Hostname: page-component-7c8c6479df-5xszh Total loading time: 0 Render date: 2024-03-28T20:17:45.225Z Has data issue: false hasContentIssue false

Holes stabilize freely falling coins

Published online by Cambridge University Press:  21 July 2016

Lionel Vincent
Affiliation:
Department of Aerospace and Mechanical Engineering, University of Southern California, Los Angeles, CA 90089, USA
W. Scott Shambaugh
Affiliation:
Department of Aerospace and Mechanical Engineering, University of Southern California, Los Angeles, CA 90089, USA
Eva Kanso*
Affiliation:
Department of Aerospace and Mechanical Engineering, University of Southern California, Los Angeles, CA 90089, USA
*
Email address for correspondence: kanso@usc.edu

Abstract

The free fall of heavy bodies in a viscous fluid medium is a problem of interest to many engineering and scientific disciplines, including the study of unpowered flight and seed dispersal. The falling behaviour of coins and thin discs in particular has been categorized into one of four distinct modes; steady, fluttering, chaotic or tumbling, depending on the moment of inertia and Reynolds number. This paper investigates, through a carefully designed experiment, the falling dynamics of thin discs with central holes. The effects of the central hole on the disc’s motion is characterized for a range of Reynolds number, moments of inertia and inner to outer diameter ratio. By increasing this ratio, that is, the hole size, the disc is found to transition from tumbling to chaotic then fluttering at values of the moment of inertia not predicted by the falling modes of whole discs. This transition from tumbling to fluttering with increased hole size is viewed as a stabilization process. Flow visualization of the wake behind annular discs shows the presence of a vortex ring at the disc’s outer edge, as in the case of whole discs, and an additional counter-rotating vortex ring at the disc’s inner edge. The inner vortex ring is responsible for stabilizing the disc’s falling motion. These findings have significant implications on the development of design principles for engineered robotic systems in free flight, and may shed light on the stability of gliding animals.

Type
Papers
Copyright
© 2016 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Andersen, A., Pesavento, U. & Wang, Z. J. 2005 Analysis of transitions between fluttering, tumbling and steady descent of falling cards. J. Fluid. Mech. 541, 91104.CrossRefGoogle Scholar
Auguste, F., Magnaudet, J. & Fabre, D. 2013 Falling styles of disks. J. Fluid. Mech. 719, 388405.CrossRefGoogle 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, 97121.Google Scholar
Field, S. B., Klaus, M., Moore, M. G. & Nori, F. 1997 Chaotic dynamics of falling disks. Nature 388, 252254.CrossRefGoogle Scholar
Gazzola, M., Chatelain, P., van Rees, W. M. & Koumoutsakos, P. 2011 Simulations of single and multiple swimmers with non-divergence free deforming geometries. J. Comput. Phys. 230, 70397114.Google Scholar
Heisinger, L., Newton, P. K. & Kanso, E. 2014 Coins falling in water. J. Fluid. Mech. 742, 245253.CrossRefGoogle Scholar
Jafari, F., Ross, S. D., Vlachos, P. P. & Socha, J. J. 2014 A theoretical analysis of pitch stability during gliding in flying snakes. Bioinspir. Biomim. 9 (16), 025014.CrossRefGoogle ScholarPubMed
Jones, M. A. & Shelley, M. J. 2005 Falling cards. J. Fluid Mech. 540, 393425.CrossRefGoogle Scholar
Landau, L. D. & Lifschitz, E. M. 1987 Fluid Mechanics. vol. 6. Pergamon Press.Google Scholar
Lentink, D. & Biewener, A. A. 2010 Nature-inspired flight – beyond the leap. Bioinsp. Biomim. 5 (9), 040201.Google Scholar
Lentink, D., Dickson, W. B., van Leeuwen, J. L. & Dickinson, M. H. 2009 Leading-edge vortices elevate lift of autorotating plant seeds. Science 324, 14381440.Google Scholar
Michelin, S. & Llewellyn-Smith, S. G. 2009 An unsteady point vortex method for coupled fluid–solid problems. Theor. Comput. Fluid Dyn. 23, 127153.Google Scholar
Mittal, R., Seshadri, V. & Udaykumar, H. S. 2004 Flutter, tumble and vortex induced autorotation. Theor. Comput. Fluid Dyn. 17, 165170.CrossRefGoogle Scholar
Moffatt, H. K. 2013 Three coins in a fountain. J. Fluid. Mech. 720, 14.Google Scholar
Paoletti, P. & Mahadevan, L. 2011 Planar controlled gliding, tumbling and descent. J. Fluid Mech. 689, 489516.Google Scholar
Roger, R. P. & Hussey, R. G. 1982 Stokes drag on a flat annular ring. Phys. Fluids 25 (6), 915922.Google Scholar
Thielicke, W. & Stamhuis, E. J. 2014 Pivlab – towards user-friendly, affordable and accurate digital particle image velocimetry in matlab. J. Open. Res. Softw. 2 (1), e30.Google Scholar
Williamson, C. H. K. & Govardhan, R. 2004 Vortex-induced vibrations. Annu. Rev. Fluid Mech. 36, 413455.CrossRefGoogle Scholar
Willmarth, W., Hawk, N. E. & Harvey, R. L. 1964 Steady and unsteady motions and wakes of freely falling disks. Phys. Fluids 7 (2), 197208.CrossRefGoogle Scholar