Experiments in Fluids 22 (1997) 199—211 ( Springer-Verlag 1997

A hybrid digital particle tracking velocimetry techniqueE. A. Cowen, S. G. Monismith

Abstract A novel approach to digital particle tracking veloc-imetry (DPTV) based on cross-correlation digital particleimage velocimetry (DPIV) is presented that eliminates the needto interpolate the randomly located velocity vectors (typicalof tracking techniques) and results in significantly improvedresolution and accuracy. In particular, this approach allows forthe direct measurement of mean squared fluctuating gradients,and thus several important components of the turbulent dis-sipation. The effect of various parameters (seeding density,particle diameter, dynamic range, out-of-plane motion, andgradient strength) on accuracy for both DPTV and DPIV areinvestigated using a Monte Carlo simulation and optimalvalues are reported. Validation results are presented fromthe comparison of measurements by the DPTV technique ina turbulent flat plate boundary layer to laser Doppler anemo-meter (LDA) measurements in the same flow as well as directnumerical simulation (DNS) data. The DPIV analysis of theimages used for the DPTV validation is included for compa-rison.

1IntroductionIn the study of turbulent flows, particle image velocimetry(PIV) is a valuable tool capable of measuring the signifi-cant spatial structure of turbulence. Liu et al. (1991) havesuccessfully demonstrated the use of PIV to probe the structureof a turbulent channel flow. Previous experimental studies of

Received: 29 August 1994/Accepted: 31 May 1996

E. A. Cowen, S. G. MonismithEnvironmental Fluid Mechanics LaboratoryStanford University, Stanford, CA 94305-4020, USA

Correspondence to: E. A. Cowen

The authors gratefully acknowledge the support of this work bythe Fluid Dynamics Program, Office of Naval Research (ScientificDirector: Dr. E. Rood, grant N00014-94-1-0190) and the High Per-formance Computing and Communication Program, NationalScience Foundation (Scientific Director: Dr. A. Thaler, grant ASC9318166).

The authors wish to thank Jonathan Harris, whose PIV work laid thefoundation from which our technique grew and who never gave up onPTV, Bob Street, Jeff Koseff and Chris Rehmann for their steadysupply of ideas and assistance, and Bob Brown for his mechanicalwizardry. We are particularly indebted to John Crimaldi for makingthe detailed LDA measurements reported in Sect. 5.

turbulent flows have generally relied on point measurementtechniques — laser Doppler anemometry (LDA) and hot wirevelocimetry. While providing high temporal resolution, thesemethods have extremely limited spatial resolution. Untilrecently, high resolution accurate PIV techniques, such as thatof Liu et al., who report spatial resolution of 0.3 mm]1.0 mm,were based on interrogating photographic images. This processis time consuming, as the photographs must first be developedand then interrogated to determine if the experiment has beensuccessful.

Willert and Gharib (1991) demonstrated that it is possibleto use a CCD based camera to acquire digital images eliminat-ing the time consuming development and interrogation ofphotographs. Their technique (DPIV), however, must be con-sidered low resolution as they report a spatial resolution ofonly 5.2 mm]5.2 mm (we note that this is more a function oftheir chosen optical configuration and not their technique).Westerweel et al. (1996) demonstrated that DPIV can operateat resolutions on the order of conventional photographicPIV (and for that matter DNS) with comparable accuracy.They interrogated digital images at a spatial resolution of1.3 mm]1.3 mm which is comparable to that achieved by Liuet al. However, correlation based analysis does not extract themaximum amount of velocity information from an image.

Recently Keane et al. (1995)1 demonstrated that trackinga significant number of particles in a photographic image isfeasible. Essentially, they followed Guezennec and Kiritsis’(1990) algorithm and used an auto-correlation analysis esti-mate of the velocity field to guide the particle tracking algori-thm. Their results suggest an improvement in resolutionby a factor of 2.5 (to a dimensional grid spacing of about0.1 mm) over their conventional PIV approach. While theyrecognized the power of such an algorithm to improve spatialresolution, it appears that the hybrid approach can offersubstantial gains in accuracy over PIV as well. As we willdemonstrate, PTV algorithms are inherently more accuratethan correlation-based PIV algorithms since they are relativelyunaffected by the presence of displacement gradients. Webelieve this behavior is typically overlooked because the com-mon practice of interpolating the randomly located trackedvelocity vectors onto a regular grid introduces noise thatobviates the noise reduction obtained with PTV relative toPIV (Aguı and Jimenez 1987; Hesselink 1988).

1 The authors wish to thank R. J. Adrian for making us aware of hisgroup’s efforts to use correlation analysis to guide a particle trackingalgorithm and for a pre-print of their work.

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In this paper we present a DPTV technique capable of bothhigh spatial resolution and high accuracy. Our technique isable to capture the significant spatial structure of turbulence,including both mean velocity statistics and instantaneousspatial gradients. Its essential feature is the use of the randomlylocated tracked velocity vectors in place to determine all tur-bulent statistics. This results in resolution and accuracy com-parable to that found by Keane et al. (1995) under conventionalphotographic conditions but through fully digital means.Similar to Keane et al. and Guezennec and Kiritsis (1990), itrelies on a correlation analysis of images to estimate thedisplacement of particles as a function of their position. It usesthis estimate to track individual particle displacements fromwhich the instantaneous velocity vectors are determined.A major advance in our approach relative to other DPTVapproaches is the elimination of the need to interpolate therandom vector fields. Since the approach is digital, we are ableto collect greater than a thousand velocity field realizationswith relative ease and determine the turbulent statistics bybinning the random velocity vectors into small measurementvolumes, thus avoiding the need to interpolate the data2. Wehave also developed a technique for directly determining theinstantaneous gradients from the randomly located data.

In order to optimize our DPTV algorithm we used a MonteCarlo simulation to investigate the effects of various para-meters on its accuracy. We present results and conclusionsdrawn from these simulations. The technique was validatedagainst both LDA and DNS measurements of a flat plateboundary layer and the measurements of mean quantities,turbulence intensities, stresses, as well as dissipation arepresented. We include both the DPIV and the DPTV results forthis validation experiment allowing for a direct comparison ofthe two algorithms.

2PIV/PTV hybridPTV has been employed in many ways but essentially thereare two distinct types: multiply-exposed single images andsingly-exposed multiple images. Multiply-exposed single imagetechniques (e.g. Aguı and Jimenez 1987) rely on relativelysparse seeding of the flow to avoid overlapping images. Be-cause we were interested in a high-resolution technique, wechose to base our technique on singly-exposed multiple images(e.g. Hassan et al. 1992). The challenge in developing a singly-exposed multiple image PTV technique is tracking a given par-ticle through sequential images which contain a relativelyhigh density of particles. Previous PTV implementationshave relied on large search windows in the second imageof an image pair in order to track particles in the pair(Hassan et al. 1992). This necessitates low seeding densitiesto avoid pairing ambiguity. However, the size of the searchwindow can be significantly reduced if an accurate estimatecan be made of the particle’s location in the second sub-win-dow (e.g. Guezennec and Kiritsis 1990; Keane et al. 1995).

