Uncertainty propagation and truncation errors in LPT kinematics

Lagrangian Particle Tracking (LPT) has become a near-standard approach for performing accurate 3D flow measurements, thanks notably to the technical breakthroughs brought by the Iterative Particle Reconstruction (IPR: Wieneke, 2013) and Shake-the-Box (STB: Schanz et.al, 2016) procedures. These decisive progresses have triggered a number of studies relative to the eduction of flow kinematics and dynamics based on particle trajectory analyses. Novara & Scarano (2013), and others, focused on polynomial approximations of the trajectories, which analytically provide the material derivatives used to estimate pressure gradients. In particular, approximations based on second order polynomials fits of a small number of particle positions are used in commercially available softwares and among research teams as a straightforward solution to obtain the first and second order derivatives with a limited effect of the measurement noise. Additionally the analyses conducted during the 2020 LPT challenge (Leclaire, 2020 ; Sciacchitano, 2020) have addressed the performance of methodologies used by different groups with respect to second order trajectory fits for both multi-pulse and four-pulse (Novara et. al, 2016) LPT cases. On more advanced theoretical grounds, Geseman et. al (2016) have proposed the trackfit approach using penalized B-splines with considerations on the time-varying acceleration rate (i.e. jolt or jerk) and spectral content of noisy particle tracks