Indi an Journal of Eng ineering & Materi a ls Sc iences Vol. 9 , December 2002, pp. 455-463

Convection in differentially heated superposed air-water layersT

Sunil Punjabi, Anamika Sethia & K Muralidhar

Department of Mechanical Engincering, Indian Institute o f Techno logy, Kanpur 208 01 6, India

Received 12 December 2001; accepted 16 September 2002

Convec ti on in a d iffc rcnti a lly heated two- layer system consisting of air and water has been studied experimentally us ing lase r- interferometry. The cav ity used for fl ow vi sua lization is square in cross-sectio n and rectangula r in plan having d imcnsions of 447x32. 1 x32. 1 mm] Experime nts with the cav ity (half-filled with wate r, the resl being a ir) are reported here. T he fo llowing tcmpcraturc d iffe rences have been imposed across the hot and the cold walls o f the superposed layers: ( i) L'.T~ I O K, (i i) L'.T= 15 K , and ( iii ) L'.T= 18 K. T he present study is aimed at understanding the fo llowing issues: (a) the in fl uence of Ray le igh number on the steady the rmal fi e ld, and (b) !low coupling mechani sms between the layers. The prese nt in vestigati ons show that the thermal fi e ld in the !luid layers is primarily dete rmined by the temperature d iffe rence and hence Ihe Ray le igh nlllnber. Further, the two- layers are the rma lly coupled at a lower Rayle igh number, while the mechani ca l and thermal coupling bo th become significant at hi gher Ray le igh numbers.

Convection refers to heat transfe r in a fluid when enhanced by energy transport due to the fluid ve locity. I n natura l convection, the fluid motio n is set up sole ly by the buoyancy fo rces ari sing from the presence of the d iffe renti a ll y heated boundaries.

Ray leigh-Benard convec tion refers to the fl o w fie ld in a fluid layer that is confined between two para lle l infi nite thermall y conduc ting plates, and is heated from below and cooled from above. The motion is dri ven by a fixed temperature di fference. The hot f lui d expands and produces an unstable density grad ient in the fluid layer. If the density gradient is suffic ient ly st rong, the hot fluid ri ses and the cold flui d descends , thus setting up a roll pattern . The convec ti ve tl ow results in an enhanced heat transport from the hot to the cold surface .

In si ng le layer convecti o n, a tluid complete ly fill s the space between the plates and no free surface is fo rmed . Convecti ve motio n starts once the buoyancy force exceeds the viscous force. The re lative s trength of thc two fo rces is characte ri zed by a dimens io nless quantity, the Ray le igh numbe r. It is the re lative measure of the potentia l energy available in the buoya ncy fi e ld to viscous d iss ipatio n. The non­di mensional parameter Ray le igh number Ra IS

dc fined in terms of buoyancy and viscous fo rces as:

Ra ::: grav itatio nal potential energy/viscous di ss ipatio n

t prescntcd at the 28'h Nat ional Confere nce on Fluid Mechanics & Fluid Power, held at Punjab Engincering College, C handi garh, during 13- t 5 Dece mber 200 I

g{3(T/w / - Teol" )H 3

VI(

. .. (\)

Two-layer convection refers to buoyancy-dri ven movement in diffe rentia lly heated superposed hori zonta l fluid layers. The main di fference between one and two layer co nvectio n is in the appearance o f an inte rface, in turn a free surface if o ne of the flui ds is a gas . Co nvectio n in such problems can be surface tension dri ven, in additio n to be ing suppo rted by buoyancy.

In a two-l ayer system, Ray le igh-Benard convection is characteri zed by two di stinct modes of fl ow coupling between the fluid phases via the interface. T hese are respecti vely known as thermal coupling and mechanica l co upling. In thermal coupling, the recirculation patte rns in the indi vidual layers are driven by the temperature di fference appropri ate for each of the m. It is, thus, poss ibl e fo r the ro ll s in cach phase to have identical sense, clockwise or anti­c lockwi se. In mechanical coupling, the c irculation in one phase drives that in the o ther by the mechani sm of vi scosity, thus requiring that the two ro lls be oppos ite ly ori ented . Fig. I shows the two coupling modes that are poss ible in two-layer convectio n.