2 As pointed out by a reviewer the method of binning is reallyzero-order interpolation. However, due to the small length scalesassociated with the binning process, typically one sixth the lengthscales associated with a 32]32 pixel sub-window, the interpolativenoise incurred by this zero-order interpolation is minimal.

Cross-correlation based PIV is probably the most accurateform of correlation based PIV (Keane and Adrian 1992). Byemploying cross-correlation based PIV to two images, anaccurate function to estimate a particle’s position in the secondimage can be determined. Using this function the search win-dow for the particle in the second image can be minimized,allowing PTV to perform at very high seeding densities (wehave demonstrated the algorithm successfully at seeding den-sities greater than 60 particles per 32]32 sub-window insimulation).

There are two significant advantages in using PIV as a guideto track all of the particles in an image pair: increased accuracyand improved resolution. Traditionally the main argumentsagainst PTV have been that the data is randomly located andlow seeding densities are required to avoid ambiguity indetermining particle pairs (Hesselink 1988; Adrian 1991;Westerweel 1993a). Using the results of PIV as an estimate toa particle’s displacement allows a PTV algorithm to searcha small region for the particle’s pair, markedly increasingthe seeding densities at which PTV can successfully operate.Researchers have previously interpolated the randomly locateddata onto a grid in order to calculate turbulent statistics (e.g.Aguı and Jimenez 1987; Keane et al. 1995) but this resultsin interpolative errors (Aguı and Jimenez 1987; Hesselink1988).

We developed a DPTV technique that was originallyconceived as the merger of Willert and Gharib’s (1991) DPIVtechnique with the PTV method of Hassan et al. (1992). Ourmethod can be summarized as follows:

1. Precondition the images (i.e. remove any mean back-ground noise and non-uniformity).

2. Make a coarse DPIV analysis pass of two images (non-overlapping 32]32 pixel sub-windows).

3. Remove stray vectors from this vector set.4. Develop a displacement estimating function from which

the displacement at any pixel location in the first image can beestimated.

5. Threshold the images and search the result for patternsthat meet a user definition of a particle.

6. Search for particle pairs.7. Remove stray vectors from this vectors set.8. Calculate the turbulent statistics.

The specifics of each step are presented in the sections thatfollow.

2.1Preconditioning the imagesAn advantage of digital images is the ability to easily manipu-late them to remove the effects of non-ideal aspects of theimaging system. Preconditioning of the images for bothPIV and PTV algorithms is generally beneficial (Westerweel1993a; Hesselink 1988). Our pre-processing consists of ana-lyzing the entire set of images to determine the minimumvalue at each pixel. These minimum values are assembled intoa minimum image which is then subtracted from all of theoriginal images. This pre-processing removes any constantnoise source as well as the effects of non-uniform illumination(for example from the Gaussian intensity profile of a light sheet

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expanded with a cylindrical lens) while enhancing the signal-to-noise ratio of the images. Non-uniform background levelswill cause problems in the thresholding of the image to findthe particles; it is particularly important to correct for them.

2.2The DPIV passOur DPIV algorithm is a modification of the algorithmoutlined in Willert and Gharib (1991). We use non-overlap-ping 32]32 pixel sub-windows, however, the location of thesub-window in the second image is determined dynamically.Through an iterative process the DPIV algorithm determinesthe amount to shift the second image sub-window. At eachiteration the mean displacement between the sub-windows isdetermined. When the nearest integer pixel displacement isnon-zero, the second image sub-window is shifted by thisdisplacement in the appropriate direction. The process isrepeated until the nearest integer pixel displacement is zero.If the displacement does not converge in three iterations,the sub-window is flagged as invalid. In this way our DPIValgorithm accomplishes two tasks: areas with insufficientseeding density are flagged as invalid, and the sub-window inthe second image is displaced to ensure that a majority ofparticles in the first sub-window are imaged in the secondsub-window. This latter step reduces a principal sourceof mean bias associated with in-plane loss of correlation(discussed further in Sect. 3.1) by correlating mean displace-ments that are limited to ^0.5 pixels.

We note that our iterative technique to locate the secondimage sub-window does not eliminate the possibility of par-ticles with large displacements, relative to the mean (e.g. instrong shear), not residing in this sub-window; it merelyreduces it. Keane and Adrian (1992) use a different approach toensure all particles in the first sub-window are in the secondsub-window. They allow the second sub-window to be largerthan the first. This approach requires the second sub-windowto be 64]64 pixels to accommodate the FFT. We prefer ourmethod because it has slightly lower computational costs andthe important ability to detect sub-windows that are not ableto produce a valid velocity vector. A small gradient bias isincurred (a maximum of about 0.03 pixels) which will beremoved when the individual particles are tracked.

2.3Removal of stray vectorsThe PIV and the PTV algorithm will, on occasion, produceerroneous results. We developed a routine to remove strayvectors based on the statistics of the displacements beinganalyzed. The probability density function of turbulent fluc-tuations can be approximated as Gaussian (Lesieur 1987).While this assumption is tentative near boundaries of the flow,where significant skewness can occur, it turns out to be ade-quate if used only to set maximum possible deviations frommean values. Our DPTV system is designed to measure tur-bulent statistics and hence we generally work with hundredsif not thousands of velocity field measurements. We invokethe Gaussian assumption to predict the maximum expecteddeviation from the mean quantities given the sample size.For the DPIV analysis of the data this entails calculating theensemble statistics at each 32]32 sub-window location. Based

-15 -10 -5 0 5 10 15Displacement (pixels)a

-15 -10 -5 0 5 10 15Displacement (pixels)b

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nt10

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Fig. 1a, b. Velocity density functions. a DPIV pass; b DPTV pass, - - -pre-filter, — post-filter

on a sample’s size and variance, the maximum expectedfluctuation is determined and fluctuations beyond this areremoved. The statistics are then recalculated and the thre-sholding of the fluctuations is repeated. This whole processis repeated until it has converged (i.e. no velocity fluctuationslie outside the determined threshold values). The traditionalestimator of the mean, +n

i/N is not robust to skewness in the

outliers (Snedecor and Cochran 1989), therefore we use themedian value to estimate the mean in the first iteration. For allsubsequent iterations the traditional estimator of the mean isused. Typical results for a DPIV data set are shown in Fig. 1a.The filter rejected about 24% of the raw DPIV data which hasrelatively high uncertainty because the seeding density wasbelow optimal, about 4 particles per 32]32 pixel sub-window.

