Comparison between Tomographic PIV and Stereo PIV Dirk Michaelis

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Jul 10, 2008 - A very accurate volume-calibration is required for Tomographic PIV ... Volumetric self-calibration was applied using disparity maps for 7 x 7 x 3 ...
14th Int Symp on Applications of Laser Techniques to Fluid Mechanics Lisbon, Portugal, 07-10 July, 2008

Comparison between Tomographic PIV and Stereo PIV Dirk Michaelis, Bernhard Wieneke LaVision GmbH, Goettingen, Germany, [email protected], [email protected]

Abstract Tomographic PIV as a full volume 3D-3C flow field measurement technique can also be applied to thin sheets used in standard PIV. Tomographic PIV and Stereo PIV are compared with experimental data from an impinging jet in water. The field of view is 60 mm x 40 mm, the light sheet thickness is varied from 8 to 55 pixel. Images have been recorded with 4 cameras to gain results from two independent Stereo PIV systems and from Tomographic PIV using 2-4 cameras. Different error sources for both techniques are quantified. It is shown that instantaneous flow fields from Tomographic PIV deviate less from each of the two stereo systems, than the two stereo systems deviate from each other. A systematic error pattern is identified for the two stereo systems, which is not present for Tomographic PIV. Thin volume Tomographic PIV allows the calculation of multiple vector planes. Two planes can be calculated even for the thinnest light sheet (8 pixel) providing the full 3x3 strain-tensor. Thicker sheets with up to twelve vector planes allow the visualization of the small vortex structures in this experiment.

1. Introduction The evolution of PIV (particle image velocimetry) has recently gained another important step forward with the introduction of tomographic PIV by Elsinga et al. (2006a), and time resolved tomographic PIV (Michaelis et al., 2006). Four PIV techniques distinguished here are: 2D PIV, stereo PIV, dual plane stereo PIV and tomographic PIV (table 1). The amount of information is increasing from 2D2C to 3D3C, requiring more hardware. The accuracy of the techniques is limited by several sources of systematic and random errors that are summarized in table 2. The systematic errors are usually small, in the order of a few tenth of a pixel. They can nevertheless have a strong impact on the measurable dynamic velocity range. Tomographic PIV provides an additional measurement dimension, and does not show most of the systematic errors. However, the occurrence of ghost particles is unique for tomographic PIV. In this paper tomographic PIV is applied to thin volumes for two reasons, first to compare it with stereo PIV and second to investigate the possibility of dual plane or volumetric vector measurement within a normal (thin) laser light sheet to measure the full 3 x 3 strain tensor or even 3D structures.

2. Experimental setup The experiment was designed to optimize the optical conditions for stereo and tomographic PIV on one side, but to give a challenging measurement task that is able to evoke the systematic errors of the PIV techniques on the other side. A water basin (300 x 200 x 200 mm³) allowed good optical access for all four cameras. 10µm silver coated glass hollow spheres seeding particles together with Table 1. Properties of PIV techniques Measurement grid Measured vector components Cameras Double pulse laser

2D PIV 2D planar 2C 1 1

Stereo PIV 2D planar 3C 2 1

-1-

Dual plane stereo PIV 2 x 2D 2 x planar 3C 4 1-2

Tomographic PIV 3D volumetric 3C 4 (3-6) 1

14th Int Symp on Applications of Laser Techniques to Fluid Mechanics Lisbon, Portugal, 07-10 July, 2008

Table 2. Errors of PIV techniques 2D PIV Loss of particles yes Stereo error: different particles in interrogation windows of camera 1 and 2 Particle intensity variation due to laser sheet yes profile Sensitivity to background noise and contamination very high of optical components (see below) Smearing of correlation peak by out of plane yes velocity gradients Specific errors Only ‘projection’ of vectors

