## 4 Two-Particle Velocity Correlation Function
Based on the results obtained by solving the [PERSON] and [PERSON] model, [PERSON] et al. [36] proposed the following functional form for the particle velocity correlation of colliding particles:  
\[R_{E}(St_{K})=\exp\left(-\frac{\alpha St_{K}^{2}}{1+\gamma St_{K}^{\beta}}\right) \tag{8}\]  
Equation (8) uses three parameters instead of the two originally proposed by [PERSON] [30], Equation (6). The parameters take the values: \(\alpha=0.019\), \(\beta=1.725\) and \(\gamma=0.044\). Correlation (8) is shown in Figure 12 as a solid line together with our own results obtained solving the Zaichik and Alipchenkov model, represented by plus symbols. As it can be seen, \(R_{E}(St_{K})\) tends to one in the limit of very small Stokes numbers according to the Saffman and Turner limit, where velocities of neighbouring particles are spatially correlated due to the interaction with the same local fluid flow. As \(St_{K}\) increases, particle velocity correlation decreases smoothly, and in the limit of very high Stokes numbers, it tends to zero, where statistics of heavy particles satisfy the assumption of molecular chaos according to [PERSON] [45]. Results for the particle pair correlation function obtained in the present work within the frame of LBM-based DNS are also presented in Figure 12. The closed symbols correspond to the Eulerian two-particle velocity correlation function evaluated at the fixed distance \(r=3d_{p}/2\), i.e.,  
\[R_{E}(St_{K})=R\left(r=\frac{3}{2}d_{p},St_{K}\right)=\frac{3}{2k_{p}}B_{pl} \left(r=\frac{3}{2}d_{p},St_{K}\right) \tag{9}\]  
Figure 12: Two-particle velocity correlation function \(R_{E}(St_{K})\). Plus (+) symbols represent the results obtained by solving the Zaichik and Alipchenkov model [18]. Cross (\(\times\)) symbols show the results of Reference [28]. Correlation of Reference [30], Equation (6), corresponds to the purple dashed line. The rest of the symbols refer to the LBM-based DNS results obtained in this work for different average volume fractions. Closed symbols represent the Eulerian correlation evaluated at a distance \(r=3d_{p}/2\), Equation (9), and open symbols represent the obtained colliding pair velocity correlation, Equation (10) (see text for details). The dot-dashed lines fit the colliding two-particle velocity correlation functions, \(R_{c}(St_{K})\), of the different volume fraction cases studied in this work. This corresponds to a situation where the longitudinal velocity correlation between neighbouring particles is evaluated at 1.5 times the particle diameter (i.e., around 0.3 \(\eta_{K}\)) at a fixed time. Correlation has been computed at different times after the initial transient has been completed, showing stable values in all the range of Stokes numbers studied. On the other hand, open symbols represent the two-particle velocity correlation for particle pairs that have actually collided, i.e.,  
\[R_{c}(St_{K})=R\big{(}r=d_{p},St_{K}\big{)}. \tag{10}\]  
This corresponds to a situation where the longitudinal velocity correlation between neighbouring particles is computed for particle pairs that have experienced a collision along the development of the simulation. Statistics are computed based on \(10^{6}\) collision events. From Figure 12, it can be clearly seen that the \(R_{E}\) values obtained from LBM-based DNS agree reasonably well with the correlation Equation (8). Therefore, the theoretical model of [PERSON] and Alipchenkov is adequate for describing the Eulerian velocity correlation between neighbouring particles, as it could be expected. Moreover, the DNS results for different solid volume fractions show nearly no scattering of values for all the range of studied Stokes numbers. As already commented in the introduction, the decreasing of particle velocity correlation of neighbouring particles, i.e., \(R_{E}\), with increasing particle inertia is usually explained [19] in terms of an increase of the relative velocity between them, consistent with the existence of caustics and a component of random uncorrelated motion of particle velocity field.