Temperature measurement in laminar free convectiveflow using digital holography
Md. Mosarraf Hossain1,2 and Chandra Shakher1,*1Laser Applications and Holography Laboratory, Instrument Design Development Center,
Indian Institute of Technology, Delhi, New Delhi—110016, India2Currently with the Space Physics Laboratory, Vikram Sarabhai Space Center, Thiruvananthapuram-695022, Kerala, India
The problem of determining temperature distribu-tion and heat flow is of immense interest in manybranches of science and engineering [1]. In the designof heat exchangers, such as boilers, condensers, andradiators, a thorough heat-transfer analysis is essen-tial for sizing their components [1]. Also during de-sign of nuclear-reactor cores, heat-transfer analysisis required for shaping the fuel elements properlyto prevent them from burnout [1]. In aerospacetechnology, the temperature-distribution and heat-transfer problems are crucial because of weightlimitations and safety considerations, whereas inheating and air-conditioning applications for build-ings, a proper heat-transfer analysis is necessaryto estimate the amount of insulation needed to pre-vent excessive heat losses or gains [1].
In this context, the measurement of temperaturefield in convective fluid flow is of enormous impor-tance, because in the above mentioned applications,the fluid flow involving convective heat transfer iscommonly encountered. If the fluid flow is generatedpredominantly by body forces because of density var-iations, which results from temperature gradientswithin the medium, the mechanism of heat transferis called free or natural convection. In free convec-tion, heat transfer first takes place by pure conduc-tion, because initially there exists no relative motionbetween the wall and the liquid layer in its immedi-ate vicinity. When a temperature gradient is estab-lished in the fluid, it generates a density gradient,which in a gravitational field generates buoyancyforces and hence convective motion sets in [1]. Ifthe heating rate is slow, the heat source inside thetest section is maintained at constant temperature(isothermal problem), and the source size is suffi-ciently small, free laminar flow can be ensured in-side the test section [2]. In free laminar flow, fluid
particles follow definite streamlines, and propertiessuch as velocity, pressure, and temperature at anypoint do not vary with time. In free turbulent flow,however, because of chaotic crosswise eddying offluid particles between the streamlines, the proper-ties at any point vary continuously with time.Although turbulent flow is more widely found in en-gineering applications involving fluid-flow and heat-transfer problems, there are many situations inwhich laminar flow is important. For example, in li-quid-metal-type nuclear reactors, laminar flow maybe desirable to reduce pumping power, since heattransfer with liquid metals is sufficiently high [1].In heat transfer experiments, the objective often isto determine the temperature distribution through-out its volume. This requirement disqualifies the ca-lorimetric or thermocouple based measurementsbecause their use may disrupt the field under study.This is particularly true in low-scale motions, such asnatural convection, which are susceptible to instabil-ity, and in high-speed compressible flows, whereshocks may be induced by the employed probe [3].Hence there arises the need for whole-field noninva-sive techniques for the measurement of refractiveindex/temperature and its distribution.A large number of optical techniques, such as clas-
sical interferometry [4], holographic interferometry[5–7], moiré deflectrometry [8], speckle photography[9,10], speckle shearing interferometry [11], Talbotinterferometry [12,13], Lau phase interferometry[14], shearing interferometry [15] and electronic/digital speckle pattern interferometry (ESPI/DSPI)[16], have been used to measure refractive index,temperature, and temperature profiles of fluids. Itis already established that digital holographic tech-niques are fast, robust, and capable of providingmore accurate data for measurement of various phy-sical parameters in comparison to conventional holo-graphy based techniques. Digital holography can bedescribed as an optoelectronic system having thecapabilities to record optical wavefront on a CCD/CMOS sensor, store the captured wavefront in thememory of a computer, and reconstruct the wave-front numerically at any time after the recording.The possibility of numerical reconstruction of com-plex wavefronts recorded holographically opensnew possibilities in coherent imaging. Digital holo-graphy, endowed with the function of interferometry,is termed as digital holographic interferometry(DHI). DHI has been applied for measurement of var-ious physical parameters, such as shape/deformation[17–19], vibration [20,21], strain [22], refractive in-dex [23], temperature [24], material parameters suchas Young’s modulus, and Poisson ratio or thermalexpansion coefficient [25].There are some reports in the literature concern-