Stereo PIV yes yes

Dual plane PIV yes yes

Tomographic PIV no -

yes

yes

no

high

high

low

yes

yes

no

Misalignment of laser planes

Ghost particle

a 120 mJ Nd-Yag laser result in a particle intensity of about 5000 counts and a high signal to background noise ratio for the cameras. The light sheet was very close to the front glass wall or even touching it. So, there was no or only very little light diffusion by unilluminated tracer particles between the light sheet and the cameras (figure 2). A turbulent impinging jet flow was generated by a water pump with an outlet of 4 mm diameter at 24 mm distance from the glass wall. Four 12 bit 1.4 megapixel cameras recorded a field of view of 60 mm x 40 mm. The cameras where mounted on a square around the center of the measurement volume with a viewing angle of about 30 degree to the z-direction. Two stereo setups with similar viewing conditions are selected using camera 1+2 and 3+4 (figure 1). The flow in this setup is turbulent and exhibits small 3D structures and strong gradients. 5 series of 1000 images each have been recorded, where the sheet thickness was altered in a range from 8 to 55 pixel. The particle density has been decreased with increasing light sheet thickness to maintain a constant particle density in the recorded images. The number of particles varied between 42000 and 46000 particles per image (avg. 44000). The particle density was about 0.031 particles per pixel (ppp). The time separation between the two exposures has been adjusted to achieve a maximal pixel shift of about ±5 pixel but also to keep the maximal out of plane shift below 25 % of the sheet thickness, to limit the maximal loss of particles to 25 %. The resulting time separation is dt = 150 µs for the thicker sheets (35 to 55 pixel), dt = 100µs for the 16 pixel sheet and dt = 75µs

Camera 1

Camera 3

Camera 4

Camera 2

Fig. 1. Impinging jet in water: Four cameras record particle images and, in this case the calibration target in a small water basin. Laser illumination enters the aquarium from the left via a laser guiding arm and a conventional light sheet optic. Two stereo setups are composed from camera 1+2 and 3+4.

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14th Int Symp on Applications of Laser Techniques to Fluid Mechanics Lisbon, Portugal, 07-10 July, 2008

1

3

4

2

Fig. 2. Impinging jet in water: Conceptual sketch illustrating the physical dimensions and the camera arrangement.

for the 8 pixel sheet (table 4). Because of the shorter dt, the maximal pixel shift in x- and ydirection was reduced to about ± 3 pixel for the 8 pixel sheet.

3. Methods In this section, a brief description of the data analysis methods and preprocessing steps is given. 3.1 Image preprocessing Image preprocessing is applied to the images to reduce the background and to equalize the particle intensities. The image preprocessing consists of several steps that are as follows: 1. Subtract sliding minimum from 3 successive (subtract background, time filter). 2. Subtract sliding minimum over 5x5 pixel (subtract background, spatial filter). 3. Normalize intensity over a sliding area of 100 x 100 pixel to equalize low frequency spatial differences in laser sheet intensity. 4. Normalize the mean intensity from all cameras and both exposures. 5. Apply 3x3 gaussian smoothing (in general improves quality of the reconstructed volume). 6. Apply peak normalization over 5x5 pixel (maximum normalization) to equalize the intensity of stronger and weaker particles (similar to min-max filter, Westerweel, 2003).

3.2 Stereo PIV 2D-3C stereo vector fields have been computed for all recordings using different interrogation window sizes: 16 x 16, 32 x 32, 64 x 64 and 128 x 128 pixel. Multi pass calculation with an initial window size of at least 64 x 64 pixel was used. Three passes have been calculated at the final window size with 75 % overlap. 3.3 Tomographic PIV Tomographic PIV is a two step procedure (Elsinga et al., 2006a): First the particle intensity distribution is reconstructed from the camera images which will be referred as reconstructed -3-

