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

In this paper, we study the transverse vortex-induced vibrations of an elas tically 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 paramete r (m*zeta) In the classical high-(m*zeta) case, an 'initial' and 'lower' am plitude branch are separated by a discontinuous mode transition, whereas in the case of low (m*zeta), a further higher-amplitude 'upper' branch of res ponse appears, and there exist two mode transitions. To understand the existence of more than one mode transition for low (m*zet a), we employ two distinct formulations of the equation of motion, one of w hich uses the 'total force', while the other uses the 'vortex force', which is related only to the dynamics of vorticity. The first mode transition in volves a jump in 'vortex phase' (between vortex force and displacement), ph i(vortex), at which point the frequency of oscillation (f) passes through t he natural frequency of the system in the fluid, f similar to f(Nwater) Thi s transition is associated with a jump between 2S <-> 2P vortex wake modes, and a corresponding switch in vortex shedding timing. Across the second mo de transition, there is a jump in 'total phase', phi(total), at which point f similar to f(Nvacuum) In this case, there is no jump in phi(vortex), sin ce both branches are associated with the 2P mode, and there is therefore no switch in timing of shedding, contrary to previous assumptions. Interestin gly, for the high-(m*zeta) case, the vibration frequency jumps across both f(Nwater) and f(Nvacuum), corresponding to the simultaneous jumps in phi(vo rtex) and phi(total). This causes a switch in the timing of shedding, coinc ident with the 'total phase' jump, in agreement with previous assumptions. For large mass ratios, m* = O(100), the vibration frequency for synchroniza tion lies close to the natural frequency (f* = f/f(N) approximate to 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: f(lower)* = root(m* + C-A)/(m* - 0.54), which agrees very well with a wide set of experimental data. This frequency equation uncovers the existence of a critical mass ratio, where the freque ncy 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-amplit ude vibrations persist for all velocities, no matter how high, and the freq uency increases indefinitely with flow velocity. Experiments at m* < m(crit )* show that the upper-branch vibrations continue to the limits (in flow sp eed) of our facility.