ing visualization of convection flow of transparentfluids [26–28] and the measurement of temperature[27,28] field therein using DHI. In [26], a DHI deviceperforming diffusion, dissolution and convectionstudies has been demonstrated. In this paper only
a qualitative study of flow in the stationary convec-tive regime has been demonstrated. In [27], a methodfor simultaneous 3D particle-image velocimetry(PIV) and temperaturemeasurement has been inves-tigated using DHI in thermoconvective flow. In [28],simultaneous measurement of Benard-convectionflow and temperature measurement have been inves-tigated. Both of these investigations are based onphase-shifting digital in-line holography and havebeen carried out in the context of particle trackingand PIV with emphasis on flow visualization. Inthese studies, however, no methodology describingthe dependence of quantitative absolute phase ontemperature at the sensor pixels has been adopted.An appropriate methodology is important for accu-rate quantitative calculations of temperatures fromthe phase data. The phase-shifting technique em-ployed in these methods, although having the advan-tage of effective use of the pixels of the CCD, involvesextra experimental efforts, because a phase-shiftingtechnique requires recording of more than threeimages for a fully complex fields, which involvesthe disadvantage of a time lag during phase-shiftingand hologram acquisition between equal phase steps.Hence the digital holographic systems involvingphase shifting are relatively complex and becomedifficult to use for time-variant problems such as con-vection flow in industrial environment.
In this paper, a method for determination of tem-perature in steady laminar free convection of water ispresented using DHI. As is obvious from the sectionsto follow, in this method the temperature and its var-iation with height from the heater plane have beenmeasured quantitatively from the flow field of waterin steady laminar free convection. The method is re-latively simple, fast, and can be adapted easily in anindustrial environment. This method does not re-quire use of any extra experimental efforts as inphase shifting, and the setup is in a lensless Fouriertransform configuration, for which the reconstruc-tion algorithm is simple and fast [29]. In the secondpart of the experiment, turbulent free convectionflow also has been visualized.
2. Theory
A. Lensless Fourier Transform Digital Holography
There exist several approaches to digital holography,e.g., Fresnel [30], single-transform Fourier–Hartleyfringe analysis [31], and convolution [32]approaches.However, these methods suffer from the followingshortcomings:
• The angle between the two interfering beamsvaries over the sensor’s surface and so does the spa-tial frequency. Hence the Nyquist sampling theoremmay not be satisfied at each point of the sensor plane.Consequently, the full spatial bandwidth of the sen-sor cannot be used. It is important to use the entirespatial bandwidth because the lateral resolution
depends on complete evaluation of all the informa-tion available to the sensor [29].• Speckle has a pronounced affect on lateral re-
solution of the reconstructed image [29].• Reconstructing algorithms involves several
Fourier transforms or complex multiplica-tions, which slow down the reconstruction and theprocessing.
All the above difficulties can be handled effectivelyby using lensless Fourier transform digital hologra-phy (LFTDH). The LFTDH approach is based on theprinciple of lensless Fourier holography. In this con-figuration, the cross section of interest of the objectand the point reference source are kept in the sameplane. The schematic of the LFTDH setup used fortemperature measurement is shown in Fig. 1. An op-tically generated interference pattern between theobject wave and the reference wave is recorded digi-tally with the help of a CCD/image grabber card andstored in an image processing system inside a com-puter. The interference pattern in the CCD plane canbe expressed as combination of the following fourterms:
where ðXO;YOÞ, ðXH ;YHÞ, and ðXI;YIÞ Cartesian co-ordinate systems correspond to object, hologram, andimage plane, respectively; RðXH ;YHÞ and OðXH ;YHÞare complex reference wave amplitude distributionand complex object wave amplitude distribution, re-spectively, in the hologram plane; and the third andfourth terms represent the interference terms with“�” denoting the complex conjugate operator. The di-gital hologram is represented by
where IðXH ;YHÞ is sampled on the CCD aperturehaving N ×N pixels of size ΔXH and ΔYH alongthe respective coordinates; and r and s are integers(−N=2 ≤ r, s ≥ N=2). Functional dependence of thecomplex amplitudes on CCD plane coordinates andthe functional representation of sampling processhave been avoided in Eqs. (1) and (2) for conciseness.
