Inertial particles distribute in turbulence as Poissonian points with random intensity inducing clustering and supervoiding

TL;DR

This paper models the spatial distribution of inertial particles in turbulence as a Poisson process with a log-normal random intensity, revealing clustering and voiding phenomena with implications for predicting particle behavior.

Contribution

It introduces a novel Poisson point process model with log-normal intensity for inertial particle distribution in turbulence, supported by theoretical derivation and numerical validation.

Findings

Particles form a Poisson point process with log-normal intensity.

Void probability for inertial particles exceeds that of tracers.

Large voids decay log-normally with size.

Abstract

This work considers the distribution of inertial particles in turbulence using the point-particle approximation. We demonstrate that the random point process formed by the positions of particles in space is a Poisson point process with log-normal random intensity ("log Gaussian Cox process" or LGCP). The probability of having a finite number of particles in a small volume is given in terms of the characteristic function of a log-normal distribution. Corrections due to discreteness of the number of particles to the previously derived statistics of particle concentration in the continuum limit are provided. These are relevant for dealing with experimental or numerical data. The probability of having regions without particles, i.e. voids, is larger for inertial particles than for tracer particles where voids are distributed according to Poisson processes. Further, the probability of having large voids decays only log-normally with size. This shows that particles cluster, leaving voids behind. At scales where there is no clustering there can still be an increase of the void probability so that turbulent voiding is stronger than clustering. The demonstrated double stochasticity of the distribution originates in the two-step formation of fluctuations. First, turbulence brings the particles randomly close together which happens with Poisson-type probability. Then, turbulence compresses the particles' volume in the observation volume. We confirm the theory of the statistics of the number of particles in small volumes by numerical observations of inertial particle motion in a chaotic ABC flow. The improved understanding of clustering processes can be applied to predict the long-time survival probability of reacting particles. Our work implies that the particle distribution in weakly compressible flow with finite time correlations is a LGCP, independently of the details of the flow statistics.