A hybrid digital particle tracking velocimetry technique

Transcription

1 Experiments in Fluids (1997) Springer-Verlag 1997 A hybrid digital particle tracking velocimetry technique E. A. Cowen, S. G. Monismith 199 Abstract A novel approach to digital particle tracking velocimetry (DPTV) based on cross-correlation digital particle image velocimetry (DPIV) is presented that eliminates the need to interpolate the randomly located velocity vectors (typical of tracking techniques) and results in significantly improved resolution and accuracy. In particular, this approach allows for the direct measurement of mean squared fluctuating gradients, and thus several important components of the turbulent dissipation. The effect of various parameters (seeding density, particle diameter, dynamic range, out-of-plane motion, and gradient strength) on accuracy for both DPTV and DPIV are investigated using a Monte Carlo simulation and optimal values are reported. Validation results are presented from the comparison of measurements by the DPTV technique in a turbulent flat plate boundary layer to laser Doppler anemometer (LDA) measurements in the same flow as well as direct numerical simulation (DNS) data. The DPIV analysis of the images used for the DPTV validation is included for comparison. 1 Introduction In the study of turbulent flows, particle image velocimetry (PIV) is a valuable tool capable of measuring the significant spatial structure of turbulence. Liu et al. (1991) have successfully demonstrated the use of PIV to probe the structure of a turbulent channel flow. Previous experimental studies of Received: 9 August 199/Accepted: 31 May 199 E. A. Cowen, S. G. Monismith Environmental Fluid Mechanics Laboratory Stanford University, Stanford, CA , USA Correspondence to: E. A. Cowen The authors gratefully acknowledge the support of this work by the Fluid Dynamics Program, Office of Naval Research (Scientific Director: Dr. E. Rood, grant N ) and the High Performance Computing and Communication Program, National Science Foundation (Scientific Director: Dr. A. Thaler, grant ASC 93181). The authors wish to thank Jonathan Harris, whose PIV work laid the foundation from which our technique grew and who never gave up on PTV, Bob Street, Jeff Koseff and Chris Rehmann for their steady supply of ideas and assistance, and Bob Brown for his mechanical wizardry. We are particularly indebted to John Crimaldi for making the detailed LDA measurements reported in Sect. 5. turbulent flows have generally relied on point measurement techniques laser Doppler anemometry (LDA) and hot wire velocimetry. While providing high temporal resolution, these methods have extremely limited spatial resolution. Until recently, high resolution accurate PIV techniques, such as that of Liu et al., who report spatial resolution of 0.3 mm 1.0 mm, were based on interrogating photographic images. This process is time consuming, as the photographs must first be developed and then interrogated to determine if the experiment has been successful. Willert and Gharib (1991) demonstrated that it is possible to use a CCD based camera to acquire digital images eliminating the time consuming development and interrogation of photographs. Their technique (DPIV), however, must be considered low resolution as they report a spatial resolution of only 5. mm 5. mm (we note that this is more a function of their chosen optical configuration and not their technique). Westerweel et al. (199) demonstrated that DPIV can operate at resolutions on the order of conventional photographic PIV (and for that matter DNS) with comparable accuracy. They interrogated digital images at a spatial resolution of 1.3 mm 1.3 mm which is comparable to that achieved by Liu et al. However, correlation based analysis does not extract the maximum amount of velocity information from an image. Recently Keane et al. (1995)1 demonstrated that tracking a significant number of particles in a photographic image is feasible. Essentially, they followed Guezennec and Kiritsis (1990) algorithm and used an auto-correlation analysis estimate of the velocity field to guide the particle tracking algorithm. Their results suggest an improvement in resolution by a factor of.5 (to a dimensional grid spacing of about 0.1 mm) over their conventional PIV approach. While they recognized the power of such an algorithm to improve spatial resolution, it appears that the hybrid approach can offer substantial gains in accuracy over PIV as well. As we will demonstrate, PTV algorithms are inherently more accurate than correlation-based PIV algorithms since they are relatively unaffected by the presence of displacement gradients. We believe this behavior is typically overlooked because the common practice of interpolating the randomly located tracked velocity vectors onto a regular grid introduces noise that obviates the noise reduction obtained with PTV relative to PIV (Agüí and Jiménez 1987; Hesselink 1988). 1 The authors wish to thank R. J. Adrian for making us aware of his group s efforts to use correlation analysis to guide a particle tracking algorithm and for a pre-print of their work.

