## 3 Statistical Properties of Inter-Particle Collisions
### Analysis of Kinematic Properties of Colliding Particles
In this regime, the effect of increasing the particle volume fraction is hardly seen as the inter-particle collision time is still much higher than the particle Lagrangian time scale (see Figure 6b). Interestingly, for \(St_{K}=1\), the mean velocity collision angle obtained in the present work, around 0.3 radians or \(17^{\circ}\), is very close to that obtained in [PERSON] et al. [22]. For larger inertia, the relative velocity of colliding pairs overcomes the Kolmogorov velocity and reaches a maximum in the present computations for \(St_{K}=10\), the same as the velocity collision angle (around 1.17 radians or \(67^{\circ}\)),  
Figure 8: Average velocity collision angles (**a**) and relative velocities (**b**) for colliding particles as a function of particle Stokes number for the three evaluated average volume fractions. Figure 7: Mean free path over Kolmogorov length scale as a function of particle Stokes number for the three evaluated average volume fractions. which again is very similar to that reported in Reference [22]. When inertia is further increased, the relative velocity of colliding pairs decreases because particles become less responsive to fluid fluctuations [37] and the velocity collision angle tends asymptotically to \(90^{\circ}\). For these large inertias, a marginal effect of \(\alpha_{p}\) starts to be appreciated as both relative velocity and collision angle are slightly reduced for increasing collision frequency. This fact was already noticed in Reference [38] and its explanation was ascribed to collective effects in clusters. Actually, the argument is that in situations where clustering is notable, particles spend most of the time in areas of high strain ratio and low vorticity so, besides some collisional effects, they have time to adapt its motion to that of the fluid, resulting finally in a reduced average relative particle velocity and collision angle. However, such explanation would not be appropriate for large inertia particles because in those conditions, particles are distributed fairly uniformly without appreciable clustering. This finding should be further investigated by using configurations with larger volume fractions and turbulent Reynolds numbers. An interesting issue to be discussed is the distribution of the location of inter-particle collisions. As already mentioned, and illustrated in Figure 2a, particles with Stokes number around 1, experiencing appreciable clustering, tend to sample fluid locations with higher strain rate than particles with small and large Stokes numbers, where clustering is not observed. Figure 9a presents the ratio between the fluid strain rate at particle collision locations, averaged for all collisional events, \(\left\langle S^{\text{(\#p)}}\right\rangle_{p,coll}\), and the fluid strain rate at particle positions, averaged for all particles, \(\left\langle S^{\text{(\#p)}}\right\rangle_{p}\). Interestingly, inter-particle collisions at \(St_{K}\sim 1\) clearly happen in fluid regions with noticeably higher strain rates than those averaged through all the particles. The maximum occurs for \(St_{K}=1\), whereas for \(St_{K}=0.1\), \(10\), the ratios are very close to 1, as in such cases no significant preferential concentration is observed. At this point, it is necessary to mention that [PERSON] et al. [22] found that collision events in the case of particles with \(St_{K}\sim 1\) occurred in locations where the fluid dissipation was higher than the corresponding mean value experienced by all the tracked particles. Both observations are related, as in the configuration of homogeneous isotropic turbulence, both quantities, fluid strain rate and dissipation, are strongly correlated (correlation coefficient of 0.97 in present simulations). Figure 9b presents the ratio between the fluid vorticity at particle collision locations, averaged for all collisional events, \(\left\langle\omega^{\text{(\#p)}}\right\rangle_{p,coll}\), and the fluid vorticity at particle positions, averaged for all particles, \(\left\langle\omega^{\text{(\#p)}}\right\rangle_{p}\).