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Volumetric particle tracking velocimetry (PTV) uncertainty quantification

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Abstract

We introduce the first comprehensive approach to determine the uncertainty in volumetric Particle Tracking Velocimetry (PTV) measurements. Volumetric PTV is a state-of-the-art non-invasive flow measurement technique, which measures the velocity field by recording successive snapshots of the tracer particle motion using a multi-camera set-up. The measurement chain involves reconstructing the three-dimensional particle positions by a triangulation process using the calibrated camera mapping functions. The non-linear combination of the elemental error sources during the iterative self-calibration correction and particle reconstruction steps increases the complexity of the task. Here, we first estimate the uncertainty in the particle image location, which we model as a combination of the particle position estimation uncertainty and the reprojection error uncertainty. The latter is obtained by a gaussian fit to the histogram of disparity estimates within a sub-volume. Next, we determine the uncertainty in the camera calibration coefficients. As a final step, the previous two uncertainties are combined using an uncertainty propagation through the volumetric reconstruction process. The uncertainty in the velocity vector is directly obtained as a function of the reconstructed particle position uncertainty. The framework is tested with synthetic vortex ring images. The results show good agreement between the predicted and the expected RMS uncertainty values. The prediction is consistent for seeding densities tested in the range of 0.01–0.1 particles per pixel. Finally, the methodology is also successfully validated for an experimental test case of laminar pipe flow velocity profile measurement where the predicted uncertainty in the streamwise component is within 9% of the RMS error value.

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Abbreviations

\({\overrightarrow{x}}_{w}=\{{x}_{w} {y}_{w} {z}_{w}\}\) :

World coordinates or physical coordinates

\({\overrightarrow{X}}^{c}=\{{X}^{c} {Y}^{c}\}\) :

Camera image coordinates for camera \(c\)

\({F}_{{X}^{c}}, {F}_{{Y}^{c}}\) :

\({X}^{c}\) And \({Y}^{c}\) calibration mapping function for camera \(c\)

\({\overrightarrow{a}}^{c}={\left\{{a}_{i}^{c}\right\}}_{i=1 to 19}\) :

Camera \(c\) mapping function coefficient error in variable \(p\)

\({\sigma }_{p}\) :

Standard uncertainty in variable \(p\)

\(\overline{\overline{\Sigma }}_{p}\) :

Covariance matrix in variable \(p\)

\(N\) :

Number of cameras

\({N}_{cal}\) :

Number of disparity grid points

\({\overrightarrow{d}}^{c}=\{{d}_{{X}^{c}} {d}_{{Y}^{c}}\}\) :

Disparity vector estimated from ensemble of reprojection error for each camera \(c\)

\(u, v, w\) :

Velocity components in \(x, y, z\) directions, respectively

\(\left|p\right|\) :

L2-norm or magnitude of a variable \(p\)

\(\overline{\overline{C}}_{{\overrightarrow{x}}_{w}}\) :

Coefficient matrix of mapping function gradients with respect to \({\overrightarrow{x}}_{w}\) for all cameras

\(\overline{\overline{C}}_{{\overrightarrow{a}}^{c}}\) :

Coefficient matrix of mapping function gradients with respect to \({\overrightarrow{a}}^{c}\) evaluated at disparity grid points

\({\rho }_{pq}\) :

Correlation coefficient between variables \(p\) and \(q\)

\({{\overrightarrow{x}}_{w}}_{j}\) :

The world coordinates at \(j\) th time frame where \(j\) is an integer

\({\sigma }_{res}^{2}\) :

Variance in fit residual error

\(diag\overline{\overline{A}}\) :

A function which denotes the diagonal elements of matrix \(\overline{\overline{A}}\)

\(b\) :

Related to the measure of bias error or uncertainty in any variable

\(cal\) :

A variable or quantity evaluated in the calibration process

\(est\) :

Denotes the estimated value of any variable

\(true\) :

Denotes the known or designed value of any variable

PIV:

Particle Image Velocimetry

PTV:

Particle Tracking Velocimetry

IPR:

Iterative Particle Reconstruction

OTF:

Optical Transfer Function

STB:

Shake-The-Box

CRLB:

Cramer Rao Lower Bound

ppp:

Particles per pixel

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Acknowledgements

This work is supported by the National Science Foundation’s MRI program grant with award number 1725929.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Purdue University, West Lafayette, USA

    Sayantan Bhattacharya & Pavlos P. Vlachos

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  1. Sayantan Bhattacharya
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  2. Pavlos P. Vlachos
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Correspondence to Pavlos P. Vlachos.

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Bhattacharya, S., Vlachos, P.P. Volumetric particle tracking velocimetry (PTV) uncertainty quantification. Exp Fluids 61, 197 (2020). https://doi.org/10.1007/s00348-020-03021-6

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  • DOI: https://doi.org/10.1007/s00348-020-03021-6

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