Imaging of a convective field in a rectangular cavity using interferometry, schlieren and...

Optics and Lasers in Engineering 42 (2004) 469–485

Imaging of a convective field in a rectangularcavity using interferometry, schlieren

and shadowgraph

Atul Srivastava, Atanu Phukan, P.K. Panigrahi*, K. Muralidhar

Department of Mechanical Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, UP, India

Received 1 May 2003; accepted 1 February 2004

Abstract

The present study is concerned with the quantitative imaging of buoyancy-driven convection

in a fluid medium that is confined in a horizontal differentially heated rectangular cavity. The

horizontal surfaces of the cavity provide a temperature difference, for initiating convection in

the fluid. The vertical side walls are thermally insulated. Three imaging techniques, namely laser

interferometry, schlieren, and shadowgraph have been utilized. Experiments have been

conducted in a cavity of length 447 mm and 32 mm vertical height. The cavity is square in

cross-section, and the imaging direction is parallel to its longer side. Convection in air and water

have been investigated. Temperature differences in the range of 5–50 K for air and 3–10 K for

water have been employed in the experiments. Quantities of interest are the temperature profiles

in unsteadiness in the thermal field. At lower temperature differences across the fluid region,

temperatures as recorded by interferometry and schlieren are in good agreement with each

other. Further, they match the numerical predictions, as well as correlations available in the

literature. Imaging based on shadowgraph is not as satisfactory at lower temperature

differences. At larger cavity temperature differences, the shadowgraph images become clear

enough for quantitative analysis, but the flow becomes time-dependent. The three techniques

reveal similar trends in terms of the spatial distribution of temperature gradients and the time

scales of unsteadiness. The schlieren and shadowgraph are more suitable for high gradients and

interferometry is suitable for low gradients and all these three techniques are not flow

visualization tools alone but are appropriate for quantitative imaging of thermal field.

r 2004 Elsevier Ltd. All rights reserved.

Keywords: Schlieren; Interferometry; Shadowgraph; Rayleigh–Benard convection; Rectangular cavity

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*Corresponding author. Tel.: +91-512-2597686; fax: +91-512-2597408.

E-mail address: [email protected] (P.K. Panigrahi).

0143-8166/$ - see front matter r 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.optlaseng.2004.03.003

1. Introduction

The buoyancy-driven convection in a fluid medium confined in a rectangularcavity [1] has been studied in this work by employing interferometry, schlieren andshadowgraph. The fluid is heated from below, cooled from the top and has insulatingside walls. The flow pattern associated with this configuration shows a sequence oftransitions from steady laminar to unsteady flow and ultimately to turbulence.Buoyancy-driven convection finds extensive applications in engineering, rangingfrom cooling of electronic components, growth of semiconductor crystals to materialprocessing [2].Wang and Zhuang [3] applied real time laser interferometric tomography for the

measurement of three-dimensional asymmetric temperature fields generated by twoheated rods. Muralidhar et al. [4] studied the transient natural convection in a squarecavity in the intermediate Rayleigh number range with air as the working fluid usinga Mach–Zehnder interferometer. Onuma et al. [2] conducted experiments on growth

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Nomenclature

g acceleration due to gravity, m=s2

h height of the cavity, mL distance traversed by the laser beam through the test cell, mn refractive index of the fluidn0 reference value of refractive indexFo fourier number, at=h2

Nu local Nusselt number, (Eq. (8))Pr prandtl number of the fluid, n=aRa rayleigh number, gbDTh3=nat time, sT temperature, KT0 reference temperature, KDTe temperature difference between successive fringes, Ka thermal diffusivity of the fluid, m2=sb coefficient of volume expansion of the fluid, K�1

l wavelength of the laser, nmn kinematic viscosity of the fluid, m2=sr density of fluid, kg=m3

