  Motivated by systems in which droplets grow and shrink in a turbulence-driven
supersaturation field, we investigate the problem of turbulent condensation in
a general manner. Using direct numerical simulations we show that the turbulent
fluctuations of the supersaturation field offer different conditions for the
growth of droplets which evolve in time due to turbulent transport and mixing.
Based on that, we propose a Lagrangian stochastic model for condensation and
evaporation of small droplets in turbulent flows. It consists of a set of
stochastic integro-differential equations for the joint evolution of the
squared radius and the supersaturation along the droplet trajectories. The
model has two parameters fixed by the total amount of water and the
thermodynamic properties, as well as the Lagrangian integral timescale of the
turbulent supersaturation. The model reproduces very well the droplet size
distributions obtained from direct numerical simulations and their time
evolution. A noticeable result is that, after a stage where the squared radius
simply diffuses, the system converges exponentially fast to a statistical
steady state independent of the initial conditions. The main mechanism involved
in this convergence is a loss of memory induced by a significant number of
droplets undergoing a complete evaporation before growing again. The
statistical steady state is characterised by an exponential tail in the droplet
mass distribution. These results reconcile those of earlier numerical studies,
once these various regimes are considered.