Numerical reconstruction of a lensless Fourier di-gital hologram is based on the Fresnel reconstructionalgorithm. The Fresnel algorithm can be expressedby the Fresnel diffraction integral [30]
OðXI;YIÞ ¼expðikdÞ
iλd exp�−i
k2d
ðXI2 þ YI
2Þ�
×Z ZðXH ;YHÞ
IðXH ;YHÞRðXH ;YHÞ
× exp�−
k2d
ðXH2 þ Y2HÞ
�
× exp�i2πλd ðXIXH þ YIYHÞ
�dXHdYH ; ð3Þ
whereOðXI;YIÞ is the reconstructed wave field in theimage plane, which is at a distance d from the holo-gram plane, λ is wavelength, and k is a wave vector.The discrete form of Eq. (3) is used to process the di-gital hologram along with the numerically computedreference wave to simulate the propagation of the dif-fracted wave from the microstructure of the holo-gram to reconstruct the images using computers.Equation (3) can be written as
OðXI;YIÞ ¼expðikdÞ
iλd exp�−i
�k2d
�ðXI
2 þ YI2Þ�
× Fλd−1
�IðXH ;YHÞRðXH ;YHÞ
× exp�−i
�k2d
�ðXH
2 þ YH2Þ��
; ð4Þ
where Fλd−1 indicates the two-dimensional inverse
Fourier transformation scaled by a factor 1=λd. Inthe specific geometry of lensless Fourier geometry,the effect of the spherical phase factor associatedwith the Fresnel diffraction pattern of the objectwave is eliminated by use of a spherical referencewave RðXH ;YHÞ with the same average curvature,
RðXH ;YHÞ ¼ ðconstÞ exp�i
�k2d
�ðXH
2 þ YH2Þ�: ð5ÞFig. 1. Schematic of the lensless Fourier transform digital holo-
Use of Eq.(5) in Eq. (4) results in a simpler algorithmfor lensless Fourier digital holography:
OðXI;YIÞ ¼ ðconstÞ expðikdÞiλd
× exp�−i
�k2d
�ðXI
2 þ YI2Þ�
× F−1λdfIðXH ;YHÞg: ð6Þ
The discrete representation of Eq. (6) is
Oðm;nÞ ¼ constexpðikdÞ
iλd
× exp�−iπλd
�m2
N2ΔXH2 þ
n2
N2ΔYH2
��
× IDFTfIðrΔXH ; sΔYHÞg; ð7Þ
where F−1λd is replaced by an IDFT (inverse discrete
Fourier transform),ΔXI andΔYI are sampling inter-vals in the image plane, m and n are integers(−N=2 ≤ m, n ≥ N=2), and the following relationshipsamong the sampling intervals in hologram and im-age planes have been used:
ΔXI ¼ λd=ðNΔXHÞ; ΔYI ¼ λd=ðNΔYHÞ: ð8Þ
Finally in terms of standard FFT algorithm, Eq. (7)can be written as
Oðm;nÞ ¼ constexpðikdÞ
iλd
× exp�−iπλd
�m2
N2ΔXH2 þ
n2
N2ΔYH2
��
× IFFTfIðrΔXH ; sΔYHÞg: ð9Þ
The above equation gives the digital reconstructionof the object wave field as an array of complex num-bers, and hence both the intensity as well as thephase distribution of the object wave can be calcu-lated using the following relations:
where the operators “Im” and “Re” denote the ima-ginary and real parts of a complex function. Theabove algorithm involving only one inverse Fouriertransform apart from some multiplicative constantsis fast in comparison to other approaches mentionedat the beginning of this section.When the digital hologram is reconstructed using
Eq. (9), the first two terms of the hologram according
to Eq. (2) contribute to the zero order of diffraction ofthe reference wave, also called a DC component, thethird term contributes to the twin image, and thefourth term contributes to the real image. To avoidan overlap of the three components of the recon-structed image field the digital hologram is recordedin the so-called off-axis geometry (unless a phase-shifting technique is used) with the angle θ betweenthe two interfering beams sufficiently large to ensureseparation between the real and the twin images.However, θ must not exceed a value such that thespatial frequency of the interferogram does notexceed the resolution of the CCD. The DC imagedisturbs the reconstructed field by affecting the dy-namic range of the display. Removal of the DC com-ponent enhances the signal-to-noise ratio (SNR) inthe reconstructed image. The twin image reducesthe size of the real image area in the total recon-structed field. Suitable filtering techniques can beused for removal of the DC component and the twinimage [33].