r0 reference value of density, kg=m3

y nondimensional temperature, ðT � TcÞ=ðTh � TcÞ

Subscripts

c coldh hoto reference value

A. Srivastava et al. / Optics and Lasers in Engineering 42 (2004) 469–485470

of barium nitrate and K-alum crystals using Schlieren and Mach–Zehnderinterferometry and studied the effect of buoyancy driven convection and forcedflow on the microtopography of the crystal surface during its growth from anaqueous solution. Kosugi et al. [5] applied the schlieren technique to experimentallyrecord gas temperature profiles in the shock region of excimer lasers and correlatedthem to the xenon concentration in a helium gas. Tanda and Devia [6] implemented anovel schlieren technique without the need of intensity measurements for the studyof two-dimensional free convective heat transfer from heated plates and fromvertical parallel channels to the surrounding fluid. The authors made use of anopaque thin filament to measure the loci of the images of points deflecting the lightby the same amount. Schopf et al. [7] adopted a shadowgraph approach for the studyof convective flow in a water-filled square cavity which was differentially heated andcooled from the opposing vertical sidewalls. Lewis et al. [8] conducted a quantitativestudy of temperature distribution in a laser ignition experiment wherein an ethylene-air mixture was heated by a high power laser, and the hot gas was illuminated with acollimated laser beam using shadowgraph. To cross-check the results obtained fromthe shadowgraph experiments, the authors determined the temperature field in thesimilar test study using Rayleigh light scattering and demonstrated a good agreementbetween the results obtained from these two approaches. Agrawal et al. [9] havedescribed a color schlieren approach for temperature field measurements in gas jetsusing white light.The existing literature on the refractive index based optical methods shows that

out of the three methods, interferometry has been predominantly applied forqualitative as well as quantitative analysis of the thermal fields in buoyancy-drivenconvection problems. Schlieren finds its potential applications towards qualitativevisualization of the flow field and to a limited extent, it has been applied toquantitative studies. Shadowgraph has been extensively used for qualitative imaging.The three techniques, however, have not been jointly compared against a benchmarkexperimental configuration.The present work compares interferometry, schlieren and shadowgraph techniques

during the measurement of temperature in buoyancy-driven convection in arectangular cavity. A limited comparison with numerical simulation is alsopresented. The cavity is 447 mm long with a 32� 32 mm2 cross-section. The topand bottom walls of the cavity are maintained at uniform temperatures at all times inan unstably stratified configuration. Fluid media considered are air and water.Temperature differences of 5–50 K for air and 3–10 K for water have been employedin the experiments. Over the range of temperature differences considered, flow wasseen to become progressively unsteady, finally becoming turbulent.

2. Apparatus and instrumentation

The apparatus used to study the buoyancy-convection phenomenon in ahorizontal fluid layer is shown schematically in Fig. 1. The cavity is 447 mm longwith a square cross-section of edge 32 mm: The test cell consists of three sections

ARTICLE IN PRESSA. Srivastava et al. / Optics and Lasers in Engineering 42 (2004) 469–485 471

namely the top plate, the fluid layer enclosed in a cavity and the bottom plate. Thetop and bottom walls of the cavity are made from 3 mm thick aluminium plates. Theflatness of these plates was manufacturer-specified to be within 70:1 mm and wasfurther improved during the fabrication of the apparatus. The central portion of theexperimental apparatus is the test section containing the fluid medium. The sidewalls of the cavity were made of a 10 mm plexiglas sheet. In turn the plexiglas sheetwas tightly wrapped with a thick bakelite padding in order to insulate the test sectionwith respect to the atmosphere. The height of the test section was 32 mm and wasmeasured to be uniform to within 70:1 mm. Optical window was provided in thedirection of propagation of the laser beam. It was held parallel to the longestdimension of the cavity for recording the projected convective field in the form oftwo-dimensional images. The apparatus was enclosed in a larger chamber made ofthermocole to minimize the influence of external temperature variations. The roomtemperature during the experiments was a constant to better than 70:5�C over a10–12 h period. The thermal fields in air stabilized over a period of 5–6 h: In water,dynamically steady patterns were realized in 1–2 h:

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Water

447 mm

Laser

Water

Water Constant temperature bath(Hot)

Water

Constant temperature bath(cold)

Baffles(Bakelite)

Cavity

Insulation(Perspex)

Air gap

Insulation

Hot wall

Window frame

Cold wall

Optical window

z

y

x Th

Tc

g

32 mm

Fig. 1. Schematic drawing of the test cell.