2 00 In this paper we present a DPTV technique capable of both high spatial resolution and high accuracy. Our technique is able to capture the significant spatial structure of turbulence, including both mean velocity statistics and instantaneous spatial gradients. Its essential feature is the use of the randomly located tracked velocity vectors in place to determine all turbulent statistics. This results in resolution and accuracy comparable to that found by Keane et al. (1995) under conventional photographic conditions but through fully digital means. Similar to Keane et al. and Guezennec and Kiritsis (1990), it relies on a correlation analysis of images to estimate the displacement of particles as a function of their position. It uses this estimate to track individual particle displacements from which the instantaneous velocity vectors are determined. A major advance in our approach relative to other DPTV approaches is the elimination of the need to interpolate the random vector fields. Since the approach is digital, we are able to collect greater than a thousand velocity field realizations with relative ease and determine the turbulent statistics by binning the random velocity vectors into small measurement volumes, thus avoiding the need to interpolate the data. We have also developed a technique for directly determining the instantaneous gradients from the randomly located data. In order to optimize our DPTV algorithm we used a Monte Carlo simulation to investigate the effects of various parameters on its accuracy. We present results and conclusions drawn from these simulations. The technique was validated against both LDA and DNS measurements of a flat plate boundary layer and the measurements of mean quantities, turbulence intensities, stresses, as well as dissipation are presented. We include both the DPIV and the DPTV results for this validation experiment allowing for a direct comparison of the two algorithms. PIV/PTV hybrid PTV has been employed in many ways but essentially there are two distinct types: multiply-exposed single images and singly-exposed multiple images. Multiply-exposed single image techniques (e.g. Agüí and Jiménez 1987) rely on relatively sparse seeding of the flow to avoid overlapping images. Because we were interested in a high-resolution technique, we chose to base our technique on singly-exposed multiple images (e.g. Hassan et al. 199). The challenge in developing a singlyexposed multiple image PTV technique is tracking a given particle through sequential images which contain a relatively high density of particles. Previous PTV implementations have relied on large search windows in the second image of an image pair in order to track particles in the pair (Hassan et al. 199). This necessitates low seeding densities to avoid pairing ambiguity. However, the size of the search window can be significantly reduced if an accurate estimate can be made of the particle s location in the second sub-window (e.g. Guezennec and Kiritsis 1990; Keane et al. 1995). As pointed out by a reviewer the method of binning is really zero-order interpolation. However, due to the small length scales associated with the binning process, typically one sixth the length scales associated with a 3 3 pixel sub-window, the interpolative noise incurred by this zero-order interpolation is minimal. Cross-correlation based PIV is probably the most accurate form of correlation based PIV (Keane and Adrian 199). By employing cross-correlation based PIV to two images, an accurate function to estimate a particle s position in the second image can be determined. Using this function the search window for the particle in the second image can be minimized, allowing PTV to perform at very high seeding densities (we have demonstrated the algorithm successfully at seeding densities greater than 0 particles per 3 3 sub-window in simulation). There are two significant advantages in using PIV as a guide to track all of the particles in an image pair: increased accuracy and improved resolution. Traditionally the main arguments against PTV have been that the data is randomly located and low seeding densities are required to avoid ambiguity in determining particle pairs (Hesselink 1988; Adrian 1991; Westerweel 1993a). Using the results of PIV as an estimate to a particle s displacement allows a PTV algorithm to search a small region for the particle s pair, markedly increasing the seeding densities at which PTV can successfully operate. Researchers have previously interpolated the randomly located data onto a grid in order to calculate turbulent statistics (e.g. Agüí and Jiménez 1987; Keane et al. 1995) but this results in interpolative errors (Agüı and Jiménez 1987; Hesselink 1988). We developed a DPTV technique that was originally conceived as the merger of Willert and Gharib s (1991) DPIV technique with the PTV method of Hassan et al. (199). Our method can be summarized as follows: 1. Precondition the images (i.e. remove any mean background noise and non-uniformity).. Make a coarse DPIV analysis pass of two images (nonoverlapping 3 3 pixel sub-windows). 3. Remove stray vectors from this vector set.. Develop a displacement estimating function from which the displacement at any pixel location in the first image can be estimated. 5. Threshold the images and search the result for patterns that meet a user definition of a particle.. 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 that follow..1 Preconditioning the images An advantage of digital images is the ability to easily manipulate them to remove the effects of non-ideal aspects of the imaging system. Preconditioning of the images for both PIV and PTV algorithms is generally beneficial (Westerweel 1993a; Hesselink 1988). Our pre-processing consists of analyzing the entire set of images to determine the minimum value at each pixel. These minimum values are assembled into a minimum image which is then subtracted from all of the original images. This pre-processing removes any constant noise source as well as the effects of non-uniform illumination (for example from the Gaussian intensity profile of a light sheet

3 expanded with a cylindrical lens) while enhancing the signalto-noise ratio of the images. Non-uniform background levels will cause problems in the thresholding of the image to find the particles; it is particularly important to correct for them.. The DPIV pass Our DPIV algorithm is a modification of the algorithm outlined in Willert and Gharib (1991). We use non-overlapping 3 3 pixel sub-windows, however, the location of the sub-window in the second image is determined dynamically. Through an iterative process the DPIV algorithm determines the amount to shift the second image sub-window. At each iteration the mean displacement between the sub-windows is determined. When the nearest integer pixel displacement is non-zero, the second image sub-window is shifted by this displacement in the appropriate direction. The process is repeated 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 DPIV algorithm accomplishes two tasks: areas with insufficient seeding density are flagged as invalid, and the sub-window in the second image is displaced to ensure that a majority of particles in the first sub-window are imaged in the second sub-window. This latter step reduces a principal source of mean