## 4 Two-Particle Velocity Correlation Function
In fact, the connection between the decreasing of particle velocity correlation and the increase of the modulus of relative velocity (or vice versa) can be easily cast in mathematical terms. The average of the modulus of relative velocity between two particles is computed as:  
\[\langle|\mathbf{w}|^{2}\rangle=\langle\mathbf{w}\cdot\mathbf{w}\rangle=\langle(\mathbf{u}_{p2} -\mathbf{u}_{p1})\cdot(\mathbf{u}_{p2}-\mathbf{u}_{p1})\rangle=\big{\langle}\big{|}\mathbf{u}_ {p1}\big{|}^{2}\rangle+\big{\langle}\big{|}\mathbf{u}_{p2}\big{|}^{2}\rangle-2 \langle\mathbf{u}_{p1}\cdot\mathbf{u}_{p2}\rangle.\]  
Now, because the mean particle velocity in isotropic homogeneous turbulence is zero, \(\big{\langle}\big{|}\mathbf{u}_{p2}\big{|}^{2}\big{\rangle}=\big{\langle}\big{|} \mathbf{u}_{p1}\big{|}^{2}\big{\rangle}=2k_{p}\), where \(k_{p}\) is the fluctuating kinetic energy of the particles. Finally, solving for the two-particle velocity correlation, it is obtained:  
\[\langle\mathbf{u}_{p1}\cdot\mathbf{u}_{p2}\rangle=2k_{p}-\frac{1}{2}\langle|\mathbf{w}|^{ 2}\rangle, \tag{11}\]  
where \(\mathbf{u}_{p1}\) and \(\mathbf{u}_{p2}\) are the velocities of particles 1 and 2 respectively, \(\mathbf{w}=\mathbf{u}_{p2}-\mathbf{u}_{p1}\) is their relative velocity and \(\langle\ \rangle\) denotes the averaging operator. As Equation (11) is a general relation, a reduction in the two-particle velocity correlation is accompanied by an increase in the modulus of relative velocity, and an increase of this quantity implies a decrease of the correlation. To support this affirmation, Figure 13 shows the modulus of radial relative velocity (RRV), defined as \(w_{r}=\mathbf{w}\cdot\mathbf{r}/|\mathbf{r}|_{t}\) between colliding particles, divided by the Kolmogorov velocity, \(u_{K}\), as a function of the particle Stokes number. It is clear that it increases with particle inertia, experiencing a rapid growth for Stokes numbers roughly between 0.5 and 2. The strong increase of the radial relative velocity for \(St_{K}>0.5\) is connected with the strong enhancement of the inter-particle collision rate due to sling caustics occurrence. For larger Stokes numbers, of more than 10, a decrease of the RRV is expected because the particles become slower and less responsive to the fluid turbulence. Such results are in close agreement with those of Figure 2 in Reference [18]. The behaviour of the colliding particles velocity correlation function, \(R_{c}\), in Figure 12 deserves attention. At small values of particle inertia, here \(St_{K}=0.1\), there is no difference with the Eulerian correlation as the collision events are governed by the fluid turbulent shear. As soon as inertia increases, \(R_{c}\) adopts significantly lower values than \(R_{E}\); for instance, in Figure 12, at \(St_{K}=0.5\), the difference between the two correlations is already appreciable. Such differences are increasing with augmenting particle Stokes number. The reason for the smaller values of \(R_{c}\), regarding \(R_{E}\), is that in three-dimensions, not all particle pairs that are close enough to interact do really collide, it is necessary that they approximate each other with the right radial relative velocity to have the interaction. In fact, the \(R_{c}\) curves cannot be obtained as \(R_{E}\big{(}r=d_{p}\big{)}\) because the RRV's are different, as already remarked in Reference [16]. The reason is that in the Eulerian case, the average of the RRV is zero, as it was implied in the derivation of Equation (7), while in the colliding particle case, this RRV is effectively negative. As a result, the modulus of the particle relative velocity is higher in the colliding case than in the Eulerian case which, due to Equation (10), has the consequence that the values of \(R_{c}\) are smaller than those of \(R_{E}\). As the modulus of the RRV increases with Stokes number (Figure 13), the distance between both curves augments with particle inertia. Moreover, the results of the recent study of [PERSON] et al.