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Falling styles of disks

Published online by Cambridge University Press:  19 February 2013

Franck Auguste ,
Jacques Magnaudet  and
David Fabre
Franck Auguste
Affiliation:
Université de Toulouse, INPT, UPS, IMFT (Institut de Mécanique des Fluides de Toulouse), Allée Camille Soula, F-31400 Toulouse, France
Jacques Magnaudet*
Affiliation:
Université de Toulouse, INPT, UPS, IMFT (Institut de Mécanique des Fluides de Toulouse), Allée Camille Soula, F-31400 Toulouse, France CNRS, IMFT, F-31400 Toulouse, France
David Fabre
Affiliation:
Université de Toulouse, INPT, UPS, IMFT (Institut de Mécanique des Fluides de Toulouse), Allée Camille Soula, F-31400 Toulouse, France
*
†Email address for correspondence: [email protected]
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Abstract

We numerically investigate the dynamics of thin disks falling under gravity in a viscous fluid medium at rest at infinity. Varying independently the density and thickness of the disk reveals the influence of the disk aspect ratio which, contrary to previous belief, is found to be highly significant as it may completely change the route to non-vertical paths as well as the boundaries between the various path regimes. The transition from the straight vertical path to the planar fluttering regime is found to exhibit complex dynamics: a bistable behaviour of the system is detected within some parameter range and several intermediate regimes are observed in which, although the wake is unstable, the path barely deviates from vertical. By varying independently the body-to-fluid inertia ratio and the relative magnitude of inertial and viscous effects over a significant range, we set up a comprehensive map of the corresponding styles of path followed by an infinitely thin disk. We observe the four types of planar regimes already reported in experiments but also identify two additional fully three-dimensional regimes in which the body experiences a slow horizontal precession superimposed onto zigzagging or tumbling motions.

JFM classification

Nonlinear Dynamical Systems: Bifurcation Nonlinear Dynamical Systems: Nonlinear Dynamical Systems
Type
Papers
Information
Journal of Fluid Mechanics , Volume 719 , 25 March 2013 , pp. 388 - 405
Copyright
©2013 Cambridge University Press