bias associated with in-plane loss of correlation (discussed further in Sect. 3.1) by correlating mean displacements that are limited to 0.5 pixels. We note that our iterative technique to locate the second image sub-window does not eliminate the possibility of particles with large displacements, relative to the mean (e.g. in strong shear), not residing in this sub-window; it merely reduces it. Keane and Adrian (199) use a different approach to ensure all particles in the first sub-window are in the second sub-window. They allow the second sub-window to be larger than the first. This approach requires the second sub-window to be pixels to accommodate the FFT. We prefer our method because it has slightly lower computational costs and the important ability to detect sub-windows that are not able to produce a valid velocity vector. A small gradient bias is incurred (a maximum of about 0.03 pixels) which will be removed when the individual particles are tracked..3 Removal of stray vectors The PIV and the PTV algorithm will, on occasion, produce erroneous results. We developed a routine to remove stray vectors based on the statistics of the displacements being analyzed. The probability density function of turbulent fluctuations 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 adequate if used only to set maximum possible deviations from mean values. Our DPTV system is designed to measure turbulent statistics and hence we generally work with hundreds if not thousands of velocity field measurements. We invoke the Gaussian assumption to predict the maximum expected deviation from the mean quantities given the sample size. For the DPIV analysis of the data this entails calculating the ensemble statistics at each 3 3 sub-window location. Based Count a Count b Displacement (pixels) Displacement (pixels) 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 expected fluctuation is determined and fluctuations beyond this are removed. The statistics are then recalculated and the thresholding of the fluctuations is repeated. This whole process is repeated until it has converged (i.e. no velocity fluctuations lie outside the determined threshold values). The traditional estimator of the mean, n i /N is not robust to skewness in the outliers (Snedecor and Cochran 1989), therefore we use the median value to estimate the mean in the first iteration. For all subsequent iterations the traditional estimator of the mean is used. Typical results for a DPIV data set are shown in Fig. 1a. The filter rejected about % of the raw DPIV data which has relatively high uncertainty because the seeding density was below optimal, about particles per 3 3 pixel sub-window. This vector filtering process is first done on the entire DPIV data set which is then used to construct the displacement estimator functions for each image pair. The DPTV algorithm is applied and this data set is filtered in the same manner. We divide the images into small two-dimensional measurement areas and bin the randomly located vectors into these areas. The areas are sized to allow the assumption of homogeneous statistics within the measurement area. This assumption is easily checked by looking at the variation of statistics in the neighboring measurement volumes. Enough images are collected to allow each binned measurement area to contain thousands of velocity vectors. Typical results for a DPTV data set are shown in Fig. 1b. The filtered DPIV data is clearly providing an excellent estimate of the particle displacements 01

4 0 as only about 0.5% of the data is rejected by the filter. Note Fig. 1a, b were calculated from the same data set (the validation data discussed in Sect. 5).. The displacement estimating function Using the DPIV data we develop a displacement estimating function based on the Hardy multiquadratic scheme (Hassan et al. 199; Hardy 1971), a global interpolation scheme given by V i (x, y) (u i, v i ) (1) u (x, y ) a 1 d () i i i j ij j 1 v (x, y ) b 1 d (3) i i i j ij j 1 where d ij (x i x j ) (y i y j ) () V i the velocity vector at coordinates x i, y i with components u i, v i N the number of vectors to be used in the interpolation x i,j, y i,j the location of the vectors a j, b j constants to be determined Using the determined u i and v i at x i and y i the a j and b j are solved for. Given a j and b j and the location of the original data the velocity at any point (u i, v i ) can be determined. We choose this scheme over other schemes because it exactly reproduces the original data and because using non-overlapping subwindows leaves N relatively small, so that we can afford to use the order N3 calculation implied by the inversion of Eqs. (1) (). If computational expense becomes a concern it may prove necessary to switch to an order N process such as adaptive Gaussian window (see Spedding and Rignot 1993)..5 The PTV algorithm Our PTV method can be summarized as follows: 1. Threshold the two images at a user-defined value above the local mean intensity of the image. This local mean intensity is determined by calculating the mode of each image along a column (or row) perpendicular to the direction of illumination. The modes (a function of the direction of the optical light path) are smoothed using a uniformly weighted six nearest neighbor kernel.. 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 clustering the binary l s into particles based on their proximity to other binary 1 s. A pixel is considered part of a particle if it is adjacent horizontally or vertically to another pixel with value 1. Each pixel is given a number corresponding to the particle of which it is a part.. The location of each particle is determined with a Gaussian sub-pixel fit estimator (this is expanded upon in the Sect. 3.). 5. Particles in the first image are paired with particles in the second image. The position of a particle in the second image is estimated using the above global interpolation scheme and a9 9 sub-window is centered on each position. A local correlation is performed and the estimate of the second image particle location is refined based on this determined displacement. Finally a 3 3 pixel sub-window (found to be adequately large for estimates based on the refined DPIV results) is centered at the estimated position in the second image. If, and only if, exactly one image two particle is located in this 3 3 pixel sub-window it is considered the pairing particle.. Determining statistics, including instantaneous gradients The stray vector filter described in Sect..3 is applied to the entire data set of tracked particles. The statistics on the valid vectors are calculated at each measurement bin. A differentiating capability of full-field methods is the ability to measure instantaneous velocity gradients, and thus mean squared gradients, yet these are never reported. The reason for this is likely the increased noise level that results from differentiating velocity data with uncertainty in it. If the uncertainty (or noise) in the data could be filtered out, or at least reduced to low levels, it would be trivial to calculate accurate mean squared gradients for DPIV or DPTV type data. We have developed a dynamic process for calculating the spatial derivatives of randomly located PTV data. Because we are interested in turbulent phenomenon the smallest length scale that needs to be