Optimal stirring strategies for passive scalar mixing

TL;DR

This paper investigates optimal stirring strategies to enhance passive scalar mixing, establishing theoretical limits and deriving flow fields that maximize mixing efficiency based on the $H^{-1}$ norm, with numerical validation.

Contribution

It introduces a framework for determining optimal stirring flows that maximize mixing efficiency using the $H^{-1}$ norm, including theoretical bounds and numerical methods.

Findings

Derived absolute limits on mixing as a function of time.

Identified flow fields that maximize decay of the mixing measure.

Validated strategies through numerical implementation and comparison.

Abstract

We address the challenge of optimal incompressible stirring to mix an initially inhomogeneous distribution of passive tracers. As a quantitative measure of mixing we adopt the $H^{-1}$ norm of the scalar fluctuation field, equivalent to the (square-root of the) variance of a low-pass filtered image of the tracer concentration field. First we establish that this is a useful gauge even in the absence of molecular diffusion: its vanishing as $t --> \infty$ is evidence of the stirring flow's mixing properties in the sense of ergodic theory. Then we derive absolute limits on the total amount of mixing, as a function of time, on a periodic spatial domain with a prescribed instantaneous stirring energy or stirring power budget. We subsequently determine the flow field that instantaneously maximizes the decay of this mixing measure---when such a flow exists. When no such `steepest descent' flow exists (a possible but non-generic situation) we determine the flow that maximizes the growth rate of the $H^{-1}$ norm's decay rate. This local-in-time optimal stirring strategy is implemented numerically on a benchmark problem and compared to an optimal control approach using a restricted set of flows. Some significant challenges for analysis are outlined.