14th Int Symp on Applications of Laser Techniques to Fluid Mechanics Lisbon, Portugal, 07-10 July, 2008

volume. Then an adaptive multi pass 3D correlation algorithm is applied to calculate the vector field. Three different approaches of correlation are used in this paper: first single plane correlation, second dual plane correlation and third multi plane correlation. The single plane calculation is used to compare the results of stereo and tomographic PIV. The depth of the interrogation volume covers the complete reconstructed volume. The dual plane approach delivers two independent vector planes and already allows the calculation of gradients in the z-direction. It works without interrogation volume overlap in the z-direction. In the multi plane approach, the z-direction is treated the same way as the x- and y-direction, with the same interrogation window size and overlap. 3.4 Planar and Volumetric Self-Calibration A very accurate volume-calibration is required for Tomographic PIV (calibration error < 0.1 pixel is desirable). A single view of a dual plane calibration target was used for the initial calibrations. Then, first the plane with z = 0 mm is aligned with the laser light sheet by planar self-calibration (Wieneke, 2005), then the remaining calibration errors are reduced by volumetric self-calibration (Wieneke, 2008). The planar self-calibration procedure originally proposed for stereo PIV has been extended for a four camera setup. The principle is the same as in stereo PIV. However a disparity map is calculated for all camera pairs (1/2, 1/3, 1/4, 2/3, 2/4, 3/4) and the correction transformation is calculated using all disparity maps. Planar self-calibration was not possible for the recordings with 45 and 55 pixel laser sheet thickness. There was no clear correlation peak detectable. Instead, the correlation peak showed multiple maxima. For these recordings a self-calibration from a thinner sheet (8 pixel) was used. The initial average disparity of about 3 pixel was reduced to 0.2 pixel in all cases (table 3). Volumetric self-calibration was applied using disparity maps for 7 x 7 x 3 sub-volumes. The remaining average calibration error was 0.02 pixel or smaller for all recordings. Table 3. Initial and remaining disparity from self-calibration and volume-self-calibration. Planar Self-Calibration Volumetric Self-Calibration Initial error Remaining error Sheet thickness Initial disparity Remaining disparity [pixel] [pixel] [pixel] [pixel] [pixel] 3.5 0.2 0.10 0.01 8 2.8 0.2 0.16 0.02 16 3.2 0.2 0.14 0.01 35 not possible 0.04 0.01 45 not possible 0.04 0.01 55

4. Results 4.1. Stereo PIV An instantaneous vector field from stereo PIV is displayed in figure 3, left, and average vector fields in figure 4. The fields show a strong out of plane component in the center (jet core) and a radial divergend flow in the outer regions. A systematic difference between stereo system one and two and tomographic PIV is visible in the outer regions (figure 4): the stereo systems show inconsistent positive (green) and negative (light blue) z-components, whereas tomographic PIV shows a small, homogeneous negative z-component, consistent with the flow expected in this region. The stereo systems also show spurious z-components in the average fields, resulting from constant image distortions like dirt on the glass wall or reflecting air bubbles at the water pump in the background. Since there is no redundancy in stereo PIV, a constant distortion in a single 2C vector field already leads to a systematic error in the average field. In tomographic PIV however, a distortion in a single camera image can be compensated by the remaining three cameras. The spurious distortions visible for stereo system one and two are not visible for tomographic PIV. -4-

14th Int Symp on Applications of Laser Techniques to Fluid Mechanics Lisbon, Portugal, 07-10 July, 2008

Fig. 3. Vectors (u,v) and color coded w- component of instantaneous vector fields for a light sheet thickness of 8 pixel and 32 x 32 pixel interrogation window size: left: stereo PIV, right: single plane tomographic PIV (in x-direction only every fourth vector is displayed).

Fig. 4. Vectors (u,v) and color coded w- component of the average vector fields (N=998) for a light sheet thickness of 8 pixel and 16 x 16 pixel interrogation window size: Top left: tomographic PIV full range for w (Vz). The out of plane velocity maximum in the center of the average field is wavg. max. = 0.67 m/s (1 m/s corresponding to a shift of 1.6 pixel). Top right: tomographic PIV with displayed w-range restricted to ± 0.1 m/s. Bottom left: stereo system 1, restricted wrange. Bottom right: stereo system 2, restricted w-range. Only every eighth vector is displayed.