The config uratio n of inte rest in the present work has fundamental as well as practica l importance. Applicatio ns of two-layer convection can be seen in la rge sca le geophysical studies and c lass i fied techno log ies such as crystal growth .

Zeren and Reyno lds I conducted analytical studies o n two- layer syste m and predicted that the stabili ty

456 INDIAN.I . ENG. MATER. SCI.. DECEMBER 2002

C~ o

- --T HERMAL

A IR

INT E RFACE

W AT E R

o o M ECHANIC A L

Fig. I - SchcJ1lati c represelllation or thcn na l anel mechanica l coupl ing in two- layer convec tion

limits are dependent on the properti es of the fluid s. Instability was also seen to vary with the total depth of the layer and the rati o of the layer heights. Rasenal el 0 1.

2 developed a mathemati cal model to investi gate the onset of convecti on in two layers o f immi scible fluids. The two superposed fluids w ith combinations of ethy lene glyco l-oil and ethy lene glycol-decane were selec ted. The authors showed that the convecti on in the two layers may occur in the form of either viscously or thermall y coupled moti ons. Dijkstra3 numericall y studi ed pattern selec ti on in small aspect rati o containers. T he author has conc luded that the no-s lip sidewall s greatl y influence the mUltipli city o f st.abl e steady pattern s. Prakash and Koster" ex perimentall y studi ed two-dimension convection in a system or two immi sc ible liquids. For now visuali zation, rea l- time holograph ic i nterfcrometry was used. T he authors concl uded that thermal and mechanical coupling mech ani sms were poss ible between the flui d layers, depending on the Iluid properti es and the cav ity Ray leigh number. A ndereck el 01 .5 presented experimental results for two immi scible flu id layers, dri ven by a verti ca l temperature gradient. They observed ti me dependent variations in the nature of coupling between two­layers. Schatz el 0 1

6 worked w ith a two- layer system consisting 01' sili cone oil (Pr=8 1) and air. The shadowgraph technique was used fo r now visuali zati on. T he authors reported on the transition between hexagonal and square patterns in the convection planforms. It was conc luded that the transition occurred as the heati ng rate was increased. Johnson el 0 1.7 conducted experiments to study the crfcc t of layer height on the pattern formation in si l icone oil -air system. For fl ow visuali zation, an inrrametric camera was used. The camera measures the infrared radi ati on emitted by sil icone oil and thus determi nes th e temperature distribution at the interface. Mishra el al .R addressed thc problem of Rayleigh- Henard convection in a horizontal sin31c layer of air at Ra=34,800. The authors interpreted

their res ult~-, to show that the flow in the tluid layer was associated with a buoyant plume ri sing from the hot surface. The fluid was seen to cool at the top boundary while descending all around the central plume.

The flui d phases considered in the present work arc air and water. Experiments have been conducted with increasing values of the cav i ty temperature difference. The convection pattern in the cav ity has been imaged using a laser interferometer. The objectives o f the present work are to examine the influence o f Ray leigh number on the steady thermal fi eld. temperatures attained at the interface, heat fluxes at the solid wall s, and the nature o f fl ow coupling between the two layers. An order-of -magnitude ana lys is shows surface tension effec ts to be o f secondary importance in the present work.

Experimental Pl'Ocedure The test cell con s i s t ~; of three sections namely the

top tank , the tes t secti on and the bottom lank . The cavity has a square cross-secti on o f edge 32 .1 mm and a length o f 447 mill , thus giving an aspect ratio of around 14. To a first approximation, th is represents a near two dimensional cav ity in the sense that gradients are dominant in the verti ca l directi on, but not al ong the cavity length. The two-dimensional approx imation of the thermal fi eld is known to be unreali sti c at very high Ray leigh numbers (Mishra el 0 1.

