# A modal analysis of the behavior of inertial particles in turbulence

## TL;DR

This paper provides an analytical expression for the Lyapunov exponents predicting inertial particle clustering behavior in turbulence, explaining the non-monotonic trend and the transition from clustering to dispersion.

## Contribution

The study introduces an analytical model for Lyapunov exponents that accurately predicts inertial particle clustering and crossing in turbulence, supported by simulations.

## Key findings

- Lyapunov exponents predict clustering at St=O(1)
- Trajectory crossing occurs in hyperbolic flows with Q<0
- Clustering rate is bounded and highly nonlinear in hyperbolic flows

## Abstract

The clustering of small heavy inertial particles subjected to Stokes drag in turbulence is known to be minimal at small and large Stokes number and substantial at $\rm St = \mathcal O(1)$. This non-monotonic trend, which has been shown computationally and experimentally, is yet to be explained analytically. In this study, we obtain an analytical expression for the Lyapunov exponents that quantitatively predicts this trend. The sum of the exponents, which is the normalized rate of change of the signed-volume of a small cloud of particles, is correctly predicted to be negative and positive at small and large Stokes numbers, respectively, asymptoting to $\tau Q$ as $\tau \to 0$ and $\tau^{-1/2} |Q|^{1/4}$ as $\tau \to \infty$, where $\tau$ is the particle relaxation time and $Q(\tau)$ is the difference between the norm of the rotation- and strain-rate tensors computed along the particle trajectory. Additionally, the trajectory crossing is predicted only in hyperbolic flows where $Q<0$ for sufficiently inertial particles with a $\tau$ that scales with $|Q|^{-1/2}$. Following the onset of crossovers, a transition from clustering to dispersion is predicted correctly. We show these behaviors are not unique to three-dimensional isotropic turbulence and can be reproduced closely by a one-dimensional mono-harmonic flow, which appears as a fundamental canonical problem in the study of particle clustering. Analysis of this one-dimensional canonical flow shows that the rate of clustering, quantified as the product of the Lyapunov exponent and particle relaxation time, is bounded by $-1/2$, behaving with extreme nonlinearity in the hyperbolic flows and always remaining positive in the elliptic flows. These findings, which are stemmed from our analysis, are corroborated by the direct numerical simulations.