Fig. 5. Average stereo reconstruction error [pixel]. Sheet thickness 35 pixel. Interrogation window size 16 x 16 pixel. Stereo system 1 (left) and stereo system 2 (right) with projection of the line that connects the camera centers. -5-

14th Int Symp on Applications of Laser Techniques to Fluid Mechanics Lisbon, Portugal, 07-10 July, 2008

Table 4. Mean stereo reconstruction error SRE (+/- RMS) in pixel for stereo systems 1 and 2. Interrogation window size Time Stereo Sheet 16 x 16 32 x 32 64 x 64 128 x 128 thickness separation system dt [pixel] 75 µs 1 8 0.12 ± 0.11 0.07 ± 0.05 0.04 ± 0.03 0.02 ± 0.02 2 0.12 ± 0.11 0.07 ± 0.05 0.04 ± 0.03 0.02 ± 0.01 100 µs 1 16 0.18 ± 0.17 0.11 ± 0.09 0.06 ± 0.05 0.03 ± 0.02 2 0.18 ± 0.16 0.10 ± 0.09 0.05 ± 0.04 0.03 ± 0.02 150 µs 1 35 0.33 ± 0.31 0.23 ± 0.22 0.14 ± 0.14 0.09 ± 0.08 2 0.32 ± 0.30 0.23 ± 0.22 0.14 ± 0.13 0.09 ± 0.08 150 µs 1 45 0.35 ± 0.33 0.25 ± 0.24 0.15 ± 0.15 0.10 ± 0.09 2 0.34 ± 0.32 0.24 ± 0.24 0.14 ± 0.14 0.09 ± 0.08 150 µs 1 55 0.37 ± 0.34 0.25 ± 0.25 0.15 ± 0.15 0.09 ± 0.08 2 0.36 ± 0.34 0.25 ± 0.25 0.14 ± 0.14 0.08 ± 0.07

Figure 5 and table 4 display the mean stereo reconstruction error (SRE) for the two stereo systems 1 and 2. The stereo reconstruction error is a measure for the quality of the vector fields. It results from the calculation of a single three-component stereo vector from two two-component vectors. In general, the stereo reconstruction error is a lower limit for the overall measurement error. In figure 5, SRE shows a systematic relation to the viewing direction of the cameras: SRE increases with the distance from the connection line of the two stereo camera centers. The SRE’s (avg. and RMS) are very similar for the two independent stereo systems: the difference is less or equal to 0.01 pixel for all window sizes and sheet thicknesses. SRE depends on the window size and on the sheet thickness: The smaller the window size and the thicker the light sheet is, the higher is the SRE. The lowest SRE is 0.02 pixel, for 8 pixel light sheet thickness and 128 x 128 pixel window size SRE. The highest SRE is 0.37 pixel, for 55 pixel sheet thickness and 16 x 16 pixel window size. The correlation coefficient (table 5) is a measure that describes how well the particles in the interrogation windows match after deformation of the original images with the vector field. Under ideal conditions a correlation coefficient of 1.0 would indicate a perfect match. The correlation coefficient is reduced by image noise, out of plane gradients, loss of particles and in plane accelerations inside the interrogation window. The correlation coefficient decreases with increasing window size, indicating flow structures too small to be resolved with bigger interrogation windows. At the same time the correlation decreases with increasing sheet thickness. The flow gradients in the z-direction smear and diminish the correlation peak for thicker sheets. Moreover, the time separation dt was bigger for thicker sheets, so that the flow structures had more time to become dissimilar. The percentage of spurious vectors or outlier, is displayed in table 6. It is calculated from the number of vectors that have been remove by the universal outlier detection filter (Westerweel et al. 2005) during vector postprocessing. There are overall only very few outlier: In the majority of cases there are fewer than 2.5% outlier, with only three exceptions: for 16 x 16 correlation window and sheet thickness greater or equal to 35 pixel the amount of outlier is greater than 5% for stereo PIV. Table 7 shows the difference between vector fields from stereo system 1 and system 2 (RMSS1,S2). A difference vector field is calculated for each of the 998 vector fields for each recording. Then the total RMS is calculated from all 998 difference vector fields. The RMS is about 0.5 pixel, when the window size is about two times the sheet thickness. The difference between system 1 and 2, calculated by the difference of the average fields (N=998), is much lower than the differences from table 7: e.g. the RMS of the difference from the average fields is only 0.023 pixel, for the recording with 8 pixel sheet thickness and 32 x 32 correlation -6-

