an:01563409
Zbl 0988.76027
Govardhan, R.; Williamson, C. H. K.
Modes of vortex formation and frequency response of a freely vibrating cylinder
EN
J. Fluid Mech. 420, 85-130 (2000).
00070737
2000
j
76D17 76-05
vortex phase; upper-branch vibration; total phase; transverse vortex-induced vibrations; elastically mounted rigid cylinder; DPIV technique; mode transition; vortex shedding; vibration frequency; lower-branch vibration; critical mass ratio
The authors study transverse vortex-induced vibrations of an elastically mounted rigid cylinder in a fluid flow. To determine the vorticity field, for the first time in such free-vibration investigation the authors employ the DPIV technique simultaneously with force and displacement measurements. There exist two distinct types of response in such systems, depending on whether one has a high or low combined mass-damping parameter \(m^*\zeta\). In the classical high-\(m^*\zeta\) case, `initial' and `lower' amplitude branches are separated by a discontinuous mode transition, whereas in the case of low \(m^*\zeta\), a further high-amplitude `upper' branch of response appears, and there exist two mode transitions. To understand the existence of more than one mode transition for low \(m^*\zeta\), the authors give 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), \(\varphi_{\text{vortex}}\), at which point the frequency of oscillation \(f\) passes through the natural frequency of the system in the fluid, \(f\sim f_{N \text{water}}\). This transition is associated with a jump between \(2S\leftrightarrow 2P\) vortex wake modes, and with a corresponding switch in vortex shedding timing. Across the second mode transition, there is a jump in `total phase', \(\varphi_{\text{total}}\), at which poin \(f\sim f_{N \text{vacuum}}\). In this case, there is no jump in \(\varphi_{\text{vortex}}\), since both branches are associated with the \(2P\) mode, and therefore there is no switch in timing of shedding, contrary to previous situations. It is noted that for the high-\(m^*\zeta\) case, the vibration frequency jumps across both \(f_{N\text{water}}\) and \(f_{N\text{vacuum}}\), corresponding to the simultaneous jumps in \(\varphi_{\text{vortex}}\) and \(\varphi_{\text{total}}\). This causes a switch in the timing of shedding, coincident with the `total phase' jump, in agreement with previous assumptions. For large mass ratios, \(m^*=O(1)\), the vibration frequency for synchronization lies close to the natural frequency \((f^*=f/f_N \approx 1.0)\), but as mass is reduced to \(m^*= O(1)\), \(f^*\) can reach remarkably large values. The authors derive an expression for the frequency of lower-branch vibration, \(f^*_{\text{lower}} =\sqrt{{m^*+C_A\over m^*-0.54}}\), which agrees very well with a wide set of experimental data. This frequency equation uncovers the existence of a critical mass ratio, where the frequency \(f^*\) becomes large: \(m^*_{\text{crit}}=0.54\). When \(m^*<m^*_{\text{crit}}\), 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 \(m^*< m^*_{\text{crit}}\) show the beginnings of this high-amplitude upper-branch, persisting to the limits of experimental facility, yielding vibration frequencies in excess of 4 times the natural frequency.
F.Kaplanski (Tallinn)