resolved is the Kolmogorov scale, η (ν3/ε)1/. In fact, this scale generally does not need to be resolved in order to fully characterize the flow. Using the proposed universal spectrum of Pao (195) to integrate the dissipation spectrum, (kη)φ(kη), one can show that 99% of the dissipation takes place for k 5.5 η 1 (as shown in Fig. ). Our algorithm determines derivatives by identifying each image one particle, say at (x 1, y 1 ), and constructing a small measurement area in the following image centered at (x 1 cη, y 1 ) and (x 1, y 1 cη) where c is a constant set to 5.5 (the sampling frequency required to measure 99% of the dissipation). A pictorial representation of this process is shown in Fig. 3. If a single particle is found in either of these two small measurement areas, with typical side length of about 0.05η, the appropriate derivative is calculated. The process is dynamic because η is estimated directly from the data based on the estimation for the dissipation ε c ν[ ( u / x) ( v / y) ( u / y) ( v / x)] (5) where c is an O(1) constant that takes into account the missing mean squared gradients. The first iteration requires that the user start the algorithm with an estimate of η which is taken as a constant throughout the domain for this iteration. Fortunately, since η depends weakly on ε (η ε 1/), this process converges relatively quickly to final estimates of the gradients (and η). As with the PTV vectors, the gradient quantities themselves are arrayed irregularly in space, and at lower densities than the velocity vectors. The in-plane mean squared gradients are determined in the same manner as the PTV statistics, namely the randomly located gradients are binned into small measurement areas and the mean squares are determined. The vorticity and divergence of the velocity field can be determined

5 (k η) Φ(k η) kη Fig.. Fraction of total dissipation as a function of resolved wavenumber dissipation, fraction of total dissipation Fraction of total dissipation Gradient (pixels/pixel) Fig.. Monte Carlo simulations of four sub-pixel estimators rms error: DPIV three point center-of-mass, DPTV three point center-of-mass, DPIV five point center-of-mass, DPTV five point center-of-mass, DPIV Gaussian, DPTV Gaussian, DPIV parabolic, DPTV parabolic 03 Fig. 3. Schematic drawing of the dynamic derivative algorithm in a similar manner by calculating both the / x and / y gradients when vectors randomly occur in the x and y components of the stencil shown in Fig. 3 simultaneously. 3 The Monte Carlo simulations To investigate the important parameters of the PIV and PTV processes, we developed a numerical model to simulate them. This approach has been taken by many researchers, including Westerweel (1993a), Keane et al. (1995), Keane and Adrian (199) and Guezennec and Kiritsis (1990). Our model generates images with randomly located, uniformly distributed particles that have a Gaussian intensity profile (which we acknowledge to be somewhat ideal). Gaussian noise can be superimposed on the entire image. The parameters in the 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. ) has a background noise standard deviation of about counts, where a count is defined as one least significant bit, so we held this value for all of our tests. We fixed the correlation subwindow size at a relatively small (Prasad et al. 199) 3 3 pixels used by Willert and Gharib (1991). This sub-window size was chosen because it allows significant spatial resolution on a CCD camera without significant loss of accuracy (Westerweel 1993b). Westerweel argues that it is the bandwidth of the spectral density that we are interested in since we are determining only the position of the particles (given by their low wave-number components) and not their exact shape (which would require the sampling of the high wave number range as well). His analysis shows that for particles in the μm range sampled in a 1 mm sub-window that sub-window sample rates of 3 3 to pixels are sufficient to obtain high levels of accuracy. Since we do not require that our DPIV analysis of the data have particularly high accuracy ( 0.5 pixels as an upper bound is sufficient), 3 3 pixels is more than adequate. The model enables us to apply an X and a Y displacement as well as displacement gradients X i / x j, i, j 1,. Using this model, we were able to systematically identify the effects of each of the parameters on the accuracy of our DPTV and DPIV processes. In order to insure that the statistics generated from the simulations converged, sub-windows were simulated for each set of parameter combinations (i.e. each data point shown in Figs. and 5a f). 3.1 Error type and structure Before reviewing the results of our simulations it is important to consider the type of error expected in PIV and PTV measurements. PIV relies on using the displacement correlation peak as a measure of displacement. However, when a gradient exists in the particle displacement field the second image sub-window can not contain all of the particles in the first image sub-window and the result is an in-plane loss-ofcorrelation (Adrian 1988; Keane and Adrian 199; Westerweel 1993a, b). This causes the correlation peak to be a biased estimate of the displacement. It is biased toward lower displacements since smaller displaced particle pairs have a higher probability of remaining in the second image subwindow. PTV does not suffer this type of mean bias since individual particles are tracked and a particle s image is not affected by displacement gradients. After Keane and Adrian, (199), we will refer to this as gradient biasing.

6 a Particle diameter (pixels) b Dynamic range (counts) Particle Diameter 18 = 1.8 pixels c Number of particles (per 3 3 sub-window) Vectors per 3 3 sub-window Vectors per 3 3 sub-window Vectors per 3 3 sub-window e Gradient (pixels/pixel) 0. f 0.0 Interpolation rms Dynamic range (counts) Vectors per 3 3 sub-window d Out-of-plane fraction (pixels/pixel) Vectors per 3 3 sub-window 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. DPIV mean error, DPTV mean error, DPIV rms error, DPTV rms error, DPIV valid vectors, DPTV valid vectors Both DPIV and DPTV rely on sub-pixel fit estimators to locate the centers of correlation peaks and particle images, respectively, to sub-pixel accuracy. Various subpixel fit estimators have been proposed (e.g. Willert and Gharib 1991; Adrian 1988), however, to date none of them provide an unbiased estimate (Westerweel 1993a). Following Westerweel, we will call these tracking biases. Both PIV and PTV suffer from tracking bias as both techniques rely on sub-pixel fit estimators. The goal becomes the minimization of this bias through optimal particle size and dynamic range. Finally, the imaging process is subject to random noise from various sources (e.g. light quantization, CCD dark current, particle blocking, etc.). This noise results in a random uncertainty in locating both the correlation peak and particle images. This uncertainty can be thought of as the rms fluctuation in the position returned by an algorithm for particles displaced exactly the same amount repeatedly. We will refer to this as rms error. The total error is the sum of all of these errors. In the results discussed below we plot the rms error and mean error, the sum of the gradient bias and tracking bias described above.

