Abstract
Considering turbulent clouds containing small heavy particles, we investigate the reverse effect of particle collision, in particular collision-&-coagulation, on particle clustering and relative motion. We perform various cases of direct numerical simulation (DNS) of coagulating particles in isotropic turbulent flow and find that, due to collision-coagulation, the radial distribution functions (RDF) fall-off dramatically at scales r ∼ 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 contribution from turbulent-fluctuations absent in earlier mean-field theories. We show numerically that the theory accurately account for the DNS results. We also proposed a phenomenological model for the MRV which is accurate when calibrated using 4th moments of the fluid velocities. We uncover a paradox: the unjustified accuracy of the differential version of the theory. Our result demonstrate strong coupling between RDF and MRV and implies that earlier isolated studies on either RDF or MRV have limited relevance for predicting particle collision rate.
Highlights
The motion and interactions of small particles in turbulence has fundamental implications for atmospheric clouds, 15 it is relevant to the time-scale of rain formation in warm-clouds (Falkovich et al, 2002; Wilkinson et al, 2006; Grabowski and Wang, 2013) [a similar problem applies to planet formation in astrophysics (Johansen et al, 2007)]
We do not have definitive answers to basic questions such as how to calculate particle collision rate from basic turbulence-particle parameters 25 and what is the exact relation between collision and particle clustering and/or motions, for, as we shall see, our work reveals that collision-coagulation causes profound changes in both mean radial-component of relative particle velocity (MRV) and radial distribution function (RDF), questioning earlier understanding of the problem (RDF is a metric of the degree of particle clustering)
110 We compute the radial dis5 tribution functions (RDF) via g(r) = Npp(r)/[ 12 N (N − 1)δVr/V ], where Npp(r) is the number of particle pairs found to be separated by distance r, δVr is the volume of a spherical shell of radius r and infinitesimal thickness δr, Figure 1 shows the RDFs obtained for particles of different Stokes numbers and sizes
Summary
The motion and interactions of small particles in turbulence has fundamental implications for atmospheric clouds, 15 it is relevant to the time-scale of rain formation in warm-clouds (Falkovich et al, 2002; Wilkinson et al, 2006; Grabowski and Wang, 2013) [a similar problem applies to planet formation in astrophysics (Johansen et al, 2007)]. The focus of our study is on the effect of particle collision-coagulation in the simplest fluid dynamic setting so that its implication and fundamental 75 interaction with turbulence can be fully understood before moving on to more complex settings in future works For this reason, we choose not to include hydrodynamic interactions and gravitational settling in the dynamic of the particles (this implies that if practical applicability is of concern, the current results only applies to cloud particles with gravitational terminal velocities that are small compared to the velocity scale of the smallest turbulent eddies, e.g. particle of size 50μm in atmospheric clouds).
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