Abstract
We present a novel experiment on interfacial wave dynamics in orbitally shaken cylindrical vessels containing two and three fluid layers. The experiment was designed as a hydrodynamical model for liquid metal batteries. It is intended to shed new light into some aspects of the very similar rotational wave motion emerging due to the metal pad roll instability, as the viscous damping behavior or the contact line dynamics. Both issues can be important to better predict instability onsets for upcoming liquid metal batteries and lab-size experiments. Different options are presented to realize stable and measurable multi-layer stratifications. We introduce a new acoustic measurement procedure allowing to reconstruct wave amplitudes also in opaque liquids by tracking ultrasonic pulse echoes reflected at the interfaces. Measurements of resonance curves and phase shifts were conducted for varying interface positions. A strong influence of the top and bottom walls was observed, considerably reducing wave amplitudes and eigenfrequencies, when the interface is getting close. Finally, measured resonance curves were successfully compared with an existing forced wave theory that we extended to two-layer interfacial waves. The comparison stresses the importance to carefully control the boundary condition at the contact line.
Graphical abstract
Similar content being viewed by others
References
Abramson HN (1966) The dynamic behavior of liquids in moving containers, with applications to space vehicle technology. Tech. Rep. SP-106, NASA
Alpresa P, Sherwin S, Weinberg P, van Reeuwijk M (2018a) Orbitally shaken shallow fluid layers. I. Regime classification. Phys Fluids 30:032107
Alpresa P, Sherwin S, Weinberg P, van Reeuwijk M (2018b) Orbitally shaken shallow fluid layers II. An improved wall shear stress model. Phys Fluids 30:032108
Batson W, Zoueshtiagh F, Narayanan R (2013) The Faraday threshold in small cylinders and the sidewall non-ideality. J Fluid Mech 729:496–523
Bojarevics V, Romerio MV (1994) Long waves instability of liquid metal-electroyte interface in aluminium electrolysis cell: a generalization of Sele’s criterion. Eur J Mech B Fluids 13:33–56
Bojarevics V, Tucs A (2017) MHD of large scale liquid metal batteries. Light Metals 2017:687–692
Bouvard J, Herreman W, Moisy F (2017) Mean mass transport in an orbitally shaken cylindrical container. Phys Rev Fluids 2:084801
Case KM, Parkinson WC (1956) Damping of surface waves in an incompressible liquid. J Fluid Mech 2(2):172–184
Davidson PA, Lindsay RI (1998) Stability of interfacial waves in aluminium reduction cells. J Fluid Mech 362:273–295
Ducci A, Weheliye H (2014) Orbitally shaken bioreactors–viscosity effects on flow characteristics. AIChE J 60:11
Evans J, Ziegler D (2007) The electrolytic production of aluminum. In: Bard A, Stratmann M (eds) Electrochemical Engineering, Encyclopedia of Electrochemistry, vol 5, Wiley-VCH, Weinheim, pp 224–265, volume editors: Macdonald, D.D. and Schmuki, P
Gerbeau JF, Le Bris C, Lelièvre T (2006) Mathematical methods for the magnetohydrodynamics of liquid metals. Numerical mathematics and scientific computation. Oxford University Press, New York
Henderson DM, Miles JW (1994) Surface-wave damping in a circular cylinder with a fixed contact line. J Fluid Mech 275:285–299
Horstmann GM, Weber N, Weier T (2018) Coupling and stability of interfacial waves in liquid metal batteries. J Fluid Mech 845:1–35
Hua J, Rudshaug M, Droste C, Jorgensen R, Giskeødegård NH (2018) Numerical simulation of multiphase magnetohydrodynamic flow and deformation of electrolyte–metal interface in aluminum electrolysis cells. Metallurg Mater Trans B 49(3):1246–1266
Ibrahim RA (2005) Liquid sloshing dynamics theory and applications. Cambridge University Press, New York
Ito T, Kukita Y (2008) Interface behavior between two fluids vertically oscillated in a circular cylinder under nonlinear contact line condition. J Fluid Sci Technol 3(5):690–711
Ito T, Tsuji Y, Kukita Y (1999) Interface waves excited by vertical vibration of stratified fluids in a circular cylinder. J Nucl Sci Technol 36(6):508–521