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References

Alben, S. 2008 An implicit method for coupled flow-body dynamics. J. Comput. Phys. 227, 4912–4933.Google Scholar
Alben, S. & Shelley, M. 2005 Coherent locomotion as an attracting state for a free flapping body. Proc. Natl Acad. Sci. U.S.A. 102, 1163–1166.CrossRefGoogle ScholarPubMed
Andersen, A., Pesavento, U. & Wang, Z. J. 2005 Unsteady aerodynamics of fluttering and tumbling plates. J. Fluid Mech. 541, 65–90.CrossRefGoogle Scholar
Assemat, P., Fabre, D. & Magnaudet, J. 2012 The onset of unsteadiness of two-dimensional bodies falling or rising freely in a viscous fluid: a linear study. J. Fluid Mech. 690, 173–202.CrossRefGoogle Scholar
Auguste, F. 2010 Instabilités de sillage et trajectoires d’un corps solide cylindrique immergé dans un fluide visqueux. PhD thesis, Université Paul Sabatier, Toulouse, France, available at https://www.imft.fr/Projet-ANR-OBLIC.Google Scholar
Blanco, A. & Magnaudet, J. 1995 The structure of the axisymmetric high-Reynolds number flow around an ellipsoidal bubble of fixed shape. Phys. Fluids 7, 1265–1274.CrossRefGoogle Scholar
Chrust, M. 2012 Etude numérique de la chute libre d’objets axisymétriques dans un fluide newtonien. PhD thesis, Université de Strasbourg, France.Google Scholar
Ern, P., Risso, F., Fabre, D. & Magnaudet, J. 2012 Wake-induced oscillatory paths of bodies freely rising or falling in fluids. Annu. Rev. Fluid Mech. 44, 97–121.Google Scholar
Fabre, D., Assemat, P. & Magnaudet, J. 2011 A quasi-static approach to the stability of the path of heavy bodies falling within a viscous fluid. J. Fluids Struct. 27, 758–767.CrossRefGoogle Scholar
Fabre, D., Auguste, F. & Magnaudet, J. 2008 Bifurcation and symmetry breaking in the wake of axisymmetric bodies. Phys. Fluids 20, 051702.CrossRefGoogle Scholar
Fabre, D., Tchoufag, J. & Magnaudet, J. 2012 The steady oblique path of buoyancy-driven disks and spheres. J. Fluid Mech. 707, 24–36.CrossRefGoogle Scholar
Fernandes, P. C., Ern, P., Risso, F. & Magnaudet, J. 2005 On the zigzag dynamics of freely moving axisymmetric bodies. Phys. Fluids 17, 098107.Google Scholar
Fernandes, P. C., Ern, P., Risso, F. & Magnaudet, J. 2007 Dynamics of axisymmetric bodies rising along a zigzag path. J. Fluid Mech. 606, 209–223.Google Scholar
Field, S., Klaus, M., Moore, M. & Nori, F. 1997 Chaotic dynamics of falling disks. Nature 388, 252–254.CrossRefGoogle Scholar
Jeong, J. & Hussain, F. 1995 On the identification of a vortex. J. Fluid Mech. 285, 69–94.CrossRefGoogle Scholar
Lamb, Sir H. 1932 Hydrodynamics. Cambridge University Press.Google Scholar
Loewenberg, M. 1993 Stokes resistance, added mass and Basset force for arbitrarily oriented, finite-length cylinders. Phys. Fluids A5, 765–767.CrossRefGoogle Scholar
Magnaudet, J., Rivero, M. & Fabre, J. 1995 Accelerated flows past a rigid sphere or a spherical bubble. Part 1. Steady straining flow. J. Fluid Mech. 284, 97–135.Google Scholar
Mahadevan, L., Ryu, W. S. & Samuel, D. T. 1999 Tumbling cards. Phys. Fluids 11, 1–3.CrossRefGoogle Scholar
Moffat, H. K. 1964 Viscous and resistive eddies near a sharp corner. J. Fluid Mech. 18, 1–18.CrossRefGoogle Scholar
Mougin, G. & Magnaudet, J. 2002a The generalized Kirchhoff equations and their application to the interaction between a rigid body and an arbitrary time-dependent viscous flow. Intl J. Multiphase Flow 28, 1837–1851.CrossRefGoogle Scholar
Mougin, G. & Magnaudet, J. 2002b Path instability of a rising bubble. Phys. Rev. Lett. 88, 14502.Google Scholar
Mougin, G. & Magnaudet, J. 2006 Forces and torques on a zigzagging/spiralling bubble. J. Fluid Mech. 567, 185–194.Google Scholar
Natarajan, R. & Acrivos, A. 1993 The instability of the steady flow past spheres and disks. J. Fluid Mech. 254, 323–344.Google Scholar
Rivero, M., Magnaudet, J. & Fabre, J. 1991 New results on the force exerted on a spherical body by an accelerating flow. C. R. Acad. Sci. Paris Sér. II 312, 1499–1506.Google Scholar
Stringham, G. E., Simons, D. B. & Guy, H. P. 1969 The behavior of large particles falling in quiescent liquids. US Geol. Surv. Prof. Pap. 562C, 1–36.Google Scholar
Tanzosh, J. P. & Stone, H. A. 1996 A general approach for analyzing the arbitrary motion or a circular disk in a Stokes flow. Chem. Engng Commun. 150, 333–346.Google Scholar
Vandenberghe, N., Zhang, J. & Childress, S. 2004 Symmetry breaking leads to forward flapping flight. J. Fluid Mech. 506, 147–155.Google Scholar
Willmarth, W., Hawk, N. & Harvey, R. 1964 Steady and unsteady motions and wakes of freely falling disks. Phys. Fluids 7, 197–208.Google Scholar
Zhong, H., Chen, S. & Lee, C. 2011 Experimental study of freely falling thin disks: transition from planar zigzag to spiral. Phys. Fluids 23, 011702.CrossRefGoogle Scholar

Auguste et al. supplementary movie

Falling styles of discs I*=0.16; Ar=27.8 “Autorotation(=tumbling) AR regime”

Download Auguste et al. supplementary movie(Video)
Video 1.5 MB

Auguste et al. supplementary movie

Falling styles of discs I*=0.48;Ar=47.9 “Helical tumbling HA regime”

Download Auguste et al. supplementary movie(Video)
Video 1.1 MB

Auguste et al. supplementary movie

Falling styles of discs I*=0.012; Ar=47.1 “Helcial fluttering HH regime”

Download Auguste et al. supplementary movie(Video)
Video 2.2 MB

Auguste et al. supplementary movie

Falling styles of discs I*0.048; Ar=25.5 “Zigzagging (=fluttering)ZZ regime”

Download Auguste et al. supplementary movie(Video)
Video 1.5 MB

Auguste et al. supplementary movie

Falling styles of discs I*=0.0035; Ar=42.56 “Small-amplitude zigzagging A-regime(similar to regime(iii)in figure 1a)”

Download Auguste et al. supplementary movie(Video)
Video 1.9 MB

Auguste et al. supplementary movie

Falling styles of discs I*=0.0035; Ar=42.56 “Small-amplitude zigzagging A-regime(similar to regime(iii)in figure 1a)”

Download Auguste et al. supplementary movie(Video)
Video 1.6 MB