In the experiments, the thermally active surfaces were maintained at uniformtemperatures by circulating a large volume of water over them from constanttemperature baths. Temperature control of the baths was rated as 70:01�C; at thecavity location, direct measurements with a multi-channel temperature recordershowed a spatial variation of less than 70:1�C: For the upper plate, a tank likeconstruction enabled extended contact between the flowing water and the aluminiumsurface. Special arrangements were required to maintain good contact between waterand the lower surface of the plate. Aluminium baffles introduced a tortuous path toflow, thus increasing the effective interfacial contact area.For air, the temperature differences applied across the cavity walls were DT ¼

5; 10; 20; 30; 40 and 50 K: These correspond to Rayleigh numbers of 1:40� 104;2:70� 104; 5:10� 104; 8:50� 104; 1:13� 105 and 1:40� 105: In experiments withwater, the temperature differences applied were DT ¼ 3; 5; 6; 8; 10 and 13 K: Thesecorrespond to Rayleigh numbers of 2:50� 106; 4:40� 106; 5:40� 106; 7:50� 106;9:80� 106 and 13:50� 106; respectively. However for qualitative comparison,interferometric, schlieren and shadowgraph images corresponding to Ra ¼ 2:50�106 (the lowest value of Rayleigh number applied in the present experiments) havebeen shown. The values realized in the present experiments are quite large incomparison to the critical Rayleigh number of 1707 for the infinite fluid layer at theonset of convection [1].For optical measurements, a helium-neon laser (35 mW; continuous wave,

632:8 nm) has been employed as the coherent light source. A monochrome CCDcamera of spatial resolution of 768� 574 pixels was used to capture the convectivefield in the form of two-dimensional images. The camera was interfaced with apersonal computer (256 MB RAM and 866 MHz processor speed) through an 8-bitA/D card. Light intensity levels were digitized over the range of 0–255. Imageacquisition was at video rates of 31 frames=s:

2.1. Optical arrangement

AMach–Zehnder interferometer has been used for the interferometric study of theconvective field. The mirrors and beam splitters in this configuration are of 150 mmdiameter. The beam splitters have 50% reflectivity and 50% transmitivity. Themirrors are coated with 99.9% pure silver and employ a silicon dioxide layer as aprotective layer. The interferometer floats on pneumatic legs to isolate the opticsfrom external vibrations. The experiments have been carried out in the infinite fringesetting of the interferometer. In this setting, the interferogram is a set of fringes onwhich the depth-averaged temperature field is a constant.A Z-type monochrome schlieren system has been utilized to record the

temperature gradient field in the form of two-dimensional images. The opticsinvolves concave mirrors of 1:30 m focal length. Relatively large focal lengths makethe schlieren technique sensitive to the thermal gradients. The concave mirrors usedin the set up are of 200 mm diameter. The knife-edge is placed at the focal length ofthe second concave mirror. It is positioned to cut off a part of the light focussed onit, so that in the absence of any optical disturbance, the illumination on the screen is


uniformly reduced. The knife-edge is set perpendicular to the direction in which thedensity gradients are to be observed. In the present study, the gradients arepredominantly in the vertical direction, and the knife-edge has been kept horizontal.The shadowgraph images have been recorded just beyond the test cell, and ahead

of the second focussing concave mirror. The position of the screen on which theshadowgraph images are displayed plays an important role is data analysis [10]. Inaddition, beam refraction from zones of high temperature gradients can interferewith those passing through a zone of nearly constant temperature. The cameraposition was chosen so as to improve the image contrast, while extracting thedominant features of the flow field. In the experiments with air, the recordedshadowgraphs showed poor contrast at low cavity temperature differences. At highertemperature differences, shadowgraphs clearly revealed the large-scale features of theconvective field.

