## 3 Statistical Properties of Inter-Particle Collisions
### Analysis of Kinematic Properties of Colliding Particles  
Figure 7 illustrates the variation of the mean free path, \(\lambda_{p}\), with particle inertia and with particle volume fraction. The free path is computed as the distance travelled by a particle among two consecutive collisions. Particles with Stokes numbers well below one mainly interact with their closest neighbours, with whom they have shared the same fluid environment during long periods. This fact is reflected in a low relative velocity and collision angles between them (see Figure 8 below). Besides, as such particles do not experience remarkable clustering: interactions between the same pair of particles are common and only a small percentage of them undergo collisions (for \(St_{K}=0.1\) and \(\alpha_{p}=10^{-3}\), only 17% of particles are involved in collisions with a partner and the percentage is even lower for smaller values of \(\alpha_{p}\), 9% for \(5\times 10^{-4}\) and 2% for \(10^{-4}\)). This means that the majority of particles present a free path equal to the length of the trajectory described during the total measured time, which in this case was 25 eddy turnover times. As a result, small particles present high values of mean free paths. As \(St_{K}\) increases, the percentage of particles involved in collisions augments (65% for \(St_{K}=0.5\) and \(\alpha_{p}=10^{-3}\)) at the same time that the number of repeated collisions among the same pair of particles is reduced; as a consequence, the mean free path reduces. For \(St_{K}\) around unity and larger, nearly all the particles are involved in collisions, a fact favoured by clustering processes and the formation of caustics [22]. Both phenomena increase collision frequency and also reduce the value of the mean free path. Finally, for large enough \(St_{K}\), particles become nearly irresponsive to fluid fluctuations and do not undergo appreciable clustering, so the main mechanism for inter-particle collisions is the so-called Random Uncorrelated Motion [19]. In this case, the number of collisions increases slightly, and the mean free path seems to reach an asymptotic value. Obviously, the effect of decreasing \(\alpha_{p}\) is reflected in a reduction of the inter-particle collision frequency, hence an increase of the mean free path, as shown in Figure 7.  
Figure 6: Kolmogorov time scale over particle Lagrangian time scale (**a**) and inter-particle collision time over particle Lagrangian time scale (**b**) as a function of particle Stokes number for the three evaluated average volume fractions. Figure 8a shows that average velocity collision angles, defined as the angle that particle velocities form at collision, increase continuously with \(St_{K}\). The same behaviour is seen for the relative velocity at collision (Figure 8b), which is a fact related with the increased uncorrelated motion of colliding particles with growing inertia. Therefore, particles can collide coming from rather different directions, a phenomenon that is enhanced by trajectory deflection after previous collisions. As mentioned before, particles with small inertia respond very fast to the fluid velocity fluctuations and colliding pairs are usually moving with pretty similar velocities in nearly parallel trajectories [22] because they are transported by the same eddy during long time periods, as a result, their velocities are very correlated, which is reflected in both small values of relative velocity and nearly zero-velocity angle of collision. In this context, there is no preferential concentration and the increased collision frequency, i.e., by augmenting \(\alpha_{p}\), has no effect in the particle transport and collision mechanisms which are driven by the turbulent velocity fluctuations. With increasing particle inertia, preferential concentration and the sling effects come into play, with the result of increasing the collision frequency, relative velocity at contact as well as velocity angle of collision.