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

TL;DR

This paper derives bounds on the scalar dissipation scale in mixing flows with sources, revealing multiple regimes depending on flow parameters and confirming these bounds through numerical simulations.

Contribution

It introduces new upper bounds on the dissipation wavenumber, identifying distinct scaling regimes and connecting them to flow parameters and homogenization theory.

Findings

Four distinct regimes for the dissipation wavenumber scaling.

Numerical simulations verify the theoretical bounds.

Identification of a regime described by homogenization theory.

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 {\it dissipation wavenumber}, $k_d$, 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 P\'eclet number ($\Pe\gg1$) yield four distinct regimes for the scaling behavior of $k_d$, one of which corresponds to the Batchelor regime. The transition between these regimes is controlled by the value of $\Pe$ and the ratio $\rho=\ell_u/\ell_s$, where $\ell_u$ and $\ell_s$ are respectively, the characteristic lengthscales 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.