## 3 Statistical Properties of Inter-Particle Collisions
### Analysis of Kinematic Properties of Colliding Particles
As it was said previously, particles \(St_{K}\sim 1\) tend to sample fluid locations with lower than mean vorticity. In our simulations, this behaviour also happens for the smallest inertia particles, \(St_{K}=0.1\), which otherwise approach the tracers' behaviour. They are located in areas with lower than mean fluid vorticity, as already noted in Reference [22]. Looking at the locations where inter-particle  
Figure 9: **(a)** Average of sampled fluid strain rate by colliding pairs over mean sampled fluid strain rate by all the particles. **(b)** Average of sampled fluid vorticity of colliding particles over mean sampled fluid vorticity. Both variables are presented as a function of particle Stokes number for the three evaluated average volume fractions. collisions happen, \(\left<\omega^{f@p}\right>_{p,coll}\), it is found that such locations are characterised by smaller values than \(\left<\omega^{f@p}\right>_{p}\) (see Figure 9b). This observation is valid for all the Stokes numbers, although for particles with \(St_{K}\approx 10\), the two values are very close. This fact is parallel to the tendency of particle collisions to occur in areas with fluid strain rate higher than the mean value computed at particle positions, \(\left<S^{f@p}\right>_{p}\), and, therefore, could be somehow expected. However, this last observation is contrary to that presented in Reference [22], where the fluid enstrophy sampled at collision locations was higher than the average value sampled at particle positions. At the moment, the reason for such discrepancy is not clear. Finally, Figure 10 shows the ratio of the average particle turbulent energy of colliding particles, \(k_{p,coll}\), and the average particle turbulent energy of all the tracked particles, \(k_{p}\), \(k_{p,coll}/k_{p}\). In this case, colliding particles with \(St_{K}\sim 1\) have smaller kinetic energy than the average, attaining a minimum of around \(0.9\) for \(St_{K}=1\) and for all considered particle volume fractions. For particles with \(St_{K}=0.1\), the ratio is closer to \(1\), although the value for \(\alpha_{p}=10^{-4}\) is higher than for the other two volume fractions. The fact that colliding particles with intermediate inertia, \(St_{K}\sim 1\), possess on average less fluctuating energy than the mean value computed over all tracked particles could be understood as follows: as it has been shown before, inter-particle collisions tend to happen in locations with high values of fluid strain rate and, equivalently, high dissipation rate; in isotropic homogeneous turbulence, the decay of turbulent kinetic energy is larger at locations of higher dissipation, therefore, such regions possess, on average, lower values of turbulent energy. Consequently, it is expected that intermediate inertia particle collisions occur at fluid locations with lower than mean values of fluid kinetic energy, a fact that is found in the present simulations. Moreover, as it has been seen for particles of low and intermediate inertia, that \(St_{col}\) values are much larger than \(1\), so those particles have time to accommodate to the local fluid conditions between collisions, meaning that their kinetic energy will also be lower than the mean kinetic energy computed for all tracked particles. In the case of large inertia particles, the situation is different as they are not preferentially sampling regions of high dissipation, and also, the collision Stokes number is only somewhat higher than \(1\). As a result, in the case of \(St_{K}=10\), the \(k_{p,coll}/k_{p}\) ratio is clearly above \(1\) for all volume fractions, indicating in this case that colliding particle pairs possess higher fluctuating velocity than the mean. According to the random uncorrelated motion interpretation, such result is expected, as inter-particle collisions are favoured by a higher particle fluctuating velocity.