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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: [email protected]
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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

REFERENCES

Angot, P., Bruneau, C.-H. & Fabrie, P. 1999 A penalization method to take into account obstacles in incompressible viscous flows. Numer. Math. 81 (4), 497–520.CrossRefGoogle Scholar
Aref, H. 1984 Stirring by chaotic advection. J. Fluid Mech. 143, 1–21.CrossRefGoogle Scholar
Balogh, A., Aamo, O. M. & Krstic, M. 2005 Optimal mixing enhancement in 3-d pipe flow. IEEE Trans. Control Syst. Technol. 13, 27–41.Google Scholar
Blonigan, P. & Wang, Q. 2012 Least-squares shadowing for chaotic nonlinear dynamical systems. J.Comput. Phys. 34, 1–2.Google Scholar
Blumenthal, R. S., Tangirala, A. K., Sujith, R. I. & Polifke, W. 2017 A systems perspective on non-normality in low-order thermoacoustic models: full norms, semi-norms and transient growth. Int. J. Spray Comb. Dyn. 9 (1), 19–43.CrossRefGoogle Scholar
Eggl, M. F. & Schmid, P. J. 2018 A gradient-based framework for maximizing mixing in binary fluids. J. Comput. Phys. 368, 131–153.CrossRefGoogle Scholar
Eggl, M. F. & Schmid, P. J. 2020 Shape optimization of stirring rods for mixing binary fluids. IMA J.Appl. Maths, hxaa012.CrossRefGoogle Scholar
Engels, T., Kolomenskiy, D., Schneider, K. & Sesterhenn, J. 2015 FLUSI: a novel parallel simulation tool for flapping insect flight using a Fourier method with volume penalization. SIAM J. Sci. Comput. 38 (6), S03–S24.Google Scholar
Finn, M. D. & Thiffeault, J.-L. 2011 Topological optimization of rod-stirring devices. SIAM Rev. 53 (4), 723–743.CrossRefGoogle Scholar
Foures, D. P. G., Caulfield, C. P. & Schmid, P. J. 2012 Variational framework for flow optimization using seminorm constraints. Phys. Rev. E 86 (2), 026306.CrossRefGoogle ScholarPubMed
Foures, D. P. G., Caulfield, C. P. & Schmid, P. J. 2014 Optimal mixing in plane Poiseuille flow. J.Fluid Mech. 748, 241–277.CrossRefGoogle Scholar
Galletti, C., Arcolini, G., Brunazzi, E. & Mauri, R. 2015 Mixing of binary fluids with composition-dependent viscosity in a T-shaped micro-device. Chem. Engng Sci. 123, 300–310.CrossRefGoogle Scholar
Griewank, A. & Walther, A. 2000 Algorithm 799: revolve: an implementation of checkpointing for the reverse or adjoint mode of computational differentiation. ACM Trans. Math. Softw. 26 (1), 19–45.CrossRefGoogle Scholar
Gubanov, O. & Cortelezzi, L. 2010 Towards the design of an optimal mixer. J. Fluid Mech. 651, 27–53.CrossRefGoogle Scholar
Horn, R. A. & Johnson, C. R. 2012 Matrix Analysis. Cambridge University Press.CrossRefGoogle Scholar
Hou, T. Y. & Li, R. 2007 Computing nearly singular solutions using pseudo-spectral methods. J. Comput. Phys. 226, 379–397.CrossRefGoogle Scholar
Kolomenskiy, D. & Schneider, K. 2009 A Fourier spectral method for the Navier–Stokes equations with volume penalization for moving solid obstacles. J. Comput. Phys. 228 (16), 5687–5709.CrossRefGoogle Scholar
Lin, Z., Thiffeault, J.-L. & Doering, C. 2011 Optimal stirring strategies for passive scalar mixing. J.Fluid Mech. 675, 465–476.CrossRefGoogle Scholar
Liu, W. 2008 Mixing enhancement by optimal flow advection. SIAM J. Control Optim. 47 (2), 624–638.CrossRefGoogle Scholar
Marcotte, F. & Caulfield, C. P. 2018 Optimal mixing in two-dimensional stratified plane Poiseuille flow at finite Péclet and Richardson numbers. J. Fluid Mech. 853, 359–385.CrossRefGoogle Scholar
Mathew, G., Mezic, I., Grivopoulos, S., Vaidya, U. & Petzold, L. 2007 Optimal control of mixing in stokes fluid flows. J. Fluid Mech. 580, 261–281.CrossRefGoogle Scholar
Mathew, G., Mezic, I. & Petzold, L. 2005 A multiscale measure for mixing. Physica D 211, 23–46.CrossRefGoogle Scholar
Orsi, G., Galletti, C., Brunazzi, E. & Mauri, R. 2013 Mixing of two miscible liquids in T-shaped microdevices. Chem. Engng Trans. 32, 1471–1476.Google Scholar
Ottino, J. M. 1989 The Kinematics of Mixing: Stretching, Chaos, and Transport. Cambridge University Press.Google Scholar
Paul, E. L., Atiemo-Obeng, V. A. & Kresta, S. M. 2003 Handbook of Industrial Mixing: Science and Practice. Wiley-Blackwell.CrossRefGoogle Scholar
Pekurovsky, D. 2012 P3dfft: a framework for parallel computations of fourier transforms in three dimensions. SIAM J. Sci. Comput. 34 (4), C192–C209.CrossRefGoogle Scholar
Spencer, R. & Wiley, R. 1951 The mixing of very viscous liquid. J. Colloid Sci. 6 (2), 133–145.CrossRefGoogle Scholar
Sturman, R., Ottino, J. M. & Wiggins, S. 2006 The Mathematical Foundations of Mixing. Cambridge University Press.CrossRefGoogle Scholar
Thiffeault, J.-L. 2012 Using multiscale norms to quantify mixing and transport. Nonlinearity 25 (2), R1.CrossRefGoogle Scholar
Uhl, V. 2012 Mixing: Theory and Practice. Elsevier.Google Scholar
Vermach, L. & Caulfield, C. P. 2018 Optimal mixing in three-dimensional plane Poiseuille flow at high Péclet number. J. Fluid Mech. 850, 875–923.CrossRefGoogle Scholar

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