# Scale Dependence of Multiplier Distributions for Particle Concentration, Enstrophy and Dissipation in the Inertial Range of Homogeneous Turbulence

## TL;DR

This study uses DNS data to analyze how cascade multipliers for particle concentration, enstrophy, and dissipation depend on scale and local Stokes number, providing insights into turbulent clustering at high Reynolds numbers.

## Contribution

It introduces a method to correct small particle number effects and demonstrates scale-dependent multiplier PDFs that collapse when scaled with local Stokes number, advancing turbulence cascade modeling.

## Key findings

- Multiplier PDFs are scale dependent but collapse with local Stokes number.
- Dissipation and enstrophy multiplier PDFs reach an asymptote at small scales.
- Predicted RDFs agree with high-Reynolds-number inertial range theory.

## Abstract

Turbulent flows preferentially concentrate inertial particles depending on their stopping time or Stokes number, which can lead to significant spatial variations in the particle concentration. Cascade models are one way to describe this process in statistical terms. Here, we use a direct numerical simulation (DNS) dataset of homogeneous, isotropic turbulence to determine probability distribution functions (PDFs) for cascade multipliers, which determine the ratio by which a property is partitioned into sub-volumes as an eddy is envisioned to decay into smaller eddies. We present a technique for correcting effects of small particle numbers in the statistics. We determine multiplier PDFs for particle number, flow dissipation, and enstrophy, all of which are shown to be scale dependent. However, the particle multiplier PDFs collapse when scaled with an appropriately defined local Stokes number. As anticipated from earlier works, dissipation and enstrophy multiplier PDFs reach an asymptote for sufficiently small spatial scales. From the DNS measurements, we derive a cascade model that is used it to make predictions for the radial distribution function (RDF) for arbitrarily high Reynolds numbers, $Re$, finding good agreement with the asymptotic, infinite $Re$ inertial range theory of Zaichik and Alipchenkov [New Journal of Physics 11, 103018 (2009)]. We discuss implications of these results for the statistical modeling of the turbulent clustering process in the inertial range for high Reynolds numbers inaccessible to numerical simulations.