Abstract

This work considers the distribution of discrete inertial particles in turbulence. We demonstrate that even for weak inertia the distribution can be strongly different from the Poisson distribution that holds for tracers. We study the cases of weak inertia or strong gravity where single-valued particle flow holds in space. In these cases, the particles distribute over a random multifractal attractor in space. This attractor is characterized by fractal dimensions describing scaling exponents of moments of number of particles inside a ball with size much smaller than the viscous scale of turbulence. Previous studies used a continuum approach to the moments which requires having a large number of particles below the viscous scale. This condition often does not hold in practice; for instance, for water droplets in clouds there is typically one droplet per viscous scale. This condition is also hard to realize in numerical simulations. In this work, we overcome this difficulty by deriving the probability pl(k) of having k particles in a ball of small radius l for which the continuum approximation may not hold. We demonstrate that the random point process formed by positions of particles' centers in space is a Poisson point process with log-normal random intensity (the so-called log Gaussian Cox process or LGCP). This gives pl(k) in terms of the characteristic function of a log-normal distribution from which the moments are derived. This allows finding the correlation dimension relevant for statistics of particles' collisions. The case of zero number of particles provides the statistics of the size of voids—regions without particles—that were not studied previously. The probability of voids is increased compared to a random distribution of particles because preferential concentration of inertial particles implies voids in the deserted regions. Thus voids and preferential concentration are different reflections of the same phenomena. In the limit of tracers with zero inertia, described by the Poisson distribution, the typical void size is of the order of the mean interparticle distance. However, at small but finite inertia, the void size is larger than the mean interparticle distance by a factor that diverges in the continuum limit of infinite number of particles, manifesting strong deviations from the Poisson distribution. Thus, voids are very sensitive to inertia and limits of zero inertia and continuum may not commute. Further, the tail of the probability distribution of the void size is log normal, in contrast with faster than exponential decay of the Poisson distribution. Remarkably, at scales where there is no clustering (so pair-correlation function of concentration splits in the product of averages), there can still be an increase of the void probability so that turbulent voiding is stronger than clustering. These results can find applications in the long-time survival probability of reacting particles where the particle survives in the void of predators. The demonstrated double stochasticity (Poisson with random intensity) of the distribution originates in the two-step formation of fluctuations of the number of particles inside the volume under observation. First, turbulence randomly brings uncorrelated particles below the viscous scale with Poisson-type probability. Then, turbulence compresses the particles inside the observation volume. We confirm the predictions by numerical observations of inertial particle motion in a chaotic Arnold-Beltrami-Childress flow. Our work implies that the particle distribution in arbitrary weakly compressible flow with finite time correlations is a LGCP.