## 1 Introduction
Particles of intermediate inertia (of similar relaxation time scales to those relevant flow time scales) experience two phenomena which lead to an enhancement of the collision rates. The first is the so-called preferential concentration effect, whereby particles of non-negligible inertia tend to concentrate at specific locations on the flow field characterised by lower than mean values of fluid vorticity and higher than mean values of strain rate. The second is known as the sling effect, by which particles experience interactions with strong intermittent turbulent events, causing modification of their movement [3]. The consequence is that particle trajectories cross thereafter in certain regions of the flow, called pockets by [PERSON] et al. [21], where the chance of collisions is remarkably augmented. Quite a number of studies (see Reference [22] and references therein) have been focused on these topics to determine an adequate collision frequency formulation in case of intermediate inertia particles. [PERSON] and [PERSON] [23] and later [PERSON] and [PERSON] [24] developed a generalised model for inter-particle collision frequency taking into account particle inertia which is formulated in dependence of the relative velocity of particle pairs. Moreover, several early numerical and theoretical studies looked at the effects of particle diameter, relaxation time and number density on collision frequency [15; 18]. Particularly studied have been the particle-turbulent fluid interaction phenomena that lead to the formation of clusters in zones of low vorticity and high strain rate [25], favouring the occurrence of inter-particle collisions. Such effect, described by the radial distribution function, and the relative velocity between particle pairs at contact point are the usual two elements employed for writing the collision kernel, defined usually as the collision frequency over the particle number density. The various mechanisms for enhancing inter-particle collision frequency at the intermediate inertia range were syrcetised in a common framework by [PERSON] et al. [26]. Starting from the phase space of particles (composed by their position and velocities), the particle clustering process can be thought of as the confluence of trajectories towards an attractor in the phase space which experiences a dynamical evolution. The features of the resulting fractal set as well as its statistical characteristics depend on the nature of particle-fluid turbulence interaction and the particle Stokes number, \(St_{K}\) (defined here as the quotient between the average particle response time and the Kolmogorov time scale of the flow). In this picture, the increase of relative velocity among neighbouring particles arises from the attractor folding in the velocity subspace, where the folded region boundaries are denominated caustics [27]. As a consequence, the particle velocity field becomes multivalued in the coordinate space, a fact that has two implications for collisions: on the one hand, particles with very different velocities are able to cross their trajectories and, on the other hand, particle number density can grow without bound at the limits of caustics. This phenomenon, distinct from that of preferential concentration, may be responsible for the observed fast increase on collision frequency happening at \(St_{K}\)\(\sim\)0.5. In fact, it has been shown [28] that the caustics mechanism for inter-particle collisions prevails when \(St_{K}>0.75\). On the other hand, when practical two-phase flow computations for industrial or environmental applications are pursued, a direct detection of inter-particle collisions along the two-phase flow domain is no longer affordable. Statistical-like inter-particle collision models based on a Lagrangian frame offer an attractive alternative in such case. In this context, the two-particle probability density function (pdf) has proven suitable for predicting two-particle statistics and particle-pair dispersion in homogeneous isotropic turbulence [18].