3. Uncertainty and measurement errors

Errors in the experimental data are associated with misalignment of the apparatuswith respect to the light beam, noise generated at different stages of the experiment,including the imperfections of the optical components, and the intrinsic uncertaintyin the convection process itself. All experiments were conducted several times toestablish the repeatability of the convective patterns, for the parameter range ofinterest. In experiments with air, the time-dependent movement of the fringes duringinterferometry and the time-wise change in intensity for schlieren were not severeenough at the lower end of the Rayleigh number range. Specifically, the agreementbetween the present experiments with published correlations and numericalsimulation was satisfactory. For RaX5:1� 104; unsteadiness was a source ofuncertainty in the local temperature profiles as well as the wall heat transfer rates. Aseries of cross-checks have been enforced to validate the quantitative results obtainedfrom the experimental images. These include predicting the temperature of the coldtop wall by starting from the lower hot wall, as well as comparing steady heattransfer rates across the cavity with published correlations. For example, aninterferometry experiment with hot and cold temperatures of 300.5 and 305:5 Kyielded the cold wall temperature as 300:2 K: For temperatures of 300 and 320 K,the hot wall temperature was predicted to be 319:65 K: In schlieren, temperatures of300 and 320 K were predicted as 301 and 319 K: The errors were considerablysmaller at lower temperature differences across the cavity where the flow field wasclearly steady.

4. Data analysis

Optical measurement of temperature in fluids is possible because of a uniquerelationship between refractive index and density of transparent media. Inapplications involving buoyant convection, changes in pressure are small, and

ARTICLE IN PRESSA. Srivastava et al. / Optics and Lasers in Engineering 42 (2004) 469–485474

density correlates directly with temperature. The gradient of refractive index withrespect to temperature depends on the fluid medium as well as the wavelength ofthe incident light [11]. The three optical methods considered in the present studydiffer in the manner in which refractive index is used for image formation. Thedetermination of the temperature field in the fluid medium from the respectiveimages of interferometry, schlieren and shadowgraph is discussed in the presentsection.

4.1. Interferometry

Following Goldstein [11], let nðr; zÞ and Tðr; zÞ be the refractive index andtemperature fields, respectively, in the test region and n0 and T0 be the referencevalues of n and T ; respectively, as encountered by the reference beam. Let L be thegeometrical path length covered by the test and the reference beams. The opticalpath difference between the test beam and the reference beam is:

DPL ¼dn

dT

Z L

0

ðTðr; zÞ � T0Þ ds:

The integral is evaluated along the path of a light ray given by the coordinate s: Thefringes seen on the interferograms are locus of points having the same optical pathdifference. Hence on any given fringe the optical path difference DPL is a constantand Z L

0

ðTðr; zÞ � T0Þ ds ¼DPL

dn=dT¼ constant:

The integralR L

0 Tðr; zÞ ds is defined as %TL; where %T is the average value of Tðr; zÞ overthe length L of the laser beam through the test cell. Thus interferometry yields theline integral of the function Tðr; zÞ: In a purely two dimensional thermal field, thisequation can be simplified to the form

DTe ¼l=L

dn=dT; ð1Þ

where DTe is the temperature change per fringe shift. For the cavity employed in theexperiments, it was calculated to be 0:8 K for air and 0:05 K in water.The above analytical treatment assumes that changes in the optical path length

owing to refraction effects are negligible.

