Imaging through turbulent media has been frequent-ly studied in the literature. The resolution of astro-nomical and satellite images, for example, is severelydegraded by random temperature variations in theatmosphere, and algorithms have been developedto improve the quality of such images. Turbulentair can also be a problem when using ground-basedinterferometric measurement techniques, such asdigital holography, to capture long time sequencesof a moving or deforming object. In this case, two ad-ditional complications arise. First of all, the absolutevalue of the phase distortion in the object wave be-comes important. Also, since the object is continu-
ously moving or deforming, it is not possible tocapture a long sequence of images and use “ensembleaveraging” techniques such as speckle imaging [1,2].One way of reducing the effects of phase noise in sucha measurement would be to use a digital spatio-temporal (ST) noise reduction filter, but this re-quires knowledge of the ST statistics of the phasefluctuations. Yamauchi and Hibino have studiedthe temporal statistics of the phase fluctuations ininterferometric measurements performed on a smalllens mold during processing in a grinding machine[3]. In this case, the measured object was so smallthat there was almost no spatial variation in thenoise due to air fluctuations. When measuring largerobjects, the spatial variations in the noise cannot beignored [4], and full ST statistics are required. Thisarticle describes a method for measuring ST phasestructure and covariance functions for the case of
temperature fluctuations, that have at least station-ary time increments, using high-speed digitalholographic interferometry. Nonstationary fluctua-tions, such as those generated during heat molding,are difficult to deal with and will be excluded. Themethod is applied to the relatively simple case ofquasihomogeneous/isotropic turbulence generatedin a wind tunnel for which the Kolmogorov theory ap-plies. Thus, it is possible to verify the method againstthe theory. The article is organized as follows: Sec-tion 2 describes both the measurement setup andthe wind tunnel for creating the refractive index fluc-tuations. Section 3 contains a short summary of the-oretical expressions for phase structure functions ina locally inhomogeneous/isotropic medium. Section 4describes how the phase structure and covariancefunctions are estimated from the measured phasevolume. Section 5 shows some of the measurementresults and a comparison with the theory of Sec-tion 3. Section 6 discusses what conclusions can bedrawn from this work.
2. Experimental Setup
There are many ways to generate a turbulent tem-perature variation in a flow of air. A common ap-proach is to force the air through a grid, whichtends to give turbulence with good homogeneityand isotropy properties. The problem with grid tur-bulence is that it is difficult to produce turbulencewith an inertial subrange in their energy spectra.This is because the Reynolds number associated withthe grid needs to be very high, which, according tothe work of Corrsin [5], is difficult to achieve in a con-ventional smaller sized wind tunnel. A way aroundthis problem is to use an active grid, which has beendone by many authors [6]. Another way of achievinga turbulent flow is to stir an otherwise laminar flowin a random manner using different kinds of moving“wings” upstream of the test section, as done byMakita [7]. Turbulence can also be produced in thebulk air flow inside a wind tunnel, provided theReynolds number is high enough. This is not an idealsolution because a very long test section is requiredfor the turbulence to get reasonably homogeneous,and this also means that large turbulent boundarylayers will form. But as noted by Strohbehn [8],the temperature fluctuations tend to homogenizeat a faster rate than the velocity fluctuations, and,therefore, this approach is not as bad as it mightseem. Indeed, Magee and Welsh [9] have succeededin constructing a small desktop flow channel with avery short test section that produced relativelyhomogeneous refractive index disturbances withan inertial subrange covering about an order of mag-nitude in spatial frequencies. In this article we haveused a wind tunnel with a test section of about 1m togenerate boundary layers that are large enough tocause significant violation of the frozen turbulencehypothesis and, thus, nontrivial ST statistics.Figure 1 shows a sketch of the experimental setup.
