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

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), φ vortex , at which point the frequency of oscillation (f) passes through the natural frequency of the system in the fluid, f ∼ f Nwater . This transition is associated with a jump between 2S ↔ 2P vortex wake modes, and a corresponding switch in vortex shedding timing. Across the second mode transition, there is a jump in 'total phase', φ total , at which point f ∼ f Nvacuum . In this case, there is no jump in φ vortex , 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 f Nwater and f Nvacuum , corresponding to the simultaneous jumps in φ vortex and φ total . 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/f N ≈ 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: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.

Nonlinear Fluid Damping In Structure-Wake Oscillators In Modeling Vortex-Induced Vibrations

“…[18] in air and water, --and --DNS of Newman and Karniadakis [6] . In (b), experimental data: + Feng = 100 Re [16] , and Khalak and Williamson [19] for lower and upper branches, Govardhan and Williamson [17] From Fig.4, the prediction of the response amplitude obtained by the coupled system with the NFD is closer to experimental data than that obtained from DNS at or by the coupled system with the LFD, especially at small mass-damping or Skop-Griffin parameters. This difference between the LFD and NFD can only explained by the weaker contribution of the NFD, than that of the LFD, on the structural motion, especially at small mass ratios, despite of the fact that the structure damping is weaker than the fluid damping.…”

“…Of course, the response amplitude is quite limited. [17] , and = 0.0052 . + experimental data from Govardhan and Williamson [17] , --model with the LFD, is the slope of f , and 0.174 is from the experimental result [17] Besides, other features of response can be observed in comparison with experimental data.…”

“…[17] , and = 0.0052 . + experimental data from Govardhan and Williamson [17] , --model with the LFD, is the slope of f , and 0.174 is from the experimental result [17] Besides, other features of response can be observed in comparison with experimental data. It is noted in Fig.5(b) that the maximum amplitude in the NFD model appears at about as observed in experiment, rather than at points slightly higher than as indicated by the LFD model.…”

“…Let us now consider dynamic behaviors of the coupled system at a very small mass ratio, below the critical mass ratio introduced by Govardhan and Williamson = 0.542 crit m [17] . The computational parameters are and = 0.52 m = 0.0052 [2,17] .…”

“…The computational parameters are and = 0.52 m = 0.0052 [2,17] . And, , 0 = 64.1 In experiments, the range of lock-in at such low mass ratio would be quite different from that observed at higher mass ratio.…”