This vector filtering process is first done on the entire DPIVdata set which is then used to construct the displacementestimator functions for each image pair. The DPTV algorithmis applied and this data set is filtered in the same manner. Wedivide the images into small two-dimensional measurementareas and bin the randomly located vectors into these areas.The areas are sized to allow the assumption of homogeneousstatistics within the measurement area. This assumption iseasily checked by looking at the variation of statistics inthe neighboring measurement volumes. Enough images arecollected to allow each binned measurement area to containthousands of velocity vectors. Typical results for a DPTV dataset are shown in Fig. 1b. The filtered DPIV data is clearlyproviding an excellent estimate of the particle displacements

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as only about 0.5% of the data is rejected by the filter. NoteFig. 1a, b were calculated from the same data set (the validationdata discussed in Sect. 5).

2.4The displacement estimating functionUsing the DPIV data we develop a displacement estimatingfunction based on the Hardy multiquadratic scheme (Hassanet al. 1992; Hardy 1971), a global interpolation scheme given by

Vi(x, y)\(ui , vi) (1)

ui(xi , yi)\N+j/1

ajJ1]d2ij (2)

vi(xi , yi)\N+j/1

bjJ1]d2ij (3)

where

dij\J(xi[xj)2](yi[yj)2 (4)

Vi\the velocity vector at coordinates x

i, y

iwith components u

i, v

iN\the number of vectors to be used in the interpolation

xi, j

, yi, j\the location of the vectors

aj, b

j\constants to be determined

Using the determined ui and vi at xi and yi the aj and bj aresolved for. Given aj and bj and the location of the original datathe velocity at any point (u

i, v

i) can be determined. We choose

this scheme over other schemes because it exactly reproducesthe original data and because using non-overlapping sub-windows leaves N relatively small, so that we can afford to usethe order N3 calculation implied by the inversion of Eqs.(1)—(4). If computational expense becomes a concern it mayprove necessary to switch to an order N2 process such asadaptive Gaussian window (see Spedding and Rignot 1993).

2.5The PTV algorithmOur PTV method can be summarized as follows:

1. Threshold the two images at a user-defined value abovethe local mean intensity of the image. This local mean intensityis determined by calculating the mode of each image alonga column (or row) perpendicular to the direction of illumi-nation. The modes (a function of the direction of the opticallight path) are smoothed using a uniformly weighted sixnearest neighbor kernel.

2. Store the thresholded image as a binary image.3. Identify and label the particles in each image. This is

achieved by scanning the binary image line by line and clus-tering the binary l’s into particles based on their proximity toother binary 1’s. A pixel is considered part of a particle if it isadjacent horizontally or vertically to another pixel with value 1.Each pixel is given a number corresponding to the particle ofwhich it is a part.

4. The location of each particle is determined witha Gaussian sub-pixel fit estimator (this is expanded upon in theSect. 3.2).

5. Particles in the first image are paired with particles in thesecond image. The position of a particle in the second image isestimated using the above global interpolation scheme and

a 9]9 sub-window is centered on each position. A local cor-relation is performed and the estimate of the second image par-ticle location is refined based on this determined displace-ment. Finally a 3]3 pixel sub-window (found to be adequatelylarge for estimates based on the refined DPIV results) is cen-tered at the estimated position in the second image. If, andonly if, exactly one image two particle is located in this 3]3pixel sub-window it is considered the pairing particle.

2.6Determining statistics, including instantaneous gradientsThe stray vector filter described in Sect. 2.3 is applied to theentire data set of tracked particles. The statistics on the validvectors are calculated at each measurement bin.

A differentiating capability of full-field methods is the abilityto measure instantaneous velocity gradients, and thus meansquared gradients, yet these are never reported. The reasonfor this is likely the increased noise level that results fromdifferentiating velocity data with uncertainty in it. If the un-certainty (or noise) in the data could be filtered out, or atleast reduced to low levels, it would be trivial to calculateaccurate mean squared gradients for DPIV or DPTV type data.

We have developed a dynamic process for calculating thespatial derivatives of randomly located PTV data. Because weare interested in turbulent phenomenon the smallest lengthscale that needs to be resolved is the Kolmogorov scale,g\(l3/e)1/4. In fact, this scale generally does not need to beresolved in order to fully characterize the flow. Using theproposed universal spectrum of Pao (1965) to integrate thedissipation spectrum, (kg)2U(kg), one can show that 99%of the dissipation takes place for k\5.5 g~1 (as shown inFig. 2).

Our algorithm determines derivatives by identifyingeach image one particle, say at (x

1, y

1), and constructing

a small measurement area in the following image centered at(x

1]cg , y

1) and (x

1, y

1]cg) where c is a constant set to 5.5

(the sampling frequency required to measure 99% of thedissipation). A pictorial representation of this process is shownin Fig. 3. If a single particle is found in either of these two smallmeasurement areas, with typical side length of about 0.05g, theappropriate derivative is calculated. The process is dynamicbecause g is estimated directly from the data based on theestimation for the dissipation

e\cel[2 (Lu@/Lx)2]2(Lv@/Ly)2](Lu@/Ly)2](Lv@/Lx)2] (5)

where ce is an O(1) constant that takes into account the missingmean squared gradients. The first iteration requires that theuser start the algorithm with an estimate of g which is takenas a constant throughout the domain for this iteration.Fortunately, since g depends weakly on e (gPe~1/4), thisprocess converges relatively quickly to final estimates of thegradients (and g).

As with the PTV vectors, the gradient quantities themselvesare arrayed irregularly in space, and at lower densities thanthe velocity vectors. The in-plane mean squared gradientsare determined in the same manner as the PTV statistics,namely the randomly located gradients are binned into smallmeasurement areas and the mean squares are determined. Thevorticity and divergence of the velocity field can be determined

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Fig. 2. Fraction of total dissipation as a function of resolvedwavenumber - - - dissipation, — fraction of total dissipation

Fig. 3. Schematic drawing of the dynamic derivative algorithm

in a similar manner by calculating both the L/Lx and L/Lygradients when vectors randomly occur in the x and ycomponents of the stencil shown in Fig. 3 simultaneously.

3The Monte Carlo simulationsTo investigate the important parameters of the PIV andPTV processes, we developed a numerical model to simulatethem. This approach has been taken by many researchers, inclu-ding Westerweel (1993a), Keane et al. (1995), Keane andAdrian (1992) and Guezennec and Kiritsis (1990). Our modelgenerates images with randomly located, uniformly distribu-ted particles that have a Gaussian intensity profile (whichwe acknowledge to be somewhat ideal). Gaussian noise canbe superimposed on the entire image. The parameters inthe model include background noise intensity, particle size,dynamic range, particle density, and out-of-plane motions.

