## 5 Conclusions  
In this paper, the effect of particle inertia and void fraction, and therefore inter-particle collisions, on the colliding particle-pair velocity correlation in homogeneous isotropic turbulence has been addressed by using direct numerical simulations based on the Lattice Boltzmann Method. Three particle volume fractions have been considered, \(\alpha_{p}=0.01\%,\ 0.05\%,\ 0.1\%\), in the dilute regime. In the range of void fractions studied, collisions have a marginal impact on particle transport, with the aerodynamic interaction with fluid turbulence being the main mechanism that drives it. This is reflected in the fact that collision Stokes number has been always larger than one, keeping the dilute flow conditions. A main result of the performed study is that collision locations of particles of intermediate inertia, \(St_{K}\sim 1\), happens in areas where the fluid strain rate and dissipation are higher than the corresponding averaged values at particle positions. Connected with this fact, the average kinetic energy of colliding particles of \(St_{K}\sim 1\) is lower than the value averaged over all particles. This finding was explained by the fact that the fluid kinetic energy at collision positions is lower than the mean value and, because \(St_{col}\gg 1\), particles have enough time to accommodate to the fluid environment between consecutive collisions. In case of the highest inertia, \(St_{K}=10\), particles do not experience noticeable clustering in areas of high strain rate and in that case, inter-particle collisions are favoured by high kinetic energy; as a result, the colliding particle kinetic energy is higher than the corresponding averaged value over all particles. Later, the Eulerian and colliding two-particle velocity correlation functions were computed in the same homogeneous isotropic turbulence configuration. As a result, the Eulerian function was well reproduced by the Zaichik and Alipchenkov [37] model, but the colliding particles correlation showed remarkably smaller values due to the role played by the mean radial relative velocity. In particular, it was demonstrated that the colliding pair correlation cannot be obtained as a limiting value of the Eulerian correlation. Moreover, it has been confirmed that the colliding particles velocity correlation increases with particle volume fraction, a fact related with the decreasing of the average relative velocity between the colliding particles and attributed to the particle dynamics inside clusters. However, the decreasing of radial relative velocity with increasing \(\alpha_{p}\) was also found for large particle inertia, where no appreciable clustering was observed. Therefore, further studies are needed to confirm if the same phenomenon happens for a wider set of conditions than those presented here, such as larger volume fractions and turbulent Reynolds numbers. The obtained results in the present study can be used to support inter-particle collision stochastic models (e.g., Reference [31]) for use in general-purpose two-phase flow Euler-Lagrange codes. However, it is necessary to study the dependence of such correlations on turbulent Reynolds number and in a larger range of particle volume fractions. Conceptualisation, S.L., M.E. and M.S. ; methodology, S.L. ; software, M.E. ; validation, M.E. and S.L. ; formal analysis, S.L. ; investigation, M.E. and S.L. ; resources, M.S. ; data curation, S.L. and M.E. ; writing--original draft preparation, S.L. and M.E. ; writing--review and editing, S.L. ; visualisation, M.E. ; supervision, S.L. and M.S. ; project administration, M.S. ; funding acquisition, M.S. and S.L. All authors have read and agreed to the published version of the manuscript. This research was funded by the German Research Council (DFG), grant number SO 204-33/1-3 (Priority program 1273 \"Colloidal Process Engineering\"). The financial support by the German Research Council (DFG) through Grants SO 204-33/1-3 (Priority program 1273 \"Colloidal Process Engineering\") is gratefully acknowledged.