## 3 Statistical Properties of Inter-Particle Collisions
### Analysis of Time Scales Related with Collisions
This fact was demonstrated in Reference [39], where the cases with and without inter-particle collisions were compared: the differences between both cases in the considered clustering measures were marginal for \(\alpha_{p}<0.1\%\). As a consequence, the increase in collision frequency with \(\alpha_{p}\) is expected to have only a moderate effect on the statistical quantities computed for colliding particle events, such as relative velocity, two-particle velocity correlation function or strain rate at collision locations, among others. The importance of collisional effects on the turbulence-induced particle transport can be illustrated by plotting the behaviour of the particle Lagrangian integral time scale, \(\tau_{L}^{p}\), versus particle inertia. \(\tau_{L}^{p}\) is computed as the integral of the particle Lagrangian autocorrelation function and it measures the time scale in which particle velocities are correlated in time along their trajectories [38]. Figure 6a shows the curves of the quotient \(\tau_{K}/\tau_{L}^{p}\) depending on particle Stokes number, \(St_{K}\), for the three considered values of \(\alpha_{p}\). A decreasing trend with increasing \(St_{K}\) is observed, which is due to the growing of the time scale \(\tau_{L}^{p}\) with particle inertia, reflecting the fact that more inertial particles are less affected by turbulent fluctuations and that the particle motion becomes more correlated. As a consequence, large inertia particles tend to experience a ballistic motion between consecutive collisions. Regarding the effect of particle volume fraction on \(\tau_{L}^{p}\), it is found that the increase in the number of collisions has only a marginal effect on the particle Lagrangian autocorrelation timescale, where, nevertheless, a small decrease of this quantity can be observed with increasing \(\alpha_{p}\) (i.e., increase in \(\tau_{K}/\tau_{L}^{p}\)), as expected. This means that the particle motion becomes slightly more decorrelated as a consequence of the increasing collision rate and the resulting faster change of particle velocity. Figure 5: Collision Stokes number, \(St_{col}=\tau_{c}/\left<\tau_{p}\right>\), as a function of particle Stokes number, \(St_{K}\), for the three evaluated average volume fractions. Figure 6b presents the behaviour of \(\tau_{c}/\tau_{L}^{p}\) versus particle inertia. The same as in Figure 6a, a decreasing tendency is found for this variable, which is related to the fact that \(\tau_{L}^{p}\) increases and \(\tau_{c}\) decreases with growing \(St_{K}\). For all the computed cases, the quotient \(\tau_{c}/\tau_{L}^{p}\) is above unity, meaning that inter-particle collision time is larger than the particle Lagrangian time scale. Similar to what happened with \(St_{cal}\), the curves for the different particle volume fractions decrease pretty fast with increasing inertia and run quasi parallel. Moreover, the values of \(\tau_{c}/\tau_{L}^{p}\) are always much larger than those of \(\tau_{K}/\tau_{L}^{p}\), which means that the main mechanism of particle velocity change is the interaction between particle inertia and underlying turbulence. Inter-particle collisions have a marginal effect modulating such primary mechanism.