The CCD camera with which we typically work (see Sect. 4)has a background noise standard deviation of about 2 counts,where a count is defined as one least significant bit, so we heldthis value for all of our tests. We fixed the correlation sub-window size at a relatively small (Prasad et al. 1992) 32]32pixels used by Willert and Gharib (1991). This sub-windowsize was chosen because it allows significant spatial resolu-tion on a CCD camera without significant loss of accuracy

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Fig. 4. Monte Carlo simulations of four sub-pixel estimators rmserror: — h— — DPIV three point center-of-mass, —j— DPTV three pointcenter-of-mass, — r— —DPIV five point center-of-mass, —r— DPTVfive point center-of-mass, — n— — DPIV Gaussian, —m— DPTV Gaussian,— s— — DPIV parabolic, —d— DPTV parabolic

(Westerweel 1993b). Westerweel argues that it is the band-width of the spectral density that we are interested in sincewe are determining only the position of the particles (givenby their low wave-number components) and not their exactshape (which would require the sampling of the high wavenumber range as well). His analysis shows that for part-icles in the 10—50 lm range sampled in a 1 mm2 sub-windowthat sub-window sample rates of 32]32 to 64]64 pixels aresufficient to obtain high levels of accuracy. Since we do notrequire that our DPIV analysis of the data have particularlyhigh accuracy (^0.5 pixels as an upper bound is sufficient),32]32 pixels is more than adequate. The model enablesus to apply an X and a Y displacement as well as displacementgradients LXi/Lxj , i , j\1, 2. Using this model, we wereable to systematically identify the effects of each of the para-meters on the accuracy of our DPTV and DPIV processes. Inorder to insure that the statistics generated from thesimulations converged, 56 250 32]32 sub-windows weresimulated for each set of parameter combinations (i.e. eachdata point shown in Figs. 4 and 5a—f).

3.1Error type and structureBefore reviewing the results of our simulations it is impor-tant to consider the type of error expected in PIV and PTVmeasurements. PIV relies on using the displacement correla-tion peak as a measure of displacement. However, when a gra-dient exists in the particle displacement field the second imagesub-window can not contain all of the particles in the firstimage sub-window and the result is an in-plane loss-of-correlation (Adrian 1988; Keane and Adrian 1992; Westerweel1993a, b). This causes the correlation peak to be a biasedestimate of the displacement. It is biased toward lowerdisplacements since smaller displaced particle pairs havea higher probability of remaining in the second image sub-window. PTV does not suffer this type of mean bias sinceindividual particles are tracked and a particle’s image is notaffected by displacement gradients. After Keane and Adrian,(1992), we will refer to this as ‘‘gradient biasing’’.

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Both DPIV and DPTV rely on sub-pixel fit estimatorsto locate the centers of correlation peaks and particleimages, respectively, to sub-pixel accuracy. Various sub-pixel fit estimators have been proposed (e.g. Willert andGharib 1991; Adrian 1988), however, to date none of themprovide an unbiased estimate (Westerweel 1993a). FollowingWesterweel, we will call these ‘‘tracking biases’’. Both PIVand PTV suffer from tracking bias as both techniques relyon sub-pixel fit estimators. The goal becomes the minimiza-tion of this bias through optimal particle size and dynamicrange.

Fig. 5a–f. Monte Carlo simulations a Particle size; b dynamic range;c seeding density d out-of-plane fraction: e gradient magnitude;f interpolation. h DPIV mean error, j DPTV mean error, — n— —DPIV rms error, —m— DPTV rms error, - - s— - - DPIV valid vectors,- - d- - DPTV valid vectors

Finally, the imaging process is subject to random noise fromvarious sources (e.g. light quantization, CCD dark current,particle blocking, etc.). This noise results in a randomuncertainty in locating both the correlation peak and particleimages. This uncertainty can be thought of as the rms fluctu-ation in the position returned by an algorithm for particlesdisplaced exactly the same amount repeatedly. We will refer tothis as ‘‘rms error’’.

The total error is the sum of all of these errors. In the resultsdiscussed below we plot the rms error and mean error, the sumof the gradient bias and tracking bias described above.

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3.2The optimal sub-pixel fit algorithmTo improve the resolution and accuracy of our CCD basedimplementation we required an algorithm capable of trackingdisplacements to sub-pixel accuracy. Sub-pixel determinationof displacements are calculated using intensity variationinformation over the correlation peak in PIV and over theparticle image in PTV. Willert and Gharib (1991) report thata three-point exponential curve fit outperforms a centroidbased fit while R.D. Keane concludes from his simulationsthat in the presence of gradients the center-of-mass methodperforms best (private communication to Prasad et al. 1992).Westerweel (1993a) performed a detailed analytical and exper-imental study of three sub-pixel fit estimators: center-of-mass, parabolic, and Gaussian. He found that, in general,center-of-mass sub-pixel fit schemes are strongly biasedtowards integer displacements. His work showed that theGaussian sub-pixel fit scheme performed best. We investigatedthe same sub-pixel fit schemes sensitivity to displacementgradients and our results are shown in Fig. 4. While Wester-weel confines his investigation of the center-of-mass estimatorto a three point model we also investigate the five pointmodel, in particular to see if this may help to improvethe results under highly strained conditions for PIV. Theparameters used for our simulations were: particle diameter\2.4 pixels, dynamic range\200 counts, 12 particles per32]32 sub-window.

We find for both correlation peak-fitting (PIV) and particleimage fitting (PTV) that the Gaussian sub-pixel fit performs thebest. We note that we are primarily interested in relativelysmall particle image diameters (1.5—3 pixels), our Monte Carlosimulations have indicated that the optimal sub-pixel fitalgorithm may be a function of particle image diameter. Notsurprisingly, the three point center-of-mass estimator out-performs the five point estimator for particle image fitting(the particle diameter was 2.4 pixels) while the opposite holdsfor correlation peak fitting. The apparent reduced rms error ata gradient of 3% for the center-of-mass sub-pixel estimatorresults from the near-integer displacement when interrogatinga 3% displacement gradient with a 32 pixel interrogationspacing. It was pointed out above that these schemes arestrongly biased toward integer displacements and henceperform better when displacements are restricted to near-integer. We recognize that the choice of Gaussian particleintensity profiles skews the analysis in favor of the Gaussiansub-pixel estimator, however, given Westerweel’s (1993a)analytic work and the reality that particles imaging near thediffraction limit of an optical system have intensity profilesthat are well approximated by a Gaussian we feel the results areuseful. Given our work and that of Westerweel we choose to usethe Gaussian sub-pixel estimator for both the correlation peakand particle image fits in our DPTV algorithm. For referencethe Gaussian sub-pixel estimator is

xi\(m2

2[m21) ln(c2/c3)[(m2

3[m22) ln(c1/c2)

2[ (m2[m1) ln(c2/c3)[(m3[m2) ln(c1/c2)](6)

where

xi\the sub-pixel estimate of position

m1, m

2, m

3\the pixel locations of the left of peak, peak, and

right of peak intensityc1, c

2, c

3\the intensity value at m

1, m

2, m

3We note that it may be possible to improve the performance

of correlation peak fitting under highly strained conditions(displacement gradients greater than 3%) through the use ofasymetric, two dimensional peak fits (Westerweel et al. 1995).This type of fit may also give modest improvements to particlecenter peak fits, particularly for non-spherical particles.

