A three-dimensional velocimetry approach using a combination of tomographic reconstruction and triangulation for double-frame particle tracking

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1 A three-dimensional velocimetry approach using a combination of tomographic reconstruction and triangulation for double-frame particle tracking Thomas Fuchs *, Rainer Hain, Christian J. Kähler Institute of Fluid Mechanics and Aerodynamics, Bundeswehr University Munich, Germany * Correspondent author: thomas.fuchs@unibw.de Keywords: tomographic reconstruction, 3D-PTV ABSTRACT Volumetric flow velocimetry has drawn increasing attention in the past years. Nowadays, capable hardware and software enables time-resolved tracking even at large seeding concentrations. This allows for high spatial resolution measurements without bias errors due to strong velocity gradients. Bias errors resulting from acceleration and curvature of the particle trajectory can be compensated. However, hardware restrictions still limit time-resolved flow measurements to rather small velocities and low magnifications. For aerod ynamics this is a major drawback, since most often higher flow velocities are of interest. Here, double-pulse PIV and PTV stud ies are still more common. In this double-frame volumetric measurement approach, the well-established techniques tomographic reconstruction and 3D-PTV are employed. A 3D fit of the reconstructed particles in the volume is used to predict the sensor locations of the corresponding particle images. Therefore, multiple uses of the particle images can be detected, and the amount of ghost particles can be reduced to a minimum. The analysis of synthetic as well as experimental data sets proves the capability of the combined tomographic 3D-PTV approach to derive 3D flow fields from double-frame data sets. 1. Introduction Quantitative three-dimensional (3D) flow information is desirable for the understanding of complex flows. Tomographic particle image velocimetry (tomographic PIV) has become a powerful tool to capture volumetric flow fields and the method has been widely applied to measure flows, at particle per pixel values around N ppp = 0.05 (Scarano (2013)). With high seeding concentrations good spatial resolutions can be achieved, such that it is possible to capture small flow structures. However, cross-correlation leads to a spatial averaging of the velocity field. Flows with strong velocity gradients, such as boundary layer flows, shear flows, and wake flows, are biased due to this averaging (Kähler et al. (2012a), Kähler et al. (2012b)). Particle tracking velocimetry (PTV) provides a means to overcome this drawback, since it tracks individual particles with sub-pixel accuracy. Employing the tomographic reconstruction, a straightforward approach is to determine the spatial particle locations by means of a 3D

2 Gaussian fit of the voxel intensities. The such determined particle locations can then be tracked (Schröder et al. (2011)). However, with an increasing seeding concentration, a rising number of ghost particles are reconstructed. Additional temporal information can help to the ghost particles. A recent tracking approach using a combination of tomography and triangulation is the so-called Shake-the-Box (STB) algorithm (Schanz et al. (2016)). It allows for the timeresolved measurement of densely seeded flows. Since STB does not require the reconstruction of a voxel volume, the hard drive storage space is much smaller compared to tomographic reconstruction methods. However, in aerodynamics it is not always feasible to obtain time-resolved data sets. Reasons for this lie in the limitations of the recording rates of the cameras and in the repetition rates of the lasers. To allow for flow measurements at higher velocities and at higher magnifications, this paper introduces a processing scheme, which is capable of measuring flows with strong velocity gradients using double-frame recordings in 3D. To meet these capabilities, the well-established methods tomographic reconstruction, triangulation, and particle tracking are combined. The tomographic reconstruction is used to find corresponding particle images on the different sensors and to detect multiple uses of the particle images. In the following, the spatial particle location is determined by means of triangulation. A stand-alone use of the triangulation approach, known as 3D-PTV, would limit the seeding densities, due to ambiguous particle image correspondences. The detailed procedure of the combined tomographic 3D-PTV processing scheme is outlined in the following. It is applied to a synthetic and an experimental data set. 2. Processing procedure The first step of the processing scheme is the tomographic reconstruction. Both, the multiplicative algebraic reconstruction technique (MART) and the multiplicative line-of-sight (MLOS) reconstruction are suitable, while at higher seeding concentrations the MART algorithm is favorable, as will be outlined later. The reconstructed volumes are binarized using an intensity threshold, following the estimation of the spatial particle locations, x = (x, y, z), from their center of mass. A more accurate 3D Gaussian fit is not necessary, since the reconstructed locations are only used as a predictor. It is not necessary to store the reconstructed volumes, since the information on the spatial particle locations is sufficient for the further processing steps. With the help of the camera matrices, P i, the particle coordinates x are then mapped back to camera sensors by simple multiplication:

3 X i = P i x (1) where X i = (X i, Y i ) denotes the predicted location of the originating particle image on camera sensor i. Now, these predicted particle image locations need to be associated with actual particle images, which have to be detected first. To find the actual particle image locations an 2D Gaussian fit is applied to the preprocessed recordings. Employing the implemented fit functions in the evaluation software DaVis by LaVision, this is a straightforward and fast procedure. In the following, the fitted particle image locations are matched with the predicted sensor locations from the tomographic reconstruction. If a particle image is associated with multiple reconstructed particle locations it is rejected entirely and not considered for triangulation anymore. Multiple matches are a result of ghost particles and overlapping particle images. To perform the triangulation for the particle location determination, it is necessary to find uniquely matching particle images on at least two sensors. If this is the case, the spatial particle location is determined by means of the so-called optimal triangulation method (Hartley and Sturm (1997)). The knowledge of which particle images are used for triangulation is essential for doubleframe tracking. Ghost particles cause spurious velocity vectors. The effect of ghosts on the result for cross-correlation techniques might not be as strong, but it certainly affects tracking methods. This also answers the question of why the particles are not tracked directly from the 3D fit of the tomographic reconstruction. In the latter case, the ghost particles can only be eliminated using additional temporal information, which is not available for double-frame recordings. The final step of the processing procedure is to apply a tracking algorithm to derive the velocity field. However, to eliminate remaining outliers as thoroughly as possible a probabilistic tracking procedure is used, which takes the motion of surrounding particles into account (Cierpka et al. (2013)). At this point it is emphasized that the tracking step is an important part of the proposed processing procedure, since it is a powerful tool to minimize the number of remaining outliers. 3. Synthetic data analysis To assess the performance of the combined tomographic 3D-PTV approach, synthetic data sets, with different particle per pixel values, N ppp, are processed according to the outlined procedure. The sets are generated using the DaVis 8 software from LaVision, yielding a volume size of voxels. The synthetic illumination has a Gaussian profile going down to e 1 at the edges of the volume. The synthetic particle images have an intensity of I = 512 ± 100 counts and a diameter of D = 2 ± 0.5 pixel. A 3D fit of the reconstructed volume serves as a reference set.

4 Fig. 1 provides the percentage of correct reconstructions, i.e. within a deviation of 1 voxel in space. With increasing N ppp values, the percentage of correct reconstructions decreases. For the combined tomographic 3D-PTV approach, the correct reconstructions drop below 60 % at higher N ppp values. The amount of reconstructions is also dependent on the tomographic reconstruction method. Using MART reconstruction, as denoted by the filled markers and the solid lines, the amount of reconstructions is larger than for the MLOS reconstruction, denoted by the hollow markers and the dashed lines. The reason for this difference is the larger share of ghost particles in the MLOS reconstruction, favoring ambiguities in the particle image matching. However, for the reference set, denoted by diamonds, the amount of correct reconstructions stays on a high level, yielding values above 95%. Fig. 1 Overview of correctly, i.e. within one voxel radius, reconstructed particles locations relative to the number of true particles

5 Fig. 2 Overview of the amount of ghost particles relative to the number of true particles Fig. 3 Effective particles per pixel values

6 The percentage of ghost particles, relative to number of true particles, is given in Fig. 2, where the ordinate has logarithmic scale. A 3D Gaussian fit of the reconstructed volume, using MART, can yield a ghost particle percentage of more than 100%, relative to the amount of true particles. This large amount of ghost particles does not allow for accurate double-frame particle tracking. However, when utilizing the tomographic reconstruction as a predictor, the fraction of ghost particles is decreased significantly, such that it yields percentages below 1.5% for the MART reconstruction predictor, even at larger N ppp values. The amount of ghosts is slightly larger for the MLOS predictor, up to 2.1% at N ppp Along with sophisticated tracking and outlier detection algorithms, this small share of ghost particles allows for reliable and accurate flow field estimations from double-frame recordings. To yield a high spatial resolution, it is a major goal for particle imaging techniques to reach high seeding concentrations. However, it is also obvious that increasing seeding concentrations result in a larger number of overlapping particle images, raising the uncertainty in estimating the displacement vector (Cierpka and Kähler (2012)). In this combined tomographic 3D-PTV approach not all particles can be employed for the flow velocity estimation, as illustrated by the share of correct reconstructions in Fig. 1. Therefore, Fig. 3 shows the effective particle per pixel values, N ppp,eff, denoting the actual amount of particles that contribute to the velocity estimation. Using the MART predictor, values of up to N ppp,eff = can be reached. The performance of the MLOS predictor is lower, yielding a value of up to N ppp,eff = There is a saturation value, i.e. a maximum N ppp,eff value, where an increasing share of ghost particles does not allow for a reliable prediction anymore. As a consequence, less sophisticated reconstruction techniques and thicker volumes lower N ppp,eff. 4. Time-resolved vs. double-frame tracking To prove the feasibility of the combined tomographic 3D-PTV approach for the measurement of real flows, it is applied for estimating the near-wall flow profile in an adverse pressure gradient (APG) region of a turbulent boundary layer experiment (Reuther et al. (2015)). The experimental set-up, as shown in Fig. 4, comprised four PCO dimax S4 high speed cameras, each equipped with a 50 mm Zeiss macro objective lens and a 2 teleconverter. The measurement volume, yielding a size of mm³, was illuminated using a Quantronix high speed laser. In total, images in 3 subsets were recorded at a frequency of 10.2 khz.