The light source is a continuous Nd:YAG laser with
an output power of about 0:4W. This amount ofpower is necessary to reach the very low exposurestimes necessary to freeze the motion of the air inthe wind tunnel. Light from the laser is split intoa reference beam and an object beam using a half-wave plate (λ=2) and a thin film polarizer (P) so thatthe relative intensity of the two beams can be ad-justed. The object beam is expanded into a thin sheet,parallel to the direction of the air flow, using two cy-lindrical lenses and collimated using the positivelens (L1). The collimated light sheet is sent througha pair of rotatable slits, 85mmwide, in the channel toan optical diffuser plate on the other side of the chan-nel, which is imaged onto the detector of a high-speedcamera (Redlake MotionPro HS-X3). Because of therandom refractive index variations in the channel,there will be random variations in the phase overthe light sheet incident on the diffuser plate. Thesmallest spatial variations in phase on the diffuserplate will correspond to the smallest refractive indexinhomogeneities. It is important to adjust the widthof the light sheet so that the imaging system can re-solve these variations, and this is what, in practice,will limit the maximum width of the light sheet.Because the random intensity variations, due tothe speckles in the object light, create points witha very uncertain phase, it is necessary to averagethe phase over a region that contains several speck-les. This means that the resolution of the imagingsystem will decrease, and, thus, the maximum widthof the light sheet will be smaller. The reference lightis sent through a fiber (F) and expanded using a ne-gative lens (L3). Reconstruction of the holograms isdone using the Fourier filtering method [10,11].
The wind tunnel shown in Fig. 2 is homebuilt andcomposed of the following parts: First is a settlingchamber (S) made of straws and a metallic grid that
Fig. 1. Sketch of the experimental setup: λ=2, half-wave plate; P,thin film polarizer; CL1 and CL2, cylindrical lenses; L1, L2, andL3, ordinary single lenses; D, diffuser plate; BS, beam splittercube; and CMOS, high-speed camera.
Fig. 2. Sketch of the wind tunnel used to produce the turbulentair: S, settling chamber; C, contraction cone; HW, heated canthalwires inside the test section—T; RS, rotatable slits; DI, air diffuser;and F, fan.
breaks down large turbulent eddies in the outsideair. Next is a contraction cone (C), where the crosssection of the flow is gradually decreased to that ofthe test section (10 × 40 cm). At the start of the testsection is a small strip of sandpaper to aid the tran-sition to a turbulent flow. This will also speed up thetransition from laminar to turbulent boundary layergrowth. Next to the sandpaper are five heatedcanthal wires (HW), which provide an initial largescale temperature variation that is successively bro-ken down by the velocity fluctuations. After the testsection is an air diffuser (DI) to lower the air velocityat the fan. The object light sheet is sent through apair of rotatable slits (RS) that are positioned90 cm from the inlet of the test section. The widthof the slits is 85mm, which limits the size of the ob-ject light sheet. In our experiments, the light sheetwas aligned with the direction of the flow. The flowis driven by a commercial 135W ventilation fan(F) with a speed range of 450 to 1260 rpm, corre-sponding to a velocity range of 2:7–8:2m=s in the testsection. The optical components are mounted on anair damped optical table with the channel running10 cm above it, without being in direct contact toavoid excessive transfer of vibrations.
3. Theory
Light propagation in locally homogeneous/isotropicmedia has been extensively studied in the literature[12–14]. The following is meant as a short summaryof the theoretical results that are important in thisarticle. Figure 3 shows the geometry of the light pro-pagation problem.An initially plane wave enters a region of refrac-
tive index fluctuations, described by nðx; y; z; tÞ, andpropagates a distance L through the disturbed re-gion. The phase distribution ϕðx; y; tÞ finally appearsin the xy plane. It is fortunate that the structurefunction of ϕðx; y; tÞ is the same, regardless of thelength of propagation [15]. This means that it is pos-sible to imagine the light rays as being perfectlystraight, as in Fig. 3. In noninterferometric imaging