8).

The fluid layer is confined by two aluminium pl ates of thickness 3 mm above and below. The two longer side wall s are made o f perspex and bakelite sheets which act as insulating wall s. The two square sides are co vered by opti cal windows to permit the passage of the light beam. The windows are made of fused sili ca, since it has excellent surface properti es such as parall el ism, transparency and surface f i ni sh. T he hot and the cold surfaces are mai ntained at un iform temperatures by circulating water continuously over them. In a model experiment, the top wall is coo led to 16°C, whil e the bottom wall is heated to 26°C, the

ambient temperature being 22°C. Both wall s have bee:l maintained at the respective temperatures to within ±O.2°C during the ex periments. The temperature difference across the cavi ty is kep t at 10 K , wi th a tolerance or ±O.2 K. Experiments mainta ining temperatu re di fferences of 15 K and 18 K have also been conducted . Fig. 2 shows the schematic diagram of the test-cell. All experiments have been continued for 4 h to confirm that steady state has

PUNJAB I el al.: CONVECTION IN DIFFERENTIALLY HEATED SUPERPOSED AIR-WATER LAYERS 457

Fig. 2- Schelllatic of the test-ce ll

reached in the fluid layers. Although, the flow field gets nominall y stabilized in 2 11 , the ex perimel1l has continued for an additional 2 h for detecting changes in the fringe field. As the volume of the test-cell is small , the thermal fi elds in the hot and the cold surfaces and the flow fi elds in the fluid layers all approach steady state together. In thi s respect, the thermal loading of the two-layer system may be categorized as continuous, as against a sudden loading commonly employed in numeri cal calculations. The effect of the manner of appl ying the temperature difference on the convection pattern is as yet unclear.

For the tcmperature fi eld measurement in the fluid medium, the primary instrument employed is the Mach-Zehnder interfero meter. It empl oys a 35 mW He-Ne laser and 150 mm diameter optics. Interferograms are recorded using a CCD camera with a 512x512 pixel reso lution. The camera is interfaced with a PC through an 8-bit AID card which di gitizcs light intensity levels over range of 0-255 . Image acquisition is at video rates (50 images/scc). For measurements in liquids, a reference chamber is required to he incl uded with thc interferometer to compensate for refracti ve changes under isothermal conditions.

K-type thermocoupl es (74 gage) were used to monitor the temperature of th e surfaces of the cavity and the ambient temperature throughout th e experiment. They were connected to a 30-channel recorder. A II the ex peri ments were performed in the infinite fringe setting of the interferometer. When a thermal disturbance is introduced in the path of the test beam, fringes appear in the field-of-view that represent isotherms. The al ignment of the

interferometer In the infinite frin ge setting IS individually carried out to capture interferograms for the air ancl the water sides. For thi s purpose, the reference chamber is also half-filled with water, the rest being air at atmospheric pressure. The temperature drop per frin ge shift, /)'T,. can be calcu lated from first principles as:

(AI L) (dn I dT)

... (2)

where ), is the wavelength of laser beam, dl/ldT is the rate of change of refracti ve index with temperature ancl L is the effective length of the test cell.

The methodology adopted in the present work is based on the analysis of the image obtained from ex periments. The image invariably contains superimposed noise. Image process ing operati ons compri sing of : (a) filterin g, (b) image enhancement, and (c) thinning, were adopted to remove the noise as well as to improve the quality of the image. Filtering utilizes the Fast Fourier transform (FFT) al gorithm to remove noise and retains information in the image. Image enhancement technique is subseq uentl y req uired to improve the quality of the image that has become blurred owing to loss of contrast after filtering. Fringe thinning is a process of ex tracti on of minimum intensities in the dark frin ge bands. Calculations for the local and the average Nusse lt number, and the temperature di stribution have been accomplished using the thinned frin ges of the i nterferogram.

