Proposal and Testing for a Fiber-Optic-Based Measurement of Flow Vorticity

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Proposal and testing for a fiber-optic-basedmeasurement of flow vorticity

Shenghong Yao, Penger Tong, and Bruce J. Ackerson

A fiber-optic arrangement is devised to measure the velocity difference, dv~l !, down to small separationl. With two sets of optical fibers and couplers the new technique becomes capable of measuring onecomponent of the time- and space-resolved vorticity vector v~r, t!. The technique is tested in a steadylaminar flow, in which the velocity gradient ~or flow vorticity! is known. The experiment verifies theworking principle of the technique and demonstrates its applications. It is found that the new techniquemeasures the velocity difference ~and hence the velocity gradient when l is known! with the same highaccuracy and high sampling rate as laser Doppler velocimetry does for the local velocity measurement.It is nonintrusive and capable of measuring the velocity gradient with a spatial resolution as low as ;50mm. The successful test of the fiber-optic technique in the laminar flow with one optical channel is animportant first step for the development of a two-channel fiber-optic vorticity probe, which has wide usein the general area of fluid dynamics, especially in the study of turbulent flows. © 2001 Optical Societyof America

OCIS codes: 120.7250, 060.2420, 060.1810, 030.7060.

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1. Introduction

A quantity of fundamental interest in the study offluid turbulence is the time- and space-resolved vor-ticity vector v~r, t! 5 = 3 v~r, t!, where v~r, t! is thelocal flow velocity. However, the determination ofvorticity, i.e., the measurement of rotation, is ex-tremely difficult because of its severe requirementson spatial resolution and accuracy during simulta-neous measurement of velocity differences. Onemethod of probing flow vorticity is to measure veloc-ity components and their gradients simultaneously,with hot-wire anemometry.1,2 Multiple sensors ~fouror more! must be used in a small region in order torealize a central difference approximation. The mul-tisensor probe is intrusive, however, and requires adelicate calibration procedure. Various image tech-niques, such as particle imaging, fluorescence imaging,and holographic imaging, have also been used to visu-alize the vortex structures in turbulent flows.1,2

These techniques display a two-dimensional ~2D! vor-icity map, but their applications are often limited to

The authors are with the Department of Physics, OklahomaState University, Stillwater, Oklahoma 74078. The e-mail ad-dress for P. Tong is [email protected].

Received 13 October 2000; revised manuscript received 20 Feb-ruary 2001.

0003-6935y01y244022-06$15.00y0© 2001 Optical Society of America

4022 APPLIED OPTICS y Vol. 40, No. 24 y 20 August 2001

ow-speed flows with limited spatial and temporal res-lution. Some of them provide only qualitative infor-ation.The technique of particle image velocimetry3,4

~PIV! has been used increasingly in recent years toobtain the velocity field map, from which one calcu-lates the vorticity field of the area of interest. Withthe PIV technique one captures two consecutive 2Dimages of the seed particles, using a CCD camerasituated normal to an illuminating light sheet. Spa-tial correlations between the two images are thencalculated to obtain information about the displace-ment of each particle, from which one obtains thevelocity map. Whereas the current PIV technique isadequate for the study of vortex structures in low-speed flows, its spatial and temporal resolutions arerather limited to further resolve the small vortexstructures in turbulent flows at high Reynolds num-bers. In fact, the accuracy of the vorticity measure-ment by PIV has not been examined in great detail,because of the lack of an alternative vorticity probe.Other methods of measuring flow vorticity are citedin two recent reviews by Foss and Wallace.1,2 Al-though many attempts have been made to measureone or more components of the time- and space-resolved vorticity vector v~r, t!, an accurate and re-liable vorticity probe that can be used widely in thegeneral area of fluid dynamics is not yet available.

