Reynolds number effects on mixing due to topological chaos

TL;DR

This study investigates how the Reynolds number influences fluid mixing efficiency in 2D laminar flows, revealing that topological chaos can enhance mixing within specific Reynolds number ranges, with complex behaviors observed for different stirring protocols.

Contribution

It provides a numerical analysis of the Reynolds number's impact on mixing efficacy of topologically complex versus simple stirring protocols in laminar flows.

Findings

Pseudo-Anosov protocol outperforms finite-order protocols within certain Reynolds ranges.

Effective mixing area increases with Reynolds number for finite-order protocols.

Pseudo-Anosov protocol shows non-monotonic mixing behavior as Reynolds number varies.

Abstract

Topological chaos has emerged as a powerful tool to investigate fluid mixing. While this theory can guarantee a lower bound on the stretching rate of certain material lines, it does not indicate what fraction of the fluid actually participates in this minimally mandated mixing. Indeed, the area in which effective mixing takes place depends on physical parameters such as the Reynolds number. To help clarify this dependency, we numerically simulate the effects of a batch stirring device on a 2D incompressible Newtonian fluid in the laminar regime. In particular, we calculate the finite time Lyapunov exponent (FTLE) field for three different stirring protocols, one topologically complex (pseudo-Anosov) and two simple (finite-order), over a range of viscosities. After extracting appropriate measures indicative of both the amount of mixing and the area of effective mixing from the FTLE field, we see a clearly defined Reynolds number range in which the relative efficacy of the pseudo-Anosov protocol over the finite-order protocols justifies the application of topological chaos. More unexpectedly, we see that while the measures of effective mixing area increase with increasing Reynolds number for the finite-order protocols, they actually exhibit non-monotonic behavior for the pseudo-Anosov protocol.