B. Methodology Used for Temperature Measurement
In this method, two light beams are allowed to pene-trate the test section containing distilled water attwo different times, one without heat transfer andthe other during the steady heat transfer in laminarfree convection. To begin with, one interferencepattern formed by superposition of the referencebeam and the light transmitted through the waterin the quiescent state is recorded. A square shapedelectrical heater kept inside the test section is thenswitched on to initiate the free convection. To ensurefree laminar convection, the heating plate is main-tained at a constant temperature, and its size ischosen sufficiently short such that turbulent flowis never triggered. Once the steady laminar freeconvection is achieved, it can be assumed (neglectingthe end effect) that the refractive index, and hencethe temperature variation above the heated plate,is one dimensional and normal to the plate [16].A second exposure is then taken in the presence offree laminar convection. In general, the differencein optical path lengths of the two light wavesregistered between two exposures can be expressedas [34]
ΔℓOptðx; yÞ ¼ ℓ½nðx; yÞ − nc�; ð12Þ
where ℓ is the length of the test section traversed bythe light, nðx; yÞ is the two-dimensional refractive in-dex distribution in a plane normal to direction of pro-pagation of light (z axis) during the heat transfer, andnc is the constant refractive index throughout the vo-lume of the test section in the quiescent state. In thederivation of Eq. (12) the following has been as-sumed [34]:
a. The light beam enters the test section nor-mally and is not deflected during its traversalthrough it.
b. The refractive index distribution is two dimen-sional and does not change along the path of propa-gation of light.
In the particular case of free laminar convection,Eq. (12) becomes
ΔℓOptðyÞ ¼ ℓ½nðyÞ − nc�; ð13Þ
where y is in a direction perpendicular to both thedirection of propagation of light and the plane ofthe heating plate. Hence the phase differenceΔΦ be-tween the two light waves reaching the sensor can beexpressed as
ΔΦðyÞ ¼ 2πλ ΔℓOptðyÞ ¼
2πλ ℓ½nðyÞ − nc�: ð14Þ
For water as a test fluid, the equation by Tilton andTaylor can be used in order to get the relationshipbetween temperature (in absolute scale) and refrac-tive index [34]:
nðyÞ ¼ 1:33711 − 9:3784 × 10−6TðyÞ − 2:1726
× 10−6TðyÞ2: ð15Þ
Combining Eqs. (14) and (15), we get the followingquadratic equation in TðyÞ:
ATðyÞ2 þ BTðyÞ þ C ¼ 0; ð16Þ
with A ¼ 2:1726 × 10−6, B ¼ 9:3784 × 10−6, and Cgiven by
C ¼�λΔΦ2πℓ − ATc
2− BTc
�; ð17Þ
where Tc is the constant temperature throughout thevolume of the water in the quiescent state. FromEq.(17), C is calculated using the experimentally
obtained unwrapped phase difference ΔΦ and theconcerned experimental parameters. Equation (16)is then solved to obtain the unknown temperatureswithin the flow field. A flow chart of the methodologyused for temperature determination is shown inFig. 2.The phase difference ΔΦ phases of the individual
reconstructed wave fields ϕ1ðm;nÞ and ϕ2ðm;nÞ cor-responding to the quiescent state and the laminar