Kelley D, Weier T (2018) Fluid mechanics of liquid metal batteries. Appl Mech Rev 70(2):020801
Kim H, Boysen DA, Newhouse JM, Spatocco BL, Chung B, Burke PJ, Bradwell DJ, Jiang K, Tomaszowska AA, Wang K, Wei W, Ortiz LA, Barriga SA, Poizeau SM, Sadoway DR (2013) Liquid metal batteries: past, present, and future. Chem Rev 113(3):2075–2099
Klöckner W, Diederichs S, Büchs J (2014) Orbitally shaken single-use bioreactors. Adv Biochem Eng Biotechnol 138:45–60
Lukyanov A, El G, Molokov S (2001) Instability of MHD-modified interfacial gravity waves revisited. Phys Lett A 290:165–172
Miles JW (1991) The capillary boundary layer for standing waves. J Fluid Mech 222:197–205
Moisy F, Bouvard J, Herreman W (2018) Counter-rotation in an orbitally shaken glass of beer. EPL 122:34002
Molokov S (2018) The nature of interfacial instabilities in liquid metal batteries in a vertical magnetic field. EPL 121:44001
Molokov S, El G, Lukyanov A (2011) Classification of instability modes in a model of aluminium reduction cells with a uniform magnetic field. Theor Comput Fluid Dyn 25(5):261–279
Pedcenko A, Molokov S, Bardet B (2017) The effect of “wave breakers” on the magnetohydrodynamic instability in aluminum reduction cells. Metallurg Mater Trans B 48(1):6–10
Pedchenko A, Molokov S, Priede J, Lukyanov A, Thomas PJ (2009) Experimental model of the interfacial instability in aluminium reduction cells. Eur Lett 88(2):24001
Reclari M (2013) Hydrodynamics of orbital shaken bioreactors. PhD thesis, École Polytechnique Fédérale De Lausanne
Reclari M, Dreyer M, Tissot S, Obreschkow D, Wurm FM, Farhat M (2014) Surface wave dynamics in orbital shaken cylindrical containers. Phys Fluids 26:052104
Sele T (1977) Instabilities of the metal surface in electrolytic alumina reduction cells. Metall Trans B 8(4):613–618
Sneyd AD (1992) Interfacial instabilities in aluminium reduction cells. J Fluid Mech 236:111–126
Sneyd AD, Wang A (1994) Interfacial instability due to MHD mode coupling in aluminium reduction cells. J Fluid Mech 263:343–359
Troy CD, Koseff JR (2005) The viscous decay of progressive interfacial waves. Phys Fluids 18:026602
Tucs A, Bojarevics V, Pericleous K (2018a) Magnetohydrodynamic stability of large scale liquid metal batteries. J Fluid Mech 852:453–483
Tucs A, Bojarevics V, Pericleous K (2018b) The nature of interfacial instabilities in liquid metal batteries in a vertical magnetic field. EPL 124:24001
Weber N, Beckstein P, Galindo V, Herreman W, Nore C, Stefani F, Weier T (2017a) Metal pad roll instability in liquid metal batteries. Magnetohydrodynamics 53(1):129–140
Weber N, Beckstein P, Herreman W, Horstmann GM, Nore C, Stefani F, Weier T (2017b) Sloshing instability and electrolyte layer rupture in liquid metal batteries. Phys Fluids 29:054101
Weheliye H, Yianneskis M, Ducci A (2012) On the fluid dynamics of shaken bioreactorsflow characterization and transition. AIChE J 59:1
Weheliye WH, Cagney N, Rodriguez G, Micheletti M, Ducci A (2018) Mode decomposition and lagrangian structures of the flow dynamics in orbitally shaken bioreactors. Phys Fluids 30:033603
Weier T, Bund A, El-Mofid W, Horstmann GM, Lalau CC, Landgraf S, Nimtz M, Starace M, Stefani F, Weber N (2017) Liquid metal batteries—materials selection and fluid dynamics. IOP Conf Ser: Mater Sci Eng 228:012013
Zikanov O (2018) Shallow water modeling of rolling pad instability in liquid metal batteries. Theor Comput Fluid Dyn 32(3):325–347
Acknowledgements
The authors would like to thank Peggy Jähnigen and Kerstin Eckert for the realization of extensive interfacial tension measurements. Fruitful discussions with Kerstin Eckert, Bernd Willers, Norbert Weber and Wietze Herreman on several aspects of acoustic measurement techniques and contact line dynamics are gratefully acknowledged.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Horstmann, G.M., Wylega, M. & Weier, T. Measurement of interfacial wave dynamics in orbitally shaken cylindrical containers using ultrasound pulse-echo techniques. Exp Fluids 60, 56 (2019). https://doi.org/10.1007/s00348-019-2699-0
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00348-019-2699-0