Vortex-induced vibration of a streamwise and transversely rotating sphere

We investigate the effect of streamwise and transverse rotation on the wake behind an elastically mounted sphere. Simulations are performed at a Reynolds number $Re=500$ over a range of reduced velocity $2\le U^{\ast }\le 12$ , considering a low and high rotational speed (0.2 and 1), keeping the mass ratio $m^{\ast }=2$ . Streamwise rotation yields a structural response akin to the non-rotating case, while transverse rotation triggers induced vibration at lower $U^{\ast }$ and sustains it across a wider range. Like the non-rotating case, the streamwise rotating sphere exhibits synchronous, high-amplitude vibration across the entire $U^\ast$ range, whereas for low transverse rotation, it is confined to $5\le U^{\ast }\le 6$ . Cross-stream displacement of the sphere remains unaffected by streamwise rotation with increasing $U^{\ast }$ . In contrast, it monotonically increases due to transverse rotation, driven by the Magnus force, as supported by our theoretical and numerical estimations. While the spiral shedding mode dominates at $\Omega _{x}=0.2$ , twisted hairpin and twisted spiral modes emerge as the rotation rate is increased. On the other hand, we observe the hairpin (HP) mode, as seen in the non-rotating case, for low transverse rotation. The HP mode gives rise to the ring vortical mode at the far wake, and with an increase in $U^\ast$ , the wake shows small-scale stretched threads and reconnected bridgelets. Wake fluctuations increase with a streamwise rotation that saturates at higher $U^{\ast }$ during synchronisation, while desynchronisation at dimensionless transverse rotation rate $\Omega _{z}=1$ induces intermittent low-amplitude vibration via the Magnus effect. Space–time reconstruction at the near wake shows an undisturbed helical vortex core at $\Omega _{x}=0.2$ and $U^{\ast }=5$ , which bifurcates at $\Omega _{x}=1$ owing to the centrifugal-induced distortion. At $\Omega _{x}=1$ and $U^{\ast }=5$ , the phase difference between $(y, C_{y})$ and $(z, C_{z})$ exhibits in-phase synchrony with occasional phase slips. The wake vortex remains unaffected by the transverse rotation of the sphere; however, a streamwise rotating sphere couples the wake, leading to a rotational lock-in. The wake rotation shifts from anti-clockwise to clockwise sense earlier (in $U^\ast$ ) at a lower rotation rate. The reduced velocity is seen to have a favourable effect on the transfer of the sphere’s rotational inertia onto the wake as the measured penetration depth increases with $U^{\ast }$ . Insights from the present research will aid in understanding complex flow interactions in rotational systems, improving efficiency, stability and control in modern engineering applications.