On the joint observability of flow fields and particle properties from Lagrangian trajectories: evidence from neural data assimilation

TL;DR

The paper addresses whether Eulerian flow fields and unknown particle properties can be jointly inferred from Lagrangian trajectories. It introduces NIPA, a neural-implicit data-assimilation framework that couples a coordinate-flow model with per-particle kinematics-constrained trackers, enforcing both Navier–Stokes and (extended) Maxey–Riley dynamics. Across three regimes—tracer-limited turbulent boundary layers, inertial particles in HIT, and compressible shock-driven cone–cylinder flow—it demonstrates empirical existence proofs of joint observability, showing that both flow states and particle properties (e.g., diameter, density) can be recovered, with performance governed by seeding density, localization noise, and Stokes number. The results provide practical guidelines for experimental design and highlight the potential to broaden LPT's applicability to multiphase and high-speed flows, including tasks such as in situ drag-law calibration and particle-property inference. Overall, NIPA offers a physics-informed, data-driven route to reconstruct complex disperse flows from limited LPT data, with measurable gains in accuracy and new insight into the information content of Lagrangian measurements.

Abstract

We numerically investigate the joint observability of flow states and unknown particle properties from Lagrangian particle tracking (LPT) data. LPT offers time-resolved, volumetric measurements of particle trajectories, but experimental tracks are spatially sparse, potentially noisy, and may be further complicated by inertial transport, raising the question of whether both Eulerian fields and particle characteristics can be reliably inferred. To address this, we develop a data assimilation framework that couples an Eulerian flow representation with Lagrangian particle models, enabling the simultaneous inference of carrier fields and particle properties under the governing equations of disperse multiphase flow. Using this approach, we establish empirical existence proofs of joint observability across three representative regimes. In a turbulent boundary layer with noisy tracer tracks (St to 0), flow states and true particle positions are jointly observable. In homogeneous isotropic turbulence seeded with inertial particles (St ~ 1-5), we demonstrate simultaneous recovery of flow states and particle diameters, showing the feasibility of implicit particle characterization. In a compressible, shock-dominated flow, we report the first joint reconstructions of velocity, pressure, density, and inertial particle properties (diameter and density), highlighting both the potential and certain limits of observability in supersonic regimes. Systematic sensitivity studies further reveal how seeding density, noise level, and Stokes number govern reconstruction accuracy, yielding practical guidelines for experimental design. Taken together, these results show that the scope of LPT could be broadened to multiphase and high-speed flows, in which tracer and measurement fidelity are limited.

Figures (24)

• Figure 1: Neural DA solver architecture. Eulerian flow fields are represented by one or more neural networks, while each particle is modeled with a Lagrangian kinematics-constrained track (KCT) model. Flow fields, particle kinematics, and particle properties (when $St > 0$) are jointly inferred from data under the governing equations.
• Figure 2: Illustration of a KCT (kinematics-constrained track) model. Colored trajectories show particle histories output by the KCT, with colors denoting particle speed. Dotted lines indicate the ground truth. Embedding \ref{['equ: particle advection']} as a hard constraint ensures that tracks pass through specified particle positions (dots), regardless of velocity adjustments. Hence, the $\theta_i$ parameters are unconstrained.
• Figure 3: Standard deviations of particle position, velocity, and acceleration errors along the $x$- and $z$-axes under varying noise levels. Joint flow--particle estimation is applied at three inter-particle spacings ($\delta = 9l_\nu$, $18l_\nu$, $36l_\nu$). Standard deviations are normalized by viscous units. Joint estimation consistently yields the lowest errors, with some dependence on seeding density.
• Figure 4: Comparison of exact, raw, filtered, and jointly estimated tracks and their pointwise velocity errors at the lowest ($\sigma_\mathrm{x} = 0.09l_\nu$) and highest ($\sigma_\mathrm{x} = 0.9l_\nu$) noise levels for the densest seeding case ($\delta = 9l_\nu$). Only 200 tracks are shown for visual clarity. Colors indicate the $v_3$ velocity or its error. Raw-track velocity errors at high noise are downscaled fivefold for visualization. Joint estimation accurately reconstructs track geometries and velocities across noise levels, while finite-difference and filtering methods show large errors, especially near boundaries and in regions of high-acceleration.
• Figure 5: Close-up of a representative track under the high-noise condition. The true track is shown in black; noisy data as red dots; filtered track in green; and the jointly estimated track in blue. While filtering improves the raw data, joint estimation better recovers the true trajectory.