through turbulent media, there is a distinct advan-tage of using structure functions instead of correla-tion functions to describe the phase fluctuationsbecause the turbulent structures that are large com-pared to the pupil of the imaging system do not causeany significant image degradation. These largestructures are the ones that are most dependenton the geometry of the turbulent flow. By subtractingthe phase fluctuations in two nearby points [seeEq. (1)], the effect of the structures that are much lar-ger than the separation of these two points will beapproximately cancelled out and the fluctuations willbe more homogeneous and isotropic. Therefore, mostof the theoretical work on phase fluctuations beingpublished has concentrated on these structure func-tions. It will be assumed that the mean of ϕðx; y; tÞchanges slowly with time compared with the randomfluctuations [ϕðx; y; tÞ is said to have stationary timeincrements] and also compared to all time differencesused in the ST structure function. Even though thestrength of the turbulence can change over time, it iscustomary, at least in the atmospheric literature, toexclude the time dependence of the phase structurefunction. In this case the spatial structure functionSDϕ can be expressed as [12]
where hi is the expectation value operator. In the caseof a locally homogeneous/isotropic flow with constantfluctuation strength, SDϕ will only depend on ρ, thedistance between points P1 and P2, and have a sim-ple functional form independent of the flow geometry[12]:
Here k is the vacuum wavenumber of the light and l0is the inner scale of the refractive index fluctuations(roughly speaking, the size of the smallest inhomo-geneities). The region l0 ≪ ρ ≪ L0 is called the iner-tial subrange because for turbulent eddies of this size,inertial forces dominate over viscous forces. The re-gion ρ ≪ l0 is called the dissipative range because tur-bulent eddies of this size are rapidly dispersed byviscous forces. The structure constantC2
n is ameasureof the strength of the refractive index fluctuations. Incases where the fluctuation strength varies slowlywith time, it is customary to include this time depen-dence inC2
n [13].Here it is assumed thatC2n is constant
along the whole propagation path. If this is not true,C2
n must be replaced by an integral over the propaga-tion path [15]. In our experiments it is the first equa-tion in (2) that will be used to validate the method. Inthe same way as in Eq. (1), the ST phase structurefunction STDϕ can be expressed as
Fig. 3. Light propagation geometry. The refractive index fluctua-tions in the turbulent region of size L are nðx; y; z; tÞ, and ϕðx; y; tÞare the phase fluctuations of the wave in the detector plane. ρ isthe vector between two general points P1 and P2 in the detectionplane with coordinates ðx1; y1Þ and ðx2; y2Þ.
Assuming for the moment that the Taylor hypoth-esis of frozen turbulence holds perfectly, the refrac-tive index inhomogeneities will simply move acrossthe region of detection in the xy plane at the bulk flowvelocity V0 without evolving or mixing. The effect ofthe V0z velocity component parallel to the light pro-pagation direction in Fig. 3 is simply to decrease thecorrelation between the phase fluctuations at P1 andP2 because during time τ, some turbulent eddies willhave drifted out of the propagation path being re-placed by others. As argued by Tatarskii [13], thisdecorrelation effect is usually not important becausefor long propagation lengths it takes a very long timedifference before the effect becomes noticeable, and,by that time, movement perpendicular to the flowwill in general be so large that the fluctuations atthe two points will be fully decorrelated anyway.However, when doing experiments where the obser-vation area is large and propagation lengths arerather small (e.g., to study turbulent mixing in a freeflow), this effect can become important. Ignoring theeffect of the flow parallel to the light propagationdirection, the ST phase structure function is
STDϕðρ; τÞ ¼ SDϕðjρ − V0⊥ · τjÞ; ð4Þ
where V0⊥ is the component of the bulk flow velocityin the xy plane (detector plane) in Fig. 3, and ρ is thevector from ðx1; y1Þ to ðx2; y2Þ. The Taylor hypothesiscan, of course, never be strictly valid because therewill always be a small random velocity fluctuationin the flow (otherwise there would be no random tem-perature variations to begin with). When there ismixing of the turbulent eddies, the ST structure func-tion STDϕðV0⊥ · τ; τÞ will not be zero for all τ, as sug-gested by Eq. (4). Physically STDϕðV0⊥ · τ; τÞ can beinterpreted as the structure function with zero spa-tial separation calculated in a coordinate system thatmoves along with the flow. In the case of a channelflow, there are two sources of mixing: the determinis-tic mixing due to a nonuniform bulk velocity and therandom velocity fluctuations. Once the ST structurefunction has been measured, it is easy to determinethe velocity component V0⊥ by finding the separationρmin that minimizes Dϕðρ; τÞ for a given τ:
V0⊥ ¼ ρmin
τ : ð5Þ
In noninterferometric imaging, the phase struc-ture function is sufficient information to determinethe image degradation (long and short exposure op-tical transfer function of the atmosphere). But in aninterferometric measurement, the lower spatial fre-quencies of the refractive index turbulence are justas important as the higher frequencies, and, hence,it would be desirable to determine the phase covar-iance function instead. Unfortunately, these coarserefractive index variations are highly dependenton the flow geometry and, therefore, usually nothomogeneous and isotropic, which means that the
