Lagrangian particle calculations of distributive mixing: Limitations and applications

A popular computational scheme to assess distributive mixing is to advect Lagrangian particles from one or more small initial clusters, divide the mixing domain into bins, and count the number of particles in each bin. Either the variance among bin counts, or an entropy associated with the bin counts, measures the progress of mixing. We analyze the numerical and physical limitations of such measures, and examines their behavior in chaotic laminar mixing flows. In a time-periodic flow each mixing measure reaches a limiting value, which is either controlled by the number of Lagrangian particles and the number of bins (a numerical limit), or by the presence of islands in the flow (a physical limit). We provide analytical expressions for both limits. The same statistical concepts are used to analyze Poincaré sections of chaotic flows, resulting in a calculation that will detect all islands that cover one or more bins. This provides a way to search for protocols that are highly chaotic (i.e., contain very small islands), and are thus very effective at mixing. These concepts are applied in sample calculations of the sine flow. The T=1.6T=1.6 sine flow, known to be highly chaotic, has small but easily detected islands. Among highly chaotic flows, the location of the initial cluster of Lagrangian particles is as important to the rate of distributive mixing as the choice of flow protocol.