free convection state of water are calculated usingEq. (11). The phase values calculated remainwrapped in the range ð−π;þπÞ radians, correspond-ing to the principal value of the arctan function,but the actual phase may range over an intervalgreater than 2π. The interference phase distributionsare calculated by modulo 2π subtraction:
ΔΦðm;nÞ ¼8<:
Φ2ðm;nÞ −Φ1ðm;nÞ þ 2π if Φ2ðm;nÞ −Φ1ðm;nÞ < −πΦ2ðm;nÞ −Φ1ðm;nÞ − 2π if Φ2ðm;nÞ −Φ1ðm;nÞ ≥ þπΦ2ðm;nÞ −Φ1ðm;nÞ else
: ð18Þ
Alternatively, a compact and fast way of calculatingthe interference phase is to calculate the argument ofthe product of the reconstructed wavefront when theobject is in the quiescent state and to calculate theconjugate of the wavefront when the object is in la-minar free convection. The discontinuous interfer-ence phase is indefinite to an additive integermultiple of 2π. The discontinuities are corrected toconvert the phase modulo 2π into a continuous phase
Fig. 2. Flow chart of the algorithm used for measurement of tem-perature in laminar free convection flow using digital holographicinterferometry.
by phase unwrapping [35], which consists of addingor subtracting suitable integer multiples of 2π tophase values at all points. Since the key to reliablephase unwrapping is the ability to accurately detectthe real 2π phase jumps (not the jumps due to noise),suitable filtering techniques need to be applied to thephasemap to improve the phase fringes, preserve theborders, and remove the isolated discontinuities andthe false phase fringes before application of theunwrapping procedure.
3. Experiment
A. Tools and Experimental Parameters andConditions Used
The schematic of the setup used in this experiment isshown in Fig. 1. A He–Ne laser (power ∼30mW) isused in the experiment. Output from the laser is di-vided into two parts using a beam splitter. One partis expanded and filtered using a microscope objectiveand spatial filter assembly and collimated using acollimating lens to produce the object beam. Anothermicroscope objective and spatial filter assembly ex-pands and filters the other part of the output to yieldthe point reference source, which lies in the sameplane as the cross section of interest of the object.A test section made of glass of inner dimensions of6 cm × 3 cm × 6:5 cm having optically flat windowswith flatness ~λ=6 and thickness 8:5mm is used inthe experiment. A 2 cm × 2 cm square shaped electri-cal heater is used as the heat source. The heater isfabricated by two copper plates, each 2mm thick, be-tween which a suitable electrical heating element issandwiched. A regulated DC power supply (ABLAB,model L3202) is used to apply the desired stablevoltage to the heater. One copper-constantanthermocouple along with a thermocouple indicator(Chowdhary Product, model CI-DT-05, range 0–199:5 °C) is used for an independent temperaturemeasurement. At the start of the experiment roomtemperature was 297:7 °K (24:7 °C). The CMOS(model SI-1280F) sensor employed in the experimenthas 1280 × 1022 pixels with the size of each pixelbeing 6:7 μm × 6:7 μm. The distance of the 8:6mm ×6:9mm sensor chip is kept at a distance 41 cm fromthe plane containing the object and the point refer-ence source. Separation of the point reference fromthe illuminated portion of the cell is 3 cm. The recon-struction of the digital holograms and calculationof phase was done in the MATLAB environment(version 6.5).
