Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-24T19:55:08.815Z Has data issue: false hasContentIssue false
  • English
  • Français

Vortex-induced vibration of a rising and falling cylinder

Published online by Cambridge University Press:  19 August 2010

M. HOROWITZ  and
C. H. K. WILLIAMSON
M. HOROWITZ
Affiliation:
144 Upson Hall, Cornell University, Ithaca, NY 14853, USA
C. H. K. WILLIAMSON*
Affiliation:
144 Upson Hall, Cornell University, Ithaca, NY 14853, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this study, we investigate the dynamics of a freely rising and falling cylinder. This is, in essence, a vortex-induced vibration (VIV) system comprising both transverse (Y) and streamwise (X) degrees-of-freedom (d.o.f.), but with zero spring stiffness and zero damping. This problem represents a limiting case among studies in VIV, and is an extension of recent research of elastically mounted bodies having very low spring stiffness, as well as bodies with very low mass and damping. We find that if the mass ratio (where m* = cylinder mass/displaced fluid mass) is greater than a critical value, m*crit = 0.545, the body falls or rises with a rectilinear trajectory. As the mass ratio is reduced below m*crit = 0.545, the cylinder suddenly begins to vibrate vigorously and periodically, with a 2P mode of vortex formation, as reported in the preliminary study of Horowitz & Williamson (J. Fluids Struct. vol. 22, 2006, pp. 837–843). The similarity in critical mass between freely rising and elastically mounted bodies is unexpected, as it is known that the addition of streamwise vibration can markedly affect the response and vortex formation in elastically mounted systems, which would be expected to modify the critical mass. However, we show in this paper that the similarity in vortex formation mode (2P) between the freely rising body and the elastically mounted counterpart is consistent with a comparable phase of vortex dynamics, strength of vortices, amplitudes and frequencies of motion and effective added mass (CEA). All of these similarities result in comparable values of critical mass. The principal fact that the 2P mode is observed for the freely rising body is an interesting and consistent result; based on the previous VIV measurements, this is the only mode out of the known set {2S, 2P, 2T} to yield negative effective added mass (CEA < 0), which is a condition for vibration of a freely rising body. In this paper, we deduce that there exists only one possible two degree-of-freedom elastically mounted cylinder system, which can be used to predict the dynamics of freely rising bodies. Because of the symmetry of the vortex wake, this system is one for which the natural frequencies are fNX = 2fNY. Although this seems clear in retrospect, previous attempts to predict critical mass did not take this into account. Implementing such an elastic system, we are able to predict vibration amplitudes and critical mass (m*crit = 0.57) for a freely rising cylinder in reasonable agreement with direct measurements for such a rising body, and even to predict the Lissajous figures representing the streamwise–transverse vibrations for a rising body with very small mass ratios (down to m* = 0.06), unobtainable from our direct measurements.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 662 , 10 November 2010 , pp. 352 - 383
Copyright
Copyright © Cambridge University Press 2010