7 y Fig. 4 Experimental set-up: The measurement volume with a size of mm³ (x y z) lies in the adverse pressure gradient (APG) region of a turbulent boundary layer experiment, set up in the Atmospheric Wind Tunnel Munich (AWM). The streamwise coordinate is x, the wallnormal coordinate is y, and the spanwise coordinate is z. The data sets are processed using a double-frame tracking approach as well as a time-resolved tracking approach for a better comparison. Due to the low particle per pixel value of N ppp < 0.01, a MLOS predictor was used for the processing. Fig. 5 shows the averaged velocity profile binned in wall-normal direction with a bin width of 0.02 mm, at a free stream velocity of U = 10 m/ s. The average velocity profile can be resolved very close to the wall, such that the first averaged velocity value has a distance of 0.01 mm from the wall. The friction velocity yields u τ = m/ s. Using viscous scales with the constants κ = 0.41 and β = 5.0. The double-frame data slightly overestimate the measured velocities relative to the time-resolved data, where only tracks longer than 5 time steps were considered. Without using the temporal information, the double-frame tracking procedure still seems to be able to estimate the mean flow velocity quite accurately. However, the analysis of the Reynolds stresses gives a better idea of the performance of the double-frame tracking. Fig. 6 shows the estimated Reynolds stresses, again for bin width of 0.02 mm in wall-normal direction. In the region near to the wall, i.e. for y + < 2, the double-frame data shows the strongest relative deviations from the time-resolved data. Generally, the doubleframe data overestimates the Reynolds stresses. However, the estimated Reynolds stresses give rise to the assumption that also double-frame 3D-PTV can yield reliable results for N ppp,eff values slightly below 0.01, while with a lower accuracy than the time-resolved data. It has to be noted that the measurement only comprised a total measurement time of three seconds. More statistically independent data would be required for a comprehensive analysis of the flow.

8 Fig. 5 Mean flow velocity, binned in wall-normal direction with a bin width of 0.02 mm. Fig. 6 Reynolds stresses, binned in wall-normal direction with a bin width of 0.02 mm. Blue points: time-resolved (tr) tracking data; Red points: double-frame (df) tracking data.

9 5. Conclusion A tomographic predictor enables the use of 3D-PTV for measuring flows with effective particle per pixels values of up to N ppp,eff = 0.033, shown for a synthetic data set. The combined 3D imaging approach limits the fraction of ghost particles to values below 2.1%, allowing for double-frame particle tracking. Since the method employs tracking algorithms to determine the flow velocities, it is suitable to resolve strong velocity gradients, as proven by the measurement of a turbulent boundary layer flow measured at a N ppp,eff value of slightly below Acknowledgments The investigations were conducted as part of the joint research programme AG Turbo 2020 in the frame of AG Turbo. The work was supported by the Bundesministerium für Wirtschaft und Technologie (BMWi) as per resolution of the German Federal Parliament under grant number 03ET2013M. The authors gratefully acknowledge AG Turbo and MTU Aero Engines AG for their support and permission to publish this paper. The responsibility for the content lies solely with its authors. References Cierpka C, Kähler CJ (2012) Particle imaging techniques for volumetric three-component (3D3C) velocity measurements in microfluidics. J Vis 15:1-31. Cierpka C, Lütke B, Kähler CJ (2013) Higher order multi-frame particle tracking velocimetry. Exp Fluids 54:1533. Hartley RI, Sturm P (1997) Triangulation. Comput Vis Image Und 68: Kähler CJ, Scharnowski S, Cierpka C (2012a) On the resolution limit of digital particle image velocimetry. Exp Fluids 52: Kähler CJ, Scharnowski S, Cierpka C (2012b) On the uncertainty of digital PIV and PTV near walls. Exp Fluids 52: Reuther N, Scharnowski S, Hain R, Schanz D, Schröder A, Kähler CJ (2015) Experimental investigation of adverse pressure gradient turbulent boundary layers by means of large-scale

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