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The mean radial component of relative velocity (MRV) between pairs of inertial particles is studied, where the particles are advected by turbulent flow and undergo collision-and-coagulation. A previously proposed phenomenological model of MRV for low-inertia particles \citep{saw2022intricate} is corrected (improved) and shown to produce better predictions of the MRV as a function of particle separation distance $r$. Using direct numerical simulation (DNS), the relationship between the MRV and particle/turbulent parameters is studied. For particles with near-zero Stokes numbers ($St$), the MRV is roughly independent of $St$. At larger $St$, the magnitude of MRV increases with $St$, particularly when $St>0.2$. Assuming that the relative particle velocities are derived from fluid velocity differences associated with a nominal resonant length scale, an empirical relation between $St$ is obtained: $d+\alpha St^{\beta}$, where $\beta\approx1.86$. Coupled with this empirical result, the aforementioned MRV model could be extended to predict MRV for any finite $St$, and we show that the predictions are accurate against the DNS results. Our results also suggest that the extended model could also accurately account for possible Reynolds number ($Re_{\lambda}$) effect by simply allowing $\alpha$ and $\beta$ to be functions of $Re_{\lambda}$. Additionally, when the particle diameter is smaller than the Kolmogorov length scale, the MRV for particles with the same St is independent of the particle diameter. The analysis under different Reynolds numbers ($Re_{\lambda}=84,124,189$) reveals that for particles with $St\ll1$, the MRV is $Re_{\lambda}$-independent. For larger $St$, $Re_{\lambda}$ dependence is observed such that the coefficients $\alpha$ and $\beta$ decrease with $Re_{\lambda}., Comment: 12 pages, 1 table, 7 figures

We perform direct numerical simulation (DNS) to study the clustering of small, heavy, monodisperse particles subject to collision-coagulation in turbulent flow (i.e., colliding particles always coagulate (coalesce) into large ones). We find that collision-coagulation causes the radial distribution function (RDF) of the particles to decrease strongly at particle separation distances $r$ close to the particle diameter $d$. However, the RDF do not decrease indefinitely but approach a finite value in the limit of $r\to d$. We study how the characteristics of this "depletion zone" relate to the particle Stokes number (St), particle diameter, and the Reynolds number of the turbulent flow. A collision-induced modulation factor $\gamma_{c}$ is defined to represent the degree of RDF depletion due to collisions-coagulation. In the region where $\gamma_c(r)$ is a quasi-power-law, the corresponding power-law exponent $\tilde{c}_1$ only depends weakly on $St$. The overall trend of $\tilde{c}_1$ with respect to $St$ is similar to that of the classical power-law exponent $c_{1}$ appearing in the RDF of non-colliding particles, i.e., the exponent increase at small $St$, peak around $St \approx 0.7$, and decrease thereafter. The same qualitative trend is also observed for the limiting values of $\gamma_c$ at $r\to d$. A complementary investigation on the Stokes number trend of the full RDF in the depletion zone is conducted. The slope of RDF appears constant for $St\ll1$ but is changing when $St$ is getting large. The position where the RDF starts to decrease is found to be $St$-dependent. The depletion zone is insensitive to the flow Reynolds number and $\gamma_c$ of different $Re_{\lambda}$ overlap. With changing particle diameter $d$, the reduction of RDF occurs at scales that shift accordingly and always starts at around $2.4d-3d$. The shape of $\gamma_c(r)$ is independent of changes in $d$.

Considering turbulent clouds containing small inertial particles, we investigate the effect of particle collision, in particular collision-coagulation, on particle clustering and particle relative motion. We perform direct numerical simulation (DNS) of coagulating particles in isotropic turbulent flow in the regime of small Stokes number ($St=0.001-0.54$) and find that, due to collision-coagulation, the radial distribution functions (RDFs) fall-off dramatically at scales $r \sim d\,\,$ (where $d$ is the particle diameter) to small but finite values, while the mean radial-component of particle relative velocities (MRV) increase sharply in magnitudes. Based on a previously proposed Fokker-Planck (drift-diffusion) framework, we derive a theoretical account of the relationship among particle collision-coagulation rate, RDF and MRV. The theory includes contributions from turbulent-fluctuations absent in earlier mean-field theories. We show numerically that the theory accurately accounts for the DNS results (i.e., given an accurate RDF, the theory could produce an accurate MRV). Separately, we also propose a phenomenological model that could directly predict MRV and find that it is accurate when calibrated using fourth moments of the fluid velocities. We use the model to derive a general solution of RDF. We uncover a paradox: the past empirical success of the differential version of the theory is theoretically unjustified. We see a further shape-preserving reduction of the RDF (and MRV) when the gravitational settling parameter ($S_g$) is of order $O(1)$. Our results demonstrate strong coupling between RDF and MRV and imply that earlier isolated studies on either RDF or MRV have limited relevance for predicting particle collision rate.