<p>Polynomial expansion optical flow velocimetry (PExFlow) is an image-based method for estimating velocity fields in particle-laden flows. The method employs a local polynomial expansion of image intensities to retrieve displacements, using a coarse-to-fine multiscale strategy combined with Gaussian-weighted neighborhood integration to maintain numerical stability in the presence of noise and velocity gradients. A systematic exploration of the parameter space and comparison over synthetic benchmark cases from the FLUID database was conducted for PExFlow, as well as for reference methods including cross-correlation particle image velocimetry (PIV), Horn–Schunck, Lucas–Kanade, TV-L1 and RAFT-PIV. For each algorithm, average optimal parameter sets were identified through global maximization of the mean determination coefficient (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\overline{R^2}$</EquationSource> <EquationSource Format="MATHML"><math> <mover> <msup> <mi>R</mi> <mn>2</mn> </msup> <mo>¯</mo> </mover> </math></EquationSource> </InlineEquation>) and the comparative assessment was performed using complementary metrics: determination coefficient (<InlineEquation ID="IEq2"> <EquationSource Format="TEX">$R^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>R</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>), cosine similarity (<i>S</i>) and execution time under controlled hardware conditions. Finally, in addition to pointwise comparisons against analytical ground truth, the methods were evaluated on an experimental quasi-two-dimensional turbulent flow generated by Lorentz forcing. Spectral analysis was used to examine the recovery of Kolmogorov (<InlineEquation ID="IEq3"> <EquationSource Format="TEX">$k^{-5/3}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>k</mi> <mrow> <mo>-</mo> <mn>5</mn> <mo stretchy="false">/</mo> <mn>3</mn> </mrow> </msup> </math></EquationSource> </InlineEquation>) and Kraichnan (<InlineEquation ID="IEq4"> <EquationSource Format="TEX">$k^{-3}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>k</mi> <mrow> <mo>-</mo> <mn>3</mn> </mrow> </msup> </math></EquationSource> </InlineEquation>) scaling regimes, providing a physics-based consistency check in the absence of pixel-level ground truth. The results show that PExFlow provides a favorable balance between accuracy, robustness to parameter variation and computational efficiency for the flow conditions investigated. The method is distributed as a freely available open-source application with a user-friendly interface, facilitating its implementation and accessibility in fluid mechanics experimental data analysis.</p>

Optical flow method based on polynomial expansion for particle-laden fluid velocimetry

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发表日期：

2026年5月23日

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M. Flores,
F. Rinderknecht,
T. Gallot,
L. G. Sarasua,
Y. Abraham,
N. Barrere,
D. Freire Caporale

摘要

Polynomial expansion optical flow velocimetry (PExFlow) is an image-based method for estimating velocity fields in particle-laden flows. The method employs a local polynomial expansion of image intensities to retrieve displacements, using a coarse-to-fine multiscale strategy combined with Gaussian-weighted neighborhood integration to maintain numerical stability in the presence of noise and velocity gradients. A systematic exploration of the parameter space and comparison over synthetic benchmark cases from the FLUID database was conducted for PExFlow, as well as for reference methods including cross-correlation particle image velocimetry (PIV), Horn–Schunck, Lucas–Kanade, TV-L1 and RAFT-PIV. For each algorithm, average optimal parameter sets were identified through global maximization of the mean determination coefficient ( $\overline{R^2}$ $\bar{R^{2}}$ ) and the comparative assessment was performed using complementary metrics: determination coefficient ( $R^2$ $R^{2}$ ), cosine similarity (S) and execution time under controlled hardware conditions. Finally, in addition to pointwise comparisons against analytical ground truth, the methods were evaluated on an experimental quasi-two-dimensional turbulent flow generated by Lorentz forcing. Spectral analysis was used to examine the recovery of Kolmogorov ( $k^{-5/3}$ $k^{- 5 / 3}$ ) and Kraichnan ( $k^{-3}$ $k^{- 3}$ ) scaling regimes, providing a physics-based consistency check in the absence of pixel-level ground truth. The results show that PExFlow provides a favorable balance between accuracy, robustness to parameter variation and computational efficiency for the flow conditions investigated. The method is distributed as a freely available open-source application with a user-friendly interface, facilitating its implementation and accessibility in fluid mechanics experimental data analysis.