×

zbMATH — the first resource for mathematics

Mixing enhancement in binary fluids using optimised stirring strategies. (English) Zbl 07235486
Summary: 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.
MSC:
76T99 Multiphase and multicomponent flows
76F25 Turbulent transport, mixing
Software:
revolve
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Angot, P., Bruneau, C.-H. & Fabrie, P.1999A penalization method to take into account obstacles in incompressible viscous flows. Numer. Math.81 (4), 497-520. · Zbl 0921.76168
[2] Aref, H.1984Stirring by chaotic advection. J. Fluid Mech.143, 1-21. · Zbl 0559.76085
[3] Balogh, A., Aamo, O. M. & Krstic, M.2005Optimal mixing enhancement in 3-d pipe flow. IEEE Trans. Control Syst. Technol.13, 27-41.
[4] Blonigan, P. & Wang, Q.2012Least-squares shadowing for chaotic nonlinear dynamical systems. J.Comput. Phys.34, 1-2.
[5] Blumenthal, R. S., Tangirala, A. K., Sujith, R. I. & Polifke, W.2017A 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.
[6] Eggl, M. F. & Schmid, P. J.2018A gradient-based framework for maximizing mixing in binary fluids. J. Comput. Phys.368, 131-153. · Zbl 1392.76019
[7] Eggl, M. F. & Schmid, P. J.2020Shape optimization of stirring rods for mixing binary fluids. IMA J.Appl. Maths, hxaa012.
[8] Engels, T., Kolomenskiy, D., Schneider, K. & Sesterhenn, J.2015FLUSI: a novel parallel simulation tool for flapping insect flight using a Fourier method with volume penalization. SIAM J. Sci. Comput.38 (6), S03-S24. · Zbl 1348.76106
[9] Finn, M. D. & Thiffeault, J.-L.2011Topological optimization of rod-stirring devices. SIAM Rev.53 (4), 723-743. · Zbl 1290.76031
[10] Foures, D. P. G., Caulfield, C. P. & Schmid, P. J.2012Variational framework for flow optimization using seminorm constraints. Phys. Rev. E86 (2), 026306.
[11] Foures, D. P. G., Caulfield, C. P. & Schmid, P. J.2014Optimal mixing in plane Poiseuille flow. J.Fluid Mech.748, 241-277. · Zbl 1416.76036
[12] Galletti, C., Arcolini, G., Brunazzi, E. & Mauri, R.2015Mixing of binary fluids with composition-dependent viscosity in a T-shaped micro-device. Chem. Engng Sci.123, 300-310.
[13] Griewank, A. & Walther, A.2000Algorithm 799: revolve: an implementation of checkpointing for the reverse or adjoint mode of computational differentiation. ACM Trans. Math. Softw.26 (1), 19-45. · Zbl 1137.65330
[14] Gubanov, O. & Cortelezzi, L.2010Towards the design of an optimal mixer. J. Fluid Mech.651, 27-53. · Zbl 1189.76168
[15] Horn, R. A. & Johnson, C. R.2012Matrix Analysis. Cambridge University Press.
[16] Hou, T. Y. & Li, R.2007Computing nearly singular solutions using pseudo-spectral methods. J. Comput. Phys.226, 379-397. · Zbl 1310.76127
[17] Kolomenskiy, D. & Schneider, K.2009A Fourier spectral method for the Navier-Stokes equations with volume penalization for moving solid obstacles. J. Comput. Phys.228 (16), 5687-5709. · Zbl 1169.76045
[18] Lin, Z., Thiffeault, J.-L. & Doering, C.2011Optimal stirring strategies for passive scalar mixing. J.Fluid Mech.675, 465-476. · Zbl 1241.76361
[19] Liu, W.2008Mixing enhancement by optimal flow advection. SIAM J. Control Optim.47 (2), 624-638. · Zbl 1158.76043
[20] Marcotte, F. & Caulfield, C. P.2018Optimal mixing in two-dimensional stratified plane Poiseuille flow at finite Péclet and Richardson numbers. J. Fluid Mech.853, 359-385. · Zbl 1415.76215
[21] Mathew, G., Mezic, I., Grivopoulos, S., Vaidya, U. & Petzold, L.2007Optimal control of mixing in stokes fluid flows. J. Fluid Mech.580, 261-281. · Zbl 1275.76088
[22] Mathew, G., Mezic, I. & Petzold, L.2005A multiscale measure for mixing. Physica D211, 23-46. · Zbl 1098.37067
[23] Orsi, G., Galletti, C., Brunazzi, E. & Mauri, R.2013Mixing of two miscible liquids in T-shaped microdevices. Chem. Engng Trans.32, 1471-1476.
[24] Ottino, J. M.1989The Kinematics of Mixing: Stretching, Chaos, and Transport. Cambridge University Press. · Zbl 0721.76015
[25] Paul, E. L., Atiemo-Obeng, V. A. & Kresta, S. M.2003Handbook of Industrial Mixing: Science and Practice. Wiley-Blackwell.
[26] Pekurovsky, D.2012P3dfft: a framework for parallel computations of fourier transforms in three dimensions. SIAM J. Sci. Comput.34 (4), C192-C209. · Zbl 1253.65205
[27] Spencer, R. & Wiley, R.1951The mixing of very viscous liquid. J. Colloid Sci.6 (2), 133-145.
[28] Sturman, R., Ottino, J. M. & Wiggins, S.2006The Mathematical Foundations of Mixing. Cambridge University Press. · Zbl 1111.37300
[29] Thiffeault, J.-L.2012Using multiscale norms to quantify mixing and transport. Nonlinearity25 (2), R1. · Zbl 1325.37007
[30] Uhl, V.2012Mixing: Theory and Practice. Elsevier.
[31] Vermach, L. & Caulfield, C. P.2018Optimal mixing in three-dimensional plane Poiseuille flow at high Péclet number. J. Fluid Mech.850, 875-923. · Zbl 1415.76304
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.