Three-dimensional flow past a rotating cylinder

Three-dimensional computations are carried out for a spinning cylinder placed in a uniform flow. The non-dimensional rotation rate is varied in the range $0.0\leqslant {\it\alpha}\leqslant 5.0$. A stabilized finite element method is utilized to solve the incompressible Navier–Stokes equations in primitive variables formulation. Linear stability analysis of the steady state shows the existence of several new unstable three-dimensional modes for $200\leqslant \mathit{Re}\leqslant 350$ and $4.0\leqslant {\it\alpha}\leqslant 5.0$. The curves of neutral stability of these modes are presented in the $\mathit{Re}{-}{\it\alpha}$ parameter space. For the flow at $\mathit{Re}=200$ and rotation rate in the ranges $0.0\leqslant {\it\alpha}\leqslant 1.91$ and $4.34\leqslant {\it\alpha}\leqslant 4.7$, the vortex shedding, earlier reported in two dimensions and commonly referred to as parallel shedding, can also exist as oblique shedding. In this mode of shedding, the vortices are inclined to the axis of the cylinder. In fact, parallel shedding is a special case of oblique shedding. It is found that the span of the cylinder plays a significant role in the time evolution of the flow. Of all the unstable eigenmodes, with varied spanwise wavenumber, only the ones whose integral number of wavelengths fit the span length of the cylinder are selected to grow. For the flow at $\mathit{Re}=200$, two steady states exist for $4.8\leqslant {\it\alpha}\leqslant 5.0$. While one of them is associated with unstable eigenmodes, the other is stable to all infinitesimal perturbations. In this regime, irrespective of the initial conditions, the fully developed flow is steady and devoid of any instabilities.