On Heavy Particle Dispersion in Turbulent Shear Flows: 3-D Analysis of the Effects of Crossing Trajectories

In the approaches used to predict the dispersion of discrete particles moving in a turbulent flow, the effects of crossing trajectories due to gravity (or any other external force field) are generally accounted for by modifying the integral time scales according to the well-known analysis of Csanady (J Atmos Sci 20:201–208, 1963). Here, an alternative theoretical analysis of the time correlation of the fluid velocity fluctuations along a particle trajectory is presented and applied in a turbulent shear flow. The study is carried out in the frame of three-dimensional Langevin-type stochastic models, where the main unknowns are the drift tensor components rather than the conventional integral time scales of the fluid seen by the particles. Starting from a model for the space-time velocity covariance tensor of the turbulence under the assumption of homogeneous shear flow, the various components of the time correlation tensor of the fluid seen are expressed in the asymptotic case of large mean relative velocity (between the particles and the flow) compared to the particle velocity fluctuations. In order to provide comparison with the generally used expressions arising from isotropic turbulence assumption, we examine also the conventional integral time scales of the fluid seen in the directions parallel and perpendicular to the mean relative velocity. The most prominent deviations from isotropic turbulence are observed when the external force field is in the direction of the mean velocity gradient: in this case the loss of correlation in the mean flow direction is significantly lower than expected in a uniform flow, an observation that is in qualitative agreement with the few available data.