Statistical steady state in turbulent droplet condensation

TL;DR

This paper develops a stochastic model for turbulent droplet condensation that accurately predicts droplet size distributions and demonstrates a rapid convergence to a steady state characterized by exponential mass distribution tails.

Contribution

The paper introduces a novel Lagrangian stochastic model for turbulent droplet condensation that aligns well with numerical simulations and explains the rapid approach to steady state.

Findings

Model reproduces droplet size distributions from simulations.

Droplets quickly reach a steady state independent of initial conditions.

Steady state features an exponential tail in droplet mass distribution.

Abstract

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.