Recently, a new fiber-optic technique5,6 capable ofmeasuring the velocity difference, dv~l ! 5 v~r 1 l ! 2

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v~r!, down to small separation l, was developed. Inthe experiment two single-mode polarization-preserving fibers collect the scattered light with thesame polarization and momentum transfer vector qby seed particles but from two spatially separatedlocations in the flow. The collected signals fromthese two regions interfere when combined by use ofa fiber-optic coupler such that the resultant light de-tected by a photomultiplier becomes modulated at afrequency equal to the difference in Doppler shifts:Dv2 5 q z v1 2 q z v2 5 q z dv~l !. Because themagnitude of q is known for a given scattering geom-etry, the measured Dv2 becomes directly proportionalto the velocity difference, dv~l !. A similar methodwas also developed independently by Otugen et al.7and by Zimmerli et al.8

In this paper we show that, with a different opticalarrangement and signal-processing scheme, thefiber-optic technique can be used to measure twovelocity-gradient components at four closely spacedpositions in the x–y plane, and the difference of thetwo velocity gradients gives the z component of v~r,t!. With the new technique one will be able to devisea compact and robust vorticity probe consisting of twosets of optical fibers and receiving optics. This probewill be much like a commercial fiber-optic laser Dopp-ler velocimetry ~LDV! probe and has all the advan-tages of the LDV probe. The new technique has theadvantages of high spatial resolution, fast temporalresponse, and ease of use. It is nonintrusive andcapable of measuring the flow vorticity with a spatialresolution as low as ;50 mm.

2. Working Principle and Experimental Setup

Figure 1 shows the scattering geometry for the mea-surement of dv~l ! in two different directions. In thexperiment we use two single-mode polarization-reserving fibers to collect light scattered by the seedarticles. Both fibers collect the scattered light withhe same polarization and momentum-transfer vec-or, q 5 ks 2 ki ~i.e., at the same scattering angle a!,

but from two spatially separated locations in the flow.The collected signals from these two regions ~i.e.,

Fig. 1. Laser light scattering arrangements for the measurementave vector; ks, scattered wave vector; a, scattering angle; and q

rom two different particles with a separation l ! in-erfere when combined by use of a fiber-optic coupleruch that the resultant light detected by a photomul-iplier becomes modulated by a frequency Dv2 equal

to the difference in Doppler shifts. As shown in Fig.~a!, q is in the x direction, and Dv2 5 q z v~y2! 2 q z

v~y1! 5 q@vx~y2! 2 vx~y1!#. The magnitude of q isgiven by q 5 ~4pnyl!sin~ay2!, where n is the refrac-tive index of the fluid, l is the wavelength of theincident light, and a is the same scattering angle foreach receiving fiber. From the measured Dv2, oneobtains the velocity gradient ]vxy]y . @vx~y2! 2vx~y1!#yl when the beam separation l 5 y2 2 y1 isknown.

Similarly, the scattering geometry shown in Fig.1~b!, which can be obtained by rotation of the scat-tering geometry shown in Fig. 1~a! about the z axis by0°, gives the velocity gradient ]vyy]x . @vy~x2! 2

vy~x1!#yl. With a complete arrangement of four op-tical fibers and two couplers, one can simultaneouslymeasure the two velocity gradients at four closelyspaced locations in the x–y plane and obtain the zcomponent of the local vorticity

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It should be mentioned that Fig. 1 shows only theforward-scattering mode ~a , 90°! for the vorticitymeasurement. With a slightly different optical ar-rangement one can also use the backward-scatteringmode to measure the velocity gradients.

To test the technique, we conduct vorticity mea-surements in a laminar tunnel flow, in which the flowvorticity is known. Because the flow field is quasi2D, the local vorticity becomes equal to the velocitygradient, vz 5 2]vxy]y . 2dvx~l !yl~]vyy]x 5 0!. In

e velocity difference dv~l ! in two different directions. ki, incident2 ki.

of th5 ks

20 August 2001 y Vol. 40, No. 24 y APPLIED OPTICS 4023

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the experiment, water seeded with a small amount ofpolymer latex spheres circulates through a 1-m-longsegment of square tubing with an inner width D 5.48 cm. The square tubing, made of transparentlexiglas, admits the incident laser beams and allows

or observation of the scattered light. The measur-ng point is in the midplane of the square tubing and2 cm downstream from the entrance of the longubing. Undesirable velocity fluctuations are damp-ned in a 5 cm 3 5 cm 3 10 cm chamber filled with

straws of 5.5-mm diameter, and then the flow is fedthrough a square contraction section ~4:1 by area! of16 cm in length. The Reynolds number used in theexperiment is defined as Re 5 DUyn, where U is theaverage flow velocity across the square tunnel and nis the fluid viscosity. Local velocity measurementsshow that velocity fluctuations at the center of thetunnel are less than 3% of the mean flow. Becausethe seed particles are small and their densitymatches well with that of water, they follow the localflow.

