Flow past a spinning circular cylinder placed in a uniform stream is investigated via two-dimensional computations. A stabilized finite element method is utilized to solve the incompressible Navier-Stokes equations in the primitive variables formulation. The Reynolds number based on the cylinder diameter and free-stream speed of the flow is 200. The non-dimensional rotation rate, α (ratio of the surface speed and freestream speed), is varied between 0 and 5. The time integration of the flow equations is carried out for very large dimensionless time. Vortex shedding is observed for α < 1.91. For higher rotation rates the flow achieves a steady state except for 4.34 < α < 4.70 where the flow is unstable again. In the second region of instability, only one-sided vortex shedding takes place. To ascertain the instability of flow as a function of α a stabilized finite element formulation is proposed to carry out a global, non-parallel stability analysis of the two-dimensional steady-state flow for small disturbances. The formulation and its implementation are validated by predicting the Hopf bifurcation for flow past a non-rotating cylinder. The results from the stability analysis for the rotating cylinder are in very good agreement with those from direct numerical simulations. For large rotation rates, very large lift coefficients can be obtained via the Magnus effect. However, the power requirement for rotating the cylinder increases rapidly with rotation rate.

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SUMMARYThe e ect of location of the lateral boundaries, of the computational domain, on the critical parameters for the instability of the ow past a circular cylinder is investigated. Linear stability analysis of the governing equations for incompressible ows is carried out via a stabilized ÿnite element method to predict the primary instability of the wake. The generalized eigenvalue problem resulting from the ÿnite element discretization of the equations is solved using a subspace iteration method to get the most unstable eigenmode. Computations are carried out for a large range of blockage, 0:005 6 D=H 6 0:125, where D is the diameter of the cylinder and H is the lateral width of the domain. A non-monotonic variation of the critical Re with the blockage is observed. It is found that as the blockage increases, the critical Re for the onset of the instability ÿrst decreases and then increases. However, a monotonic increase in the non-dimensional shedding frequency at the onset of instability, with increase in blockage, is observed. The increased blockage damps out the low-frequency modes giving way to higher frequency modes. The blockage is found to play an important role in the scatter in the data for the non-dimensional vortex shedding frequency at the onset of the instability, from various researchers in the past.

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