## 1 Introduction
Besides fundamental theoretical meaning, the two-particle pdf also has sound implications in developing models for predicting two-phase practical flows. For instance, modelling of inter-particle collisions in the frame of the Euler-Lagrange method for the numerical calculation of two-phase flows has so far been based mainly on two approaches: a direct simulation (deterministic model) and a stochastic model based on concepts of the kinetic theory of gases. The most straightforward approach to account for inter-particle collisions is the direct detection approach, as it is usually performed in Direct Numerical Simulation (DNS), including the present study. This requires that all the particles have to be tracked simultaneously through the flow field. Thereby, the occurrence of collisions between any pair of particles can be judged based on their positions and relative motion during one time-step. This process is however very time consuming and its computational cost is, at least, proportional to \(N_{p}\) (total number of tracked particles) at each Lagrangian time step. For instance, a recent study by [PERSON] et al. [29] mentions that the computational time needed for the direct detection approach is two orders of magnitude higher than that for a stochastic approach. Once a collision occurs, the change in translational and angular particle velocities is determined by solving the equations for the conservation of linear and angular momentum in connection with [PERSON]'s law of friction. When the duration of the collision process is negligibly small compared to the time of collisionless motion (i.e., hard sphere collision), the size of the colliding particles is not too different, and the ratio of solid particle density to the fluid density is much larger than unity, fluid dynamic effects during the collision process can be neglected and the collision efficiency may be assumed as 100% [30]. Stochastic inter-particle collision models [29; 31; 32], on the other hand, rely on the generation of fictitious collision partners and the calculation of the collision probability according to kinetic theory. The advantage of such models is that they do not require information on the location of the surrounding particles and hence they are also applicable if a sequential tracking of the particles is adopted, as it is usually done when applying the Euler-Lagrange approach to stationary flows. However, unsteady flows employing simultaneous tracking may also be conducted using the stochastic collision model [33]. During each time step of the trajectory calculation of the considered particle, a fictitious second particle is generated. The properties of this fictitious particle are randomly sampled from local distribution functions. The generation of the fictitious particle fluctuating velocities requires taking into account the correlation with the velocity of the considered particle due to turbulence. The degree to which the particle fluctuating velocities are correlated depends on their response to the turbulent fluctuations. [PERSON] [31] compared his model computations with the Large Eddy Simulations (LES) published in Reference [34] and proposed an exponentially decreasing correlation, function of particle Stokes number, based on a trial-and-error basis. Such correlation has been employed successfully for predicting the effect of inter-particle collisions in pneumatic conveying and particle separation processes [9; 10; 11; 35]. [PERSON] et al. [36] developed a more rigorous way to determine the Eulerian particle-pair velocity correlation function between colliding particles in homogeneous isotropic turbulence, employing the model proposed in Zaichik and Alipchenkov [37] (or its refined version, [PERSON] et al. [18]), showing good agreement with previous data [19]. Precisely, the adequacy of such correlation is evaluated and discussed in the present contribution in light of the new DNS results obtained in this work.