Developments and difficulties in predicting the relative velocities of inertial particles at the small-scales of turbulence

TL;DR

This paper improves theoretical predictions of inertial particle relative velocities in turbulence's dissipation range using a modified model based on backward-in-time dispersion theory, but highlights limitations at sub-Kolmogorov scales.

Contribution

The paper develops a modified Pan & Padoan model incorporating backward-in-time dispersion theory to better predict inertial particle velocities in turbulence.

Findings

Modified model improves velocity predictions in the dissipation range.

Model overpredicts velocities for particle separations less than the Kolmogorov length.

Failure to predict correct scale-invariant forms of velocity structure functions.

Abstract

In this paper, we use our recently developed theory for the backward-in-time (BIT) relative dispersion of inertial particles in turbulence (Bragg \emph{et al.}, Phys. Fluids 28, 013305, 2016) to develop the theoretical model by Pan \& Padoan (J. Fluid Mech. 661 73, 2010) for inertial particle relative velocities in isotropic turbulence. We focus on the most difficult regime to model, the dissipation range, and find that the modified Pan \& Padoan model (that uses the BIT dispersion theory) can lead to significantly improved predictions for the relative velocities, when compared with Direct Numerical Simulation (DNS) data. However, when the particle separation distance, $r$, is less than the Kolmogorov length scale, $\eta$, the modified model overpredicts the DNS data. We explain how these overpredictions arise from two assumptions in the BIT dispersion theory, that are in general not satisfied when the final separation of the BIT dispersing particles is $<\eta$. We then demonstrate the failure of both the original and modified versions of the Pan \& Padoan model to predict the correct scale-invariant forms for the inertial particle relative velocity structure functions in the dissipation regime. It is shown how this failure, which is also present in other models, is associated with our present inability to correctly predict not only the quantitative, but also the qualitative behavior of the Radial Distribution Function in the dissipation range when $St=\mathcal{O}(1)$