Figure 2 shows the experimental setup. We gen-erate the two incident laser beams by passing anincident beam from a Nd:YVO4 laser with a powerrange of 0.5–2 W and wavelength l 5 532 nmhrough a Bragg cell ~not shown!. Only the zeroth-rder ~unshifted! and the first-order ~40-MHzrequency-shifted! outgoing beams are used in thexperiment, and all the higher-order beams arelocked. The two laser beams are adjusted to havehe same intensity and are directed to the measuringoint by a lens ~ f 5 7 cm!, which is positioned suchhat the two outgoing beams become parallel at theeasuring point. Beam profile measurements at

he measuring point show that the two beams areeparated by a distance l 5 0.42 mm and that theiameter of each beam is s . 38 mm. The value of l

is fixed in the experiment, because it is sufficient forthe vorticity measurement in the laminar flow.

It is evident that the spatial resolution of thevelocity-gradient ~or vorticity! measurement is de-pendent on the separation l between the two incidentbeams. An incident laser beam becomes two diverg-ing beams ~one of them is frequency shifted! whenpassing through the Bragg cell, which acts like anoptical grating. If the distance between the beam-

Fig. 2. Experimental setup for the measurement of the velocitygradient in the horizontal direction over a vertical distance l.


splitting point ~inside the Bragg cell! and the lens isset to be equal to the focal length f, the two divergingbeams become parallel after passing through thelens. We find that the value of l, which is propor-ional to f, can be readily reduced to 0.1 mm by use oflow-magnification objective ~ f . 2 cm! as the colli-

mating lens. The value of l can be further reduced to50 mm by use of a 253 objective. Clearly, this spa-tial resolution will be adequate for most turbulentvorticity measurements.

A graded-index lens is installed at the front end ofeach input fiber for better light collection. With anextra focusing lens ~ f 5 7 cm!, the two input fiberscollect the scattered light from two very small scat-tering spots of size 0.12 mm.5 In the experiment thescattering angle a is fixed at 47°. One of the outputfibers is connected to a photomultiplier tube, whoseanalog output is fed into a LDV signal processor~Model IFA655, TSI Incorporated!. An oscilloscopeis also connected to the photomultiplier tube for di-rect viewing of the output signals. The other outputfiber is connected to a He–Ne laser, which is used foroptical alignment. With the reversed He–Ne lightcoming out of the input fibers, we can directly observethe scattering volume viewed by each input fiber.

For the new technique to measure Dv2 accurately,ne requirement is that the residence time of the seedarticles in each measuring volume, i.e., the beamrossing time tr 5 syv# , should be kept much longer

than the cross-beat time ~Dv2!21. This will allow theLDV signal processor to detect enough oscillations~say, ten! in each burst signal ~see Fig. 3 below!. Asn the standard LDV, this condition is satisfied by

eans of shifting the frequency of an incident beam.hen the shift frequency V0 is set to be much larger

than Dv2, the above requirement becomes trV0 .. 1or sV0 .. v# . With a maximum frequency shift of 40

Hz available for many Bragg cells used in LDV, thisondition is satisfied for most turbulent flows. Therequency shift also allows us to measure small fluc-uations of the velocity difference near zero and toetermine the sign of the velocity gradient, which isssential for the vorticity measurement in turbulentows. This frequency shift discriminates the self-eat signals ~between the particles in the same mea-uring volume or laser beam! from the desired signal.ecause the cross-beat frequency between the parti-les in the two measuring volumes is shifted to arequency range much higher than the self-beat fre-uency, we can readily remove the self-beat signalsy using a high-pass filter.