covariance function will depend on absolute coordi-nates. Since there is no simple theory for the coarserefractive index variation, the covariance functionwill be expressed only in terms of the measuredphase distribution ϕðx; y; tÞ. To enable the estimationof the covariance function from a single measuredphase volume, it must be assumed that ϕðx; y; tÞ iswide sense stationary in time with zero mean
For the simple case of a homogeneous/isotropic re-fractive index turbulence, the relationship betweenthe covariance and the structure functions is
Cϕðρ; τÞ ¼12· ½Dϕð∞;∞Þ −Dϕðρ; τÞ�; ð7Þ
where ρ is the distance between the points ðx1; y1Þand ðx2; y2Þ. It should be mentioned that Vecherinet al. [16–18] have successfully used homogeneousand isotropic Gaussian ST covariance functions toapproximately describe the full three-dimensionaltemperature field of a limited region of the atmo-sphere for recording acoustic tomography images oftemperature inhomogeneities. Acoustics is, in manyaspects, a more complicated field because, due to therelatively long wavelengths, it is never possible touse geometrical approximations as in Fig. 3.
4. Estimation of Statistics from the MeasuredPhase Volume
The procedure for estimating the spatial structurefunction, ST structure function, and covariance func-tion for the phase fluctuations is illustrated in Fig. 4.When estimating the spatial phase structure func-tion, the camera can be run at a relatively low framerate so that the phase noise in the different imagesare fairly independent. This will reduce the amountof data necessary to get a good estimate. The randomphase contribution from the diffuser plate is compen-sated for by capturing a reference image before thewind tunnel is switched on and then multiplyingall of the measured complex amplitude distributionswith the complex conjugate of this reference image.Figure 4 shows a sketch of part of a measured phase
Fig. 4. Part of a measured phase volume where A, B, C, and D arephase elements at different points in the image.
volume. The spatial phase structure function wasfirst estimated using Eq. (1), i.e., it was not initiallyassumed that the flow was locally homogeneous/isotropic. For a pair of points (AB), this means thatonly temporal averaging was performed. Since it isonly the phase difference between the two points thatis important, it is possible to use a two-dimensionalspatial unwrapping of each image separately. In ourexperiment, we used only a sheet of light, and, hence,the phase volume had only one spatial dimension,and a simple one-dimensional unwrapping betweenthe points was sufficient, as illustrated by the solidbox around the AB point pair. If, as in our measure-ment, the fluctuations turn out to be locally homoge-neous/isotropic, then all pairs of point with the sameseparation, e.g., AB, CD, AC, and BD, can be used inthe averaging.The ST structure and covariance functions need to
be estimated from a measurement in which thephase fluctuations are resolved in time. Here anotherprocedure was chosen to remove the random phasecontribution from the diffuser plate. The time se-quence at each measurement point is unwrapped in-dependently and then adjusted against each other byremoving the average value of each sequence. This ispossible because the measurement time is muchlonger than the longest characteristic phase fluctua-tion time. This method was chosen because it bothremoves the effects of the diffuser plate and the needto perform a difficult and time-consuming three-dimensional unwrapping. When studying self-convective flows, where the fluctuation times can bevery long, it is necessary to use the previous methodfor removing the effects of the diffuser plate togetherwith a real three-dimensional unwrapping proce-dure. The procedure for estimating the ST structurefunction is then completely analog to the spatial caseexcept that one of the phase elements in Fig. 4 is ta-ken from a later image. As mentioned in connectionwith Eq. (6), the covariance cannot always be esti-mated from a single phase volume, such as that inFig. 4. Because the measured phase ϕðx; y; tÞ is usual-ly not homogeneous/isotropic, it needs to be widesense stationary in time, otherwise no averagingat all is possible in the phase volume. This is a rea-sonable assumption for a stationary channel flowsuch as that used in our experiment, but it does nothold for the very important class of self-convectiveflows. However we have previously, with some suc-cess, used a method of local time averaging [19] toestimate the temporal correlation of the phase fluc-tuations in such a flow, and this method can easily beextended to the ST case. Even in a relatively stablechannel flow, there will be slow variations in the sta-tistics of the refractive index fluctuations because theair entering the wind tunnel is not of perfect uniformtemperature. It also takes quite a long time for theheated part of the channel to reach thermal equili-brium, which can cause a slow net heating of theair through the channel. To decrease the time varia-tions in the phase data due to these effects, the data
