Optimal stirring strategies for passive scalar mixing

By Zhi Lin, Jean-Luc Thiffeault, Charles R. Doering. Journal of Fluid Mechanics, 2010

156 citations recorded by the source corpus.

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.

Subfields: Computer Networks and Communications, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/d4dcb53cd9d1fbdca77f46b6ccfac74b44dceb61

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Citations in the corpus

Frontiers of chaotic advection (2014, methodology): “Lin et al. (2011b) advocated the use of an H−1 norm, but H−1/2 seems to have better physical motivation, as it incorporates the idea of equal distributions in equal volumes for the perfect mixture.”
Exponential self-similar mixing and loss of regularity for continuity equations (2014, background): “This fact means that it is possible to have perfect mixing in finite time, as already observed in [10] and established in [11] for s = 0, building on an example from [7, 5].”
Enhanced Dissipation, Hypoellipticity, and Anomalous Small Noise Inviscid Limits in Shear Flows (2015, methodology): “8) The H−1 norm is often used to quantify mixing [29, 38] as it provides a kind of averaged measure of the characteristic length-scale of the solution.”
Using multiscale norms to quantify mixing and transport (2011, methodology): “This is a strategy that has been used by many authors for the case of variance (q = 0) [24] and for negative Sobolev norms [42, 54, 58], as we discussed in section 5.”