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We study the statistical properties of ensembles of polymers in isotropic turbulence numerically in the one-way coupling regime. A linear polymer chain passively convected by turbulence is modeled by a line of beads, each of which is connected by a finitely extensible nonlinear elastic force and subject to Brownian motion. We find that when the Weissenberg number Wi(η)<1, the polymer chain has a coiled configuration, while for Wi(η)>10, it remains stretched for a much longer time than the typical time scale of the fluctuating turbulent velocity gradient. Various statistical quantities characterizing the ensemble of polymers, such as the mean, variance, autocorrelation time, and probability density function of the end-to-end vector distance, indicate that the coil-stretch transition occurs at Wi(η)=3-4. We also find that this trend is insensitive to the number of beads N(b) ( N(b)=20 or N(b)=2), provided that the parameters in the model with a small number of beads are properly generated from the one with a large number of beads (i.e., using the formula of Jin and Collins). Finally, the Wi(η) effects on the alignment of the end-to-end vector versus the principal axis of the rate of strain tensor and on the polymer elongation are examined from the viewpoint of local flow topology.

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To consider the growth of cloud droplets by condensation in turbulence, the Fokker-Planck equation is derived for the droplet size distribution (droplet spectrum). This is an extension of the statistical theory proposed by Chandrakar and coauthors in 2016 for explaining the broadening of the droplet spectrum obtained from the "Π-chamber", a laboratory cloud chamber. In this Fokker-Planck equation, the diffusion term represents the broadening effect of the supersaturation fluctuation on the droplet spectrum. The aerosol (curvature and solute) effects are introduced into the Fokker-Planck equation as the zero flux boundary condition at R 2 = 0, where R is the droplet radius, which is mathematically equivalent to the case of Brownian motion in the presence of a wall. The analytical expression for the droplet spectrum in the steady state is obtained and shown to be proportional to R exp (-cR 2), where c is a constant. We conduct direct numerical simulations of cloud droplets in turbulence and show that the results agree closely with the theoretical predictions and, when the computational domain is large enough to be comparable to the Π-chamber, agree with the results from the Π-chamber as well. We also show that the diffusion coefficient in the Fokker-Planck equation should be expressed in terms of the Lagrangian autocorrelation time of the supersaturation fluctuation in turbulent flow.