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Mixing enhancement in binary fluids using optimised stirring strategies

Published online by Cambridge University Press:  24 July 2020

M. F. Eggl Open the ORCID record for M. F. Eggl [Opens in a new window]  and
Peter J. Schmid Open the ORCID record for Peter J. Schmid [Opens in a new window]
M. F. Eggl*
Affiliation:
Department of Mathematics, Imperial College London, LondonSW7 2AZ, UK
Peter J. Schmid
Affiliation:
Department of Mathematics, Imperial College London, LondonSW7 2AZ, UK
*
†Email address for correspondence: meggl@princeton.edu
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Abstract

Mixing of binary fluids by moving stirrers is a commonplace process in many industrial applications, where even modest improvements in mixing efficiency could translate into considerable power savings or enhanced product quality. We propose a gradient-based nonlinear optimisation scheme to minimise the mix-norm of a passive scalar. The velocities of two cylindrical stirrers, moving on concentric circular paths inside a circular container, represent the control variables, and an iterative direct–adjoint algorithm is employed to arrive at enhanced mixing results. The associated stirring protocol is characterised by a complex interplay of vortical structures, generated and promoted by the stirrers’ action. Full convergence of the optimisation process requires constraints that penalise the acceleration of the moving bodies. Under these conditions, considerable mixing enhancement can be accomplished, even though an optimum cannot be guaranteed due to the non-convex nature of the optimisation problem. Various challenges and extensions of our approach are discussed.

JFM classification

Flow Control: Mixing enhancement Mathematical Foundations: Computational methods
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 899 , 25 September 2020 , A24
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

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Eggl and Schmid supplementary movie 1

Mixing optimisation based on only energy constraints for the stirrers. The time horizon for applying control is Tcontrol = 1: Shown are iso-contours of the passive scalar. The optimisation algorithm includes information over a time window of Tinfo = 8:

Download Eggl and Schmid supplementary movie 1(Video)
Video 34.2 MB

Eggl and Schmid supplementary movie 2

Mixing optimisation based on only energy constraints for the stirrers. The time horizon for applying control is Tcontrol = 8: Shown are iso-contours of the passive scalar. The optimisation algorithm includes information over a time window of Tinfo = 8:

Download Eggl and Schmid supplementary movie 2(Video)
Video 18.1 MB

Eggl and Schmid supplementary movie 3

Mixing optimisation based on energy and velocity constraints for the stir- rers. The time horizon for applying control is Tcontrol = 1: Shown are iso-contours of the passive scalar. The optimisation algorithm includes information over a time window of Tinfo = 8:

Download Eggl and Schmid supplementary movie 3(Video)
Video 33 MB

Eggl and Schmid supplementary movie 4

Mixing optimisation based on energy and velocity constraints for the stir- rers. The time horizon for applying control is Tcontrol = 8: Shown are iso-contours of the passive scalar. The optimisation algorithm includes information over a time window of Tinfo = 8:

Download Eggl and Schmid supplementary movie 4(Video)
Video 34.1 MB

Eggl and Schmid supplementary movie 5

Mixing optimisation based on energy, velocity and acceleration constraints for the stirrers. The time horizon for applying control is Tcontrol = 1: Shown are iso-contours of the passive scalar. The optimisation algorithm includes information over a time window of Tinfo = 8:

Download Eggl and Schmid supplementary movie 5(Video)
Video 29.4 MB

Eggl and Schmid supplementary movie 6

Mixing optimisation based on energy, velocity and acceleration constraints for the stirrers. The time horizon for applying control is Tcontrol = 8: Shown are iso-contours of the passive scalar. The optimisation algorithm includes information over a time window of Tinfo = 8:

Download Eggl and Schmid supplementary movie 6(Video)
Video 27.2 MB