3.3Particle sizeTo investigate the effect of particle size on error, we generatedimages with an average seeding density of twelve particlesper 32]32 pixel sub-window, no out-of-plane motions, anda dynamic range of 200 counts. A linear displacement gradientof 3% (0.03 pixels/pixel) was applied to the data. The results ofthis simulation are shown in Fig. 5a. Note that the right axis isdiscontinuous for the Figs. 5a—e. The mean error for DPTV isessentially zero. For DPIV it is also essentially constant but ata bias of about [0.03 pixels. This is principally due to thegradient bias effects on the PIV algorithm. The rms error forDPTV drops off initially with increasing particle diameter,reaching a minimum error at a particle diameter of about 2.4pixels before increasing again for larger diameters. The DPIVrms error’s functional dependence on particle diameter issimilar to DPTV’s, however, the error is roughly twice as largeand reaches its minimum closer to a diameter of 3.0 pixels.

The significant increase in resolution is clearly reflected inthe number of valid vectors found per 32]32 pixel sub-windowfor each technique. As expected, DPIV produces very close toone vector per sub-window. However, DPTV resolves nearly anorder of magnitude greater valid vector density with a particlediameter of close to 2 pixels. It is not surprising that the validvector density decays strongly with increasing diameter asparticle image overlapping increases with increasing diameter.DPIV does not suffer this loss of valid vector density withincreasing particle diameter nearly as severely since overlap-ping particle images still contribute to the correlation whilethey are discarded by the DPTV algorithm.

Reconciling the desire for a high valid vector density withthe need for low error, the optimal particle size for DPTVis taken to be between 1.8—2.4 pixels. Our results for DPIVindicate an optimal particle size in the broad range of 2.0to nearly 4.0 pixels. This is larger than the slightly less than2.0 pixels suggested by Prasad et al. (1992) for the center-of-mass estimator and 1.0 pixels determined by Westerweel(1993a) for Gaussian and parabolic estimators. However,both Prasad et al. and Westerweel looked at linear displace-ments in the absence of shear. In contrast, our results suggestthat the presence of a gradient increases the optimal particlesize for the correlation analysis. This is not surprising as thecorrelation algorithm will be more robust for larger particlessince there will be a higher probability of sub-window particlecorrelations in the presence of gradients for larger particles.

3.4Dynamic rangeFixing the particle diameter at 2.4 pixels we varied dynamicrange, defined as the difference in mean maximum particle

205

intensity level and the mean background intensity level, toinvestigate its effect on error. From experience gatheredworking with several traditional 8-bit CCD cameras we findthat typically we are able to produce dynamic ranges of about100—150 counts. As Fig. 5b shows, the accuracy of DPTV isstrongly tied to the dynamic range while DPIV accuracy isrelatively independent of dynamic range, in agreement withWesterweel (1993a). As expected, the DPTV mean error isessentially zero while the DPIV mean error again shows thenegative bias due to the mean gradient. The DPTV rms errordecays monotonically with increasing dynamic range, while fordynamic ranges above 50 counts the DPIV rms error isessentially constant at a level greater than twice the asymptoticlimit of the DPTV rms error. The valid vector density for bothDPIV and DPTV is essentially independent of dynamic range.

This simulation suggests that the optimal dynamic rangefor DPTV is at minimum 200 counts with clear gains out toa dynamic range of 500. These results lead us to the conclusionthat while 7-bit cameras are sufficient to minimize error forDPIV, true 10-bit or better cameras (potential dynamic rangeof 1024) offer significant advantages over lower bit depthdevices for DPTV. We note that we have been working withrather pedestrian particles, Pliolite VT-AC-L, which sell foraround two dollars per pound. It is possible to more effectivelyuse the dynamic range of a camera by imaging fluorescentparticles (Willert 1992) and bandpass filtering the scatteredlight. This may allow dynamic ranges in excess 200 to beobtained with an 8-bit camera, however, these particles areexpensive (about $500 per pound) and our experimentalfacilities are large, O(10 m3), making a 10-bit camera the lessexpensive alternative.

3.5Seeding densityMaintaining the particle diameter of 2.4 pixels and dynamicrange of 200, we varied the particle seeding density, definedas the mean number of particles per 32]32 sub-window.Figure 5c shows that, as expected, the DPTV mean error isessentially zero while the DPIV mean error shows the negativebias discussed above. The DPTV rms error increases nearlylinearly with increasing seeding density but remains belowthe rms error for DPIV for densities up to 25 particles persub-window. DPTV rms error is half that of DPIV for seedingdensities below 14 particles per sub-window. The DPIV rmserror decays monotonically with increasing seeding densityand can be driven down to the level of optimal DPTV for verylarge seeding densities (greater than 40 particles per sub-window). We have assumed, however, moderate velocitygradients and zero out-of-plane motion which is tenuousin turbulent flows. It is shown in the next section that foreven moderate out-of-plane motions DPIV rms error increasessignificantly relative to DPTV rms error.

The DPTV valid vector density initially increases almostlinearly with increasing seeding density but this growth ratedecays, a result of particle image overlapping. The dashed linesare the results under the same conditions but with a particlediameter of 1.8 pixels. Clearly much larger valid vectordensities can be achieved at relatively small rms error cost ifnecessary. Our simulation suggests that the optimal seedingdensity for DPTV is 5—20 particles per sub-window (balancing

the desire for greater valid vector density with low error). OurDPIV simulations are in agreement with Keane and Adrian(1992), suggesting that seeding density should be greater than10 particles per 32]32 sub-window. For seeding densitiesgreater than 10 particles per sub-window the DPIV valid vectordensity is essentially constant at 1.0.