7 3. The optimal sub-pixel fit algorithm To improve the resolution and accuracy of our CCD based implementation we required an algorithm capable of tracking displacements to sub-pixel accuracy. Sub-pixel determination of displacements are calculated using intensity variation information over the correlation peak in PIV and over the particle image in PTV. Willert and Gharib (1991) report that a three-point exponential curve fit outperforms a centroid based fit while R.D. Keane concludes from his simulations that in the presence of gradients the center-of-mass method performs best (private communication to Prasad et al. 199). Westerweel (1993a) performed a detailed analytical and experimental study of three sub-pixel fit estimators: center-ofmass, parabolic, and Gaussian. He found that, in general, center-of-mass sub-pixel fit schemes are strongly biased towards integer displacements. His work showed that the Gaussian sub-pixel fit scheme performed best. We investigated the same sub-pixel fit schemes sensitivity to displacement gradients and our results are shown in Fig.. While Westerweel confines his investigation of the center-of-mass estimator to a three point model we also investigate the five point model, in particular to see if this may help to improve the results under highly strained conditions for PIV. The parameters used for our simulations were: particle diameter. pixels, dynamic range 00 counts, 1 particles per 3 3 sub-window. We find for both correlation peak-fitting (PIV) and particle image fitting (PTV) that the Gaussian sub-pixel fit performs the best. We note that we are primarily interested in relatively small particle image diameters (1.5 3 pixels), our Monte Carlo simulations have indicated that the optimal sub-pixel fit algorithm may be a function of particle image diameter. Not surprisingly, the three point center-of-mass estimator outperforms the five point estimator for particle image fitting (the particle diameter was. pixels) while the opposite holds for correlation peak fitting. The apparent reduced rms error at a gradient of 3% for the center-of-mass sub-pixel estimator results from the near-integer displacement when interrogating a 3%displacement gradient with a 3 pixel interrogation spacing. It was pointed out above that these schemes are strongly biased toward integer displacements and hence perform better when displacements are restricted to nearinteger. We recognize that the choice of Gaussian particle intensity profiles skews the analysis in favor of the Gaussian sub-pixel estimator, however, given Westerweel s (1993a) analytic work and the reality that particles imaging near the diffraction limit of an optical system have intensity profiles that are well approximated by a Gaussian we feel the results are useful. Given our work and that of Westerweel we choose to use the Gaussian sub-pixel estimator for both the correlation peak and particle image fits in our DPTV algorithm. For reference the Gaussian sub-pixel estimator is x i (ξ ξ 1 )ln(γ /γ 3 ) (ξ 3 ξ )ln(γ 1 /γ ) [(ξ ξ 1 )ln(γ /γ 3 ) (ξ 3 ξ )ln(γ 1 /γ )] where x i the sub-pixel estimate of position () ξ 1, ξ, ξ 3 the pixel locations of the left of peak, peak, and right of peak intensity γ 1, γ, γ 3 the intensity value at ξ 1, ξ, ξ 3 We 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 of asymetric, two dimensional peak fits (Westerweel et al. 1995). This type of fit may also give modest improvements to particle center peak fits, particularly for non-spherical particles. 3.3 Particle size To investigate the effect of particle size on error, we generated images with an average seeding density of twelve particles per 3 3 pixel sub-window, no out-of-plane motions, and a dynamic range of 00 counts. A linear displacement gradient of 3% (0.03 pixels/pixel) was applied to the data. The results of this simulation are shown in Fig. 5a. Note that the right axis is discontinuous for the Figs. 5a e. The mean error for DPTV is essentially zero. For DPIV it is also essentially constant but at a bias of about 0.03 pixels. This is principally due to the gradient bias effects on the PIV algorithm. The rms error for DPTV drops off initially with increasing particle diameter, reaching a minimum error at a particle diameter of about. pixels before increasing again for larger diameters. The DPIV rms error s functional dependence on particle diameter is similar to DPTV s, however, the error is roughly twice as large and reaches its minimum closer to a diameter of 3.0 pixels. The significant increase in resolution is clearly reflected in the number of valid vectors found per 3 3 pixel sub-window for each technique. As expected, DPIV produces very close to one vector per sub-window. However, DPTV resolves nearly an order of magnitude greater valid vector density with a particle diameter of close to pixels. It is not surprising that the valid vector density decays strongly with increasing diameter as particle image overlapping increases with increasing diameter. DPIV does not suffer this loss of valid vector density with increasing particle diameter nearly as severely since overlapping particle images still contribute to the correlation while they are discarded by the DPTV algorithm. Reconciling the desire for a high valid vector density with the need for low error, the optimal particle size for DPTV is taken to be between 1.8. pixels. Our results for DPIV indicate an optimal particle size in the broad range of.0 to nearly.0 pixels. This is larger than the slightly less than.0 pixels suggested by Prasad et al. (199) for the centerof-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 displacements in the absence of shear. In contrast, our results suggest that the presence of a gradient increases the optimal particle size for the correlation analysis. This is not surprising as the correlation algorithm will be more robust for larger particles since there will be a higher probability of sub-window particle correlations in the presence of gradients for larger particles. 3. Dynamic range Fixing the particle diameter at. pixels we varied dynamic range, defined as the difference in mean maximum particle 05