4.2. Schlieren

Image formation in a schlieren measurement is due to the deflection of light beamtowards the regions of higher refractive index. In order to recover quantitativeinformation from a schlieren image, the angle of deflection as a function of positionin the transverse x–y plane should be determined. The relative intensity, DI ¼ID � Ik; where ID is the intensity with the thermal disturbance in the path of the lightbeam and Ik is the intensity with the knife-edge in place but without the thermal


disturbance is [10]

DI

Ik¼

Da

ak¼

a� f

ak;

where, ak is the dimension of the beam perpendicular to the knife edge after the lightbeam has been cut off by the knife-edge, Da is the displacement of the light beam inthe vertical direction above the knife edge in the presence of thermal disturbance anda is the angle of deflection. From the principles of ray-optics [10], the elementalangular deflection, da of the light ray in a variable refractive index region can beshown to be

da ¼1

n

qn

qydz ¼

qðln nÞqy

dz;

where z is the direction of propagation of light, normal to the x–y plane. Hence, thetotal angular deflection of a light ray emerging from an apparatus of length L is

a ¼Z L

0

qðln nÞqy

dz: ð2Þ

For a two dimensional field in the x–y plane, the integrand is independent of z andthe above equation can be readily integrated in the direction normal to the knife edge.

4.3. Shadowgraph

In the shadowgraph approach, the reduction in the light intensity owing to beamdivergence is monitored. The relationship between the change in intensity and therefractive index field in the test cell can be developed by ray tracing techniques [7]. Ifthe screen is placed close to the apparatus, certain linearizing approximations can beadopted and the mathematical development can be simplified. The relationshipbetween the refractive index field and the change in light intensity can be shown tobe [7,10]:

DI

I0¼ D

q2gðnÞqx2

þq2gðnÞqy2

� �; ð3Þ

where

gðnÞ ¼Z L

0

ln n dz

D is the distance between the exit plane of the test cell and the screen.

4.4. Correlations

Correlations of steady average heat transfer rate are expressed in dimensionlessform in terms of the Nusselt number

Nu ¼�h

Th � Tc

qT

qy

��y¼0;h

: ð4Þ

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With this definition, the local and average Nusselt numbers at the top and thebottom walls of the cavity can be calculated. The average Nusselt number for an air-filled cavity has been compared with the following experimental correlation [1]:

Nu ¼ 1þ 1:44 1�1708

Ra

� �þ

Ra

5830

� �1=3

�1

" #: ð5Þ

Eq. (5) is applicable in air over a wide range of Rayleigh numbers ðRaÞ and ispractically independent of the cavity aspect ratio.

5. Numerical simulation

Numerical simulation of buoyant convection in a square cavity has beenperformed with FLUENT, a commercial software package. For calculations inair, the flow was taken to be laminar, though unsteady. In water, the flow was takento be turbulent, and the k–emodel was additionally employed. The dimensions of thetest cell in the simulation were identical to those in the experiments. The boundaryconditions at the upper and lower surfaces were taken to be prescribed temperatures,while all other boundaries were thermally insulated. Fluid velocities at all surfaces ofthe enclosure were set to zero. A uniform three dimensional grid of 25� 25� 354was found to be an optimum with respect to resolving the flow details near theboundaries and CPU time. The time step was based on the stability criterion fornumerical integration.

6. Results and discussion

Experiments have been conducted in the present work to image buoyancy-drivenconvection in a rectangular cavity. The cavity has a square cross-section. Thetemperature difference maintained across the horizontal surfaces of the cavity leadsto unstable density gradients in the fluid medium. In turn, fluid motion in the form ofclosed loops is initiated in the cavity. The configuration studied here, calledRayleigh–Benard convection is a problem of fundamental as well as practicalimportance.

6.1. Comparison of the three optical techniques

A direct comparison of the images seen in interferometry, schlieren andshadowgraph is presented here. The temperature difference across the cavity was10 K in the experiments, while the fluid medium in the cavity was air. Thecorresponding Rayleigh number was calculated to be 6� 104: The respective imagesare compared on the right column of Fig. 2. The relative spacing of the fringes yieldsthe temperature profile in interferometry. For schlieren and shadowgraph, theinformation is contained in the relative intensity variation. The respective thermal


properties recovered are the local values of the first derivative and the Laplacian oftemperature. These quantities have been plotted for the mid-plane of the cavity onthe left side of Fig. 2. The individual data points are specific to the mid-plane of the

