## 2 Context and Summary of Numerical Simulations
The response time scale of the particle is defined as:  
\[\tau_{p}=\tau_{pS}\frac{24}{c_{D}Re_{p}}, \tag{1}\]  
where \(\tau_{pS}=\rho_{p}d_{p}^{2}/18\mu\) is the Stokesian particle relaxation time, \(\mu\) is the carrier fluid dynamic viscosity, \(c_{D}\) is the particle drag coefficient which depends on the particle Reynolds number, \(Re_{p}=\rho d_{p}\big{|}\mathbf{u}-\mathbf{u}_{p}\big{|}/\mu\), whose expression extended for Reynolds number higher than 0.5 is given in Reference [40] as: \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Symbol** & **Value** & **Units** \\ \hline Fluid density & \(\rho\) & \(1.17\) & kg/m\({}^{3}\) \\ Kinematic fluid viscosity & \(\
u\) & \(1.47\times 10^{-5}\) & m\({}^{2}\)/s \\ Fluctuating velocity & \(u^{\prime}\) & \(2.75\times 10^{-2}\) & m/s \\ Dissipation rate & \(\epsilon\) & \(2.92\times 10^{-3}\) & m\({}^{3}\)/s \\ Integral length scale & \(L_{int}/L\) & \(0.11\) & – \\ Integral time scale & \(T_{int}/\tau_{k}\) & \(4.83\) & – \\ Taylor microscale & \(\lambda_{T}/L_{int}\) & \(0.54\) & – \\ Kolmogorov length scale & \(\eta_{K}/\Delta x\) & \(1.02\) & – \\ Kolmogorov time scale & \(\tau_{K}/\Delta t\) & \(208.5\) & – \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Turbulence characteristics of the Lattice Boltzmann Method (LBM)-based Direct Numerical Simulation (DNS). Figure 1: Non-dimensional three-dimensional (3D) energy spectrum \(\varepsilon_{x}\) of the turbulent flow field of the present DNS. Experimental points correspond to Comte-Bellot and Corrsin [41] and the black solid line indicates the \(-5\)/3 scaling. \[c_{D}= \begin{cases}\frac{24}{R_{cp}}&Re_{p}\leq 0.5\\ \frac{24}{R_{cp}}\big{(}1+0.15R\rho_{p}^{0.687}\big{)}&0.5<Re_{p}<10000\,,\\ 0.44&Re_{p}\geq 1000\end{cases}  \tag{2}\]  
whereas the Kolmogorov time microscale is expressed as:  
\[\tau_{K}=\bigg{(}\frac{\
u}{\varepsilon}\bigg{)}^{1/2}. \tag{3}\]  
In this study, variation of particle Stokes number is achieved by modifying its density according to the values shown in Table 2.  
For quantifying the importance of inter-particle collisions on disperse phase transport, a collision Stokes number can be defined as \(St_{col}=\tau_{c}/\big{\langle}\tau_{p}\big{\rangle}\), where \(\tau_{c}\) is the mean inter-particle collision time, i.e., the average time between two consecutive collisions. As mentioned in Reference [42], \(St_{col}\) allows drawing a boundary between dilute and dense flow conditions: if \(\tau_{c}\) is larger than \(\big{\langle}\tau_{p}\big{\rangle}\), the flow is ditute (i.e, \(St_{col}>1\)), and dense otherwise. In the case of dilute flows, the main particle transport mechanism is driven by its interaction with the turbulence, but in case of dense systems, collisions play a role in particle motion. Although previous works have considered the effect of particle number density on inter-particle collision frequency (e.g., Reference [15]), it was in the article of [PERSON] and [PERSON] [38] where a systematic study of the effect of particle volume fraction on collision statistics was performed. The employed DNS resolution of that work was low, only \(64^{3}\), and therefore the grid size was a factor two larger than the Kolmogorov length scale. Nevertheless, some interesting results were found: as expected, inter-particle collision frequency augmented with increasing volume fraction and Stokes number, the Lagrangian integral time scale of particles decreased with volume fraction but increased with \(St_{K}\) due to the decorrelation of particle motion induced by the increase in the number of collisions, also, the particle pair relative velocity at collision rose with growing \(St_{K}\) but it decreased for higher values of \(\alpha_{p}\), with this last effect being more pronounced for large inertia particles. Such findings are revised in the next section with the results obtained in the present, more refined, DNS. Finally, our previous work ([PERSON] et al.