3. Results and Discussion

Figure 3 shows the typical beat signal from the ana-log output of the photomultiplier tube. The signalstrongly resembles the burst signal in the standardLDV. The only difference is that the signal shown inFig. 3 results from the beating of the scattered lightwith the same q but from two different particles sep-arated by a distance l. In the standard LDV, how-ever, the Doppler signal is produced by the beating ofthe scattered light from the same particle but with

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two different values of q. Figure 3 reveals that thecross-beat signal is large enough that the LDV signalprocessor can be used to obtain the instantaneousvelocity gradient dvx~l, t!yl in real time.

Figure 4~a! shows the time-averaged velocity-gradient ~or vorticity! profiles across the channelheight y at Re 5 510 and 1020. The solid and the

Fig. 3. Oscilloscope trace of a typical beat burst between twomoving particles. The signal was obtained in the near-wall regionat Re 5 1020.

Fig. 4. ~a! Measured velocity-gradient profile across the channelheight y at Re 5 1020 ~filled circles! and Re 5 510 ~open circles!.

he solid and the dashed curves are the calculated velocity gradi-nts from the measured velocity profiles shown in ~b!. ~b! Mea-ured velocity profile at Re 5 1020 ~filled circles! and Re 5 510open circles!. The solid and the dashed curves are the polyno-

mial fits ~up to y4! to the data points.

dashed curves are the calculated velocity gradientsusing the polynomial fits ~up to y4! of the measuredlocal velocity profiles shown in Fig. 4~b!. Both theelocity measurement and the velocity-gradient mea-urement are conducted with the LDV signal proces-or. In Fig. 4 each data point is averaged over a timenterval of 200 s with more than 3000 readings. It iseen from Fig. 4~a! that the measured velocity-radient profiles agree well with the calculated onesoth in magnitude and in sign. Figure 4 thus dem-nstrates that the new fiber-optic technique worksell. The small deviations shown in Fig. 4~b! be-

ween the solid curve and the measured velocity gra-ient at Re 5 1020 arise from the poor accuracy of theolynomial fit to the velocity profile in the centralegion of the flow. The velocity gradient in this re-ion is small when compared with the local velocityuctuations. This is especially true at high Reyn-lds numbers.The measurements shown in Fig. 4 reveal an im-

ortant advantage of the new technique. Because iteasures the velocity difference directly from the

ross-beat between two moving particles, the accu-acy of the velocity gradient ~and hence vorticity!easurements becomes as high as the local velocityeasurement with LDV. By using the best LDV

ignal processor available on the market, one canesolve the beat frequency up to 1 out of 216, which is

the highest accuracy among all the velocimetry meth-ods.9 Another way of obtaining the velocity gradientis to measure the local velocities at two closely spacedlocations individually and then find the velocity dif-ference by data subtraction. This is the usual ap-proach employed by many vorticity measurementmethods, such as the multisensor hot-wire anemom-etry, PIV, and the multipoint LDV.1,2 With thesemethods, however, the measured velocity differencesmay incur relatively large uncertainties in certainturbulent flows, in which the mean velocity v# is muchlarger than the velocity difference dv. Under thiscircumstance the relative error is magnified by a fac-tor proportional to 2v#ydv. The error bar may becomeeven larger for higher-order statistics. Because theoptics of the new technique is similar to that of LDV,we expect its cost to be approximately the same asthat for LDV.

Since the measurement of the velocity gradientrequires two particles for optical mixing, particleseeding becomes important in determining the datarate. Figure 5 shows the measured data rate S as afunction of the particle-number density n for threedifferent particle sizes. The measurements are per-formed in the near-wall region ~2 mm from the topwall!, where the mean velocity is v# . 3.3 cmys. It isseen that the measured S increases rapidly with n forsmall values of n and then reaches a maximum valueSm at a concentration nm, which depends on the par-ticle size. The maximum data rate achieved withthe current experimental condition is in the range of500–640 countsys, varying slightly with particle size.The data rate reaches its maximum when there is


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always at least one pair of particles in the measuringvolume at any given time. Further increase in par-ticle concentration introduces the possibility of hav-ing more than one pair of particles in the measuringvolume at a given time. Evidently, signals frommore than one pair of particles cause additional in-terference, resulting in a decrease in the signal-to-noise level ~and hence the data rate S!.

