## 4 Two-Particle Velocity Correlation Function  
As it was previously mentioned, it is necessary to account for the correlation of the motion of neighbouring particles and their preferential concentration when dealing with the description of inter-particle interactions in practical two-phase flow computations. As commented in the introduction section, in this context, Lagrangian-based stochastic collision models offer an alternative to the costly, and often impracticable, deterministic collision algorithms. In such stochastic models, along the trajectory of each particle, a fictitious collision partner is generated at each time step. The properties of the fictitious particle such as diameter or velocities are sampled randomly from previously constructed local distributions, and the probability of collision is computed according to kinetic theory. However, in this process, the fluctuating velocity correlation of both particles has to be taken into account [29; 32] and such correlation depends on the particle response to the turbulent environment. In the original paper of [PERSON] [31], it was modelled by the following decaying exponential:  
\[R=\exp\Bigl{(}-0.55 St_{L}^{0.4}\Bigr{)}, \tag{6}\]  
where \(St_{L}\) is the Stokes number based on the fluid Lagrangian integral time scale. Coefficients appearing in Equation (6) were obtained after comparison with LES computations of Reference [34] by trial and error. 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], and it is shown in Figure 12. A more rigorous way to determine the particle-pair velocity correlation function between colliding particles, \(R(St_{K})\), was developed in [PERSON] et al. [36]. In such study, the model proposed in Zaichik and Alipchenkov [37] (or its refined version, [18]) developed for homogeneous isotropic turbulence was solved to build the Eulerian particle-pair velocity correlation function between colliding particles, \(R_{E}(St_{K})\). The conclusions obtained in Reference [36] are revised in this section in light of the new DNS results obtained. The information relevant to the estimation of the correlation function of particle-pair fluctuating velocities can be extracted from the solution of the Zaichik and Alipchenkov model. To this end, they define the Eulerian two-particle velocity correlation as:  
\[B_{pij}(\mathbf{r})=\langle u_{pi}(\mathbf{x},t)u_{pj}(\mathbf{x}^{\prime},t)\rangle=\frac {2}{3}k_{p}\delta_{ij}-\frac{2}{3}S_{pij}(\mathbf{r}), \tag{7}\]  
where \(k_{p}\) is the kinetic energy of particles, \(S_{pij}\) is the particle structure function of the relative velocity between two particles and \(\mathbf{r}\) is the distance between the two particles [37]. Expression (7) is obtained assuming that there is a balance of net radial inward and outward fluxes implying a zero mean radial relative velocity. The longitudinal two-particle velocity correlation function is defined as \(R(r)=3B_{pll}(r)/2k_{p}\) in terms of the longitudinal space correlation, \(B_{pll}\), which depends on the separation distance and Stokes number. As pointed out by [PERSON] and [PERSON] [37] and discussed in depth in [PERSON] et al. [19], in contrast to the well-known result for fluid turbulence, this correlation for inertial particles is not tending towards unity in the limit of \(r\to 0\). This specific feature of the spatial correlation indicates that as a consequence of the inertial bias in their trajectories, the velocities of particles are not completely correlated, even at \(r=0\). In the case of colliding particles, the distance between them tends to zero (obviously it cannot be less than the sum of the radii of both particles), so the colliding particle-pair velocity correlation could be expected to be approximated by the behaviour near the origin of the longitudinal space correlation, \(R_{E}=3B_{pl}(r\to 0)/2k_{p}\), which clearly depends on particle inertia.