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Fig. 2. Comparison of data recovered from the three optical techniques (left column). The corresponding

experimental images are shown in the right column. Ra ¼ 6� 104: The solid line in the left column is

representative of the trend seen at any vertical section of the image.


cavity, while the solid line indicates the overall trend. The shaded circles of the leftcolumn indicate the gradients calculated from interferometric data (for schlieren)and from schlieren data (for shadowgraph). A good overall match is a confirmationof the result that schlieren is a derivative of the interferometric field, and theshadowgraph in turn is the derivative of the schlieren. The appearance of densefringes near the horizontal walls is indicative of high temperature gradients at theselocations. This is brought in the schlieren image, as well as the data points. Thecentral region is a zone of nearly constant temperature, where the gradients are closeto zero. Thus, the schlieren images and interferograms correlate quite well with eachother. These also correlate with the shadowgraph, once it is realized that in thisapproach, light is redistributed over the image. In a shadowgraph image, light fromthe region close to cold top wall deflects towards the lower hot wall, where the lightintensity shows a maximum. Thus, high gradients are represented in a shadowgraphby regions of very low as well as very high light intensity. In the central core, thechange in light intensity with respect to the initial setting is small. Thus the Laplacianof temperature in this region is practically zero. The thermal lensing effect thatdistorts the shape of the cavity cross-section is most visible in the shadowgraph.

6.2. Convection in an air-filled cavity

Images of convection in air as recorded by the three optical techniques arediscussed in the present section. Quantitative analysis of the thermal fields and wallheat transfer rates is reported for the lower range of cavity temperature differences(and hence Rayleigh number). Clear shadowgraph images could not be recorded forsmall temperature differences, and have not been shown. At higher temperaturedifferences, the field was seen to become unsteady. The discussion for larger cavitytemperature differences is based on qualitative comparison of the three imagingtechniques. Comparison with the numerically simulated thermal field is reportedwhere appropriate.

6.2.1. Low Rayleigh numbers

Fig. 3 shows the steady state interferometric and schlieren images for the lowerrange of Rayleigh numbers (1:3� 104oRao5� 104). At Ra ¼ 1:3� 104; theinterferogram has only a few fringes. The number of fringes in the experimentswas uniformly found to be consistent with the estimate ðTh � TcÞ=DTe: The numberof fringes increase with Rayleigh number, along with the fringe density near thehorizontal walls. With respect to the schlieren images, it can be seen that the increasein light intensity is distributed over the cavity cross-section at the lowest Rayleighnumber. As Rayleigh number increases, the brightness is limited to the wall region,and its size progressively diminishes. The schlieren image clearly brings out aboundary-layer type of flow structure in the cavity.A comparison of the local steady dimensionless temperature profiles in the cavity

as obtained from interferometry, schlieren, and numerical simulation is presented inFig. 4. Temperature profiles at two column locations (x=h ¼ 1

4 and 34) have been

considered. The comparison has been presented for the three Rayleigh numbers


referred in Fig. 3. The shape of the profile, a characteristic of buoyancy-drivenconvection is reflected in all the three approaches. The slopes of the individual curvesnear the walls give a measure of the wall heat flux. Under steady conditions, theaverage wall heat flux is also equal to the energy transferred across any horizontalplane of the cavity. The comparison between the experiments and simulation is seento be good. Schlieren measurements marginally compare better with numericalsimulation, as against interferometry. This is because in interferograms, informationabout the thermal field is available only at the fringes. Constructing a complete

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Fig. 3. Interferometric and schlieren images for the lower range of Rayleigh numbers in air.


temperature profile requires interpolation between fringes, and is a source of error.In addition, the number of fringes at low Rayleigh numbers is also small.Average heat transfer rates at the hot and the cold walls of the cavity have been

calculated in terms of Nusselt number, Eq. (4), and presented in Table 1. Further,they have been compared against the experimental correlation of [1] (Eq. (5)). Theaverage Nusselt numbers predicted by the two optical techniques match well witheach other, with a maximum difference of 75%; that occurs at the lowest Rayleighnumber ðRa ¼ 1:40� 104Þ of the present work. The hot and cold wall Nusseltnumbers are also reasonably close, indicating a good energy balance in the datareduction procedure. The discrepancies are higher in interferometry at this Rayleighnumber owing to the formation of just a few fringes in the cavity.