8 0 intensity level and the mean background intensity level, to investigate its effect on error. From experience gathered working with several traditional 8-bit CCD cameras we find that typically we are able to produce dynamic ranges of about counts. As Fig. 5b shows, the accuracy of DPTV is strongly tied to the dynamic range while DPIV accuracy is relatively independent of dynamic range, in agreement with Westerweel (1993a). As expected, the DPTV mean error is essentially zero while the DPIV mean error again shows the negative bias due to the mean gradient. The DPTV rms error decays monotonically with increasing dynamic range, while for dynamic ranges above 50 counts the DPIV rms error is essentially constant at a level greater than twice the asymptotic limit of the DPTV rms error. The valid vector density for both DPIV and DPTV is essentially independent of dynamic range. This simulation suggests that the optimal dynamic range for DPTV is at minimum 00 counts with clear gains out to a dynamic range of 500. These results lead us to the conclusion that while 7-bit cameras are sufficient to minimize error for DPIV, true 10-bit or better cameras (potential dynamic range of 10) offer significant advantages over lower bit depth devices for DPTV. We note that we have been working with rather pedestrian particles, Pliolite VT-AC-L, which sell for around two dollars per pound. It is possible to more effectively use the dynamic range of a camera by imaging fluorescent particles (Willert 199) and bandpass filtering the scattered light. This may allow dynamic ranges in excess 00 to be obtained with an 8-bit camera, however, these particles are expensive (about $500 per pound) and our experimental facilities are large, O(10 m3), making a 10-bit camera the less expensive alternative. 3.5 Seeding density Maintaining the particle diameter of. pixels and dynamic range of 00, we varied the particle seeding density, defined as the mean number of particles per 3 3 sub-window. Figure 5c shows that, as expected, the DPTV mean error is essentially zero while the DPIV mean error shows the negative bias discussed above. The DPTV rms error increases nearly linearly with increasing seeding density but remains below the rms error for DPIV for densities up to 5 particles per sub-window. DPTV rms error is half that of DPIV for seeding densities below 1 particles per sub-window. The DPIV rms error decays monotonically with increasing seeding density and can be driven down to the level of optimal DPTV for very large seeding densities (greater than 0 particles per subwindow). We have assumed, however, moderate velocity gradients and zero out-of-plane motion which is tenuous in turbulent flows. It is shown in the next section that for even moderate out-of-plane motions DPIV rms error increases significantly relative to DPTV rms error. The DPTV valid vector density initially increases almost linearly with increasing seeding density but this growth rate decays, a result of particle image overlapping. The dashed lines are the results under the same conditions but with a particle diameter of 1.8 pixels. Clearly much larger valid vector densities can be achieved at relatively small rms error cost if necessary. Our simulation suggests that the optimal seeding density for DPTV is 5 0 particles per sub-window (balancing the desire for greater valid vector density with low error). Our DPIV simulations are in agreement with Keane and Adrian (199), suggesting that seeding density should be greater than 10 particles per 3 3 sub-window. For seeding densities greater than 10 particles per sub-window the DPIV valid vector density is essentially constant at Out-of-plane motions Single-camera particle imaging techniques are inherently two-dimensional measurement techniques while turbulence is fully three-dimensional and thus it is important to consider the robustness of DPTV and DPIV to out-of-plane motions. Maintaining the previously determined optimal parameter settings: particle diameter of. pixels, dynamic range of 00, and 1 particles per sub-window, we varied the out-of-plane fraction, where 0.0 corresponds to all particles in the first image remaining in the second image and 1.0 corresponds to no original particles remaining in the second image. Figure 5d demonstrates that DPTV error is relatively insensitive to out-of-plane fraction, while the valid vector density drops linearly as expected. In constrast, DPIV is extremely sensitive to out-of-plane fraction, with rms error doubling for fractions of about This agrees with Keane et al. (1991) who find that the valid detection probability is a strong function of out-of-plane fraction for a constant in-plane displacement. They recommend keeping the out-of-plane fraction below We find that the presence of an in-plane gradient requires a more severe restriction for DPIV (3 3 sub-window); out-of-plane fraction should be kept to below This is significant since all of the above simulations have been carried out at zero out-of-plane fraction, almost never a characteristic of turbulent flows of interest. Typically, turbulence intensities can exceed 10% near walls or in moderate shear conditions. Therefore, the simulations of rms error discussed above must be considered low estimates for DPIV yet reasonable estimates for DPTV. 3.7 Gradients Another significant advantage of DPTV over DPIV is its increased robustness in the presence of gradients. Figure 5e shows that while DPIV error (both rms and mean) increases sharply with gradient strength, DPTV error is nearly independent of gradient strength. Since the DPTV algorithm only requires that the DPIV estimate of position be good to 0.5 pixels, the DPTV algorithm is capable of tracking nearly constant valid vector density up to a displacement gradient of about 7%. This compares with a nearly linear increase in DPIV rms error of pixels/%. Thus it is not recommended that DPIV be used for displacement gradients in excess of about 3% which agrees with the parameters set by Keane and Adrian (199) while 7% is the limit for DPTV. 3.8 Interpolation error As discussed above, previous researchers have relied on interpolating randomly located PTV vectors onto a uniform grid in order to calculate turbulent statistics. We used the AGW interpolator with the optimal parameters suggested by Agüı

