Stirring up trouble: Multi-scale measures of mixing for steady scalar sources

We study the evolution of passive scalar fields maintained by steady but spatially inhomogeneous sources and sinks, and stirred by an incompressible flow. The effectiveness of a flow field to enhance mixing over molecular diffusion is measured by the suppression of the space-time averaged scalar variance, the gradient variance (stressing small scales), and the inverse gradient variance (focusing on large scale fluctuations). Ratios of these variances without stirring to the corresponding variances with stirring provide non-dimensional measures of the "mixing efficiency" of the flow on different scales. In this work we derive rigorous estimates on these multi-scale mixing efficiencies for a variety of source distributions with general stirring flows including statistically homogeneous and isotropic velocity fields, and compare them with direct numerical simulations and exact calculations for sample problems.

This is joint work with Matt Finn and Jean-Luc Thiffeault from Imperial College, and Tiffany Shaw from the University of Toronto.