Journal of Fluid Mechanics



Modes of vortex formation and frequency response of a freely vibrating cylinder


R. GOVARDHAN a1 and C. H. K. WILLIAMSON a1
a1 Sibley School of Mechanical and Aerospace Engineering, Upson Hall, Cornell University, Ithaca, NY 14853-7501, USA

Abstract

In this paper, we study the transverse vortex-induced vibrations of an elastically mounted rigid cylinder in a fluid flow. We use simultaneous force, displacement and vorticity measurements (using DPIV) for the first time in free vibrations. There exist two distinct types of response in such systems, depending on whether one has a high or low combined mass–damping parameter (m*ζ). In the classical high-(m*ζ) case, an ‘initial’ and ‘lower’ amplitude branch are separated by a discontinuous mode transition, whereas in the case of low (m*ζ), a further higher-amplitude ‘upper’ branch of response appears, and there exist two mode transitions.

To understand the existence of more than one mode transition for low (m*ζ), we employ two distinct formulations of the equation of motion, one of which uses the ‘total force’, while the other uses the ‘vortex force’, which is related only to the dynamics of vorticity. The first mode transition involves a jump in ‘vortex phase’ (between vortex force and displacement), φvortex, at which point the frequency of oscillation (f) passes through the natural frequency of the system in the fluid, f [similar] fNwater. This transition is associated with a jump between 2S [leftrightarrow A: l&r arrow] 2P vortex wake modes, and a corresponding switch in vortex shedding timing. Across the second mode transition, there is a jump in ‘total phase’, phis;total , at which point f [similar] fNvacuum. In this case, there is no jump in φvortex, since both branches are associated with the 2P mode, and there is therefore no switch in timing of shedding, contrary to previous assumptions. Interestingly, for the high-(m*ζ) case, the vibration frequency jumps across both fNwater and fNvacuum, corresponding to the simultaneous jumps in φvortex and φtotal. This causes a switch in the timing of shedding, coincident with the ‘total phase’ jump, in agreement with previous assumptions.

For large mass ratios, m* = O(100), the vibration frequency for synchronization lies close to the natural frequency (f* = f/fN [approximate] 1.0), but as mass is reduced to m* = O(1), f* can reach remarkably large values. We deduce an expression for the frequency of the lower-branch vibration, as follows:

formula here

which agrees very well with a wide set of experimental data. This frequency equation uncovers the existence of a critical mass ratio, where the frequency f* becomes large: m*crit = 0.54. When m* < m*crit, the lower branch can never be reached and it ceases to exist. The upper-branch large-amplitude vibrations persist for all velocities, no matter how high, and the frequency increases indefinitely with flow velocity. Experiments at m* < m*crit show that the upper-branch vibrations continue to the limits (in flow speed) of our facility.

(Received December 22 1999)
(Revised April 21 2000)



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