The authors study the condensational growth of cloud droplets in homogeneous isotropic turbulence by means of a large-eddy simulation (LES) approach. The authors investigate the role of a mean updraft velocity and of the chemical composition of the cloud condensation nuclei (CCN) on droplet growth. The results show that a mean constant updraft velocity superimposed onto a turbulent field reduces the broadening of the droplet size spectra induced by the turbulent fluctuations alone. Extending the authors' previous results regarding stochastic condensation, the authors introduce a new theoretical estimation of the droplet size spectrum broadening that accounts for this updraft velocity effect. A similar reduction of the spectra broadening is observed when the droplets reach their critical size, which depends on the chemical composition of CCN. The analysis of the square of the droplet radius distribution, proportional to the droplet surface, shows that for large particles the distribution is purely Gaussian, while it becomes strongly non-Gaussian for smaller particles, with the left tail characterized by a peak around the haze activation radius. This kind of distribution can significantly affect the later stages of the droplet growth involving turbulent collisions, since the collision probability kernel depends on the droplet size, implying the need for new specific closure models to capture this effect.