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Bounding the scalar dissipation scale for mixing flows in the presence of sources

Published online by Cambridge University Press:  28 October 2011

A. Alexakis*
Affiliation:
Laboratoire de Physique Statistique, CNRS UMR 8550, Ecole Normale Supérieure, 24 rue Lhomond, Paris, 75005, France
A. Tzella*
Affiliation:
Laboratoire de Météorologie Dynamique, CNRS UMR 8539, Ecole Normale Supérieure, 24 rue Lhomond, Paris, 75005, France
*
Email addresses for correspondence: alexakis@lps.ens.fr, tzella@lmd.ens.fr
Email addresses for correspondence: alexakis@lps.ens.fr, tzella@lmd.ens.fr

Abstract

We investigate the mixing properties of scalars stirred by spatially smooth, divergence-free flows and maintained by a steady source–sink distribution. We focus on the spatial variation of the scalar field, described by the dissipation wavenumber, , that we define as a function of the mean variance of the scalar and its gradient. We derive a set of upper bounds that for large Péclet number () yield four distinct regimes for the scaling behaviour of , one of which corresponds to the Batchelor regime. The transition between these regimes is controlled by the value of and the ratio , where and are, respectively, the characteristic length scales of the velocity and source fields. A fifth regime is revealed by homogenization theory. These regimes reflect the balance between different processes: scalar injection, molecular diffusion, stirring and bulk transport from the sources to the sinks. We verify the relevance of these bounds by numerical simulations for a two-dimensional, chaotically mixing example flow and discuss their relation to previous bounds. Finally, we note some implications for three-dimensional turbulent flows.

Type
Papers
Copyright
Copyright © Cambridge University Press 2011

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