Volumetric particle tracking velocimetry (PTV) uncertainty quantification
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RESEARCH ARTICLE
Volumetric particle tracking velocimetry (PTV) uncertainty quantification Sayantan Bhattacharya1 · Pavlos P. Vlachos1 Received: 27 November 2019 / Revised: 6 May 2020 / Accepted: 22 July 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
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.
* Pavlos P. Vlachos [email protected] 1
Department of Mechanical Engineering, Purdue University, West Lafayette, USA
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Graphic abstract
Abbreviations x�⃗w = {xw yw zw } World coordinates or physical coordinates c X�⃗ = {X c Y c } Camera image coordinates for camera c FX c , FY c X c And Y c calibration mapping function for camera c { } c a�⃗ = aci i=1to19 Camera c mapping function coefficient error in variable p 𝜎p Standard uncertainty in variable p
Σp Covariance matrix in variable p
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N Number of cameras Ncal Number of disparity grid points c d�⃗ = {dX c dY c } Disparity vector estimated from ensemble of reprojection error for each camera c u, v, w Velocity components in x, y, z directions, respectively |p| L2-norm or magnitude of a variable p
Experiments in Fluids
(2020) 61:197
C⃗xw Coefficient matrix of mapping function gradients with respect to
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