Errors in the ex perimental data are associated with mi sali gnment of the apparatus with respect to the li ght beam, image processing operations including filtering, thinning and assigning temperature of fringes . Errors related to refract ion effects in case of water have been found to bc hi gh and affcct the air-water interface locati on. All ex periments have been conducted scveral times to es tabli sh the repeatability of the fringe patterns. For all experiments, the plate­averaged Nusselt number has been found to be in good agree ment with publi shed correlations. Interface temperature calcu lated from the top and the bottom surfaces are within ± I %. The average Nussclt number of the cavity on the water and the air side have been compared to those reported for single layer convec ti on. Uncertainty in compari son is expected since in a single fluid layer the top and boltom boundaries arc solid walls , while in the present case, one of the bounda ries is an interface. For the air side,

458 INDIA J. ENG. MATER. SCI. , DECEMBER 2002

the Nussclt number matches the correlat ion to within ±3% and on water side it is w ithin ±IO%. The Nusselt number correlations employed in the present work are given in re lat ions 4 and5.

Results and Discussion Experiments have been conductcd in a two- layer

system consisting of 50% air present abovc 50% water, both contained inside the sq uare cav ity. T he tcmperature di f ferences imposed between hot (bottom) and cold (top) sidcs fo r thc expcriments arc: ( i) .6 1'=10 K, ( ii ) .61'=15 K , and (iii) .6T= 18 K . The results obtained rrom ex periments arc in thc form of rri ngc patterns that are representati ve of buoyancy­drivcn flows in the two Iluid phases. I f the fi eld is two di mcn;., · ona l, the fri ngcs represent iso therms (constant tcmperature contours) owing to the infinite fringe sct! i ng of the interferometer. I n three di mensions, thc fringe pattern s are indicat i ve o f the line intcgral s of thc thermal field (Mishra e/ al. s).

Results have been presented in terms of the indi\'idual Ray leigh numbers in the fluid phases . These Ray leigh numbers are calculated on thc basis of the temperature difference o f the neares t wa ll wi th the average interface temperature and the thermo-physical properti es of the fluids evaluated at the respective average temperatures. Local as well as the average

usselt numbers on the hot and the cold walls have been calculated to understand the energy transfer mechanisms at thcse locati ons. These have been supplemented by the plots of the temperature cii stri buti on al ong the coordinate parallel to rhe hcight of the cavity.

The wa ll heat transfer rates have been determined in terms of the usselt number:

Nil = T ~T r ~: llr=().11 lim cold .

... (3)

where H is the height o f thc cav ity. The average usselt number 1'01' each or the plates has also been

compared with the experimental correlation reported by Gebhart e l cil.<J. For air, the correlat ion is given by:

[ ] [( )

111 1 1708 Ra ' II = 1 + 1.44 1 - -- + -- - I

Ra 5830 ... (4)

For water, the con'elarion is given by:

[ 1708] [( R )1 13 1 Nil = 1 + 1.44 I - -- + _(_I - I Ra 5830

+ 2 (Ra)' n [I 140

In (Ra )'n] 140

. .. (5)

Results have been interpreted to understand the now coupling mechan ism, a spec ial feature of two­layer convect ion. The coupling has been class ified as therma l or mechanica l depending on thc sign o f the respective ro ll s in the ai r and the water phases.

Flow characteristics at t1T = .10 K imposed across the cav it~'

I n the present set o f ex peri ments, a temperature di fference of 10 K is imposcd between the lower and the upper surfaces . This CrllTcsponds to a cavity

average temperature of 2 1°C, Using energy balance princip les, it is poss ible to estimate the interrace temperature by using the corre lat ions, Eq ~ (4) and (5) . The interface temperature for the presen t ex peri ment,

thus, has been found 10 be 25.79°C. The Rayleigh number in air and water layers can now be cal cu\att'd to be 4, 185 and /6,4 13, respectively. Thi s indicates that the driving buoyancy potential in the water layer is around 4 times larger than in air. It is also to be noted that these Ray leigh numbers rail in the stabl e regime of the transition diagram discussed by Kri shnamurti ICJ Accordi ngly, one can expect the therma l fields in the n u id layer to be t'vvo­dimensional. In the experimcnts, it has been observed that convection develops earli er in the lower water layer, due to a higher Rayleigh number as compared to the upper air layer. The interferograll1 depicted ill Fig. 3 shows steady two-dimensional f low patterns in the two-layer system. The fringes in the water phase

