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Abstract The evolution of micrometer-sized droplets in clouds is studied with focus on the “size-gap” regime of 15–40-μm radii, where condensation and differential sedimentation are least effective in promoting growth. This bottleneck leads to inaccurate growth models, and turbulence can potentially rectify disagreement with in situ cloud measurements. The role of turbulent collisions, mixing of droplets, and water vapor fluctuations in crossing the size gap has been analyzed in detail. Collisions driven by the coupled effects of turbulent shear and differential sedimentation are shown to grow drizzle sized droplets. Growth is also promoted by turbulence-induced water vapor fluctuations, which maintain polydispersity during the initial-condensation-driven growth and facilitate subsequent growth by differential-sedimentation-driven coalescence. The collision rate of droplets is strongly influenced by noncontinuum hydrodynamics, and so the size evolution beyond the condensation regime is found to be very sensitive to the mean-free path of air. Turbulence-induced inertial clustering leads to a moderate enhancement in the growth rate, but the intermittency of the turbulent shear rate does not change the coalescence rate significantly. The coupled influence of all these phenomena is evaluated by evolving a large number of droplets within an adiabatically rising parcel of air using a Monte Carlo scheme that captures turbulent intermittency and mixing. Significance StatementThis study is directed toward improving descriptions of the microphysical determinants of the time for rain formation in clouds. Existing models predict significantly longer times than the tens of minutes observed in warm clouds. There is a growing body of evidence that turbulence plays a key role in resolving this discrepancy. We incorporate accurate turbulent collision dynamics and assess the interplay of the various underlying physical factors facilitating growth to rain-sized droplets. Our study, in addition to providing important insight into cloud microphysics, will pave the path to the next generation of large-scale rain cloud evolution studies. 

The collisions in a dilute polydisperse suspension of sub-Kolmogorov spheres with negligible inertia settling in a turbulent flow and interacting through hydrodynamics including continuum breakdown on close approach are studied. A statistically significant decrease in ideal collision rate without gravity is resolved via a Lagrangian stochastic velocity-gradient model at Taylor microscale Reynolds number larger than those accessible by current direct numerical simulation capabilities. This arises from the difference between the mean inward velocity and the root-mean-square particle relative velocity. Differential sedimentation, comparable to the turbulent shear relative velocity, but minimally influencing the sampling of the velocity gradient, diminishes the Reynolds number dependence and enhances the ideal collision rate i.e. the rate without interactions. The collision rate is retarded by hydrodynamic interactions between sphere pairs and is governed by non-continuum lubrication as well as full continuum hydrodynamic interactions at larger separations. The collision efficiency (ratio of actual to ideal collision rate) depends on the relative strength of differential sedimentation and turbulent shear, the size ratio of the interacting spheres and the Knudsen number (defined as the ratio of the mean-free path of the gas to the mean radius of the interacting spheres). We develop an analytical approximation to concisely report computed results across the parameter space. This accurate closed form expression could be a critical component in computing the evolution of the size distribution in applications such as water droplets in clouds or commercially valuable products in industrial aggregators. 

Collisions in a dilute polydisperse suspension of spheres of negligible inertia interacting through non-continuum hydrodynamics and settling in a slow uniaxial compressional flow are studied. The ideal collision rate is evaluated as a function of the relative strength of gravity and uniaxial compressional flow and it deviates significantly from a linear superposition of these driving terms. This non-trivial behaviour is exacerbated by interparticle interactions based on uniformly valid non-continuum hydrodynamics, that capture non-continuum lubrication at small separations and full continuum hydrodynamic interactions at larger separations, retarding collisions driven purely by sedimentation significantly more than those driven purely by the linear flow. While the ideal collision rate is weakly dependent on the orientation of gravity with the axis of compression, the rate including hydrodynamic interactions varies by more than $$100\,\%$$ with orientation. This dramatic shift can be attributed to complex trajectories driven by interparticle interactions that prevent particle pairs from colliding or enable a circuitous path to collision. These and other important features of the collision process are studied in detail using trajectory analysis at near unity and significantly smaller than unity size ratios of the interacting spheres. For each case analysis is carried for a large range of relative strengths and orientations of gravity to the uniaxial compressional flow, and Knudsen numbers (ratio of mean free path of the media to mean radius).