Fri March 1st 2013
Seminar Optimal Stirring for Passive Scalar Mixing
Charles R. Doering


We address the challenge of optimal incompressible stirring to mix an initially inhomogeneous distribution of passive tracers. As a measure for mixing we adopt the H^{-1} norm of the scalar fluctuation field. This 'mix-norm' is equivalent to (the square root of) the variance of a low-pass filtered image of the tracer concentration field, and is a useful gauge even in the absence of molecular diffusion. This mix-norm's vanishing as time progresses is evidence of the stirring flow's mixing property in the sense of ergodic theory. For the case of a periodic spatial domain with a prescribed instantaneous energy or power budget for the stirring, we determine the flow field that instantaneously maximizes the decay of the mix-norm, i.e., the instantaneous optimal stirring --- when such a flow exists. When no such 'steepest descent' stirring exists, we determine the flow that maximizes that rate of increase of the rate of decrease of the norm. This local-in-time stirring strategy is implemented computationally on a benchmark problem and compared to an optimal control approach utilizing a restricted set of flows.


Journal of Fluid Mechanics 675, 465-476 (2011);
Journal of Mathematical Physics 53, 115611 (2012).
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The 10th Complex Motion in Fluids 2021
Max Planck Gesellschaft
Centre for Scientific Computing