Eddy diffusivity of quasi-neutrally-buoyant inertial particles

TL;DR

This paper develops a first-principles perturbative approach to compute large-scale transport properties like eddy diffusivity and terminal velocity of quasi-neutrally-buoyant inertial particles in incompressible flows, without assuming scale separation.

Contribution

It introduces a novel perturbative method around the added-mass factor near unity, deriving explicit formulas for inertial particle transport properties in general and parallel flows.

Findings

Explicit formulas for terminal velocity and eddy diffusivity.

Analytic solutions for parallel flows.

Insights into particle behavior at Stokes numbers near unity.

Abstract

We investigate the large-scale transport properties of quasi-neutrally-buoyant inertial particles carried by incompressible zero-mean periodic or steady ergodic flows. We show how to compute large-scale indicators such as the inertial-particle terminal velocity and eddy diffusivity from first principles in a perturbative expansion around the limit of added-mass factor close to unity. Physically, this limit corresponds to the case where the mass density of the particles is constant and close in value to the mass density of the fluid which is also constant. Our approach differs from the usual over-damped expansion inasmuch we do not assume a separation of time scales between thermalization and small-scale convection effects. For general incompressible flows, we derive closed-form cell equations for the auxiliary quantities determining the terminal velocity and effective diffusivity. In the special case of parallel flows these equations admit explicit analytic solution. We use parallel flows to show that our approach enables to shed light onto the behavior of terminal velocity and effective diffusivity for Stokes numbers of the order of unity.