Ergodic and non-ergodic clustering of inertial particles

TL;DR

This paper develops a new series expansion theory to analyze clustering of inertial particles in mixing flows, capturing both ergodic and non-ergodic mechanisms and their interplay at finite Kubo and Stokes numbers.

Contribution

The paper introduces a novel series expansion approach in Kubo number to unify and analyze ergodic and non-ergodic clustering mechanisms of inertial particles.

Findings

The theory accurately predicts clustering for Ku < 0.2 across St values.

Ergodic 'multiplicative amplification' significantly influences clustering at Ku ~ St ~ 1.

Non-ergodic 'centrifuge' effect dominates at very small St.

Abstract

We compute the fractal dimension of clusters of inertial particles in mixing flows at finite values of Kubo (Ku) and Stokes (St) numbers, by a new series expansion in Ku. At small St, the theory includes clustering by Maxey's non-ergodic 'centrifuge' effect. In the limit of St to infinity and Ku to zero (so that Ku^2 St remains finite) it explains clustering in terms of ergodic 'multiplicative amplification'. In this limit, the theory is consistent with the asymptotic perturbation series in [Duncan et al., Phys. Rev. Lett. 95 (2005) 240602]. The new theory allows to analyse how the two clustering mechanisms compete at finite values of St and Ku. For particles suspended in two-dimensional random Gaussian incompressible flows, the theory yields excellent results for Ku < 0.2 for arbitrary values of St; the ergodic mechanism is found to contribute significantly unless St is very small. For higher values of Ku the new series is likely to require resummation. But numerical simulations show that for Ku ~ St ~ 1 too, ergodic 'multiplicative amplification' makes a substantial contribution to the observed clustering.