Modal analysis of the behavior of inertial particles in turbulence subjected to Stokes drag

Mahdi Esmaily and Ali Mani
Phys. Rev. Fluids 5, 084303 – Published 12 August 2020
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Abstract

The clustering of small heavy inertial particles subjected to Stokes drag in turbulence, which is the concern of present study, is known to be minimal at small and large Stokes number and substantial at St=O(1). This nonmonotonic 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 τQ as τ0 and τ1/2|Q|1/4 as τ, where τ is the particle relaxation time and Q(τ) is the difference between the norm of the rotation- and strain-rate tensors computed along the particle trajectory, which remains negative at all τ in turbulent flows. Additionally, the trajectory crossing is predicted only in hyperbolic flows where Q<0 for sufficiently inertial particles with a τ 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. This model problem also confirms that dispersion in a hyperbolic flow will occur only if particles' trajectory cross. These findings, which stem from our analysis, are corroborated by the direct numerical simulations.

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  • Received 4 April 2020
  • Accepted 8 July 2020

DOI:https://doi.org/10.1103/PhysRevFluids.5.084303

©2020 American Physical Society

Physics Subject Headings (PhySH)

Fluid Dynamics

Authors & Affiliations

Mahdi Esmaily1,2,* and Ali Mani2,3

  • 1Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, New York 14850, USA
  • 2Center for Turbulence Research, Stanford University, Stanford, California 94305, USA
  • 3Department of Mechanical Engineering, Stanford University, Stanford, California 94305, USA

  • *Corresponding author: me399@cornell.edu

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Issue

Vol. 5, Iss. 8 — August 2020

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