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Optimal stirring strategies for passive scalar mixing

Published online by Cambridge University Press:  10 March 2011

ZHI LIN ,
JEAN-LUC THIFFEAULT  and
CHARLES R. DOERING
ZHI LIN
Affiliation:
Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, MN 55455, USA
JEAN-LUC THIFFEAULT
Affiliation:
Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA
CHARLES R. DOERING*
Affiliation:
Department of Mathematics, Department of Physics and Center for the Study of Complex Systems, University of Michigan, Ann Arbor, MI 48109, USA
*
†Email address for correspondence: doering@umich.edu
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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 → ∞ 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.

JFM classification

Mathematical Foundations: Mathematical Foundations Mixing: Mixing Nonlinear Dynamical Systems: Nonlinear Dynamical Systems
Type
Papers
Information
Journal of Fluid Mechanics , Volume 675 , 25 May 2011 , pp. 465 - 476
Copyright
Copyright © Cambridge University Press 2011

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References

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