B. Experimental Method
The schematic of the setup for flow visualization andtemperature measurement is shown in Fig. 1.Figure 3 shows the heater inside the test section.The test section is filled with the distilled waterand allowed to reach the quiescent state. A colli-mated object beam of diameter 1 cm is passedthrough the water over the central portion of the hea-ter in such a way that the circular patch of light
illuminates the region of water under study. Thejunction of a copper-constantan thermocouple isplaced just above the plane of the heater and nearthe periphery of the light without obstructing it. Thisis to compare the measured values of temperaturesby using DHI and that of the thermocouple. The in-terference pattern formed between the referencebeam and the light transmitted through the waterin the quiescent state is recorded on a CMOS sensorand stored in a computer. The heater is thenswitched on using the regulated DC power supplyat a rate of 0:3W=cm2 to ensure slow heating. Aftera momentary heat transfer by pure conduction first,this way of heating initiates free convection withinthe water. After steady laminar flow is attained,the second exposure is recorded. The illuminationbeam, the reference beam, and the test section con-taining the water are kept undisturbed between twoexposures. In the second part of the experiment, freeturbulent convection flow is visualized. In the firstexposure, a digital hologram for the quiescent stateof water is recorded. Then the rate of power supply issuddenly increased to 5W=cm2, such that from thevery beginning the flow becomes turbulent. In thesecond exposure again a digital hologram is recordedin the presence of turbulence in water.
The digital holograms are reconstructed usingEq. (9). In the off-axis recording configuration of thisexperiment, all three image components are wellseparated. To remove the DC component from the re-constructed wavefront the hologram is filtered in thefollowing way:
i. A suitable Gaussian filter transfer function (G),which acts as a low pass filter, is formed.
ii. G is multiplied with a Fourier transform of theoriginal hologram (I); that is, G × FTðIÞ is calculated,where FT denotes the Fourier transform.
iii. An inverse Fourier transform of G × FTðIÞ isperformed; that is, FT−1½G × FTðIÞ� is calculated.Let us denote IDC ¼ FT−1½G × FTðIÞ�.
iv. IDC is substrated from the original hologram I;that is, I − IDC is calculated. This constitutes the fil-tered hologram.
Reconstruction of the filtered hologram gives thewhole image minus the zero DC component. Afterremoval of the DC component, the twin image fromthe whole image is removed by using a proper
Fig. 3. Schematic of the arrangement with the heater showninside the test section.
combination of transparent and opaque win-dow sizes.The interference phase is determined by calculat-
ing the argument of the product of the zero orderfiltered reconstructed wavefront when the object isin the quiescent state and by calculating the conju-gate of the zero order filtered wavefront when theobject is in laminar free convection. The inherentspeckle noise is reduced from the interference phasemap by 5 × 5 median filtering. The filtered interfer-ence phase is unwrapped to get the actual continuousphase distribution needed for evaluation of thetemperature.
4. Results and Discussion
Figure 4(a) shows the real interference phase mapcalculated from a pair of digital holograms after zeroorder removal, median filtering, and twin image re-moval. The first hologram is recorded in the quies-cent state (room temperature) of water, while thesecond hologram is recorded in a state approachingthe steady laminar free convection. Using a similarprocedure, the phase map is obtained for the steadylaminar free convection, where the first hologram isthe same as that in the quiescent state, and the sec-ond one is recorded after steady laminar free convec-tion is reached. The real phase map obtained insteady laminar free convection is shown in Fig.4(b). As shown in Fig. 4(b), the temperature variationis one dimensional and normal to the plane of theheater. Figure 5 shows the two-dimensional un-wrapped phase map of Fig. 4(b). Figure 6 showsthe one-dimensional phase map drawn along the lineAB from Fig. 4(b). The one-dimensional unwrappedphase map obtained from Fig. 5 along the line CDis shown in Fig. 7. The phase difference data ofFig. 7 is used to calculate the temperature throughthe use of Eqs. (16) and (17). Temperature has beenmeasured at ten points along the line CD as shown inFig. 5, including the points of phase maxima markedby P1, P2, P3, P4, and P5 as shown in Fig. 6. For ex-ample, consider the determination of temperature atthe second maxima as shown in Fig. 4(b). The dis-tance of this second maximum from the surface ofthe heater is 25 pixels, i.e., 0:95mm, because the im-age pixel size ΔYI, as calculated by using Eq. (8), is∼38 μm. The value of phase difference at P2 of Fig. 7
is 13:13314 rad, which is found from Fig. 7. The valueof temperature calculated by using Eqs. (16) and (17)at point P2 is 302:01655 K (29:01655 °C). The tem-peraturemeasured by a thermocouple independentlyin this region was 302:6 K (29:6 °C). The experimen-tally measured temperature differs by 0:58° fromthat measured by the thermocouple. The accuracyof measurement in this method depends on the pre-cision with which the phase is evaluated, and tem-perature in the quiescent state of water ismeasured. The phase measurement accuracy in digi-tal holography is∼0:067° (λ=5370) for 8bit digital ho-lograms [36]. Hence the measurement accuracy ofthis method can be expected to be reasonably high.Errors in phase evaluation may arise because ofspeckle noise and the signal discretization processby the subsequent electronics. Hence a high-resolu-tion sensor, appropriate signal processing, and aproper unwrapping algorithm may improve the mea-surement accuracy of the method. This method, withvideo rate data acquisition, high reconstruction rate,availability of direct interference phase, and largeprocessing capability, can make the measurementof the temperature field and the study of convective
Fig. 4. (a) Interference phase map obtained in a state approach-ing the steady laminar free convection and (b) interference phasemap obtained in the steady laminar free convection flow.