Fig. 3- Slcady Slale inleri'l.:rogram for a ir-waleI' syslcm ill a squarc cav it y (leill peralur::: d iffercnce = I () K)

PUNJAB I el (Ii.: CONVECTION IN DIFFERENTIALLY HEATED SUPERPOSED AIR-WATER LAY ERS 459

can be seen to be dense, even though the overall temperature difference here is quite small. This is because the temperature drop per fringe shift is much smaller for water as compared to air, the values being 0.0 16 and 1.527°C respecti vely. The higher Rayleigh number in water also indicates a more vigorous buoyant moti on. Thi s is confirmed by the greater disp lacement of the fringes, while those in air are sensibl y straight.

The interferogram in Fig . 3 was captured after an ex perimental run time of 4 h, when the flow fi eld was fully evo lved and the frin ge pattern s were quite steady. Since each frin ge is an isotherm, regions of a small fringe spacing can be associated with a large loca l diffusive heat flu x.

In the portion of the cavity fi ll ed with air, the largest loca l heat flux at the cooled cavity wall at steady state occurs at around the mid-plane of the cav ity. The fl ow develops in the form of single ro ll whi ch ri ses al ong th e left side wall and descends along the ri ght side wa ll. Thi s gives an uni cellul ar pattern in the cav ity with a sense of rotation in the clockwise direction. The strength of convecti on in air is weak, as call be co nfirmed by the dimini shed curvature and increased strai ghtness in the fringe patterns. The isotherms are densely placed near the top wa ll (the cold plate) as well as the interface in co mpari son to the central core region . The reason for thi s beha viour is the dominance of diffusive heat transfer near the wa ll s as aga inst convec ti ve heat transfer in the central region. ear the interface, the dense fringes indicate that the zone is thermall y active, despite the fac t that the water layer is not stri ctl y a so lid surface.

In the water phase, the larges t loca l heat flu x at the lower cavity wa ll at steady state occ urs near the mid­plane of the cav ity. Here, the fl ow develops in the form of two counter-rotating ro ll s with fl ow

0.5 (a)

, ,

0.25 ,

, , ' .... ',:~

, 1:

, --.. 0 >,

-0.25

-0.5 '---'---'---'--_'---.1 o 0.2 0.4 0.6 0.8

e Fig. -Ja- Te: lIlpcraturc profile at th ree: dilkrcnt column ,

descending al ong the vertical axis close to the cav ity center. The recirculating flow in water ex hibits stronge; convection compared to air, as seen in the greater curvature of the frin ge patterns. The isotherms are denser near the lower hot wall in comparison to the region near the interface.

The temperature profiles in the air and water sides of the cav ity at steady state are shown in Fig. 4a. The variation of temperature field is plotted as a functi on of the vertical distance for three co lumns along the width of the cavity. The y-coordinate is measured from the interface represented as yIH=O in the pl ot with air above and water below it. The fi gure clearl y indicates the large temperature drop on the air side as compared to the water side.

Interface temperatures were calcul ated indiv iduall y from the top as well as bottom wall s. The thinned fringes for air and water were empl oyed for thi s purpose . The temperatures of each fringe was ass igned on the basis of the value !1T", [Eq. (2)]. The ex perimentall y determined interface temperature was 25.82°C, whil e the value determined from correlations was 25.79°C. Thus, the difference in the average va lue of the interface temperature bet \oveen theory and experiment is less than ± 1%.