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

REFERENCES

Aronsen, K. R. 2007 An experimental investigation of in-line and combined in-line and cross-flow vortex induced vibrations. PhD thesis, NTNU, Trondheim, Norway.Google Scholar
Bearman, P. 1984 Vortex shedding from oscillating bluff bodies. J. Fluid Mech. 16, 195–222.CrossRefGoogle Scholar
Clift, R., Grace, J. R. & Weber, M. E. 1973 Bubbles, Drops, and Particles. Academic.Google Scholar
Dahl, J. M., Hover, F. S. & Triantafyllou, M. S. 2006 Two-degree-of-freedom vortex-induced vibrations using a force assisted apparatus. J. Fluids Struct. 22, 807–818.CrossRefGoogle Scholar
Dahl, J. M., Hover, F. S., Triantafyllou, M. S., Dong, S. & Karniadakis, G. E. 2007 Resonant vibrations of bluff bodies cause multivortex shedding and high frequency forces. Phys. Rev. Lett. 99, 144503.CrossRefGoogle ScholarPubMed
Ern, P., Fernandes, P. C., Risso, F. & Magnaudet, J. 2007 Evolution of wake structure and wake-induced loads along the path of freely rising axisymmetric bodies. Phys. Fluids 19, 113–302.CrossRefGoogle Scholar
Feng, J., Hu, H. H. & Joseph, D. D. 1994 Direct simulation of initial value problems for the motion of solid bodies in a Newtonian fluid. Part 1. Sedimentation. J. Fluid Mech. 261, 95–134.CrossRefGoogle Scholar
Fernandes, P. C., Risso, F., Ern, P. & Magnaudet, J. 2007 Oscillatory motion and wake instability of freely rising axisymmetric bodies. J. Fluid Mech. 573, 479–502.CrossRefGoogle Scholar
Govardhan, R. & Williamson, C. H. K. 2000 Modes of vortex formation and frequency response of a freely vibrating cylinder. J. Fluid Mech. 420, 85–130.CrossRefGoogle Scholar
Govardhan, R. & Williamson, C. H. K. 2002 Resonance forever: existence of a critical mass and an infinite regime of resonance in vortex-induced vibration. J. Fluid Mech. 473, 147–166.CrossRefGoogle Scholar
Griffin, O. M. & Ramberg, S. E. 1982 Some recent studies of vortex shedding with application to marine tubulars and risers. J. Energy Res. Tech. 104, 2–13.CrossRefGoogle Scholar
Hartman, M. & Yates, J. G. 1993 Free-fall of solid particles through fluids. Collect. Czech. Chem. Commun. 58, 961–982.CrossRefGoogle Scholar
Horowitz, M. & Williamson, C. H. K. 2006 Dynamics of a rising and falling cylinder. J. Fluids Struct. 22, 837–843.CrossRefGoogle Scholar
Horowitz, M. & Williamson, C. H. K. 2008 Critical mass and a new periodic four-ring vortex wake mode for freely rising and falling spheres. Phys. Fluids 20, 101–701.CrossRefGoogle Scholar
Horowitz, M. & Williamson, C. H. K. 2010 The effect of Reynolds number on the dynamics and wakes of freely rising and falling spheres. J. Fluid Mech. 651, 251–294.CrossRefGoogle Scholar
Hover, F. S., Techet, A. H. & Triantafyllou, M. S. 1998 Forces on oscillating uniform and tapered cylinders in crossflow. J. Fluid Mech. 363, 97–114.CrossRefGoogle Scholar
Hu, H. H. 1995 Motion of a circular cylinder in a viscous liquid between parallel plates. Theor. Comput. Fluid Dyn. 363, 441–455.CrossRefGoogle Scholar
Isaacs, J. L. & Thodos, G. 1967 The free settling of solid cylindrical particles in the turbulent regime. Can. J. Chem. Engng 45, 150–155.CrossRefGoogle Scholar
Jauvtis, N. & Williamson, C. H. K. 2004 The effect of two degrees of freedom on vortex-induced vibration at low mass and damping. J. Fluid Mech. 509, 23–62.CrossRefGoogle Scholar
Jayaweera, K. O. L. F. & Mason, B. J. 1965 The behaviour of freely falling cylinders and cones in a viscous fluid. J. Fluid Mech. 22, 709–720.CrossRefGoogle Scholar
Jenny, M., Dušek, J. & Bouchet, G. 2004 Instabilities and transition of a sphere falling or ascending freely in a Newtonian fluid. J. Fluid Mech. 508, 709–720.CrossRefGoogle Scholar
Jeon, D. & Gharib, M. 2001 On circular cylinders undergoing two-degree-of-freedom forced motions. J. Fluids Struct. 15, 533–541.CrossRefGoogle Scholar
Jeon, D. & Gharib, M. 2004 On the relationship between the vortex formation process and cylinder wake vortex patterns. J. Fluid Mech. 519, 161–181.CrossRefGoogle Scholar
Khalak, A. & Williamson, C. H. K. 1999 Motions, forces and mode transitions in vortex-induced vibrations at low mass-damping. J. Fluids Struct. 10, 813–851.CrossRefGoogle Scholar
Magnaudet, J. & Eames, I. 2000 The motion of high Reynolds number bubbles in inhomogeneous flows. Annu. Rev. Fluid Mech. 32, 659–708.CrossRefGoogle Scholar
Marchildon, E. K., Clamen, A. & Gauvin, W. H. 1964 Drag and oscillatory motion of freely falling cylindrical particles. Can. J. Chem. Engng 42, 178–182.CrossRefGoogle Scholar
Moe, G. & Wu, Z.-J. 1990 The lift force on a cylinder vibrating in a current. J. Offshore Mech. Arctic Engng 112, 297–303.CrossRefGoogle Scholar
Morse, T. L. & Williamson, C. H. K. 2009 a The effect of Reynolds number on the critical mass phenomenon in vortex-induced vibration. Phys. Fluids 21, 045105.CrossRefGoogle Scholar
Morse, T. L. & Williamson, C. H. K. 2009 b Fluid forcing, wake modes, and transitions for a cylinder undergoing controlled oscillations. J. Fluids Struct. 25, 697–712.CrossRefGoogle Scholar
Morse, T. L. & Williamson, C. H. K. 2009 c Prediction of vortex-induced vibration response by employing controlled motion. J. Fluid Mech. 634, 5–39.CrossRefGoogle Scholar
Mougin, G. & Magnaudet, J. 2002 Wake-induced forces and torques on a zigzagging/spiralling bubble. Phys. Rev. Lett 88, 014502.CrossRefGoogle Scholar
Parkinson, G. 1989 Phenomena and modelling of flow-induced vibrations of bluff bodies. Prog. Aerosp. Sci. 26, 169–224.CrossRefGoogle Scholar
Richardson, J. F. & Zaki, W. N. 1954 Sedimentation and fluidisation. Part I. Trans. Inst. Chem. Engng 32, 35–53.Google Scholar
Sarpkaya, T. 1979 Vortex-induced oscillations. J. Appl. Mech. 46, 241–258.CrossRefGoogle Scholar
Sarpkaya, T. 1995 Hydrodynamic damping, flow-induced oscillations and biharmonic response. J. Offshore Mech. Arctic Engng 117, 232–238.CrossRefGoogle Scholar
Stringham, G. E., Simons, D. B. & Guy, H. P. 1969 The behavior of large particles falling in quiescent liquids. Tech. Rep. Paper 562C. US Geological Survey.CrossRefGoogle Scholar
Veldhuis, C., Biesheuvel, A., van Wijngaarden, L. & Lohse, D. 2005 Motion and wake structure of spherical particles. Nonlinearity 18, C1–C8.CrossRefGoogle Scholar
Williamson, C. H. K. & Govardhan, R. N. 2004 Vortex-induced vibrations. Annu. Rev. Fluid Mech. 36, 413–455.CrossRefGoogle Scholar
Williamson, C. H. K. & Roshko, A. 1988 Vortex formation in the wake of an oscillating cylinder. J. Fluids Struct. 2, 355–381.CrossRefGoogle Scholar
Willmarth, W. W., Hawk, N. E. & Harvey, R. L. 1964 Steady and unsteady motions and wakes of freely falling disks. Phys. Fluids 7, 197–208.CrossRefGoogle Scholar