14th Int Symp on Applications of Laser Techniques to Fluid Mechanics Lisbon, Portugal, 07-10 July, 2008

Table 5. Mean correlation coefficient, from 998 vector fields, for stereo systems 1 and 2 and single plane tomographic PIV (tomo). Depth of interrogation Sheet thickness Interrogation window size System window for tomo (z[pixel] 16 x 16 32 x 32 64 x 64 128 x 128 direction) [pixel] 0.851 0.824 0.806 0.775 8 stereo 1 stereo 2 0.852 0.824 0.807 0.776 32 16 35 45 55

tomo stereo 1 stereo 2 tomo stereo 1 stereo 2 tomo stereo 1 stereo 2 tomo stereo 1 stereo 2 tomo

0.862 0.802 0.802 0.794 0.694 0.694 0.640 0.674 0.673 0.596 0.668 0.668 0.566

56 80 96 120

0.846 0.754 0.754 0.769 0.601 0.601 0.599 0.584 0.584 0.567 0.580 0.580 0.538

0.835 0.730 0.731 0.753 0.559 0.559 0.574 0.545 0.545 0.542 0.545 0.545 0.521

0.817 0.687 0.687 0.729 0.491 0.491 0.541 0.485 0.486 0.514 0.496 0.496 0.499

Table 6. Mean percentage of outlier, from 998 vector fields, for stereo systems 1 and 2 and single plane tomographic PIV (tomo). Interrogation window size System 16 x 16 Sheet thickness 32 x 32 64 x 64 128 x 128 [pixel] stereo 1 0.50 % 0.01% < 0.01 % < 0.01 % 8 16 35 45 55

stereo 2 tomo stereo 1 stereo 2 tomo stereo 1 stereo 2 tomo stereo 1 stereo 2 tomo stereo 1 stereo 2 tomo

0.42 % 0.35 % 1.41 % 1.32 % 0.84 % 5.39 % 5.32 % 3.44 % 5.88 % 5.80 % 3.66 % 6.50 % 6.42 % 4.09 %

0.01% 0.01 % 0.16 % 0.15 % 0.07 % 2.21 % 2.19 % 1.03 % 2.41 % 2.38 % 0.82 % 2.52 % 2.48 % 1.12 %

< 0.01 % < 0.01 % 0.02 % 0.02 % 0.01 % 1.28 % 1.31 % 0.43 % 1.40 % 1.39 % 0.47 % 1.18 % 1.19 % 0.38 %

< 0.01 % < 0.01 % < 0.01 % < 0.01 % < 0.01 % 0.99 % 0.26 % 0.27 % 0.76 % 0.84 % 0.35 % 0.49 % 0.54 % 0.21 %

Table 7. Difference between instantaneous vector fields from stereo system 1 (camera 1 and 2) and stereo system 2 (camera 3 and 4): Total RMS of difference vector fields in pixel (RMSS1,S2, after postprocessing, from 998 vector fields each). Interrogation window size 16 x 16 32 x 32 64 x 64 128 x 128 Sheet thickness [pixel] 8 0.52 0.27 0.15 0.09 16 0.79 0.42 0.23 0.13 35 1.66 1.06 0.68 0.45 45 1.78 1.12 0.69 0.46 55 1.84 1.11 0.66 0.42

window size. This is an order of magnitude smaller than the instantaneous difference of 0.27 pixel for the same sheet thickness and window size (table 7). 4.2 Tomographic PIV: Intensity profile The intensity profile of the laser sheet in the z-direction (out of plane direction) can be estimated from the intensity of each z-plane of the reconstructed volume (figure 6). -7-

14th Int Symp on Applications of Laser Techniques to Fluid Mechanics Lisbon, Portugal, 07-10 July, 2008

Fig.6. Sheet intensity profile calculated from reconstructed volumes (without peak normalization).

