Vortex-shedding modes of a streamwise and transversely rotating sphere undergoing vortex-induced vibrations

We investigate the vortex-shedding modes of a streamwise and transversely rotating sphere undergoing vortex-induced vibration (VIV). At Reynolds number Re = 500, we have conducted direct numerical simulations for reduced velocities U*=6,8, and 11, considering dimensionless streamwise (alpha(x)) and transverse (alpha(z)) rotation rates set at alpha(x,z)=1. At alpha(x)=1, the vortex-shedding mode exhibits twisted spiral-like structures, while at alpha(z)=1, ring-like structures form through the stretching and twisting of hairpins. For alpha(x)=1, the dimensionless circulation (Gamma(x)) decreases with increasing U*, showing negative growth at U*=6. Conversely, for alpha(z)=1, Gamma(x) remains negative except at U*=8, where counterclockwise vortex rotation results in positive circulation. At alpha(z)=1, VIV is periodic for U*=6 and 8 but intermittent at U*=11. Phase analysis using the Hilbert transform reveals anti-phase synchronization in the phase between the sphere's displacements and the vortex force coefficient (Delta phi(V)) and phase slip in the phase between the sphere's displacements and the total force coefficient (Delta phi(T)) along the y direction for alpha(x)=1. At alpha(z)=1, phase slip with distinct phase-locking epochs is observed along the z direction for all U* cases. Pre-lock-in behavior with phase slip is also noted for U*=11 along the y direction.