is filtered with a high-pass frequency filter with alower cutoff of 0:5Hz. Since is takes the air 0.3–1 sto pass the channel (depending on fan speed), this fil-tering should not remove any effects on the flow ori-ginating from the geometry of the channel. Choosingan appropriate frame rate can be done by looking atthe temporal spectra of the phase fluctuations.Assuming that the Taylor hypothesis holds, thetemporal spectral density Wϕ depends only on theone-dimensional spatial spectral density Vϕ of thephase and the magnitude of the bulk flow velocityv0⊥ ¼ jV0⊥j perpendicular to the light propagationaccording to [13]
Wϕðf Þ ¼1v0⊥
· Vϕ
�f
v0⊥
�; ð8Þ
where f is the temporal frequency. The effects of tur-bulent mixing of eddies will manifest itself mostly forlow time frequencies, which means that it plays norole in estimating the bandwidth. The highest spa-tial frequency will approximately be that correspond-ing to the inner scale size (∼1=l0), which can beestimated from the spatial structure function. Hence,from Eq. (8) the highest time frequency will be
fmax ≈v0⊥l0
: ð9Þ
Since most of the energy in the spatial fluctuations isconcentrated to lower frequencies, it may not be ne-cessary to choose the sampling frequency as high as2fmax in an actual experiment. In our experiments,the sampling frequency was chosen to be just overfmax.
5. Results and Discussion
Figures 5 and 6 show the estimated spatial phasestructure functions for flow velocities of 8.2 and2:7m=s, respectively, in the test section. The struc-ture function was first estimated using Eq. (1) tocheck that the flow really was locally homogeneous/isotropic across a 15mm wide light sheet, which isaligned with the flow. The spatial resolution wasabout 0:07mm, which was also the height of thesheet. According to the work of Magee and Welsh[9], this should be well below the inner scale size.They used a channel of similar geometry and flowspeeds, and their measurements showed inner scalesizes of about 0:4–1mm. The sampling rate was100Hz, which means that the air will have movedabout 46mm between each image so that the phasefluctuations will be fairly independent. The exposuretime was 30 μs, which corresponds to a movement ofthe air of 0:25mmand 81 μm, respectively, during theexposure. This motion is below the inner scale size,and, hence, the effect of spatial averaging during theexposure should be negligible. The total number ofimages used in the averaging was 15,000. Theamount of power generated in the heated wireswas 600W and 400W, respectively. Also plotted in
Figs. 5 and 6 are curves fitted from Eq. (2) for theinertial subrange and dissipative range, respectively.The structure constant C2
n is estimated by fitting aρ5=3 function to the central part of the measuredcurve. Typical daytime near ground values of C2
n inthe outside air are in the range 10−17 < C2
n < 10−12,which is about 10 − 106 weaker than the fluctuationstrength inside the wind tunnel in our experiments[8]. It is possible, in principle, to estimate the innerscale size by using the value of C2
n and fitting a ρ2function to the first part of the measured curve. Inmany cases, such as astronomical imaging, the reso-lution of the detector might not be sufficient to allowthis, as the collecting aperture is often very largecompared to l0. Therefore, it is customary to definethe inner scale size as the value of the separationρ where the measured curve starts to deviate fromthe fitted ρ5=3 function [9]. For a small separation(ρ < l0), the measured curves tend to curve upward
instead of downward, as suggested by the dissipativerange theory [see Eq. (2)]. We believe that this devia-tion is due to the large boundary layers close to thewall of the channel, which contains more fine struc-tures than the bulk flow. The random measurementnoise in the detection process has been estimated tobe less than 10−3 (rad2) and should not affect theshape of the curves. The displacement of the turbu-lent eddies, due to the bulk flow velocity during theexposure time, should have the effect of smoothingthe spatial phase variations, so that the measuredcurves in Figs. 5 and 6 should tend to curve down-ward in the dissipative range. The random displace-ment of eddies due to the turbulent velocityfluctuations could have the opposite effect of increas-ing spatial phase variation. But since the random ve-locity fluctuations are typically only a few percent ofthe bulk velocity, this effect should be negligibleduring the short exposure time.
