Accurate estimation of velocity and acceleration, their statistics and position prediction in Lagrangian Particle Tracking

Technical Summary: Accurate Estimation of Velocity and Acceleration in Lagrangian Particle Tracking

Problem Statement
Lagrangian Particle Tracking (LPT) is a critical tool for understanding turbulent flows, yet its utility is often limited by random measurement noise inherent in particle position data. This noise significantly degrades the accuracy of derived quantities, particularly velocity and acceleration, which require differentiation of position signals. Double differentiation amplifies high-frequency noise, leading to inaccurate acceleration statistics and biased results in regions of strong inhomogeneity, such as the viscous sublayer. Furthermore, existing filtering techniques often rely on global parameters that fail to adapt to local flow conditions, creating a trade-off between noise suppression and the preservation of high-acceleration events. Additionally, the lack of precise acceleration data hinders the validation of theoretical turbulence models (e.g., Heisenberg–Yaglom relation) and limits the effectiveness of data assimilation methods like FlowFit and Physics-Informed Neural Networks (PINNs).

Methodology
The study proposes three complementary approaches to mitigate measurement noise, validated using synthetically generated particle tracks in a turbulent near-wall channel flow (DNS data at $Re_\tau = 1000$) and a cylinder wake flow dataset.

1. Adaptive Local TrackFit:
   The authors introduce an extension to the conventional TrackFit algorithm, which represents particle tracks using cubic B-splines and approximates an optimal Wiener filter. Unlike previous global approaches, this method adaptively adjusts filter parameters based on local flow conditions.

   • Mechanism: The algorithm segments tracks into sub-tracks (up to 50 time steps) and computes the mean absolute velocity and acceleration for each. An Amplitude Spectral Density (ASD) analysis is performed on these bins to determine a local cutoff frequency ($f_{cutoff}$) where the signal-to-noise ratio (SNR) equals unity.
   • Implementation: A continuous $f_{cutoff}$ field is generated via bilinear interpolation of the velocity-acceleration bins. During filtering, the regularization parameter $w$ is updated for each time step based on the local estimated velocity and acceleration, allowing the filter to respond to local inhomogeneities.
   • Prediction: This adaptive framework is also applied to the Wiener predictor used in Shake-The-Box (STB) algorithms to improve the prediction of particle positions in subsequent time steps.

2. Track Merging Strategies:
   To address noise reduction in variable-timestep LPT (e.g., VT-STB), the study assesses methods for combining statistically independent track segments of the same particle obtained from temporally shifted tracking passes.

   • Approaches: Two methods are evaluated: Track Interleaving (combining noisy position signals to recover original temporal resolution) and Track Averaging (interpolating tracks to a common timestep and averaging).
   • Principle: Averaging $N$ independent measurements reduces uncorrelated noise standard deviation by a factor of $\sqrt{N}$.

3. Statistical Noise Correction:
   A framework is introduced to separate measurement noise from physical statistics without relying on aggressive temporal filtering, which can underestimate second-order moments.

   • Mechanism: Using Gaussian error propagation, the study quantifies the noise-induced variance contribution to velocity and acceleration estimates based on the known standard deviation of position noise ($\sigma_p$) and the specific weighting coefficients of the reconstruction filter.
   • Correction: The noise-induced variance is explicitly subtracted from the measured second-order moments ($\sigma^2_{measured}$) to recover the physical variance ($\sigma^2$). This allows for the use of minimal filtering (or none) to preserve temporal resolution while correcting statistical biases.

Key Results

• Instantaneous Accuracy: The adaptive local TrackFit approach significantly reduces the Mean Absolute Error (MAE) for instantaneous velocity and acceleration, particularly in the near-wall region ($y^+ < 10$) where flow gradients are steep. In the viscous sublayer, local TrackFit reduced acceleration relative errors by up to 80% compared to global filtering. While global filtering with a lower cutoff frequency ($SNR \approx 0.1$) performed well in homogeneous regions, local adaptation provided superior performance in inhomogeneous flows.
• Position Prediction: In STB applications, the local Wiener predictor reduced the misprediction ratio (deviations > 2 pixels) by up to 55% in cylinder wake flows and 38% in channel flows compared to global calibration. This improvement is critical for preventing premature track termination in high-acceleration regions.
• Statistical Moments: The study demonstrates that aggressive filtering leads to significant underestimation of acceleration variance due to the truncation of intermittent, high-amplitude events. The proposed noise subtraction method, when applied to central difference schemes or TrackFit with high cutoff frequencies, yields accurate second-order moment estimates. For acceleration, the noise contribution was found to be substantial, and its removal was essential for reliable statistics.
• Track Merging: Track interleaving combined with local TrackFit filtering proved superior to simple averaging for high-fidelity reconstructions in the channel flow test case, as averaging followed by interpolation can distort the ASD and invalidate filter assumptions.

Significance and Claims
The paper claims that accurate estimation of instantaneous Lagrangian velocities and accelerations requires adaptive track filtering that responds to local flow conditions, whereas reliable statistical moments can be obtained by explicitly correcting for measurement noise rather than relying on aggressive smoothing.

• For Instantaneous Data: The adaptive local TrackFit enables accurate estimation of instantaneous quantities in strongly inhomogeneous flows (e.g., viscous sublayers) where global filtering introduces significant bias. This is particularly beneficial for advanced analyses requiring jerk (third derivative) and for improving the prediction steps in STB algorithms, thereby reducing track termination rates in high-acceleration regions.
• For Statistical Data: The statistical noise correction framework allows for the robust estimation of velocity and acceleration statistics without the need for strong temporal filtering. This preserves the temporal resolution necessary to resolve small-scale dynamics and avoids the underestimation of variance caused by filtering out intermittent events.
• Practical Impact: These methods facilitate more accurate flow reconstruction via data assimilation (e.g., FlowFit, PINNs) by providing better-weighted input data. They also offer a pathway to investigate discrepancies between theoretical turbulence predictions and experimental data regarding acceleration intermittency, using comparatively simple experimental setups.

The authors conclude that while the presented methods show significant promise, further systematic investigation into the interplay between predictor parametrization and specific flow characteristics is required to fully realize their potential in complex experimental scenarios.