3.6Out-of-plane motionsSingle-camera particle imaging techniques are inherentlytwo-dimensional measurement techniques while turbulenceis fully three-dimensional and thus it is important to considerthe robustness of DPTV and DPIV to out-of-plane motions.Maintaining the previously determined optimal parametersettings: particle diameter of 2.4 pixels, dynamic range of 200,and 12 particles per sub-window, we varied the out-of-planefraction, where 0.0 corresponds to all particles in the firstimage remaining in the second image and 1.0 corresponds tono original particles remaining in the second image. Figure 5ddemonstrates that DPTV error is relatively insensitive toout-of-plane fraction, while the valid vector density dropslinearly as expected. In constrast, DPIV is extremely sensitiveto out-of-plane fraction, with rms error doubling for fractionsof about 0.15. This agrees with Keane et al. (1991) who findthat the valid detection probability is a strong function ofout-of-plane fraction for a constant in-plane displacement.They recommend keeping the out-of-plane fraction below 0.30.We find that the presence of an in-plane gradient requiresa more severe restriction for DPIV (32]32 sub-window);out-of-plane fraction should be kept to below 0.15. This issignificant since all of the above simulations have been carriedout at zero out-of-plane fraction, almost never a characteristicof turbulent flows of interest. Typically, turbulence intensitiescan exceed 10% near walls or in moderate shear conditions.Therefore, the simulations of rms error discussed above mustbe considered low estimates for DPIV yet reasonable estimatesfor DPTV.

3.7GradientsAnother significant advantage of DPTV over DPIV is itsincreased robustness in the presence of gradients. Figure 5eshows that while DPIV error (both rms and mean) increasessharply with gradient strength, DPTV error is nearly indepen-dent of gradient strength. Since the DPTV algorithm onlyrequires that the DPIV estimate of position be good to^0.5 pixels, the DPTV algorithm is capable of tracking nearlyconstant valid vector density up to a displacement gradient ofabout 7%. This compares with a nearly linear increase in DPIVrms error of 0.04 pixels/%. Thus it is not recommended thatDPIV be used for displacement gradients in excess of about 3%which agrees with the parameters set by Keane and Adrian(1992) while 7% is the limit for DPTV.

3.8Interpolation errorAs discussed above, previous researchers have relied oninterpolating randomly located PTV vectors onto a uniformgrid in order to calculate turbulent statistics. We used the AGWinterpolator with the optimal parameters suggested by Aguı

206

and Jimenez (1987) to interpolate the simulation data (this wasthe interpolator used by Keane et al. 1995). The results areshown for the dynamic range sensitivity test in Fig. 5f. Only therms error is shown since the mean results are unaffected(indicating that AGW is an unbiased interpolator). Clearly theaccuracy gains made by using the DPTV process are more thanoffset by the interpolation noise incurred using the AGWinterpolator. We have tried tweaking the optimal parameterssuggested by Aguı and Jimenez but with little improvement.Spedding and Rignot (1993) demonstrate that a spline thinshell (STS) interpolator may perform in a superior manner toAGW. We have only worked briefly with this interpolator andwere unable to find significant improvement relative to AGWbut STS deserves further exploration.

3.9Summary of PTV findings and comparison to PIVThe results of our Monte Carlo simulation are summarized inTable 1. Looking at Figs. 5a—f and Table 1 it is apparent thatthe DPTV technique has performance characteristics that aresuperior to those of DPIV. In general DPTV eliminates meanerror and reduces rms errors to one half to one third theassociated rms DPIV error. Significantly, as we demonstrate,DPTV determines an order of magnitude more valid vectorsfrom an image than does DPIV. It should also be noted that ourPTV technique is much more robust in the face of out-of-planemotions and strong gradients than correlation-based PIV. It iscomparably insensitive to out-of-plane fraction relative to PIVand can operate accurately at better than twice the gradientstrength relative to PIV.

4Specifications of our PTV hardwareWe designed our PTV system using the above analysis. Thecenterpiece of the imaging system is the camera. In orderto measure velocity fields resolved to order 100 lm using32]32 pixel sub-windows for the correlation analysis inlaboratory scale water flows, it is necessary to capture twoimages between 3 and 20 ms apart. Since our technique needsto be digital we must use a CCD type camera. These camerastypically run at framing rates between 25 and 60 Hz (inter-image times of 17—40 ms). There are some specialty camerasthat are capable of framing rates greater than 500 Hz but theseare quite small format, not greater than 256]256 pixels. Fastercameras appear to be on the horizon. Rood (1994) states thatPrinceton Scientific Instruments is working on a one-millionframe per second CCD camera.

Table 1. Optimal values for DPTV and DPIV parameters

Parameter DPTV DPIV

Sub-pixel fit estimator Gaussian GaussianParticle diameter (pixels) 1.8—2.4 2.0—4.0Dynamic range (counts) [200 [50Seeding density (per 32]32 sub-window) 5—20 [10Out-of-plane fraction \0.5 \0.15Displacement gradient \7% \3%

When only two images are required, as in our case, there isa solution to the CCD camera’s limited framing rate (Willert1992). A few CCD chips are built in a format known as full-frame transfer — a chip which captures an image and transfersit to an on-chip storage area, typically a masked off portion ofthe CCD array. Since the transfer is accomplished on chip, itoccurs quite rapidly. Transfer times on the order of 1 ms arepossible. A camera based on a full-frame transfer chip can beused to collect two images, spaced on the order of 1 ms apart,by strobing the image area, shifting the image into on-chipstorage, and strobing the image area again. To take fulladvantage of the velocity field measurement capabilities of PIVrequires that the image area be as large as possible. Sincefull-frame transfer chips are available in formats as large as1024]2048 pixels, we chose to use a full-frame transfer CCDcamera.

The camera we are presently working with is a cooled 12-bit digital slow-scan camera manufactured by PattersonElectronics of Tustin, CA. The camera is a full-frame trans-fer camera with a transfer time of about 3.5 ms. It hasthe advantage that it can house essentially any full-frametransfer CCD chip. Because of economics we began with aTexas Instruments TC-217 chip which has an active area of1134]485 pixels. This chip has a fill factor of 100% but is nottruly a scientific-grade chip. Its electron well is not quite deepenough to support a full 12-bit dynamic range, but it is capableof dynamic ranges in excess of 11 bits. It has a pixel sizeof 7.8 lm]13.6 lm and thus has an aspect ratio of 1.74.This causes a problem in trying to maintain an ideal particlediameter since the particle will appear roughly twice thediameter in one direction as the other. Despite theselimitations the chip performs quite well as will be demon-strated in Sect. 5. Cooling the chip to [20°C reduces mostof the thermal noise associated with the chip (the rmsnoise is reduced from 3.2 counts at room temperature to1.8 counts) and any non-linearities and spatial gradients insensitivity.

We illuminate the flow with two Continuum MiniliteNd : YAG pulsed lasers. The lasers output wavelengths arefrequency doubled to 532 nm, have typical power of 10 mJ/pulse, and have typical pulse length of 5 ns. They are combinedinto a single optical path via a polarizing beam splitter anda half-wave retarder before passing through a cylindrical lensto form a 1 mm thick light sheet. The laser and camera triggersare controlled with National Instruments LabVIEW softwarerunning on a Macintosh IIfx computer.

