How rapidly is a passive scalar mixed within closed streamlines?

Abstract

The homogenization of a passive ‘tracer’ in a flow with closed mean streamlines occurs in two stages: first, a rapid phase dominated by shear-augmented diffusion over a time ≈ P ^1/3( L/ U), where the Péclet number P= LU/κ ( L,U and κ are lengthscale, velocity scale and diffusivity), in which initial values of the tracer are replaced by their (generalized) average about a streamline; second, a slow phase requiring the full diffusion time ≈ L ²/κ. The diffusion problem for the second phase, where tracer isopleths are held to streamlines by shear diffusion, involves a generalized diffusivity which is proportional to κ, but exceeds it if the streamlines are not circular. Expressions are also given for flow fields that are oscillatory rather than steady.