Us ing the temperature profil es in the fluid layers, the local Nusselt number distributi on was calcu lated at the hot and the cold wall s of the cavity. The var iation of the local Nusselt number at the respec ti ve wa ll s with respect to the coordinate parallel to the width of the cav ity is show n in Fig . 4b. The wid th averaged Nusselt numbers calculated are 1.96 and 2.97 for the top and the bottom wall s respectively. The reference values given by Gebhart et 0 1.9 are 1.85 and 3.68, respecti vely, in air and water. It is to be recalled that the correlati on for the average usselt number considers single layer in a cav ity bounded by solid wall s at top and bottom, unlike the present case,

~

Z

(b)

3 HOT WALL

2 COLD WALL

1 '----'--__ L-__ L-~L-~ o 0.2 0.4 0.6 0.8

x/W

Fig . 4b- Nussc lt nUlllber va ri a tio n at the wa ll s in a cavity (te mperature difle re nce=I O K)

460 INDIAN J. ENG. MATER. SC I , DECEMBER 2002

wh ich has an interface. With thi s factor taken into account, it can be concluded that the Nusselt number in the presen t experi ment matches we ll wi th the reference correlat ion.

The prese nce of a convect ive field in the fluid layers leads to a coupling between them. In the presen t work , the air and water layers are seen to be coupled at the interface. Here, a sin gle roll in air and two counter-rotating ro ll s in water have been observed. Thi s is depicted in Fi g. S. As the convective fl ow in each layer is dependent on the temperature dillerences across the respec ti ve fluid layers, one may cO llclude that the t\vo- Iaye rs arc thermall y coupl ed.

FloI\' characteristics at AT = 15 J( imJlosed across the cavity

In the present ex periment , a temperature difference of I:. :< was im posed between the two bound ing surraces . The cav ity average temperatu re was ca lculated to be 23 .S°C. The illterface temperature in the present experiment was found to be 30.7°C. Accordingly, the Ray leigh !lumbers calcu lated in the ai r and the water layers were 6,044 and 30,772 res pecti ve ly. These numbers indi cate that the driving buoyancy potential ill water is around 5 times larger tha n in air, lead ing to a more vigorou s buoyant moti on in wate r. It was observed in the ex periments that fl ow patterns in th e water layer had a tendency towards sli ght unstead iness . In turn , the rringes 0 11 the ai r side al so show margi nal unsteadi ness, bei ng dri ven by the ai r-water interrace. The interferog ram depicted in Fi g. 6 shows quasi-steady two-dimensiona l fl ow patterns in the two-layer system.

In the porti on or the cavity fi ll ed wi th air, the flo w develops in the rorm or a I., ing le roll. It ri ses rrom the le rt side wall and descends along th e right side wa ll . This again gives an uni ce llular pattern in the cavity wit h sense o r ro tati on in the clockwise direction.

Fig. :i- Schemalic dra wing o f coupling bel ween lwo-Iayers (le mpera lure ditTcrcllcc= 10K)

Convection in air in the present experiment has strengthened, as seen in the greater curvature in the fr inge patterns.

In the water phase, the fl ow develops in lhe form of four counter-rotating ro ll s with up-flow along the verti cal axis close to the cav ity center and down-flow near the sides . The fringe patterns show high curvature in the cen tral porti on of the roll, conrirming a stronger convective fl ow than in air.

The temperature profiles of isotherms in the air and water sides of the cavity at steady state are show n in Fig. 7a. The figure shows large temperature drop in air than in water. The interface temperature obtained from the ex periment is 30.8°C, wh ile the value from correlations is 30.7°C. The difference between two va lues is within ± I %. Local Nusselt number variat ion at the two wall s is show n in Fig. 7b. The width averaged N ussel t numbers are calcul ated: 1.94 and 3.40 for the top and bottom wall s, respecti vely. The reference values determined from the correlation are 2.04 and 4.2 1 res pectively in air and water. The diffe rences in the average Nusse lt number are seen to in crease, poss ibl y due to an increased importance of the interface in determ ining the flow patterns.