6.2.2. High Rayleigh numbers

Convection in a cavity subjected to a large temperature difference, and hence ahigh Rayleigh number is discussed in the present section. The shadowgraph imageswere quite clear in these experiments and have been included for comparison. In eachof the experiments, a true steady state was not attained, even after the passage of along period of time. Secondly, interferograms were subjected to large refraction

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y/H

θ

0 0.3 0.6 0.90.0

0.3

0.6

0.9

Ra = 13,000

∆∆

y/H

0 0.3 0.6 0.9

Ra = 26,000

y/H

0.1 0.4 0.7 1.0

Ra = 50,000

Fig. 4. Convection in an air-filled cavity. Non-dimensional temperature profiles as a function of the

vertical coordinate.

Table 1

Wall-averaged Nusselt numbers at the lower (hot) and the upper (cold) walls; Comparison of correlation

with experiments

Rayleigh number Nu (cold) Nu (hot) [1]

(Air) Interferometry Schlieren Interferometry Schlieren

1:40� 104 2.38 2.28 1.965 2.07 2.615

2:70� 104 3.32 3.5 3.13 3.1 3.03

5:10� 104 3.56 3.37 3.51 3.38 3.45

A. Srivastava et al. / Optics and Lasers in Engineering 42 (2004) 469–485 481

errors. The cavity viewed through the schlieren and shadowgraph layouts lookeddeformed, once again due to refraction. Hence, the present discussion is purelydescriptive and quantitative data has not been reported. Rayleigh numbersconsidered are Ra ¼ 8:50� 104; 1:13� 105 and 1:40� 105:Fig. 5 shows representative images of the convective field recorded after the

passage of 8–9 h of real time. The three optical techniques have been compared. AtRa ¼ 8:50� 104; the interferogram shows nearly straight dense fringes parallel to thehorizontal walls. These are indicative of near-parallel flow near the hot and the cold

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Fig. 5. Interferometric, schlieren and shadowgraph experimental images for the higher range of Rayleigh

numbers in air.


walls. The flow turns at the corners, leading further to fringe curvature andseparation. The flow field is symmetric about the centerline of the cavity. Thesetrends are well-reproduced by schlieren and shadowgraph images.The movement of the fluid medium in the cavity cross-section has the shape of a

roll. The roll size can be determined directly from the optical images since the fringesas well as the intensity fields should turn along at corners with the local velocityvector. The roll sizes of Fig. 5 have been compared in Table 2, and are found to bequite close. The three experimental techniques as well as the numerical simulationreveal a reduction in the roll size with increasing Rayleigh number.An experimental study by Krishnamurti [12] has revealed two dimensional roll