9 and Jime nez (1987) to interpolate the simulation data (this was the interpolator used by Keane et al. 1995). The results are shown for the dynamic range sensitivity test in Fig. 5f. Only the rms error is shown since the mean results are unaffected (indicating that AGW is an unbiased interpolator). Clearly the accuracy gains made by using the DPTV process are more than offset by the interpolation noise incurred using the AGW interpolator. We have tried tweaking the optimal parameters suggested by Agüí and Jime nez but with little improvement. Spedding and Rignot (1993) demonstrate that a spline thin shell (STS) interpolator may perform in a superior manner to AGW. We have only worked briefly with this interpolator and were unable to find significant improvement relative to AGW but STS deserves further exploration. 3.9 Summary of PTV findings and comparison to PIV The results of our Monte Carlo simulation are summarized in Table 1. Looking at Figs. 5a f and Table 1 it is apparent that the DPTV technique has performance characteristics that are superior to those of DPIV. In general DPTV eliminates mean error and reduces rms errors to one half to one third the associated rms DPIV error. Significantly, as we demonstrate, DPTV determines an order of magnitude more valid vectors from an image than does DPIV. It should also be noted that our PTV technique is much more robust in the face of out-of-plane motions and strong gradients than correlation-based PIV. It is comparably insensitive to out-of-plane fraction relative to PIV and can operate accurately at better than twice the gradient strength relative to PIV. Specifications of our PTV hardware We designed our PTV system using the above analysis. The centerpiece of the imaging system is the camera. In order to measure velocity fields resolved to order 100 μm using 3 3 pixel sub-windows for the correlation analysis in laboratory scale water flows, it is necessary to capture two images between 3 and 0 ms apart. Since our technique needs to be digital we must use a CCD type camera. These cameras typically run at framing rates between 5 and 0 Hz (interimage times of 17 0 ms). There are some specialty cameras that are capable of framing rates greater than 500 Hz but these are quite small format, not greater than 5 5 pixels. Faster cameras appear to be on the horizon. Rood (199) states that Princeton Scientific Instruments is working on a one-million frame per second CCD camera. Table 1. Optimal values for DPTV and DPIV parameters Parameter DPTV DPIV Sub-pixel fit estimator Gaussian Gaussian Particle diameter (pixels) Dynamic range (counts) Seeding density (per 3 3 sub-window) Out-of-plane fraction Displacement gradient 7% 3% When only two images are required, as in our case, there is a solution to the CCD camera s limited framing rate (Willert 199). A few CCD chips are built in a format known as fullframe transfer a chip which captures an image and transfers it to an on-chip storage area, typically a masked off portion of the CCD array. Since the transfer is accomplished on chip, it occurs quite rapidly. Transfer times on the order of 1 ms are possible. A camera based on a full-frame transfer chip can be used to collect two images, spaced on the order of 1 ms apart, by strobing the image area, shifting the image into on-chip storage, and strobing the image area again. To take full advantage of the velocity field measurement capabilities of PIV requires that the image area be as large as possible. Since full-frame transfer chips are available in formats as large as pixels, we chose to use a full-frame transfer CCD camera. The camera we are presently working with is a cooled 1- bit digital slow-scan camera manufactured by Patterson Electronics of Tustin, CA. The camera is a full-frame transfer camera with a transfer time of about 3.5 ms. It has the advantage that it can house essentially any full-frame transfer CCD chip. Because of economics we began with a Texas Instruments TC-17 chip which has an active area of pixels. This chip has a fill factor of 100% but is not truly a scientific-grade chip. Its electron well is not quite deep enough to support a full 1-bit dynamic range, but it is capable of dynamic ranges in excess of 11 bits. It has a pixel size of 7.8 μm 13. μm and thus has an aspect ratio of 1.7. This causes a problem in trying to maintain an ideal particle diameter since the particle will appear roughly twice the diameter in one direction as the other. Despite these limitations the chip performs quite well as will be demonstrated in Sect. 5. Cooling the chip to 0 C reduces most of the thermal noise associated with the chip (the rms noise is reduced from 3. counts at room temperature to 1.8 counts) and any non-linearities and spatial gradients in sensitivity. We illuminate the flow with two Continuum Minilite Nd : YAG pulsed lasers. The lasers output wavelengths are frequency doubled to 53 nm, have typical power of 10 mj/ pulse, and have typical pulse length of 5 ns. They are combined into a single optical path via a polarizing beam splitter and a half-wave retarder before passing through a cylindrical lens to form a 1 mm thick light sheet. The laser and camera triggers are controlled with National Instruments LabVIEW software running on a Macintosh IIfx computer. The digital images are collected and processed on an Intel i80 based processing card, manufactured by Alacron, and hosted by a Pentium based PC equipped with 3.5 gigabytes of hard drive space. The images are written to the host s hard drives and stored on DAT tapes for later analysis. 5 Validation experiment the flat-plate boundary layer To validate our DPTV technique we made measurements in a partially developed free-surface channel flow with both the DPTV technique and a two-component LDA. We also include the results from a DPIV analysis of the data set for comparison. This flow is a good approximation of a zero pressure gradient flat-plate boundary layer. The Reynolds number of the flow 07

