Inhomogeneous distribution of droplets in cloud turbulence

TL;DR

This paper analyzes how inertial particles, such as water droplets in clouds, distribute in turbulent flows under gravity, revealing a power-law spatial distribution and confirming predictions through simulations.

Contribution

It provides a theoretical framework for the spatial distribution of sedimenting inertial particles in turbulence, accounting for gravity and arbitrary particle inertia.

Findings

Pair-correlation function follows a power-law with a negative exponent.

Distribution becomes singular at zero distance, indicating clustering.

Simulation results confirm theoretical predictions for cloud droplets.

Abstract

We solve the problem of spatial distribution of inertial particles that sediment in turbulent flow with small ratio of acceleration of fluid particles to acceleration of gravity $g$. The particles are driven by linear drag and have arbitrary inertia. The pair-correlation function of concentration obeys a power-law in distance with negative exponent. Divergence at zero signifies singular distribution of particles in space. Independently of particle size the exponent is ratio of integral of energy spectrum of turbulence times the wavenumber to $g$ times numerical factor. We find Lyapunov exponents and confirm predictions by direct numerical simulations of Navier-Stokes turbulence. The predictions include typical case of water droplets in clouds. This significant progress in the study of turbulent transport is possible because strong gravity makes the particle's velocity at a given point unique.