The half maximum width of the intensity profile is used as an estimator of the light sheet thickness. Image preprocessing has an influence on the estimated thickness: The calculated sheet thickness is bigger when peak normalization is used (table 8). However it is expected that the sheet thickness calculated without peak normalization is closer to the true light laser sheet thickness, because the original particle intensity distribution is maintained. In the following the sheet thickness is always indicated as the half maximum width of the profile without peak normalization rounded to the nearest integer pixel value. The sheet intensity profiles in figure 6 do not drop down to zero intensity at the border. Instead they reach a constant level that is caused by ghost particles, which are spread over the entire reconstructed volume. The percentage of ghost particles is (Elsinga et al. 2006b): Np Ng

=

1 ppp

N −1

ApN lZ

(1)

where Np is the number of particle, Ng is the number of ghost particle, ppp is particle per pixel in the camera images, Ap is the area of one particle in pixel (calculated from the particle diameter Dp), lZ is the sheet thickness in pixel and N is the number of cameras. The percentage of ghost particles depends strongly on the particle diameter Dp. A good correspondence between the ghost-particle-level from figure 6 and the calculated values in table 9 is achieved for a particle diameter of 2.0 pixel. This corresponds well to the measured diameter in the preprocessed particle images of 1.7 to 2.3 pixel. Table 8. Sheet thickness calculated from the intensity profiles of the reconstructed volume. Sheet thickness (half maximum width) without peak normalization with peak normalization pixel mm pixel mm Sheet thickness [pixel] 7.80 0.37 9.1 0.43 8 16.16 0.76 21.6 1.01 16 35.24 1.65 40.5 1.90 35 45.29 2.13 54.7 2.57 45 55.47 2.60 69.2 3.25 55

-8-

14th Int Symp on Applications of Laser Techniques to Fluid Mechanics Lisbon, Portugal, 07-10 July, 2008

Table 9. Percentage of ghost particles (equation 1): N=4, ppp=0.031. Sheet thickness [pixel] Ghost-particle-level Calculated Np / Ng from figure 3 Dp = 1.5 pixel Dp = 2.0 pixel 2% 0.2 % 2.3 % 8 5% 0.5 % 4.6 % 16 10 % 1.0 % 10.2 % 35 16 % 1.3 % 13.1 % 45 17 % 1.6 % 16.0 % 55