10 08 based on momentum thickness (Re ) was 1300 allowing the data to be compared directly to the zero pressure gradient flat-plate boudnary layer DNS at Re 110 of Spalart (198). 5.1 Experimental facility and LDA The flow facility used was a constant head type recirculating flume fitted with a smooth bottom, described in detail in O Riordan et al. (1993). The facility is 10 m in length and the flow is driven through an inlet section by a m constant head. The inlet section is equipped with a baffle to minimize secondary flow structures developing at the inlet jet. The baffle is followed by three screens of decreasing mesh size to further homogenize the flow. The flow enters the main channel section through a.5 : 1 two-dimensional contraction. To trip the boundary layer, there is a 3 mm diameter cylindrical rod located just downstream of the contraction. The main channel is m long and is equipped with glass side walls over the center 3 m. The flume was operated at a depth of 5.5 cm which produced a free-stream velocity of 11. cm/s. All data was taken with the image area (or LDA measurement volume) centered.0 m downstream of the boundary layer trip. We defined a right-hand coordinate system such that x is taken positive in the streamwise direction and z is the vertical distance above the smooth bottom. The two component LDA system included two Dantec trackers and a Spectraphysics argon-ion laser operating at 1 W with output beam wavelength of 88 nm. The beams were electronically shifted to differentiate the two components. The optical configuration resulted in a measurement volume with in-plane dimensions of mm and an out-of-plane (transverse) length of 1 mm. Doppler frequencies determined by the trackers were sampled at 80 Hz to yield data records of between and points, depending on turbulence intensity. The LDA was used to measure the entire boundary layer profile from which the momentum thickness was determined to be 1.1 cm, giving a Reynolds number based on momentum thickness, Re, of This Re was intentionally chosen close to Spalart s (198) DNS at Re 110 to allow direct comparison. To validate the DPTV technique LDA measurements were made at the same location as the DPTV measurements. 5. DPTV calibration, data collection and processing The DPTV system was calibrated by imaging a very fine grid etched in Plexiglas that is placed in the light sheet. The grid is 10 cm 10 cm with a grid spacing of 1.00 cm. A single image was manually interrogated to determine to the nearest pixel the location of the grid elements. In this way the distance per pixel in the horizontal and vertical directions are determined to better than 0.5% (1 part in 00 pixels). The calibration returned a horizontal value of 5. μm per pixel and vertical value of 31.1 μm per pixel which gives a magnification of 0.51 and yields an image area of mm. The flow was seeded with Pliolite VT-AC-L, manufactured by Goodyear Tire & Rubber Company. Pliolite has a specific gravity of 1.03 and good reflectivity. We used a mechanical shaker and sieves to acquire particles with a diameter of between 5 μm and 75 μm. We expected that particles in this size range would follow the fluid motion which we confirm using Eq. (7), derived by Adrian (1991) for the difference between particle and fluid motion, in Sect v u ρ d ρ d r vr 3ν where v the measured particle velocity u the exact velocity per the velocity field ρ d /ρ the specific gravity of the particle d r the particle diameter vr the particle acceleration We will estimate the peak acceleration, vr, asu /Lwhere L is an appropriate length scale. In order to maintain a uniform distribution of particles across the boundary layer we seeded our entire reservoir of water, about 8 m3, with approximately 13 g of particles. Elgoboshi (199) reports that particle laden flows show evidence of two-way interaction at volume fractions as low as 10. Our volume fraction was about However, Elgoboshi s findings are for monodisperse particles while our seed is polydisperse, with a factor of 8 difference in particle volume over the diameter distribution. The number density function for particle diameter is probably skewed heavily toward the low diameter range (due to the sieving process) which would give an effective void fraction closer to Elgoboshi s cutoff. (Westerweel et al. (1995a) use a similar argument to justify a void fraction in the same range). To measure the boundary layer flow, 109 image pairs were collected. The camera was mounted in a portrait orientation (horizontal axis of CCD area in the vertical, z, direction). This was done to take advantage of the asymmetry of the pixel dimensions (Liu et al. (1991) also use an asymmetric interrogation window in their boundary layer measurements). To obtain optimal signal-to-noise ratio the maximum mean displacement in each direction should be equal to about 35 0% of the sub-window length, about 1 pixels for a 3 3 sub-window. In a boundary layer flow the dominant velocity is the mean free-stream velocity and thus to obtain maximal signal-to-noise ratio we aligned this velocity component with the long side of the pixels. The time between images, Δt, was set at ms. The area of each image above the flat plate was pixels and thus the area imaged was mm. The DPIV pass on the data was performed on a 99 8 sub-region with non-overlapping 3 3 pixel sub-windows yielding 3 measurement sites. Due to a relatively low seeding density the DPIV algorithm converged on average in only 80 sub-windows. Again due to the low seeding density the stray vector filter determined that only 75.9% of these vectors were valid (13 sub-windows). The images were thresholded such that a particle contained at least two pixels with intensity greater than 15 counts above the local mean intensity level. This process yielded a mean particle density of 3. and. particles per sub-window for the image one and image two data sets respectively (the limiting factor was the volume of water that needed to be seeded). The image two set contains more particles due to a slightly greater output intensity of the second laser. The median dynamic range for the image set was (7)