Flow coup ling betwecn air and water layers is more pronou nced in the present experiment, being linked at thc interface. Here, a si ngle ro ll in air and four coun ter-rotating ro ll s in water have been observed. The fl ow pattern is schemati ca ll y shown in Fig. 8. Con vec ti ve fl ow in each layer is dri ven by the individ ual temperature differences across the res pect ive riuid layer, indicating thermal coupling at the interface. At the same time, the conv ction ce ll s in

Fig. 6- - Steady state inlerferog ram for air-water system in a sq uare eav ilY (lemperature di ffe rence= l:i K)

PUNJABI el al.: CONVECTION IN DIFFERENTIALLY HEATED SUPERPOSED AIR-WATER LA YERS 46 1

0.5 r--==---------,

0.25

o

-0.25

-"" '"

(a)

'-' ''-,.

""~. \', , . , , , , ,

-0.5 '-_'-_'-_L.-_L.------'

o 0.2 0.4 0.6 0 .8

e Fig.7a- TemperalUre profile at three d ifferent co lumns

4

:::J3

Z

2

1

(b)

HOT WALL

COLD WALL

o 0.2 0.4 0.6 0.8

x/W

Fig. 7b- Nu,sc lt numher vari,llio n at the wa ll s in a cavity (temperature dilTc re nce= 15 K )

,vater tend to destabili ze the circ ul at ion in atr by introduci ng unsteadiness. Thi s is possible only when the two- layers are coupled mechanica ll y as well . Thcrefore, it may be concluded that the fluid layers revea l full (thermal + mechanical) coupling at the in terf<:ce.

Flull' cha r ac teristics at t\ T = 18 K imposed across the c'lvit.y

In the present experiment , a temperature dilTerence of 18 K WdS imposed betwee n the two surfaces. The cav ity average temperature was 25°C. The interface temperature was ca lculated to be :n.64°c. The Rayleigh numbers were determined in the air and the water layers to be 7,088 and 42 , 147 respect ively. This reveals that the dri ving buoyancy potential in the water layer is 6 times larger than in air. A hi gh Ray leigh number in water was see n to develop a high uns tead iness in the frin ge pa ttern s of the water layer, part icu larly in the central rcgion of the cav ity. It has been observed that the air layer also registers unsteady flow patterns, leading to increased fringe curvature. The interfcrogram depicted in Fi g. 9 shows quas i-steady two-dimensional flow patterns in the

QGOG Fig. 8- Schematic drawing of coupling between two- layers

(temperature di ffe rence= t5 K)

Fig 9- Steady state interfe rogram fur air-wate r sys te lll in a square cav ity (temperature diffe re nce= 18 K )

two-layer system, seen at the end of 4 h of experi mentation .

I n the air layer, the now develops in the fo rm of a si ngle ro ll. I t ri ses along the left side wall and descends along the right side wall. This gives an unicell ul ar pattern in the cav ity with sense of rotati on in the clockwise direction. The frin ges in the central portion are seen to switch between two states, one co mparatively long-lived in relation to the other. The dominant trend in the cavi ty as captured is shown in Fig . 9.

In the water phase, the flow deve ~ ops in the form of fo ur counter- rotating rolls with upflow close to thc cav ity center and downflow near sides. The fringes in the central portion have been seen to sw itch among 5-6 states, though only one of them is long- li ved. The dominant state has been recorded hy the camera, as show n in Fig. 9.