Dp = 2.5 pixel 13.8 % 27.7 % 60.6 % 77.9 % 95.2 %

4.3. Single plane tomographic PIV A single-plane vector field was calculated from the reconstructed volume to compare the results from stereo PIV and tomographic PIV. The depth of the interrogation volumes was constant for a given sheet thickness and covered the complete illuminated z-range (table 5, third column). So, if e.g. the correlation window size is 32 x 32 and the sheet thickness is labeled with 35 pixel, then the interrogation windows for 3D correlation had a size of 32 x 32 x 80 voxel, including tailing intensities. The volume deformation is constant along the z-direction, so that this technique can not take an advantage of deforming the interrogation windows according to the gradients in the zdirection. Similar to stereo PIV, the correlation coefficients (table 5) from 3D-correlation is a measure how well the dewarped interrogation volumes match. The average correlation coefficient shows the same tendency as in stereo PIV to decrease with increasing interrogation volume size and with sheet thickness. They are higher for tomographic PIV than for stereo PIV as long as the correlation window size is greater than the thickness of the sheet. The increasing amount of uncorrelated ghost particles for thicker sheets is diminishing the correlation value here. The percentage of outlier (table 6) for tomographic PIV is even smaller for all configurations compared to stereo PIV. All configurations show less than about 1 % of outlier, with the same exception as in stereo PIV: for 16 x 16 pixel interrogation volumes and sheet thickness of 35 pixel or thicker, the amount of outlier is 3.4 % to 4.1 %. The RMS of the difference between vector fields from tomographic PIV and stereo systems 1 (RMST,S1) and 2 (RMST,S2), (table 10), is, with no exception, smaller than the difference in between stereo systems 1 and 2 (table 7). Table 10. Difference between vector fields from tomographic PIV and stereo systems 1 (RMST,S1) and 2 (RMST,S2): Total RMS of difference vector fields in pixel (after post processing, from 998 vector fields each). Interrogation window size 16 x 16 32 x 32 64 x 64 128 x 128 Sheet thickness [pixel] Stereo system 1 8 0.43 0.23 0.13 0.08 2 0.37 0.19 0.11 0.08 1 16 0.63 0.34 0.19 0.12 2 0.58 0.30 0.17 0.11 1 35 1.31 0.84 0.55 0.41 2 1.26 0.80 0.54 0.41 1 45 1.40 0.89 0.57 0.42 2 1.35 0.85 0.55 0.41 1 55 1.44 0.87 0.54 0.37 2 1.39 0.83 0.52 0.37

4.4. Dual plane tomographic PIV Two independent vector planes are calculated in dual plane tomographic PIV. Therefore, the central 16 voxel-planes of the reconstructed volumes from the recording with 8 pixel sheet thickness were used. The interrogation windows had 75% overlap in x- and y-direction, but no overlap (0%) in zdirection. So, the size of the interrogation volumes was 32 x 32 x 8 voxel. With this dual plane -9-

14th Int Symp on Applications of Laser Techniques to Fluid Mechanics Lisbon, Portugal, 07-10 July, 2008

approach the gradients in the z-direction are approximated by a linear interpolation between the two vector planes. These gradients allow a non uniform volume deformation in the z-direction during the iterative correlation procedure. The flow gradients in the z-direction are then calculated by subtracting the two vector planes. For visualization (figure 7), the length of the difference vectors are calculated. The gradients of the mean flow are estimated by calculating the average of both vector planes and subtracting the averages. The vector length of the resulting field is displayed in figure 7 on the left. On the right hand side, the average vector length of the instantaneous difference fields is displayed. The average instantaneous gradients are clearly stronger (up to 2.4 pixel) than the mean flow gradients (up to 0.7 pixel), indicating that the main gradients result from non stationary flow. Inspecting individual difference fields, revealed local gradients of up to five pixel magnitude.

Fig. 7. Difference between first and second z-plane: Length of difference vectors. Left: difference of averaged planes (N=998), right: average of individual difference planes (8 pixel light sheet thickness, 32 x 32 x 8 pixel interrogation volumes, distance of planes = 8 pixel).

4.5. Multiple plane tomographic PIV In multi plane tomographic PIV, cubic interrogation volumes, e.g. 16 x 16 x 16 voxel, are used with 75 % overlap in all three dimensions. In this way the gradients in all three spatial directions can be approximated by the resulting vector fields, using the adaptive multi pass approach. This leads to a better correlation and higher correlation coefficients compared to the single plane approach. In the center of the reconstructed volume (figure 9 left) the correlation coefficient increases from a maximum of 0.86 (table 5) to 0.90 for the sheet with 8 pixel thickness. The effect is even stronger for the thicker sheet (55 pixel) where the correlation coefficient increased from 0.566 to a maximum of 0.74 (figure 9, right). Figure 9, right also shows an optimum of the correlation window size at 22 pixel for the thicker sheet. For bigger windows, the gradients can not be approximated accurately and for smaller windows the correlation is decreased because of poor correlation from spurious vectors. This effect is also visible in figure 10, left, where the percentage of outlier increases, when the correlation windows become too small. The divergence can be calculated from 3D-3C vector fields. It is used as an error estimator, since the flow should be divergence free. If the calculated divergence is not zero, this is due to errors in the computed vector field. The divergence is calculated from six vector components, so that the error of a single vector component, ediv, is given by the divergence divided by the square root of six (figure 10, right). The resulting error ediv is in the range of 0.1 pixel as long as the interrogation windows are not too small. It is clearly lower than the errors estimated from the difference of the vector fields from stereo PIV and tomographic PIV (table 7 and 10). The iso vorticity surfaces (figure 11) from a multi plane vector field illustrate the turbulent structures. The structures are chaotic at the core center and become radial aligned in the outer regions. Zones with w-components of opposite signs are heading and trailing the radial aligned - 10 -

