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Published online by Cambridge University Press: 30 July 2025
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.