Vortex-induced vibrations of an elastically mounted sphere with three degrees of freedom at Re = 300: hysteresis and vortex shedding modes

Fluid–structure interaction (FSI) simulations are carried out to investigate vortex-induced vibrations of a sphere, mounted on elastic supports in all three spatial directions. The reduced velocity (${U}^{\ast } $) is systematically varied in the range ${U}^{\ast } = 4\text{{\ndash}} 9$, while the Reynolds number and reduced mass are held fixed at $\mathit{Re}= 300$ and ${m}^{\ast } = 2$, respectively. In the lock-in regime, two distinct branches are observed in the response curve, each corresponding to a distinct type of vortex shedding, namely, hairpin and spiral vortices. While shedding of hairpin vortices has been observed in several previous investigations of stationary and vibrating spheres, the shedding of intertwined, longitudinal spiral vortices in the wake of a vibrating sphere is reported herein for the first time. When the wake is in the hairpin shedding mode, the sphere moves along a linear path in the transverse plane, while when spiral vortices are shed, the sphere vibrates along a circular orbit. In the spiral mode branch, the simulations reveal hysteresis in the response amplitude at the beginning of the lock-in regime. Lower-amplitude vibrations are found as the sphere sheds hairpin vortices for increasing ${U}^{\ast } $ up until the beginning of the synchronization regime. On the other hand, higher-amplitude oscillations persist for the spiral mode as ${U}^{\ast } $ is decreased from the point of the start of the synchronization. The hairpin mode is found to be unstable for the value of reduced velocity where the spiral and hairpin solution branches merge together. When this point is approached along the hairpin solution branch, the sphere naturally transitions from shedding hairpin vortices and moving along a linear path to shedding spiral vortices and moving along a circular path in the transverse plane. The spiral mode was not observed in the work of Horowitz & Williamson (J. Fluid Mech., vol. 651, 2010, pp. 251–294), who studied experimentally the vibration modes of a freely rising or falling sphere and only reported zigzag vibrations. Our results suggest that this apparent discrepancy between experiments and simulations should be attributed to the fact that, for the range of governing parameters considered in the simulations, the elastic supports act to suppress streamwise vibrations, thus subjecting the sphere to a nearly axisymmetric elasticity constraint and enabling it to vibrate transversely along a circular path.