14th Int Symp on Applications of Laser Techniques to Fluid Mechanics Lisbon, Portugal, 07-10 July, 2008

Fig. 9. Correlation coefficient as a function of z-position and interrogation window size. Left: Sheet thickness 8 pixel, interrogation volume 12x12x12 voxel. Right: sheet thickness 16 and 55 pixel.

Fig. 10. Percentage of outlier (left) and error of each vector component ediv estimated from the residual divergence (right) for 16 pixel (blue) and 55 pixel (red) light sheet thickness as a function of correlation volume size.

vortices, which show diameters below 1mm. Reducing the number of cameras for reconstruction from four to three or even only two cameras, leads to difference fields relative to the four camera solution with an RMS of 0.5 pixel for three cameras and 1.0 pixel for two cameras. Assuming the four camera solution to be the most accurate one, the deviation for three and two cameras is so high, that using less than four cameras is not recommended.

5. Conclusion Considerable differences, in the range of 0.09 to 1.84 pixel, have been detected between the measured vector fields of two independent stereo PIV systems. The differences are clearly higher than 0.1 pixel, the precision, that is often stated in the literature for stereo PIV. Possible reasons for the high deviations are the strong gradients of the turbulent flow used in this study, which leads to systematic errors in stereo PIV. Tomographic PIV on the other hand shows smaller differences to the individual results from the stereo system, than the stereo systems show between each other, even when a sub optimal single- 11 -

14th Int Symp on Applications of Laser Techniques to Fluid Mechanics Lisbon, Portugal, 07-10 July, 2008

plane approach is used. With the multi plane approach, the error estimated for tomographic PIV from the residual divergence is only about 0.1 pixel. Tomographic PIV seems to be a more accurate measurement technique than stereo PIV, when applied for thin volumes and turbulent flow.

Fig. 11. Iso vorticity surfaces (purple) from 13 vector planes calculated with 24x24x24 window size and 75 % overlap for a sheet thickness of 55 pixel. The w-component (Vz) of the central plane is color coded.

References Elsinga GE, Scarano F, Wieneke B, van Oudheusden BW (2006a) Tomographic particle image velocimetry. Exp Fluids 41:933–947 Elsinga GE, van Oudheusden BW Scarano F U (2006b) Experimental assessment of Tomographic-PIV accuracy. 13th Int Symp on Applications of Laser Techniques to Fluid Mechanics, Lisbon, Portugal Michaelis D, Poelma C, Scarano F, Westerweel J, Wieneke B (2006) A 3D time-resolved cylinder wake survey by Tomographic PIV, 12th International Symposium on Flow Visualization, Göttingen, Germany, Sept 10-14. Westerweel J. (1993) PhD thesis Delft University of Technology Westerweel J, Scarano F (2005) A universal detection criterion for the median test. Exp. Fluids 39:1096 Wieneke B (2005) Stereo-PIV using self-calibration on particle images. Exp Fluids 39:267-280 Wieneke B (2008) Volume Self-Calibration for 3D Particle Image Velocimetry. Exp Fluids accepted

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