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

The Polynomial Expansion Optical Flow Velocimetry (PExFlow) is a novel open-source image‑based approach for estimating velocity fields in particle‑laden flows. The method relies on the Optical flow hypothesis on conservation of tracer intensity between successive frames together with a locally approximates image intensities by polynomial expansion fitting around each pixel. Displacement estimates are obtained via a coarse‑to‑fine multiscale scheme, delivering high spatial resolution and effective noise suppression. We validate PExFlow using synthetic datasets generated from numerical simulations of seeded flows. Comparisons with analytical solutions and with established techniques, including cross-correlation Particle Image Velocimetry (PIV), Horn-Schunck and Lucas-Kanade optical flow, demonstrate that our achieves a marked reduction in displacement error and a substantial decrease in computational cost. Moreover, PExFlow shows low sensitivity to interrogation window size and successfully captures fine-scale turbulent features. Additionally, we compare the methods on an experimental quasi-two-dimensional turbulent flows generated by Lorentz forcing. We analyze the energy cascade dynamics to assess each method's capability to resolve both Kolmogorov and Kraichnan scaling regimes. The PExFlow method is shown to be the more accurate method on a wider spatial scale. These characteristics make PExFlow a robust and efficient alternative for high-resolution velocimetry in both laminar and turbulent regimes, easily integrable into existing experimental workflows.