Fig. 5. Spatial phase structure function with a flow velocity of 8:2m=s. The power generated in the heated wires was 600W. Also shownare theoretical curves fitted from Eq. (2) with C2
n ¼ 3:8 × 10−11 m−2=3 and l0 ≈ 0:5mm.
Fig. 6. Spatial phase structure function with a flow velocity of 2:7m=s. The power generated in the heated wires was 400W. Also shownare theoretical curves fitted from Eq. (2) with C2
Figure 7(a) shows an estimated ST phase structurefunction for three different time delays. The flow ve-locity in the test section was 4:6m=s. This velocitywas measured using a sensitive Pitot tube, but itcan also be very accurately estimated from Eq. (5).The structure function was first estimated using
Eq. (3) to verify that the flow really is locally homo-geneous/isotropic across the 56mm wide light sheet,which is aligned with the flow. In this case the spatialresolution is about 0:26mm, which might not be ade-quate to really resolve the smallest structures, butfrom Figs. 5 and 6 , we know that there is a goodagreement with the Kolmogorov theory, so that re-sult need not be repeated here. The sampling ratewas 10kHz, which is to be compared with the highesttemporal frequencies estimated as 9kHz usingEq. (9) with l0 ≈ 0:5mm and v0⊥ ¼ 4:6m=s. As men-tioned in connection with Eq. (9), this apparent un-dersampling is probably not important because mostof the fluctuation energy is contained in the largerspatial structures (lower temporal frequencies).The exposure time was 30 μs, which corresponds toamovement of the air of 0:14mm, which is well belowthe inner scale size. Because of the high samplingrate necessary to resolve the fluctuations in timecombined with the need to average over manyindependent fluctuations, the necessary number ofimages required to get a reasonable estimate be-comes very high. In this case, the total number ofimages was 104791 (limited by the camera memory).The amount of power generated in the heated wireswas 0:45kW. The width of the structure function canbe thought of as the size of the “correlation cells” inthe flow. If this size is divided by the flow speed, acharacteristic phase fluctuation time is attained.Defining the width of the correlation function asthe 1=e width, this fluctuation time is about 3msin this case. Figure 7(b) shows the phase covariancefunction estimated from the same measurement. Itwas first estimated using Eq. (6), but as it turned
out the flow was very homogeneous along the 56mmlight sheet, and so the covariance function is simply afunction of the magnitude of the separation alongthis sheet. For a homogeneous/isotropic flow, the re-lationship between the structure and covariancefunctions is given by Eq. (7). Figure 7(b) also showsthe theoretical covariance function calculated fromthe structure function in Fig. 7(a) using Eq. (7).Had the slit been rotated 90 deg, there would havebeen larger variations in the statistics along the lightsheet, and there would have been a larger differencebetween the theoretical (homogeneous/isotropic) andmeasured covariance functions. But because the fluc-tuations are wide sense stationary in time, it wouldstill have been possible to use Eq. (6) to estimate thecovariance function. Note that as mentioned inSection 4, the phase data have been high-pass fil-tered to remove the slow phase fluctuations due torandom variations in the temperature of the air en-tering the wind tunnel. These fluctuations are tooslow to significantly affect the structure functionfor the short time difference used. The covariancefunction, on the other hand, is affected by fluctua-tions of all frequencies and requires that the phasefluctuations are strictly wide sense stationary intime. Therefore, had the phase data not been high-pass filtered, there would have been a larger differ-ence between the two curves in Fig. 7(b).
