Wake structure of a transversely rotating sphere at moderate Reynolds numbers

Abstract

The uniform flow past a sphere undergoing steady rotation about an axis transverse to the free stream flow was investigated numerically. The objective was to reveal the effect of sphere rotation on the characteristics of the vortical wake structure and on the forces exerted on the sphere. This was achieved by solving the time-dependent, incompressible Navier–Stokes equations, using an accurate Fourier–Chebyshev spectral collocation method. Reynolds numbers Re of 100, 250 and 300 were considered, which for a stationary sphere cover the axisymmetric steady, non-axisymmetric steady and vortex shedding regimes. The study identified wake transitions that occur over the range of non-dimensional rotational speeds Ω* = 0 to 1.00, where Ω* is the maximum velocity on the sphere surface normalized by the free stream velocity. At Re = 100, sphere rotation triggers a transition to a steady double-threaded structure. At Re = 250, the wake undergoes a transition to vortex shedding for Ω* ≥ 0.08. With an increasing rotation rate, the recirculating region is progressively reduced until a further transition to a steady double-threaded wake structure for Ω* ≥ 0.30. At Re = 300, wake shedding is suppressed for Ω* ≥ 0.50 via the same mechanism found at Re = 250. For Ω* ≥ 0.80, the wake undergoes a further transition to vortex shedding, through what appears to be a shear layer instability of the Kelvin–Helmholtz type.

Vortex-induced vibration of a rotating sphere

“…case (e) and case ( f ), the recirculation bubble is evidently mostly suppressed. This near wake is very similar to the 'shear-layer instability' regime reported by Giacobello et al (2009), Kim (2009) and Poon et al (2014) for rigidly mounted rotating spheres at low Reynolds number (Re 1000). They reported single-sided shedding on the advancing side of the rotating sphere.…”

“…Previous studies (e.g. Giacobello, Ooi & Balachandar 2009;Kim 2009;Poon et al 2014, and references therein) have shown that the onset of the shear-layer instability wake state of a non-oscillating rotating sphere occurs beyond this α value. That wake state forms when fluid that passes the retreating side of the sphere is pushed towards the other side of the wake to form a distinctive one-sided separating shear layer, thus changing the characteristic formation and release of vortex loops that defines the non-rotating wake state.…”

“…From previous studies on rigidly mounted spheres, it is known that the transverse rotation imposes strong asymmetry in the wake, causing the loops to bend towards the advancing side (the side of the sphere moving in the direction opposite to the fluid) of the sphere due to the Magnus effect (Magnus 1853), which in consequence increases the 'lift force' towards the retreating side (the side of the sphere moving in the same direction as the fluid). Previous numerical studies by Giacobello et al (2009), Kim (2009 and Poon et al (2014) reported suppression of the vortex shedding for a certain range of rotation rates that depended on the Reynolds number. However, when α was increased, the vortex shedding resumed, although it was very different to the vortex shedding at lower α values that is associated with the 'buildup and release' of the recirculation bubbles behind the sphere.…”

The flow structures of a transversely rotating sphere at high rotation rates

“…The study by Poon et al [17,18] revealed that the orientation of the rotation axis also has a significant effect on the flow structure and hydrodynamic forces. In particular, the shear layer instability mechanism reported by Giacobello et al [11] was found at Re ¼ 300 not only for transverse rotation but also for a limited number of oblique rotation axis angles.…”

“…Numerical simulations offer a solution to this problem and by the early 1990s numerical solutions of the governing equations could produce revealing information on the flow field [8]. Subsequent studies solved for the full Navier-Stokes equations using a number of different numerical techniques such as the marker and cell method [9], immersed boundary methods [10] and spectral collocation methods [11][12][13][14][15][16]. The study by Giacobello et al [11] found that rotation significantly altered the structure of the flow field compared with the stationary case, resulting in significant changes in the hydrodynamic forces.…”