patterns bifurcating slowly to three-dimensional rolls with an increase in Rayleighnumber. The author summarized the transition sequence over a wide range ofRayleigh numbers in the form of a stability map. For Ra > 105; the stability diagramindicates unsteadiness, followed subsequently by turbulence. In the range 8:50�104oRao1:40� 105; the flow in the present study was seen to be unsteady. The timedependence was more pronounced at the two higher values of Rayleigh numberswith a definite periodicity being observed in the flow field. The time period of theoscillations was calculated by following the intensity variation (in schlieren orshadowgraph) over a small region in the cavity. To capture the time variation oftemperature, it was sufficient to focus on a portion of the cavity for schlieren andshadowgraph. In contrast, it was found necessary to record complete interferogramsas a function of time, requiring large computer memory. Hence this approach wasnot pursued. In experiments, schlieren signals were seen to be stronger in comparisonto the shadowgraph, in view of the contrast supported by the knife-edge. The timetraces of intensity were continuous and a dominant harmonic was discernible. Thedimensional time period of this harmonic has been scaled by the thermal diffusiontime ðh2=aÞ: At Ra ¼ 1:13� 105; the non-dimensional time period of oscillation wasobtained as 0.038. At Ra ¼ 1:40� 105; it was 0.015. The reduction in the time periodis a clear indicator of the appearance of smaller length scales (and higherfrequencies), in agreement with [12]. It confirms the progress of the convective fieldtowards turbulence. The corresponding values of the dimensionless time quoted byKrishnamurti [12] are E0:055 and 0.030, respectively. The difference between thetwo sets of values can be attributed to the cavity sizes, being small in the verticalplane in the present work, as opposed to a vast horizontal fluid layer in [12].

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Table 2

A comparison of roll sizes of convection seen in the three optical techniques; effect of increasing Rayleigh

number

Rayleigh number (Air) Roll size (relative to cavity height) Numerical

Interferometry Schlieren Shadowgraph

8:50� 104 0.270 0.278 0.310 0.304

1:10� 105 0.258 0.252 0.252 0.296

1:40� 105 0.210 0.242 0.220 0.256

A. Srivastava et al. / Optics and Lasers in Engineering 42 (2004) 469–485 483

6.3. Experiments in water

Thermal imaging of the test cell filled with water is reported in the present section.The vast difference in the properties of air and water leads to certain difficulties intemperature measurement. Specifically, the following difficulties are to be noted:

1. Even for modest temperature differences, the equivalent Rayleigh numbers arevery high. Consequently, the convective field reaches the unsteady, turbulentregime quite early.

2. The temperature drop per fringe shift ð¼ DTeÞ is small in water. Thus the fringedensity in interferograms is high, making data analysis difficult.

3. The sensitivity of the refractive index to temperature ð¼ dn=dTÞ is high. Thus thedeformation of the cavity size as seen on the screen is large and leads to ambiguityin data analysis.

In view of these complications, it is natural that shadowgraph should be bestsuited for temperature measurement in water. A comparison of interferometry,schlieren and shadowgraph for a temperature difference of 4 K ðRa ¼ 2:5� 106Þ ispresented in Fig. 6. The optical images are depth averages, but are functions of time.The images reflect the degree of complexities involved in quantitative analysis of theinterferometric and schlieren images quite clearly due to the presence of high fringedensity in the interferogram and highly deformed boundaries in the case of schlierenimage. Shadowgraph shows minimal deformation in the physical boundary withsystematic change in intensity inside the flow field and therefore is most amenable toquantitative analysis.

7. Conclusions

Three refractive index-based optical techniques, namely interferometry, schlierenand shadowgraph have been applied for quantitative evaluation of the convectiveflow field in a differentially heated rectangular cavity. Air and water have been

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Fig. 6. Comparison of (a) interferometry, (b) schlieren and (c) shadowgraph for buoyant convection in a

water-filled cavity at Ra ¼ 2:50� 106:


considered as working fluids. Experiments have been conducted over a wide range oftemperature differences. A limited comparison with numerical simulation is alsopresented.The following conclusions have been arrived at in the present work.

1. In low temperature gradient experiments, all the three techniques correlate wellwith one another, and match the numerical solution. Interferograms are limitedby few fringes in air, and too many fringes in water. The shadowgraph image doesnot show sufficient contrast for analysis. In this respect, the schlieren technique ismost amenable to data reduction.

2. In high gradient experiments, both schlieren and shadowgraph yield clear images.The interferograms are however corrupted by refraction errors. Schlieren andshadowgraph track the temporal response of the fluid medium in the form of thelight intensity variation.

3. In high Rayleigh number experiments with water, the flow field is turbulent.Shadowgraph images are seen to be meaningful, as against interferograms andschlieren and most suitable for quantitative analysis.

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