## 3 Statistical Properties of Inter-Particle Collisions  
As an illustration, Figure 2 shows typical snapshots of a two-dimensional slice of particle distribution in the turbulent flow field depending on their inertia. Following Reference [39], particles tend to form distinct patterns in the flow, which depend on its inertia. For instance, for the case of \(\alpha_{p}=0.1\%\), Figure 2a presents the case of \(St_{K}=1\), where well-differentiated narrow particle ropes can \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(St_{K}\) (-) & **0.1** & **0.5** & **1.0** & **1.25** & **1.5** & **2.0** & **10.0** \\ \hline \(\rho_{p}\big{(}\text{kg}/\text{m}^{3}\big{)}\) & 55.32 & 276.58 & 553.15 & 691.44 & 829.73 & 1106.30 & 5531.52 \\ \(\tau_{pS}\) (s) & 0.00715 & 0.0357 & 0.0714 & 0.0893 & 0.107 & 0.142 & 0.686 \\ \(R\epsilon_{p}\) (\(-\)) & 0.0256 & 0.109 & 0.177 & 0.203 & 0.227 & 0.266 & 0.466 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Properties of the particles considered in this study. be observed in the background coloured by fluid strain rate (average strain rate in this configuration is 61 s\({}^{-1}\)). As it is clearly noticed, particles tend to preferentially sample zones with a higher than mean fluid strain rate (green to red colours), and also, although not shown, lower than mean fluid vorticity--a view that agrees with the previous literature. Figure 2b shows, on the other hand, the distribution of particles with \(St_{K}=10\); here, no distinct clusters can be appreciated, and particles tend to be spread much more evenly than in Figure 2a, although some voids are noticed. As commented before, the reason for such distribution patterns lies in the particle-turbulence interaction: for intermediate particle inertia, \(St_{K}\sim 1\), the centrifuging effect of particles towards the rim of vortical structures as well as the sling effect impulse them to show preferential concentration; on the other side, large inertia particles are only marginally affected by its interaction with turbulent vortices and tend to describe straight trajectories (the so-called inertia filtering effect), which are only modified by collisions, resulting in a much more uniform distribution along the fluid field. Figure 3 illustrates the clustering behaviour of inertial particles as a function of their Stokes number by means of the so-called particle segregation parameter \(\Sigma_{p}\)[39]. It compares the standard deviation of the actual particle number density distribution with that of a purely random distribution (which is described by a Poisson process). The value \(\Sigma_{p}=0\) corresponds to a random particle distribution, \(\Sigma_{p}<0\) corresponds to a uniform particle distribution, while \(\Sigma_{p}>0\) indicates particle clustering. Figure 3: Particle segregation parameter, \(\Sigma_{p}\), variation versus particle Stokes number. \(\Sigma_{p}\) presents a peak for the maximum attained clustering, reached for \(St_{K}\sim 1\). Data are for volume fraction \(\alpha_{p}=0.1\%\). Figure 2: Snapshot of a two-dimensional (2D) slice of the fluid turbulent field laden with particles, where background is coloured by fluid strain rate. Particle segregation illustration for particles with intermediate (\(St_{K}=1\)) and large (\(St_{K}=10\)) inertia, (**a**) and (**b**), respectively. Particle volume fraction \(\alpha_{p}\) = 0.1%. Edge lengths \(l\) and thickness \(s\) of the image sections: \(l/\eta_{K}=128\) and \(s/\eta_{K}=2\), respectively. The diameter of the spheres is enlarged by a factor or 3 for the sake of better visibility. As it is clearly seen from Figure 3, \(\Sigma_{p}\) is positive for all Stokes numbers, showing a peak for \(St_{K}\) around one and values closer to zero for Stokes numbers of 0.1 and 10, where particles tend to be distributed randomly along the fluid domain.