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Chaotic orbits of tumbling ellipsoids

Published online by Cambridge University Press:  21 September 2020

Erich Essmann Open the ORCID record for Erich Essmann [Opens in a new window] ,
Pei Shui Open the ORCID record for Pei Shui [Opens in a new window] ,
Stéphane Popinet Open the ORCID record for Stéphane Popinet [Opens in a new window] ,
Stéphane Zaleski Open the ORCID record for Stéphane Zaleski [Opens in a new window] ,
Prashant Valluri Open the ORCID record for Prashant Valluri [Opens in a new window]  and
Rama Govindarajan Open the ORCID record for Rama Govindarajan [Opens in a new window]
Erich Essmann
Affiliation:
Institute for Multiscale Thermofluids, School of Engineering, The University of Edinburgh, EdinburghEH9 3FB, UK
Pei Shui
Affiliation:
Institute of Smart City Research, University of Science and Technology of China, Wuhu241000, PR China
Stéphane Popinet
Affiliation:
Institut Jean le Rond d'Alembert, Sorbonne Université, 75252Paris Cedex 05, France
Stéphane Zaleski
Affiliation:
Institut Jean le Rond d'Alembert, Sorbonne Université, 75252Paris Cedex 05, France
Prashant Valluri*
Affiliation:
Institute for Multiscale Thermofluids, School of Engineering, The University of Edinburgh, EdinburghEH9 3FB, UK
Rama Govindarajan*
Affiliation:
International Centre for Theoretical Sciences, TIFR, Bengaluru560089, India
*
†Email addresses for correspondence: prashant.valluri@ed.ac.uk, rama@icts.res.in
†Email addresses for correspondence: prashant.valluri@ed.ac.uk, rama@icts.res.in
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Abstract

Orbits tracked by ellipsoids immersed in inviscid and viscous environments are studied by means of Kirchhoff's equations and high resolution numerical simulations using a variant of the immersed boundary method. We explore the consequences of Kozlov and Onishchenko's theorem of non-integrability of Kirchhoff's equations to show how the fraction of phase space in chaotic orbits is sensitively determined by the body shape, fluid/solid density ratio and the fraction of initial energy in rotational motion. We show how the added mass tensor of the system is an important player in both viscous and inviscid flow, in causing chaos in a triaxial ellipsoid while acting to suppress it in a spheroid. We identify a new integral of motion for a spheroid in inviscid fluid: one component of the generalised angular momentum. A spheroid, which can never execute chaotic dynamics in inviscid flow, is shown to display chaos in viscous flow due to irregular vortex shedding. But the dynamics of the spheroid is restricted whether in viscous or in inviscid flow, unlike in the triaxial ellipsoid, due to our extra integral of motion.

JFM classification

Multiphase and Particle-laden Flows: Particle/fluid flow Nonlinear Dynamical Systems: Chaos
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 903 , 25 November 2020 , A10
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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