The digital images are collected and processed on an Inteli860 based processing card, manufactured by Alacron, andhosted by a Pentium based PC equipped with 3.5 gigabytes ofhard drive space. The images are written to the host’s harddrives and stored on DAT tapes for later analysis.

5Validation experiment – the flat-plate boundary layerTo validate our DPTV technique we made measurements ina partially developed free-surface channel flow with both theDPTV technique and a two-component LDA. We also includethe results from a DPIV analysis of the data set for comparison.This flow is a good approximation of a zero pressure gradientflat-plate boundary layer. The Reynolds number of the flow

207

based on momentum thickness (Reh) was 1300 allowing thedata to be compared directly to the zero pressure gradientflat-plate boudnary layer DNS at Reh\1410 of Spalart (1986).

5.1Experimental facility and LDAThe flow facility used was a constant head type recirculatingflume fitted with a smooth bottom, described in detail inO’Riordan et al. (1993). The facility is 10 m in length andthe flow is driven through an inlet section by a 4 m constanthead. The inlet section is equipped with a baffle to minimizesecondary flow structures developing at the inlet jet. The baffleis followed by three screens of decreasing mesh size to furtherhomogenize the flow. The flow enters the main channel sectionthrough a 6.25 : 1 two-dimensional contraction. To trip theboundary layer, there is a 3 mm diameter cylindrical rodlocated just downstream of the contraction. The main channelis 6 m long and is equipped with glass side walls over the center3 m. The flume was operated at a depth of 25.5 cm whichproduced a free-stream velocity of 11.6 cm/s. All data wastaken with the image area (or LDA measurement volume)centered 4.0 m downstream of the boundary layer trip. Wedefined a right-hand coordinate system such that x is takenpositive in the streamwise direction and z is the verticaldistance above the smooth bottom.

The two component LDA system included two Dantectrackers and a Spectraphysics argon-ion laser operating at 1 Wwith output beam wavelength of 488 nm. The beams wereelectronically shifted to differentiate the two components. Theoptical configuration resulted in a measurement volume within-plane dimensions of 0.1]0.1 mm and an out-of-plane(transverse) length of 1 mm. Doppler frequencies determinedby the trackers were sampled at 80 Hz to yield data records ofbetween 20 000 and 150 000 points, depending on turbulenceintensity. The LDA was used to measure the entire boundarylayer profile from which the momentum thickness wasdetermined to be 1.14 cm, giving a Reynolds number based onmomentum thickness, Reh, of 1300. This Reh was intentionallychosen close to Spalart’s (1986) DNS at Reh\1410 to allowdirect comparison. To validate the DPTV technique LDAmeasurements were made at the same location as the DPTVmeasurements.

5.2DPTV calibration, data collection and processingThe DPTV system was calibrated by imaging a very fine gridetched in Plexiglas that is placed in the light sheet. The grid is10 cm]10 cm with a grid spacing of 1.00 cm. A single imagewas manually interrogated to determine to the nearest pixel thelocation of the grid elements. In this way the distance per pixelin the horizontal and vertical directions are determined tobetter than 0.25% (1 part in 400 pixels). The calibrationreturned a horizontal value of 54.2 lm per pixel and verticalvalue of 31.1 lm per pixel which gives a magnification of 0.251and yields an image area of 26.3]35.3 mm.

The flow was seeded with Pliolite VT-AC-L, manufactured byGoodyear Tire & Rubber Company. Pliolite has a specificgravity of 1.03 and good reflectivity. We used a mechanicalshaker and sieves to acquire particles with a diameter of

between 45 lm and 75 lm. We expected that particles in thissize range would follow the fluid motion which we confirmusing Eq. (7), derived by Adrian (1991) for the differencebetween particle and fluid motion, in Sect. 5.3.

Dv[u D\odo

d2r DvR D36l

(7)

wherev\the measured particle velocityu\the exact velocity per the velocity fieldod/o\the specific gravity of the particle

dr\the particle diametervR \the particle acceleration

We will estimate the peak acceleration, DvR D, as u@2/L where L isan appropriate length scale.

In order to maintain a uniform distribution of particlesacross the boundary layer we seeded our entire reservoir ofwater, about 8 m3, with approximately 163 g of particles.Elgoboshi (1994) reports that particle laden flows showevidence of two-way interaction at volume fractions as lowas 10~6. Our volume fraction was about 2]10~5. However,Elgoboshi’s findings are for monodisperse particles while ourseed is polydisperse, with a factor of 8 difference in particlevolume over the diameter distribution. The number densityfunction for particle diameter is probably skewed heavilytoward the low diameter range (due to the sieving process)which would give an effective void fraction closer toElgoboshi’s cutoff. (Westerweel et al. (1995a) use a similarargument to justify a void fraction in the same range).

To measure the boundary layer flow, 1092 image pairs werecollected. The camera was mounted in a portrait orientation(horizontal axis of CCD area in the vertical, z, direction). Thiswas done to take advantage of the asymmetry of the pixeldimensions (Liu et al. (1991) also use an asymmetric inter-rogation window in their boundary layer measurements). Toobtain optimal signal-to-noise ratio the maximum meandisplacement in each direction should be equal to about35—40% of the sub-window length, about 12 pixels for a 32]32sub-window. In a boundary layer flow the dominant velocity isthe mean free-stream velocity and thus to obtain maximalsignal-to-noise ratio we aligned this velocity component withthe long side of the pixels. The time between images, Dt, was setat 6 ms. The area of each image above the flat plate was998]485 pixels and thus the area imaged was 26.3]31.0 mm2.

The DPIV pass on the data was performed on a 992]448sub-region with non-overlapping 32]32 pixel sub-windowsyielding 434 measurement sites. Due to a relatively low seedingdensity the DPIV algorithm converged on average in only 280sub-windows. Again due to the low seeding density the strayvector filter determined that only 75.9% of these vectors werevalid (213 sub-windows). The images were thresholded suchthat a particle contained at least two pixels with intensitygreater than 15 counts above the local mean intensity level.This process yielded a mean particle density of 3.6 and 4.2particles per sub-window for the image one and image two datasets respectively (the limiting factor was the volume of waterthat needed to be seeded). The image two set contains moreparticles due to a slightly greater output intensity of the secondlaser. The median dynamic range for the image set was

208

Fig. 6. Typical instantaneous fluctuating velocity field

580 counts. The DPTV pass of the data yielded 1.29 millionvectors (a mean vector density of 1183 vectors per image). Theentire algorithm took about 15 s per image pair for this dataset. We note that the DPIV cross-correlation is a fixed costof our algorithm and that the cost to find additional DPTVvectors goes as N2 as opposed to N2(log

2N)2 for correlation

based DPIV. The fluctuating components, u@, w@, of a typicalinstantaneous vector field are shown in Fig. 6. Visual inspec-tion of this field reveals about five vectors that are dubious(out of 1162). This spurious vector rate is sufficiently lowto have minimal effect on the accuracy of the turbulentstatistics (see the results presented in Sect. 5.3). If these fewspurious vectors become a concern a local median filter(Westerweel 1993a) can be applied.

