## 3 Statistical Properties of Inter-Particle Collisions
### Analysis of Time Scales Related with Collisions  
The inter-particle collision time, \(\tau_{c}\), is defined as the average of elapsed time between two consecutive collisions and it is computed as the inverse of inter-particle collision frequency (which indicates the number of collisions per unit time). Figure 4 presents such quantity for the three considered volume fractions, \(\alpha_{p}\), and it has been made non-dimensional with the Kolmogorov time scale, \(\tau_{K}\). As shown in Figure 4, inter-particle collision time is a decreasing function of the Stokes number, although for the largest Stokes numbers considered, it seems to reach an asymptotic value. This trend is similar to those presented in References [22] and [38]. Moreover, for a fixed value of \(St_{K}\), \(\tau_{c}\) decreases almost linearly with increasing volume fraction, which is verified for all the considered Stokes number range. In the case of the smallest \(St_{K}\) = 0.1, there are many repeated collisions among the same particle pair with very small radial relative velocity and nearly parallel velocities. In order to exclude possible false collision detections due to numerical errors in such small \(St_{K}\), only collision intervals larger than the Kolmogorov time scale are counted in the statistics [22]. Another point to be mentioned is that for the smallest values of \(St_{K}=0.1\) and 0.5, not all the particles are involved in collisions. In fact, for the Stokes number of 0.1, at \(\alpha_{p}=10^{-3}\), only 17% of all tracked particles are experiencing at least one collision; in this case, particles tend to be nearly homogeneously distributed along the flow field and inter-particle collisions happen between particle pairs that have shared the same flow structures during long periods. As a result, the velocities of both particles are correlated, and the collision mechanism is that of fluid velocity shear. For \(St_{K}=0.5\), where particle accumulation is already noticeable, the number of particles involved in collisions increases up to 65%, whereas for the rest of the higher \(St_{K}\), such value is over 94%. Therefore, essentially all particles are participating in collisions for the cases with intermediate and large inertia. In the next figure, the collision Stokes number, \(St_{col}\), is plotted as a function of the particle Stokes number for the three volume fractions (Figure 5). Figure 4: Non-dimensional inter-particle collision time, \(\tau_{c}/\tau_{K}\), as function of particle Stokes number, \(St_{K}\), for the three evaluated average volume fractions, \(\alpha_{p}\). It can be seen that, for all particle volume fractions and for all inertias, the flow can be considered dilute, as the collision Stokes number is always above unity. A pronounced decreasing of \(St_{col}\) is observed for the three considered volume fractions and the curves run pretty much parallel, suggesting a linear dependence of \(St_{col}\) with \(\alpha_{p}\) for a fixed particle Stokes number. The collision Stokes number decreases continuously with \(St_{K}\), following a nearly straight line in the double logarithmic diagram for the three values of particle volume fraction considered. Only for the most inertial particles did the values of \(St_{col}\) become near the dilute-dense flow conditions boundary; however, in such case, the increment of the number of collisions is mainly ascribed to the increase of the caustics contribution (or random uncorrelated motion) to inter-particle collision rate [28]. According to Figure 5, all the considered cases are in the dilute regime, where \(\left<\tau_{p}\right><\tau_{c}\); therefore, the main mechanism affecting the particles' motion is the aerodynamic transport, and particles have time to adapt to its local fluid environment between inter-particle collisions. Increasing \(\alpha_{p}\) has the effect of reducing inter-particle distance, so collisions are more frequent. However, such collisions are not dominant, they must be viewed instead as a modulating mechanism.