Universal statistics of density of inertial particles sedimenting in turbulence

TL;DR

This paper analyzes how inertial particles sedimenting in turbulence distribute on fractal sets with universal, particles-independent statistics, characterized by the Kaplan-Yorke dimension, and confirms findings with numerical simulations.

Contribution

It provides a universal description of inertial particle distribution in turbulence, linking fractal statistics to turbulence spectra and particle inertia, with numerical validation.

Findings

Particles distribute on fractal sets with log-normal statistics.

The Kaplan-Yorke dimension determines the distribution's properties.

Numerical simulations confirm theoretical predictions.

Abstract

We solve the problem of spatial distribution of inertial particles that sediment in Navier-Stokes turbulence with small ratio $Fr$ of acceleration of fluid particles to acceleration of gravity $g$. The particles are driven by linear drag and have arbitrary inertia. We demonstrate that independently of the particles' size or density the particles distribute over fractal set with log-normal statistics determined completely by the Kaplan-Yorke dimension $D_{KY}$. When inertia is not small $D_{KY}$ is proportional to the ratio of integral of spectrum of turbulence multiplied by wave-number and $g$. This ratio is independent of properties of particles so that the particles concentrate on fractal with universal, particles-independent statistics. We find Lyapunov exponents and confirm predictions numerically. The considered case includes typical situation of water droplets in clouds.