Published In

Physical Review Fluids

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Publication Date



Fluids -- Velocity and Vorticity, Incompressible Stokes equations · Mixed finite elements, Pressure-robustness


When a fluid in turbulent motion is tagged by a nonuniform concentration of ideal tracers, the mean velocity of the tracers may not match with the mean velocity of the fluid flow. This implies that conventional particle tracking velocimetry will not produce the mean flow of a turbulent flow unless the particle seeding is homogeneous. In this work, we consider the problem of mean flow estimation from a set of particle tracks obtained in a situation of nonhomogeneous seeding. To compensate the bias caused by the nonhomogeneous particle seeding, we propose a modified particle tracking velocimetry method. This method is called a time-delayed velocity and considers the velocity trajectory of a given particle shifted in time with respect to its position. We first introduce our method for an ideal advection–diffusion model and then we implement it for a turbulent channel and a turbulent jet. For both situations, we find that the velocity bias caused by the nonhomogeneous tracer concentration is reduced with a time delay introduced between position and velocity of the tracer trajectories. For the turbulent channel, the error on the mean flow estimation monotonically decreases for increasing time delays. For the turbulent jet, the error on the mean flow estimation also reduces with positive time delays but the time delay should not be too large. We interpret this limitation as a consequence of the spatial dependence of the mean flow. For the turbulent channel, this limitation does not appear because the velocity for the mean flow streamlines is constant. For both flows, the optimal time delay for the velocity bias compensation is consistent with the Lagrangian timescales of the flow. This method gives promising elements to take into account inhomogeneous seedings in velocity fields measurements for all kinds of turbulent flows and interesting perspectives to understand how Lagrangian trajectories from various sources build an Eulerian mean field.


This work was authored as part of the Contributor's official duties as an Employee of the United States Government and is therefore a work of the United States Government. In accordance with 17 U.S.C. 105, no copyright protection is available for such works under U.S. Law.



Persistent Identifier


American Physical Society