curve for different
times gives the position of the front at a particular time.
It is worth noting that, except for the observation made above for a span of few seconds, there was not
at any other period of time. any other evidence of thermal oscillations at this height. In the experiments, there
was no clear evidence of thermal oscillations in the top of the lowa strahm, however, this could have been
caused by the smallness of the fluctuations themselves.
29
I . I .
0 5 10 15 20 25
x (mm)
FIG. 7 Temperature profiles for an interval of time of 33 sec., at y== mm.
Fluctuations though depended on the region where the analysis was made. For instance in the top of
the upper stratum, at (20,96), (60,96) and (120,96) oscillations in temperature (Figure 8a) and oscillations in
velocity (Figure 8b) were cleared observed. The temperature as a function of time presented an over-shut
followed, in the last two points, for a monotonic increase of temperature and at (20,96) a second oscillation
was also observed. The velocity traces also presented an over-shut but this time followed by a monotonic
decrease. The maximum in temperature in all cases occurred few seconds before the corresponding
maximum in velocity.
In the lower stratum at (20,38), (60,38) and (120,38) the temperature traces did not present oscillations
(Figure 9a) but velocities, although of lower amplitude, were similar to those trace5 in the upper Stratum.
Vol. 28, No. 2 BUOYANCY DRIVEN FLOW OF TWO STRATIFIED LIQUIDS 229
Thermal oscillations in the experiments were, in agreement with the numerics, only detected in the upper
stratum.
--, , , , , 1 yj/ *‘, 1 , , 0 50 too 150 200 260
0 50 100 (50 200 250 I (d 1 (8)
FIG. 8 Temperature and horizontal velocity as a function of time at (20,96), (60.96) and (120,96).
8.0. .
27.6 -
(a) W 4.0. . , . , ,
3.6 -
WdOnnn
28.6 :,Jp
3.0 -
27.0 x.dOWMll 2.5
o a
L x=12Omm f dOrnrn
c 28.0 =, t.5
2.0 ;)p-+ J-20 mm
x.120mm
25.6 10
05 26.0
0.0
24.5 0 50 100 160 200 250 0 50 tO0 150 200 250
1 (N t w
FIG. 9 Tempaature and horizontal velocity as a function of time at (20,381, (60,38) and (120.38).
Figure 10 shows the thermal intrusions at t = 150 set, in a 60 degrees wall cavity; the global behavior
is similar to that observed in Figure 4a for a 90 degrees cavity. The effect of the wall inclination in the
numerics. although small, was more evident than in the experiments. For hutaxe the velocity of the thermal
intrusions were 2.2 mm/s and 1.4 mm/s, for the upper and lower strata, in the 90 degrees cavity and 3.0 mm/s
and 2.8 mm/s in the 60 degrees cavity.
230 R. TOW, F.E. Avila and J. Rojas Vol. 28, No. 2
25
FIG 10 Thermal intrusions in a 60 degrees cavity, tz 150 sec.
Conclusions
The numerical scheme, with the inclusion of the experimental initial concentration distribution and a
time dependent wall temperature, reproduced the global thermal and viscous behavior satisfactorily. The
upper thermal intrusion advanced faster than the lower one in both the 90 degrees and 60 degrees cavities.
Velocity of the thermal intrusion of both strata was larger in the 60 degrees cavity than the corresponding in
the 90 degrees cavity. That difference was not detected in the experiments mainly due to the lower spatial
resolution.
The development of the double vertical boundary layer in each stratum followed closely the
description given Patterson and Imberger [2] for square cavities. The time and length scales of the upper
thermal boundary layer were in good agreement between numerics (12 s and 3mm) and experiments (15 s
and 3mm). These numbers were slightly larger than the corresponding in the theory (5 s and 1 mm). Also the
time for the upper thermal intrusion to touch the cold wall was larger in the numerics (150 s) and in the
experiments (190 s) than in the theory (120 s). Oscillations in velocity were observed in both strata but they
were strongly dependent on the position of analysis and oscillations in temperature were only observed in the
upper stratum, in agreement with the experiments.
Vol. 28, No. 2 BUOYANCY DRIVEN FLOW OF TWO STRATIFIED LIQUIDS 231
The authors wish to thank the DGAPA and the DGEP of the Universidad National Aut6noma de
M&co for their financial support.
1. R. Tovar. F.E. Avila, J. Rojas and B. Vargas. ICHMT. 26,955 (1999)
2. J.C. Patterson and J. Imberger. J Fluid Mech. 100,65 (1980)
3. J.C. Patterson Transactions of ASME 106. 104 (1984)
4. G.N. Ivey. J. Fluid Mech. 144,389 (1984)
5. S.G. Schladow, J.C. Patterson andR.L. Street. J. Fluid Mech. 200.121 (1989)
6. S.G. Schladow. J. Fluid Me& 213.589 (1990)
7. J.C. Patterson and S.W. Armfield. J. Fluid Mech. 219,469 (1990)
8. S.W. Armfleld and J.C. Patterson. IJHiUT34,929 (1991).
9. Adaptive Research, CFD2000 Version 3. O-User Is Guide, (1997).
Received November 19, 2000