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

In this paper, we consider the development of theoretical models to predict the relative velocities of inertial particles in isotropic turbulence. In particular, we use our recently developed theory for the backward-in-time (BIT) relative dispersion of inertial particles in turbulence [Bragg et al., "Forward and backward in time dispersion of fluid and inertial particles in isotropic turbulence," Phys. Fluids 28, 013305 (2016)] to develop the theoretical model by Pan and Padoan ["Relative velocity of inertial particles in turbulent flows," J. Fluid Mech. 661, 73 (2010)]. We focus on the most difficult regime to model, the dissipation range, and find that the modified Pan and Padoan model (that uses the BIT dispersion theory) can lead to significantly improved predictions for the relative velocities, when compared with the Direct Numerical Simulation (DNS) data. However, when the particle separation distance, r, is less than the Kolmogorov lengthscale, η, 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 <η. We then demonstrate the failure of both the original and modified versions of the Pan and 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 current inability to correctly predict not only the quantitative but also the qualitative behavior of the radial distribution function in the dissipation range when St=O(1).