A three-dimensional velocimetry approach using a...

18th International Symposium on the Application of Laser and Imaging Techniques to Fluid Mechanics・LISBON | PORTUGAL ・JULY 4 – 7, 2016

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: [email protected]

Keywords: tomographic reconstruction, 3D-PTV

ABSTRACT

Volumetric flow velocimetry has d rawn increasing attention in the past years. Nowadays, capable hardware and

software enables time-resolved tracking even at large seed ing concentrations. This allows for high spatial resolution

measurements without bias errors due to strong velocity grad ients. 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 d rawback,

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 pred ict 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-d imensional (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 𝑁𝑝𝑝𝑝 = 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


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 time-

resolved 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 record ing 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 record ings 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 d ifferent 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, 𝑷𝑖, the particle coordinates 𝒙 are then mapped back to camera

sensors by simple multiplication:


𝑿𝑖 = 𝑷𝑖 ⋅ 𝒙 (1)

where 𝑿𝑖 = (𝑋𝑖, 𝑌𝑖) 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 record ings. 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 double-

frame 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 d irectly 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 record ings.

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 d ifferent particle per pixel values, 𝑁𝑝𝑝𝑝, are processed according to the outlined procedure.

The sets are generated using the DaVis 8 software from LaVision, yield ing a volume size of 800 ×

800 × 300 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 d iameter of 𝐷 = 2 ± 0.5 pixel. A 3D fit of the reconstructed volume serves as a reference set.


Fig. 1 provides the percentage of correct reconstructions, i.e. within a deviation of 1 voxel in

space. With increasing 𝑁𝑝𝑝𝑝 values, the percentage of correct reconstructions decreases. For the

combined tomographic 3D-PTV approach, the correct reconstructions drop below 60 % at higher

𝑁𝑝𝑝𝑝 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 d ifference 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 d iamonds, the amount of correct reconstructions stays on a high

level, yield ing values above 95%.

Fig. 1 Overview of correctly, i.e. within one voxel rad ius, reconstructed particles locations

relative to the number of true particles


Fig. 2 Overview of the amount of ghost particles relative to the number of true particles

Fig. 3 Effective particles per pixel values


The percentage of ghost particles, relative to number of true particles, is given in Fig. 2, where

the ord inate 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 𝑁𝑝𝑝𝑝 values. The amount of ghosts is slightly larger for

the MLOS predictor, up to 2.1% at 𝑁𝑝𝑝𝑝 ≈ 0.058. 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 record ings.

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 d isplacement 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, 𝑁𝑝𝑝𝑝,𝑒𝑓𝑓, denoting the actual amount of particles that

contribute to the velocity estimation. Using the MART predictor, values of up to 𝑁𝑝𝑝𝑝,𝑒𝑓𝑓 = 0.033

can be reached . The performance of the MLOS predictor is lower, yield ing a value of up to

𝑁𝑝𝑝𝑝,𝑒𝑓𝑓 = 0.024. There is a saturation value, i.e. a maximum 𝑁𝑝𝑝𝑝,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 𝑁𝑝𝑝𝑝,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,

yield ing a size of 8 × 6 × 2.5 mm³, was illuminated using a Quantronix high speed laser. In total,

30000 images in 3 subsets were recorded at a frequency of 10.2 kHz.


Fig. 4 Experimental set-up: The measurement volume with a size of 8 × 6 × 2.5 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 wall-

normal 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 𝑁𝑝𝑝𝑝 < 0.01,

a MLOS predictor was used for the processing. Fig. 5 shows the averaged velocity profile binned

in wall-normal d irection 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 d istance of 0.01 mm from the wall. The friction velocity yields 𝑢𝜏 = 0.2004

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 d irection. 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 double-

frame 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 𝑁𝑝𝑝𝑝,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.

y


Fig. 5 Mean flow velocity, binned in wall-normal d irection with a bin width of 0.02 mm.

Fig. 6 Reynolds stresses, binned in wall-normal d irection with a bin width of 0.02 mm. Blue

points: time-resolved (tr) tracking data; Red points: double-frame (df) tracking data.


5. Conclusion

A tomographic predictor enables the use of 3D-PTV for measuring flows with effective particle

per pixels values of up to 𝑁𝑝𝑝𝑝,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 su itable to resolve strong velocity gradients, as proven by the measurement

of a turbulent boundary layer flow measured at a 𝑁𝑝𝑝𝑝,𝑒𝑓𝑓 value of slightly below 0.01.

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 microfluid ics. 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 68:146-157.

Kähler CJ, Scharnowski S, Cierpka C (2012a) On the resolution limit of d igital particle image

velocimetry. Exp Fluids 52:1629-1639.

Kähler CJ, Scharnowski S, Cierpka C (2012b) On the uncertainty of d igital PIV and PTV near

walls. Exp Fluids 52:1641-1656.

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


PIV. 11th International Symposium on Particle Image Velocimetry, Santa Barbara, CA USA,

September 14-16.

Scarano F (2013) Tomographic PIV: principles and practice. Meas Sci Technol 24:012001.

Schanz D, Gesemann S, Schröder A (2016) Shake-The-Box: Lagrangian particle tracking at high

particle image densities. Exp Fluids 57:70.

Schröder A, Geisler R, Staack K, Elsinga GE, Scarano F, Wieneke B, Henning A, Poelma C,

Westerweel J (2011) Eulerian and Lagrangian views of a turbulent boundary layer flow using

time resolved tomographic PIV. Exp Fluids 50:1071-1091.

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