In fact, one can estimate Sm and nm by assumingthat the seed particles are uniformly distributed inthe fluid. Given that the measuring volume has acylindrical shape of s . 40 mm diameter and L 5 120

m length, we find Sm 5 v#ys . 825 countsys andnm 5 4y~ps2L! . 6.6 3 103 mm23. It is seen fromFig. 5 that the measured Sm is close to the calculatedvalue ~dashed line!. Figure 5 thus demonstratesthat, under a proper seeding condition, the data ratefor the velocity-gradient measurement is comparablewith that for the LDV measurement. It is also foundfrom Fig. 5 that the measured Sm for the particles of1.0 mm in diameter is located at nm . 1.1 3 104

mm23, which is 67% larger than the estimated value.This suggests that the scattering from some of theparticles is not counted as valid signal. The effectbecomes more pronounced for smaller particles andneeds to be studied further.

From the above measurements we find that theseed particles of 0.55 mm in diameter are more effec-tive than the other particles. They produce a largervalue of Sm . 620 countsys at nm . 1.42 3 104 mm23,which corresponds to a particle volume fraction of1.2 3 1026. This is a very small particle seeding,and the solution looks clear. For the particles of 1.0mm in diameter, however, the solution becomes some-what turbid at the larger particle concentrations. Itshould be noted that, to increase the particle numberdensity and at the same time avoid the multiple scat-tering problem, we have used the seed particles thatare 5–10 times smaller than those used in LDV ~typ-ically 5 mm in diameter!.

Fig. 5. Measured data rate S as a function of the particle-numberdensity n. The diameters of the particles used are 1.0 mm ~trian-gles!, 0.55 mm ~open circles!, and 0.4 mm ~filled circles!. Thedashed line indicates the calculated maximum data rate Sm.


4. Summary

We have devised a dual-beam, two-fiber scheme tomeasure the velocity difference, dv~l !, over a smallseparation l. With two sets of optical fibers and cou-plers, the new technique becomes capable of measur-ing two velocity-gradient components simultaneouslyat four closely spaced positions in the x–y plane, andthe difference of the two velocity gradients gives thez component of the time- and space-resolved vorticityvector v~r, t!. Because it is based on the same Dopp-ler beating principle, the new fiber-optic techniqueoperates in a manner similar to that of the standardLDV. The only difference is that the beat signal inthe new scheme, which is proportional to dv~l !, re-sults from two different particles separated by a dis-tance l ~cross beat!. In the standard LDV, however,the signal is proportional to the local velocity of theseed particle, because it is produced by the beating ofthe scattered light from the same particle ~self-beat!.

To find optimal experimental conditions for themeasurement of the velocity gradient ~and hence theflow vorticity!, we test the technique in a steady lam-inar flow, in which the velocity gradient ~or flow vor-ticity! can be calculated from the independentmeasurement of the mean velocity profile. The ex-periment verifies the working principle of the fiber-optic technique and demonstrates its applications.It is found that the new technique measures the ve-locity difference ~and hence the velocity gradientwhen l is known! with the same high accuracy andhigh sampling rate as LDV does for the local velocitymeasurement. It is nonintrusive and capable ofmeasuring the velocity gradient ~or flow vorticity!with a spatial resolution as low as ;50 mm.

The successful test of the fiber-optic technique inthe laminar flow with one optical channel is an im-portant first step for the development of a two-channel fiber-optic vorticity probe, which has wideuse in the general area of fluid mechanics, especiallyin the study of turbulent flows. Because it is basedon the Doppler beating principle, the new techniquecan be applied to both laminar and turbulent flowswithout any need for recalibration ~or any calibrationat all!. With the tested optical design and the opti-

al seeding conditions found in the experiment, itecomes straightforward to add an additional opticalhannel and build a compact fiber-optic vorticityrobe. The commercial LDV signal processor usu-lly has at least two independent input channels andhus is capable of processing two velocity-gradientignals simultaneously. The new fiber-optic tech-ique combined with the LDV electronics, therefore,an be used to perform time-series measurements ofz~r, t!. Information concerning the vorticity fieldan be obtained by means of scanning the vorticityrobe over an area of interest. We have recently setp a new turbulent tunnel flow system, and furtheresting of the technique in the turbulent flow is underay.The new fiber-optic technique should be comparedith PIV, which has been used increasingly in recent

