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

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 Péclet 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.

References in corpus (4)

• Using multiscale norms to quantify mixing and transport
• Optimal stirring strategies for passive scalar mixing
• Multiscale Mixing Efficiencies for Steady Sources
• Optimizing the Source Distribution in Fluid Mixing

Cited by in corpus (4)

• Cascades and transitions in turbulent flows
• Using multiscale norms to quantify mixing and transport
• Invariant measures for passive scalars in the small noise inviscid limit
• Weak-strong clustering transition in renewing compressible flows