# Computation of finite-time Lyapunov exponents from time-resolved particle image velocimetry data

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## Abstract

This work presents two new methods for computing finite-time Lyapunov exponents (FTLEs) from noisy spatiotemporally resolved experimentally measured image data of the type used for particle image velocimetry (PIV) or particle tracking velocimetry (PTV). These new approaches are based on the simple insight that the particle images recorded during PIV experiments represent Lagrangian flow tracers whose trajectories lend themselves to the direct computation of flow maps, and related quantities such as flow map gradients and FTLEs. We show that using this idea we can improve the reliability and accuracy of FTLE calculation through the use of either direct pathline flow map (PFM) calculation, where individual particle pathlines over a fixed period of time are used to determine the flow map, or particle tracking flow map compilation (FMC), where instantaneous tracking results are used to estimate small snapshots of the flow map which are then compiled to describe the complete flow map. Comparisons of the traditional velocity field integration (VFI) method for computing FTLE fields with these new methods show that FMC produces significantly more accurate estimates of the FTLE field for both synthetic data and experimental data especially in cases where the particle number density is low. This is because the VFI estimates particle motion while PTV directly measures particle motion and therefore generates a more accurate flow map. Overall, our results suggest that VFI is not always a reliable approach when applied to noisy experimental PIV data. For cases where particle loss between frames is minimal, the PFM can also produce better results, but the final field is susceptible to error due to the unstructured nature of the raw flow maps. When comparing the ability to match the true separatrix of a flow, FMC is shown to be a far superior method. The separatrix from FMC has an 80 % overlap with the true solution as compared to approximately 25 % for the PFM and only 1 % for the VFI method. FMC shows a significant advantage when the particle seeding is low, which is particularly relevant for applications to environmental or biological flows where adding seed particles is not always practical, and investigation of Lagrangian flow structures must rely on naturally occurring flow tracers.

## Keywords

Particle Image Velocimetry Vortex Ring Seeding Density Particle Tracking Velocimetry Lagrangian Coherent Structure## Notes

### Acknowledgments

The authors would like to thank Roderick La Foy, Amir Bozorg Magham, Shibabrat Naik, and John Charonko for their helpful conversations and for commenting on the manuscript as well as Kelley Stewart and Cassie Niebel for providing the experimental data. SDR gratefully acknowledges partial support by the National Science Foundation under Grant No. CMMI-1150456. PPV gratefully acknowledges the partial support of NIH Grant 1R21HL106276-01A1.

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