As mentioned in the introduction, the main pur-pose of estimating the ST statistics of the phase fluc-tuations is to use them in combination with an STnoise reduction filter to postprocess measurementsof object deformation/movement performed in a dis-turbed environment. When calculating, for example,the ST phase covariance function from such a mea-surement, the result will be the sum of the ST covar-iance function of the object and the air. If the objectmotion does not exactly match the bulk flow velocityof the air, the covariance functions of the air andobject will, for sufficiently large time differences,
Fig. 7. (a) ST phase structure and (b) covariance function for a flow velocity of 4:6m=s. The amount of power generated in the heated wireswas 0:45kW.
be separated. When looking at Fig. 7(b) it is obviousthat the shape of the covariance function of the air forsmall time differences could be estimated from thecorresponding covariance function for a larger timedifference. Hence the bulk flow of the air (which ex-ists even in self-convective flows) can be used to se-parate the covariance functions of the object and air.Figure 8 shows a plot of STDϕðV0⊥ · τ; τÞ as a func-
tion of time difference for a flow speed of 2:7m=s inthe test section. The largest allowable time differ-ence is limited by the flow velocity and the size ofthe observation region. The rapid growth at theend is because STDϕðV0⊥ · τ; τÞ is calculated for a gi-ven τ simply as the minimum value of STDϕðρ; τÞ overall separations ρ in the observation region. When theseparation V0⊥ · τ is larger than the size of the obser-vation region, this value will not be equal to the mini-mum value taken over all possible separations. Forsmall time delays most of the growth in STDϕðV0⊥ ·τ; τÞ is likely to be due to the rapid mixing of eddiesinside the boundary layers. The exact shape ofSTDϕðV0⊥ · τ; τÞ as a function of τ will depend onthe size distribution of eddies in the boundary layer,the variation in fluctuation strength across theboundary layer, and the velocity profile. For largetime delays the contribution from the boundarylayers to STDϕðV0⊥ · τ; τÞwill eventually settle towarda constant value (equal to twice the phase varianceover the layer). The effect of the turbulent mixing in-side the bulk flow will then be the dominant contri-bution to the growth of STDϕðV0⊥ · τ; τÞ. In this regionthe growth will depend on the magnitude of the ran-dom velocity fluctuation compared to the mean flowvelocity. Using an array of cameras, it would be pos-sible, in principle, to follow the flow for a long enoughtime to be able to see how STDϕðV0⊥ · τ; τÞ behaves inthis region. But because the range of time delays isvery limited in Fig. 8, it is difficult to draw anyquantitative conclusions from it. Finally, it wouldbe interesting in the future to investigate whetherit is possible to determine the change in shape ofSTDϕðρ; τÞ as a function of ρ with increasing time de-
lay τ from the curve in Fig. 8 alone, at least for a giventype of flow. The reason is that finding the maximumvalue on the surface STDϕðρ; τÞ for different timedelays τ can be implemented as an optimization pro-blem, which takes much less time to solve than hav-ing to calculate the entire STDϕðρ; τÞ function for allpossible separations ρ (in the observation region)and time delays τ.
6. Conclusions
Interferometric measurement techniques, such as di-gital holographic interferometry, require that thephase disturbances due to temperature inhomogene-ities in the surrounding air do not change during thetime of the measurement. This is a problem whenperforming a time-resolved measurement of themovement or deformation of an object over a timethat is longer than the fluctuation time of the inho-mogeneities. Since the object is continually changing,it is not possible to use classical image enhancementtechniques, such as speckle imaging [1,2], to reducenoise in the measurement. Using a digital noise re-duction filter is possible but requires that some sta-tistics of the phase fluctuations are known. In thisarticle we have developed a method for estimatingST phase structure and covariance functions in a tur-bulent air flow using high-speed digital holographicinterferometry.
The method has been applied to the case of locallyhomogeneous/isotropic refractive index turbulencegenerated in a small wind tunnel. The measuredstructure functions show good agreement with theKolmogorov theory, which is taken as a validationof the method. It is also shown that the ST structurefunction contains information about the bulk flow ve-locity and the turbulent mixing of eddies inside theflow. For the case of measuring an object deforma-tion/movement in the presence of refractive indexfluctuations, a method of separating the ST statisticsof the object and air is also suggested.
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