Vortex-induced vibrations of a sphere

There are many studies on the vortex-induced vibrations of a cylindrical body, but almost none concerned with such vibrations for a sphere, despite the fact that tethered bodies are a common configuration. In this paper, we study the dynamics of an elastically mounted or tethered sphere in a steady flow, employing displacement, force and vorticity measurements. Within a particular range of flow speeds, where the oscillation frequency ($f$) is of the order of the static-body vortex shedding frequency ($f_{vo})$, there exist two modes of periodic large-amplitude oscillation, defined as modes I and II, separated by a transition regime exhibiting non-periodic vibration. The dominant wake structure for both modes is a chain of streamwise vortex loops on alternating sides of the wake. Further downstream, the heads of the vortex loops pinch off to form a sequence of vortex rings. We employ an analogy with the lift on an aircraft that is associated with its trailing vortex pair (of strength $\Gamma^*$ and spacing $b^*$), and thereby compute the rate of change of impulse for the streamwise vortex pair, yielding the vortex force coefficient ($\cvortex$): \[ \cvortex = \frac{8}{\pi} {U^*_{v}}b^*( - \Gamma^*). \] This calculation yields predicted forces in reasonable agreement with direct measurements on the sphere. This is significant because it indicates that the principal vorticity dynamics giving rise to vortex-induced vibration for a sphere are the motions of these streamwise vortex pairs. The Griffin plot, showing peak amplitudes as a function of the mass–damping ($m^*\zeta$), exhibits a good collapse of data, indicating a maximum response of around 0.9 diameters. Following recent studies of cylinder vortex-induced vibration, we deduce the existence of a critical mass ratio, $m^*_{crit} {\approx} 0.6$, below which large-amplitude vibrations are predicted to persist to infinite normalized velocities. An unexpected large-amplitude and highly periodic mode (mode III) is found at distinctly higher flow velocities where the frequency of vibration ($f$) is far below the frequency of vortex shedding for a static body. We find that the low-frequency streamwise vortex pairs are able to impart lift (or transverse force) to the body, yielding a positive energy transfer per cycle.