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Dynamics of heavy and buoyant underwater pendulums

  • Varghese Mathai (a1) (a2), Laura A. W. M. Loeffen (a2), Timothy T. K. Chan (a2) (a3) and Sander Wildeman (a2) (a4)

Abstract

The humble pendulum is often invoked as the archetype of a simple, gravity driven, oscillator. Under ideal circumstances, the oscillation frequency of the pendulum is independent of its mass and swing amplitude. However, in most real-world situations, the dynamics of pendulums is not quite so simple, particularly with additional interactions between the pendulum and a surrounding fluid. Here we extend the realm of pendulum studies to include large amplitude oscillations of heavy and buoyant pendulums in a fluid. We performed experiments with massive and hollow cylindrical pendulums in water, and constructed a simple model that takes the buoyancy, added mass, fluid (nonlinear) drag and bearing friction into account. To first order, the model predicts the oscillation frequencies, peak decelerations and damping rate well. An interesting effect of the nonlinear drag captured well by the model is that, for heavy pendulums, the damping time shows a non-monotonic dependence on pendulum mass, reaching a minimum when the pendulum mass density is nearly twice that of the fluid. Small deviations from the model’s predictions are seen, particularly in the second and subsequent maxima of oscillations. Using time-resolved particle image velocimetry (TR-PIV), we reveal that these deviations likely arise due to the disturbed flow created by the pendulum at earlier times. The mean wake velocity obtained from PIV is used to model an extra drag term due to incoming wake flow. The revised model significantly improves the predictions for the second and subsequent oscillations.

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Copyright

This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.

Corresponding author

Email address for correspondence: swildeman@gmail.com

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Equally contributed authors.

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References

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Alméras, E., Mathai, V., Lohse, D. & Sun, C. 2017 Experimental investigation of the turbulence induced by a bubble swarm rising within incident turbulence. J. Fluid Mech. 825, 10911112.
Bagchi, P. & Balachandar, S. 2004 Response of the wake of an isolated particle to an isotropic turbulent flow. J. Fluid Mech. 518, 95123.
Batchelor, G. K. 2000 An Introduction to Fluid Dynamics. Cambridge University Press.
Bolster, D., Hershberger, R. E. & Donnelly, R. J. 2010 Oscillating pendulum decay by emission of vortex rings. Phys. Rev. E 81 (4), 046317.
Dong, R. G.1978 Effective mass and damping of submerged structures. Tech. Rep. California University, Livermore (USA). Lawrence Livermore Lab.
Govardhan, R. & Williamson, C. H. K. 2000 Modes of vortex formation and frequency response of a freely vibrating cylinder. J. Fluid Mech. 420, 85130.
Govardhan, R. N. & Williamson, C. H. K. 2005 Vortex-induced vibrations of a sphere. J. Fluid Mech. 531, 1147.
Hoerner, S. F. 1965 Fluid-Dynamic Drag: Practical Information on Aerodynamic Drag and Hydrodynamic Resistance. Hoerner Fluid Dynamics.
Huygens, C. 1986 Christiaan Huygens’ the Pendulum Clock, or, Geometrical Demonstrations Concerning the Motion of Pendula as Applied to Clocks. Iowa State University Press.
Konstantinidis, E. 2013 Added mass of a circular cylinder oscillating in a free stream. Proc. R. Soc. Lond. A 469 (2156), 20130135.
Koo, W. & Kim, J.-D. 2015 Simplified formulas of heave added mass coefficients at high frequency for various two-dimensional bodies in a finite water depth. Intl J. Naval Arch. Ocean Engng 7 (1), 115127.
Lienhard, J. H. 1966 Synopsis of Lift, Drag, and Vortex Frequency Data for Rigid Circular Cylinders, vol. 300. Technical Extension Service, Washington State University.
Mathai, V., Calzavarini, E., Brons, J., Sun, C. & Lohse, D. 2016 Microbubbles and microparticles are not faithful tracers of turbulent acceleration. Phys. Rev. Lett. 117 (2), 024501.
Mathai, V., Prakash, V. N., Brons, J., Sun, C. & Lohse, D. 2015 Wake-driven dynamics of finite-sized buoyant spheres in turbulence. Phys. Rev. Lett. 115 (12), 124501.
Mathai, V., Zhu, X., Sun, C. & Lohse, D. 2017 Mass and moment of inertia govern the transition in the dynamics and wakes of freely rising and falling cylinders. Phys. Rev. Lett. 119 (5), 054501.
Mathai, V., Zhu, X., Sun, C. & Lohse, D. 2018 Flutter to tumble transition of buoyant spheres triggered by rotational inertia changes. Nat. Commun. 9 (1), 1792.
Mittal, R. & Iaccarino, G. 2005 Immersed boundary methods. Annu. Rev. Fluid Mech. 37, 239261.
Naso, A. & Prosperetti, A. 2010 The interaction between a solid particle and a turbulent flow. New J. Phys. 12 (3), 033040.
Neill, D., Livelybrooks, D. & Donnelly, R. J. 2007 A pendulum experiment on added mass and the principle of equivalence. Am. J. Phys. 75 (3), 226229.
Obligado, M., Puy, M. & Bourgoin, M. 2013 Bi-stability of a pendular disk in laminar and turbulent flows. J. Fluid Mech. 728.
Stokes, G. G. 1851 On the Effect of the Internal Friction of Fluids on the Motion of Pendulums. Pitt Press Cambridge.
Tatsuno, M. & Bearman, P. W. 1990 A visual study of the flow around an oscillating circular cylinder at low Keulegan–Carpenter numbers and low Stokes numbers. J. Fluid Mech. 211, 157182.
Tatsuno, M. & Karasudani, T. 1993 Wavy mode of the streaked flow around an oscillating cylinder in a stratified fluid at rest. Fluid Dyn. Res. 11 (6), 313.
Wu, J.-S. & Faeth, G. M. 1993 Sphere wakes in still surroundings at intermediate Reynolds numbers. AIAA J. 31 (8), 14481455.
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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
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VIDEO
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Mathai et al. supplementary movie
PIV flow velocity field for a heavy pendulum with mass ratio around 2.2

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