Correlation Dimension of Inertial Particles in Random Flows

TL;DR

This paper develops a new method to calculate the correlation dimension of inertial particles in random flows using an integral over a propagator, with solutions obtained via a power series expansion in inertial strength.

Contribution

Introduces an implicit integral equation for correlation dimension and applies it to inertial particles, deriving solutions through a power series in inertial effects.

Findings

Correlation dimension can be computed from a propagator integral.

Short-time limit simplifies the calculation to a PDE solution.

Power series expansion effectively captures inertial effects.

Abstract

We obtain an implicit equation for the correlation dimension of dynamical systems in terms of an integral over a propagator. We illustrate the utility of this approach by evaluating the correlation dimension for inertial particles suspended in a random flow. In the limit where the correlation time of the flow field approaches zero, taking the short-time limit of the propagator enables the correlation dimension to be determined from the solution of a partial differential equation. We develop the solution as a power series in a dimensionless parameter which represents the strength of inertial effects.