Shake‑The‑Box: Lagrangian particle tracking at high particle image densities

Abstract

A Lagrangian tracking method is introduced, which uses a prediction of the particle distribution for the subsequent time-step as a mean to seize the temporal domain. The scheme was termed ‘Shake-The-Box’ and previously characterized as ‘4D-PTV’ due to the strong interaction with the temporal dimension. Trajectories of tracer particles, representing fluid elements, are identified at high spatial accuracy due to a nearly complete suppression of ghost particles; a temporal filtering scheme further improves on accuracy and allows for the extraction of local velocity and acceleration as derivatives of a continuous function. Exploiting the temporal information enables the processing of densely seeded flows (beyond 0.1 particles per pixel), which were previously reserved for tomographic PIV evaluations. While TOMO-PIV uses statistical means to evaluate the flow (building an ‘anonymous’ voxel-space with subsequent spatial averaging using correlation), the Shake-The-Box approach is able to individually identify and accurately describe particles in tracks at numbers of tens- or even hundreds of thousands per time-step. The locally highly accurate flow information can be used to e.g. generate high-resolution statistics of mean velocities and Reynolds stresses via ensemble averaging. The discrete particle distribution can be interpolated onto an Eulerian reference frame, using a three-dimensional grid of B-splines (‘FlowFit’). Opposed to correlation-based methods, the process evades spatial smoothing and allows for the introduction of regularizations, e.g. the penalization of divergence in the resulting vector volume. Using this scheme, derived properties – like the velocity gradient tensor or the acceleration field – can be extracted with high precision, allowing for advanced data evaluation, e.g. the extraction of pressure fields. The joint knowledge of Lagrangian and Eulerian data allows extracting a maximum of information content from the tracer distribution describing the flow. The Shake-The-Box method is outlined in detail, followed by an application to synthetic and experimental data. The former demonstrates the ability of the method to attaining high accuracy by inherently evading the formation of ghost particles and by capturing virtually the entirety of true particles, even for high seeding densities of here up to 0.125 particles per pixel. Processing an experimental dataset on a transitional jet in water demonstrates the benefits of advanced Lagrangian evaluation in describing flow details – both on small scales (by the individual tracks), as well as on larger structures (using FlowFit). Comparisons to TOMO-PIV processing for synthetic and experimental evaluations show distinct benefits in local accuracy, completeness of the solution, ghost particle occurrence, spatial resolution, temporal coherence and computational effort.