Analysis of the Correlation Dimension for Inertial Particles

TL;DR

This paper develops a new analytical approach to calculate the correlation dimension of inertial particles in complex flows, providing exact coefficients for a perturbation series and connecting stochastic process propagators with fractal clustering.

Contribution

It introduces a novel implicit equation linking the correlation dimension to a stochastic propagator, enabling perturbative expansion and exact coefficient calculation.

Findings

Derived the first 33 coefficients of the correlation dimension perturbation series.

Connected the correlation dimension calculation to a stochastic propagator formalism.

Provided a method to analyze clustering of inertial particles in complex flows.

Abstract

We obtain an implicit equation for the correlation dimension which describes clustering of inertial particles in a complex flow onto a fractal measure. Our general equation involves a propagator of a nonlinear stochastic process in which the velocity gradient of the fluid appears as additive noise. When the long-time limit of the propagator is considered our equation reduces to an existing large-deviation formalism, from which it is difficult to extract concrete results. In the short-time limit, however, our equation reduces to a solvability condition on a partial differential equation. We show how this approach leads to a perturbative expansion of the correlation dimension, for which the coefficients can be obtained exactly and in principle to any order. We derive the perturbation series for the correlation dimension of inertial particles suspended in three-dimensional spatially smooth random flows with white-noise time correlations, obtaining the first 33 non-zero coefficients exactly.