## 3 Statistical Properties of Inter-Particle Collisions
### Collision Frequency  
A way of characterising inter-particle collision probability is the collision kernel, which is equal to two times the collision frequency over the particle number density (particles per unit volume) [43]. In the literature, there are two limiting cases for the behaviour of such collision kernel with particle Stokes number.  
[PERSON] and [PERSON] [44] provided an approximation for the collision kernel of drops in atmospheric turbulence. In their analysis, the particles are small compared to the smallest length scales of the fluid flow. Hence, particles follow the turbulent structures completely. The resulting collision kernel for mono-disperse particles is given by:  
\[\Gamma_{ST}=d_{p}^{\beta}\sqrt{\frac{8\pi\epsilon}{15\
u}}. \tag{4}\]  
Another limiting case was studied by [PERSON] [45], in which the motion of heavy particles suspended in high-intensity turbulence neglecting external forces is analysed. According to the kinetic theory, the particle motion is completely uncorrelated concerning the underlying fluid motion. Under the assumption that all components of the particle fluctuating velocity are isotropic, the resulting collision kernel reads:  
\[\Gamma_{A}=4d_{p}^{2}\sqrt{\frac{2\pi}{3}k_{p}}, \tag{5}\]  
where \(k_{p}\) represents the average particle turbulent energy of all tracked particles.  
In practice, particles have finite inertia and, therefore, their collision kernel is expected to lie between both limiting values. Figure 11 presents the non-dimensional collision kernel obtained in this work as a function of the particle Stokes number for the three considered particle volume fractions. In addition, the collision kernels of Saffman and Turner, \(\Gamma_{ST}\), as well as [PERSON], \(\Gamma_{A}\), are also shown in Figure 11 for comparison.  
Let us observe that the non-dimensional Saffman and Turner kernel is actually constant but that of [PERSON] depends on particle fluctuating velocity, which varies with particle volume fraction. However, for the simulated cases, \(k_{p}\) is similar for the three values of \(\alpha_{p}\).  
As it can be readily seen in Figure 11, the computed collision kernel is between the Saffman and Turner (lower limit) and the Abrahamson kernels (upper limit), as it had to be expected. In the case of zero particle inertia, represented by the Saffman and Turner limit, particle collisions are only controlled by the smallest fluid scales. This regime is approached in the present simulations of the lowest inertia  
Figure 11: Collision kernel, normalised by the Kolmogorov velocity, and the square of particle diameter, obtained by the present simulations as a function of the particle Stokes number and for the three average particle volume fractions. The Saffman and Turner [44] and Abrahamson [45] kernels for colliding particles are also shown for comparison.  
particles, where the inter-particle collision time is determined by the Kolmogorov time scale. On the other limit, the collision kernel is governed by the particle kinetic energy, which is approached in the present simulations for the highest considered Stokes number. Eventually, for larger inertia particles, the collision kernel will decrease because they tend to be less responsive to fluid turbulence. The trend of the computed kernel in present simulations agrees qualitatively with the results of [PERSON] [46], showing a fast increase around Stokes numbers close to one and then growing much more moderately for higher Stokes numbers (here up to 10).  
Regarding the dependence of the simulated non-dimensional collision kernel with particle volume fraction, Figure 11 shows that it does not have an appreciable effect: only a marginal increase of \(\Gamma/\left(u_{K}d_{P}^{2}\right)\) is observed with growing \(\alpha_{P}\).