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years. PIV can provide a simultaneous 2D vorticitymap but with relatively low spatial and temporalresolution. Because the laser sheet used in PIV hasto be aligned along the direction of the mean flow toprevent the seed particles from moving out of theillumination plane, PIV is unable to measure thestreamwise vorticity component. The new vorticityprobe, however, measures only a local vorticity com-ponent but with high precision and much better spa-tial and temporal resolution. It is capable ofmeasuring both the streamwise and the cross-streamvorticity components. With a high data rate, thenew vorticity probe can be used to measure the vor-ticity statistics and power spectra in various turbu-lent flows.

In a typical PIV setup one needs approximately tenparticles to construct a velocity vector and ten morevelocity vectors to calculate a vorticity vector.4 As aresult, approximately 100 particles are needed forresolving an average vorticity vector in the occupiedregion. The new vorticity probe requires only fourparticles for resolving a local vorticity component.These four particles, however, are required to bepresent at four closely spaced fixed locations simul-taneously. To increase the data rate, one may per-form independent time-series measurements for eachvelocity-gradient component ~instead of synchronousmeasurements! and then apply a temporal smoothingalgorithm to the respective data sets. Clearly, thetwo types of technique are complementary and can-not be substituted by each other.

We thank F. W. Chambers for useful discussionsand for loaning us the LDV parts. The assistance ofM. Lucas and his team in fabricating the experimen-tal apparatus is gratefully acknowledged. This re-

search was supported by NASA grant NAG3-1852and by National Science Foundation ~NSF! grant

MR-9981285.

References and Note1. J. F. Foss and J. M. Wallace, “The measurement of vorticity in

transitional and fully developed turbulent flows,” in Advancesin Fluid Mechanics Measurements, M. Gad-el-Hak, ed., Vol. 45of Lecture Notes in Engineering ~Springer-Verlag, New York,1989!, pp. 263–321.

2. J. M. Wallace and J. F. Foss, “The measurement of vorticity inturbulent flows,” Annu. Rev. Fluid Mech. 27, 469–514 ~1995!.

3. R. J. Adrian, “Particle-imaging techniques for experimentalfluid mechanics,” Annu. Rev. Fluid Mech. 23, 261–304 ~1991!.

4. E. B. Arik, “Current status of particle image velocimetry andlaser Doppler anemometry instrumentation,” in Flow at Ultra-High Reynolds and Rayleigh Numbers: a Status Report, R. J.Donnelly and K. R. Sreenivasan, eds. ~Springer-Verlag, NewYork, 1998!, pp. 138–158.

5. Y. Du, B. J. Ackerson, and P. Tong, “Velocity difference mea-surement with a fiber-optic coupler,” J. Opt. Soc. Am. A 15,2433–2439 ~1998!.

6. S.-H. Yao, V. K. Horvath, P. Tong, B. J. Ackerson, and W. I.Goldburg, “Measurements of the instantaneous velocity differ-ence and the local velocity with a fiber-optic coupler,” J. Opt.Soc. Am. A 18, 696–703 ~2001!.

7. M. V. Otugen, W.-J. Su, and G. Papadopoulos, “A new laser-based method for strain rate and vorticity measurements,”Meas. Sci. Technol. 9, 267–274 ~1998!.

8. G. A. Zimmerli, K. Y. Min, and W. I. Goldburg, “Measuring theprobability distribution of velocity differences using homodynecorrelation spectroscopy,” in Optical Diagnostics for FluidsyHeatyCombustion and Photomechanics for Solids, S. S. Cha,P. J. Bryanston-Cross, and C. R. Mercer, eds., Proc. SPIE 3783,46–53 ~1999!.

9. This frequency resolution is obtained when a simple sinusoidalwave is used as the input signal. The actual frequency reso-lution of LDV for real burst signals is somewhat less.


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Proposal and Testing for a Fiber-Optic-Based Measurement of Flow Vorticity