-+62 INDIAN J. ENG. M ATER. SCI, DECEMBER 2002

-0.25

-0.5 '--_L---..J'----..J_----'_----'

o 0.2 0.4 0.6 0.8

e Fig. I Oa- TL:lllperatun: prufi le at th ree different co luillns

4

::) 3

Z

2

1

(b)

HOT WALL

COLD WALL

o 0.2 0.4 0.6 0.8 X/W

Fi g. l()b- NlI sse ltnllmbcr var iati on atthL: wall s in a cavit y ( tcmperatu re di ITcrL: ncc= I X K )

The tcmperaturc pro files shown in Fig. l Oa, indicate large temperature drop in air compared to waleI'. T he interface temperature obtai ned from

ex periments is 33 .72°C, while the va lue from the

correlation is 33.64°C. The local Nusselt IlLtmber var iation at the two wa ll s is shown in Fig. lOb. The wid th-averaged Nusselt number ca lculated for the Gtvity wal ls are 1.99 ,1I~d 3.70 for the top and bottom surfaces respectivel y. The reference val ues eva luated using the correlation are 2.1 6 and 4.50 respectively in air and water. The experimental and the reference

usse lt numbers appear to be close. However, it should be recalled that signifi can t differences in average Nus:;e lt numbers would be seen if all poss ible flow states , including the purely transient are included in the ca lcu lati on .

Flow coup ling between air and water layers in the present ex periment is quite dominant at the interface. Ilcrc. a single roll in the air and four counter-rotating ro ll s in the water layer have been observed. This interpretation is schematica lly shown in Fig. II . It has been observed that the convec ti on ce ll s in water again

Fig. II - Schemat ic draw ing of coupling bctwccn two-layers (temperatu re clilTerence= I R K )

tend to des tabili ze the patterns in air, by introducing unsteadiness. It is ev idence of mechanica l coupling bel ween the two- layers at the interface. At the same time, Ihe fl ow fi elds in indi vidual layers are driven by the respecti ve temperature differences, suggesting the coupling to be thermal in nature. Th us, one may conclude that the fluid layers are full y coupled.

Conclusions Buoyancy-dri ven convection In a differenti ally

heated cavity containing air and water was experimentally studi ed, using a Mach-Zehnder

interferometer. The flow field was mapped in terms of fringe patterns that were indicati ve of isotherms in the cavity. The following temperature differences were

imposed across the hot and cold wa lls of the cavity : (i) t.T = IO K, ( ii) t.T= 15 K , and ( iii ) t.T= 18 K . These

led to Rayleigh numbers in the range in air and in water. A steady two-dimensional convection pattern

was obtained at the lowest Ray leigh number. An increase in the Ray leigh number led to weak unsteadiness in the flow patterns. T he temperature profile across the fluid layers clearl y indicated a large temperature drop in air compared to the water phase. The interface temperature calcu lated from water side was found to be quite close to that from the air side in all cases. The average Nusselt number for the cavity showed some difference w ith the reference Nusselt number, the reason being one of the s Ir faces, namely the interface is not rigid . At the lowest Ray leigh number, the two fluid layers were seen to be thermally

coup led at their interface. At the higher Rayleigh numbers, fu ll coupling consisting of both thermal and mechanical interactions were observed in the cavity.

PUNJAB I el al.: CONVECTION IN DIFFERENTI ALLY HEATED SU PERPOS ED AIR -WATER LA YERS 463

Nomenclature /\ = Asp~c t rati o U H

dllMT = Refrac ti ve index change with temperature. K - t g = Accel~ration due to gravity. m/s" /I = I-k igh t of the cav ity, m I. = Length of the cav ity, m Nil = Nusselt number /I ilr y= O.11

I 'r

No

T 'I i

l' i\ "/" \I'

/1 , I ,

I(

/)

I) ()

/IJ 0."

= Prandt l number U/ K

,/3(T - T )/-1 -' = Ray leigh numbe r ·~ 10", ",Id

VI(

= Temperature, °C = Interface tem peratu re. DC = Tern perallire di lTerence between hot and cold surfaces. K = T~ l1l perature difference b~ t wccn successive fringes. K = Widt h of the cavity. m = Coe fti c i ~ n t of volume ex pansion, K I

= Wave l cn~th of the laser. nm = Tl,erma l 7litTusivity. m2/s = Kinematic viscosity. m"/s = DCll sity of fluid , kg/nl" = NO Il -dilllensiona l tempcra ture

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