5.3Turbulent boundary layer resultsTurbulent statistics were calculated by assuming the flow washomogeneous in the streamwise direction. The data wasbinned in the vertical over 155 lm (the average number of

vectors per bin was 6500). The friction velocity, u*\Jq0/o,

was estimated by plotting the dimensional total stress profile(the non-dimensional components are shown in Figs. 9a, b)and taking a best fit of the near-wall total stress to determine q

0.

In this manner u*

was determined to be 0.51 cm/s.Figure 7 shows the mean streamwise velocity profiles plotted

in wall coordinates along with the log-law and viscousapproximations (the von Karman constant, i, is 0.41 andC\5.5). The LDA and DPTV data show excellent agreementover the entire profile. The DPIV data looks very good too,however, a slight under-biasing is apparent throughout theprofile due to the mean velocity gradient. The DNS mean

20

15

10

5

0

U+

12 3 4 5 6 7

102 3 4 5 6 7

1002

z+

Fig. 7. Mean streamwise velocity: s DPTV, j LDA, — DNS, n DPIV- - - Log law (i\0.41, C\5.5)

3.0

2.5

2.0

1.5

1.0

0.5

0.0

u'+

,w'+

140120100806040200

z+

Fig. 8. Turbulence intensities: s u@` DPTV, d u@` LDA, - u@` DNS,. u@` DPIV, h w@` DPTV, j w@` LDA, - - w@` DNS, m w@` DPIV

streamwise profile has a lower value of C in the log-law regionwhich is probably the result of a thinner viscous sub-layer. Thisis not surprising since the bottom of our facility had severaljoints that were not perfectly smooth. The effect of this slightroughness may explain the small overshoot in the peak of thestreamwise velocity fluctuation profile (Fig. 8) relative to theDNS. The LDA measurements verify that these small variationsfrom the DNS data are real in the sense that they are truecharacteristics of the flow in our facility. Notice the highlyresolved viscous sub-layer (12 points for z`\10) as well as itsexcellent agreement with the DNS data in this region.

The turbulence intensity profiles are shown in Fig. 8. Theagreement amongst the DPTV, LDA, and DNS data is verygood for both u@ and w@ with the exception being in w@ forz`\8. This probably results from the break down of theGaussian assumption in the stray vector filter. The DPIV datais also quite good although it consistently is below the LDA andDPTV data, probably as a result of the spatial filtering effect ofthe sub-window length scale (1.7 mm and 1.0 mm in the x andz directions, respectively). We estimate DvR D in equation (10) tobe u@2/z for a boundary layer. Its maximum value is 10.6 cm/s2and occurs at z`\9. Based on this value the worst casetracking error is ^0.1%. Away from the wall tracking errordecays monotonically, dropping below ^0.01% for z`[60.

209

1.2

1.0

0.8

0.6

0.4

0.2

0.0

Vis

cous

stre

ss(+

units

)

12 3 4 5 6 7

102 3 4 5 6 7

1002

z+a1.2

1.0

0.8

0.6

0.4

0.2

0.0

Rey

nold

sS

tres

s(+

units

)

12 3 4 5 6 7

102 3 4 5 6 7

1002

z+b

Fig. 9a, b. Stress: n DPIV, j LDA, — DNS, s DPTV

The wall scaled viscous and Reynolds stress components areshown in Figs. 9a, b. The resolution of the DPTV is reducedsince the gradient in the viscous stress (k(Lu/Lz)) was mea-sured using the dynamic derivative method discussed inSect. 2.6 (this resolution was held for the Reynolds stress sothat the total stress could be calculated without interpolatingthe data). The LDA, DNS and DPTV data agreement is verygood with the exception being that the LDA has difficultyresolving the mean velocity gradient very near the wall. TheDPIV viscous data is also excellent with a similar exceptionnear the wall, however, the DPIV determined Reynolds stressesare clearly low, again suggesting a spatial filter effect of thecorrelation sub-window.

Four of the mean squared fluctuating gradients were

measured, (Lu@/Lx)2 , (Lw@/Lx)2 , (Lu@/Lz)2 , and (Lw@/Lz)2, usingthe dynamic derivative method. Spalart does not report theindividual mean squared gradients but does separate x andz components of the dissipation. In a 2-D flow a reasona-ble approximation to these components is given by: e

uu\

l[2 (Lu@/Lx)2](Lu@/Lz)2] and eww

\l[2 (Lw@/Lz)2](Lw@/Lx)2].We chose ce in Eq. (5) based on the ratio of Spalart’s(e

uu]e

vv]e

ww)/(e

uu]e

ww) to be about 1.4. This is slightly less

than the isotropic value of 1.5. The DPTV, DPIV and DNS datais shown in Fig. 10. We note that the DPTV estimate of e

uuand

eww should be beneath that of the DNS (as it is) as we haveonly used the two dominant gradient terms of the five impliedby the exact expressions for euu and eww . The DPIV estimateof the dissipation is good where the dissipation is strong(near the wall); however, the signal-to-noise ratio clearly

0.001

2

46

0.01

2

46

0.1

2

46

1

Dis

sipa

tion

(+un

its)

1 2 3 4 5 6 7 10 2 3 4 5 6 7 100 2

z+

Fig. 10. Dissipation: h euu estimate DPTV, s eww estimate DPTV,— euu DNS, — — eww DNS, j euu DPIV, d eww DPIV

becomes O(1) away from the wall. We note that the DPIVmeasurement of the fluctuating gradients did not use thedynamic derivative algorithm but instead they were deter-mined by directly differentiating the regularly griddedcorrelation data.

6ConclusionsWe have demonstrated that PIV can be used to guide PTV,producing superior results to PIV alone. We have developeda digital based PTV technique that has both high resolutionand accuracy and can be run nearly real time, allowing foroptimization of experiments as they are in progress. Thesignificant new element of our technique is the use of thetracked velocity vectors in place, avoiding the noise associatedwith interpolation. Our technique is capable of measuring theturbulence statistics, including the mean squared fluctuatinggradients. The highly resolved results for the flat-plateboundary layer compare very well with 2-D LDA measure-ments made in the same flow as well as Spalart’s DNS simula-tion. While the DPIV measurements in the same flow are quitegood, they clearly under-resolve the fluctuations associatedwith the turbulence.

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