We have performed direct numerical simulations for the flow past a streamwise and transversely rotating sphere at a Reynolds number $\mathrm{Re}=500$. To investigate the effect of low to very high rotation rates on the flow dynamics, we have varied the dimensionless streamwise $\left({\mathrm{\Omega }}_x\right)$ and transverse $\left({\mathrm{\Omega }}_z\right)$ rotation rates between 0.5 and 5. The streamwise rotation of the sphere causes twisting of the vortex structures in the wake and results in centrifugal instabilities. The wake consists of two intertwined spiral vortices at ${\mathrm{\Omega }}_x=0.5$, whereas the vortex structures form threads and bifurcate into large-scale shear layers for ${\mathrm{\Omega }}_x\ge 2$. The shedding pattern during the transverse rotation is subject to turning. The structures at the aiding side of the motion experience tearing, whereas they squeeze at the opposing side. We found single-sided shedding of the hairpin structure at ${\mathrm{\Omega }}_z=0.5$. The wake deforms into numerous small-scale vortex filaments for ${\mathrm{\Omega }}_x=2$, whereas it becomes complex, constituting stretched hairpins at ${\mathrm{\Omega }}_x=5$. During the rotation cycle of the sphere, the wake shows steady flow at ${\mathrm{\Omega }}_x=0.5$. In contrast, at ${\mathrm{\Omega }}_z=0.5$, we found one-sided shedding of hairpin vortices in the near wake, which becomes ringlike vortices in the far wake. The wake at ${\mathrm{\Omega }}_z=2$ forms randomly oriented vorticity patches, which cannot cope with the sphere's rotation, thereby retaining their shape and orientation during the rotation cycle. The line-time reconstruction of vertical velocity signals shows discontinuity in vortex formation for higher ${\mathrm{\Omega }}_{x,z}$. We have formulated a method for tracking the trajectories of the vortices. In the near wake at ${\mathrm{\Omega }}_x=0.5$, the vortex trajectories move in a circular path, whereas it settles at a larger radius in the far wake. We observe that for ${\mathrm{\Omega }}_x\ge 2$, the local rotation center of the near-wake vortices cannot cope with high rotational speeds, and the fluid particles partially obey the effects of solid-body rotation. A higher degree of mixing of the fluid particles in the near wake at ${\mathrm{\Omega }}_x=5$ decreases the localized rotation of the vortex trajectories, and the vortex centers settle along the radial lines in the far wake. We found the symmetric distribution of the vortex trajectories at all transverse planes during ${\mathrm{\Omega }}_z$ rotation. Our results indicate that a rotating sphere always experiences a positive drag irrespective of the rotational direction. The time average normalized angular velocity of the vortices at various transverse planes reveals that the near wake vortices could cope with the sphere's rotation for ${\mathrm{\Omega }}_x=0.5$. However, it decays marginally for ${\mathrm{\Omega }}_x=2$. In contrast, most of the rotational speed is lost for ${\mathrm{\Omega }}_x=5$. We have calculated the penetration depth as the downstream length for which $10\%$ of the sphere's angular speed is preserved by the downstream vortices, which reveals rapid reduction for increasing ${\mathrm{\Omega }}_x$. At various downstream locations, the normalized angular frequency of the vortices renders the asynchronous rotation of the vortex structures. We found a minimal impact of ${\mathrm{\Omega }}_z$ on the rotation of the downstream vortices. However, the wake fluctuations are triggered with increasing rotation rate, although fluctuations are lesser for ${\mathrm{\Omega }}_z$ rotation. The near wake vortices reveal low-frequency rotation for all ${\mathrm{\Omega }}_z$, and the asynchronous motion of the vortices triggers its asynchronicity for increasing ${\mathrm{\Omega }}_z$.

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• Flow-structure interactions
• Wakes & jets