Lagrangian Proper Orthogonal Decomposition

TL;DR

This paper introduces Lagrangian Proper Orthogonal Decomposition (LPOD), a modal analysis method for turbulence trajectories, demonstrating its effectiveness in reconstructing particle dynamics from limited modes in simulations and experiments.

Contribution

The paper presents LPOD, a novel modal decomposition technique for Lagrangian turbulence data, enabling efficient trajectory reconstruction and potential synthetic data generation.

Findings

Leading modes capture similar structures in simulations and experiments.

A small number of modes (~10) accurately reproduce dispersion and curvature statistics.

More modes are needed to accurately capture acceleration distribution tails.

Abstract

We introduce a modal representation for Lagrangian trajectories in turbulence, termed Lagrangian Proper Orthogonal Decomposition (LPOD). An ensemble of particle trajectories is used to construct velocity time series, which are normalized independently for each trajectory to isolate fluctuations. Principal Component Analysis is then applied to the resulting dataset, with temporal instances defining the feature space. The method is tested on trajectories from both direct numerical simulations of homogeneous isotropic turbulence and three-dimensional particle-tracking experiments, showing that the leading modes exhibit similar structures and energy distributions in both cases. Truncated reconstructions are obtained by combining modes and coefficients, rescaling the fluctuations, and integrating in time. For trajectories of the order of the integral time scale, single-particle dispersion and curvature statistics are accurately reproduced using a limited number of modes (c.a. 10), whereas capturing the tails of acceleration distributions requires a larger set (c.a. 30-60). Longer trajectories require progressively more modes for accurate reconstruction. These results suggest a possible route to data-driven generation of synthetic particle trajectories via stochastic sampling of the modal Lagrangian dynamics.