Analysis of the Correlation Dimension for Inertial Particles

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.

22 pages, 3 figures

References in corpus (7)

• The large deviation approach to statistical mechanics
• Heavy particle concentration in turbulence at dissipative and inertial scales
• Lyapunov exponents of heavy particles in turbulence
• Unmixing in Random Flows
• Stochastic suspensions of heavy particles
• Stokes trapping and planet formation
• Advective collisions

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• Fractal catastrophes
• Fractal dimensions and trajectory crossings in correlated random walks