In the present study, we investigate the computation of the Lagrangian second-order structure-function to characterise the multiscale dynamics of turbulence from measured particle trajectories. We performed time-resolved three-dimensional particle tracking velocimetry (4D-PTV) to study the anisotropic and inhomogeneous flow field of the wake behind a cylinder at a Reynolds number equal to 3900. We performed Lagrangian statistical analysis on nearly 12000 trajectories for 4000 time steps.

The dataset contains Eulerian velocity and pressure fields, and Lagrangian particle trajectories of the wake flow downstream of a smooth cylinder at a Reynolds number equal to 3900. An open source Direct Numerical Simulation (DNS) flow solver named Incompact3d was used to calculate the Eulerian field around the cylinder. The synthetic Lagrangian tracer particles were transported using a fourth-order Runge-Kutta scheme in time and trilinear interpolations in space. Trajectories of roughly 200,000 particles for two 3D sub-domains are available to the public. This dataset can be used as a test case for tracking algorithm assessment, exploring the Lagrangian physics, statistic analyses, machine learning, and data assimilation interests.

Advances in time-resolved three-dimensional Particle Tracking Velocimetry (4D-PTV) techniques have consistently revealed more accurate Lagrangian particle motions. A novel track initialization technique as a complementary part of 4D-PTV, based on local temporal and spatial coherency of neighbor trajectories, is proposed. The proposed Lagrangian Coherent Track Initialization (LCTI) applies physics-based Finite Time Lyapunov Exponent (FTLE) to build four frame coherent tracks. We locally determine Lagrangian coherent structures among neighbor trajectories by using the FTLE boundaries (i.e., ridges) to distinguish the clusters of coherent motions. To evaluate the proposed technique, we created an open-access synthetic Lagrangian and Eulerian dataset of the wake downstream of a smooth cylinder at a Reynolds number equal to 3900 obtained from three-dimensional direct numerical simulation. Performance of the proposed method based on three characteristic parameters, temporal scale, particle concentration (i.e., density), and noise ratio, showed robust behavior in finding true tracks compared to the recent initialization algorithms. Sensitivity of LCTI to the number of untracked and wrong tracks is also discussed. We address the capability of using the proposed method as a function of a 4D-PTV scheme in the Lagrangian particle tracking challenge. We showed that LCTI prevents 4D-PTV divergence in flows with high particle concentrations. Finally, the LCTI behavior was demonstrated in a jet impingement experiment. LCTI was found to be a reliable tracking tool in complex flow motions, with a strength revealed for flows with high velocity and acceleration gradients.

We present a variational assimilation technique (4D-Var) to reconstruct time resolved incompressible turbulent flows from measurements on two orthogonal 2D planes. The proposed technique incorporates an error term associated to the flow dynamics. It is therefore a compromise between a strong constraint assimilation procedure (for which the dynamical model is assumed to be perfectly known) and a weak constraint variational assimilation which considers a model enriched by an additive Gaussian forcing. The first solution would require either an unaffordable direct numerical simulation (DNS) of the model at the finest scale or an inaccurate and numerically unstable large scale simulation without parametrisation of the unresolved scales. The second option, the weakly constrained assimilation, relies on a blind error model that needs to be estimated from the data. This latter option is also computationally impractical for turbulent flow models as it requires to augment the state variable by an error variable of the same dimension. The proposed 4D-Var algorithm is successfully applied on a 3D turbulent wake flow in the transitional regime without specifying the obstacle geometry. The algorithm is validated on a synthetic 3D data-set with full-scale information. The performance of the algorithm is further analysed on data emulating large-scale experimental PIV observations.