Fig. 5. Two-dimensional unwrapped phase map of Fig. 4(b).
Fig. 6. One-dimensional interference phase drawn along the lineAB of Fig. 4(b).
heat flow faster, reliable, and almost in real time. Butin this method the problem is that the region ofimaging is severely restricted. This is because thecurrently available optoelectronic sensor offers reso-lution of not more than a few hundred lines permillimeter, and also the size of the sensor isseverely restricted.The variation of temperature with distance from
the heater plane is shown in Fig.8. The range of tem-perature variation is very small, as the region of ima-ging is quite small. Figures 9(a) and 9(b) show theturbulent free convection flow fields at two differenttimes. In both these cases in the vicinity of the hea-ter, the water suddenly becomes warmer, but thewater above and away from the heater still remainscold. Because of the difference in density, the coldwater flows in narrow zones toward the bottom,which is recognizable at the downward enhancedseparations of the phase fringes along the central
vertical line of each phase map. In contrast to it,the warm water rises in wide zones toward the toplayers. This is evident at the upward facing phasefringes on either side of the central vertical line ofthe phase maps.
As far as the convective flow visualization is con-cerned, variety of reference beam interferometershave been employed. Mach–Zehnder and holo-graphic interferometer are by far the interferometersmost often used for flow and temperature visualiza-tion [37]. Other interferometers, such as Michelsonand Twyman–Green, although popular in otherfields of applications, have found limited applicationsin these studies [37]. In these applications, DHI of-fers some advantages not attainable by the abovementioned interferometric methods. In the contextof space experimentation involving study of fluidflow, DHI removes the need to expose, develop, trans-port, and optically analyze photographic plates orfilms. The results become immediately availablefor digital storage, digital transmission, and displayon a screen [28]. Another unique aspect of digitalholography is the extraction of flow information bycomputational focusing on selected planes withinthe 3D flow field [27].
5. Conclusions
A simple method for measurement of temperaturein laminar free convection flow of water is demon-strated using digital holographic interferometry.The method is based on a lensless Fourier transformgeometry, for which the reconstruction algorithm issimple and fast. The method also does not requireany extra experimental efforts as in phase shifting,and hence with proper instrumentation the methodcan be useful in an industrial environment. In thismethod the unwrapped interference phase, whichis calculated by unwrapping of the interferencephase obtained from two digital holograms corre-sponding to two different states of water, one inthe quiescent state and the other in the laminar freeconvection, is used for measurement of temperature.The necessary link between the phase and tempera-ture of the medium is obtained by using the Tiltonand Taylor equation. Accuracy of the proposed meth-od depends mainly on the precision with whichthe phase is evaluated. Since phase measurementaccuracy in digital holography is fairly high
Fig. 7. One-dimensional unwrapped phase map obtained fromFig. 5 along the line CD.
Fig. 8. Variation of temperature with distance from plane of theheater. Temperature is calculated on ten points marked in thefigure.
Fig. 9. (a), (b) Interference phase map showing the turbulent freeconvection flow fields at two different times.
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