Hydrodynamic forces on a rotating sphere

► We study the hydrodynamic forces on a rotating sphere. ► Phase diagrams are employed to aid analysis. ► Isosurfaces of the wake structures are classified into flow regimes. ► Existence of shear layer instability is quantitatively compared. ► Oscillations amplitude and periodicity of the forces are highly correlated to the flow regimes. The wake dynamics of a rotating sphere with prescribed rotation axis angles are quantitatively analysed by carrying out numerical simulations at Reynolds numbers of Re=100, 250 and 300, non-dimensional rotational rates Ω∗=0–1 and rotation axis angles α=0, π/6, π/3 and π/2 measured from the free stream axis. These parameters are the same as those in an earlier study (Poon et al., 2010, Int. J. Heat Fluid Flow) where the instantaneous flow structures were discussed qualitatively. This study extends the findings of the earlier study by employing phase diagrams (CLx,CLy) and (CD,CL) to provide a quantitative analysis of the time-dependent behaviour of the flow structures. At Re=300 and Ω∗=0.05, the phase diagrams (CLx,CLy) show ‘saw tooth’ patterns for both α=0 and π/6. The ‘saw tooth’ pattern indicates that the flow structures comprise a higher frequency oscillation component at a Reynolds number of 300 which is not observed until Re≈800 for a stationary sphere. This ‘saw tooth’ pattern disappears as Ω∗ increases. The employment of the phase diagrams also reveals that different flow structures induce different oscillation amplitudes on both lateral force coefficients. With the exception of the vortices formed from a shear layer instability, all other flow regimes show larger fluctuations in CL than CD.