Characterizing scale dependence of effective diffusion driven by fluid flows

TL;DR

This paper investigates how effective diffusion in fluid flows depends on scale and Péclet number without assuming scale separation, introducing a new method to characterize diffusivity through spatio-temporal evolution analysis.

Contribution

A novel method is proposed to determine effective diffusivity without relying on scale separation assumptions, applicable to complex flow regimes.

Findings

Effective diffusivity varies with scale and flow type.

Discrepancies from classical limits are explained by flow structure and oscillations.

Method successfully captures scale dependence in gyre and shear flows.

Abstract

We study the scale dependence of effective diffusion of fluid tracers, specifically, its dependence on the P\'{e}clet number, a dimensionless parameter of the ratio between advection and molecular diffusion. Here, we address the case that length and time scales on which the effective diffusion can be described are not separated from those of advection and molecular diffusion. For this, we propose a new method for characterizing the effective diffusivity without relying on the scale separation. For a given spatial domain inside which the effective diffusion can emerge, a time constant related to the diffusion is identified by considering the spatio-temporal evolution of a test advection-diffusion equation, where its initial condition is set at a pulse function. Then, the value of effective diffusivity is identified by minimizing the $L_\infty$ distance between solutions of the above test equation and the diffusion one with mean drift. With this method, for time-independent gyre and time-periodic shear flows, we numerically show the scale dependence of the effective diffusivity and its discrepancy from the classical limits that were derived on the assumption of the scale separation. The kinematic origins of the discrepancy are revealed as the development of the molecular diffusion across flow cells of the gyre and as the suppression of the drift motion due to a temporal oscillation in the shear.