Primary and secondary instabilities in the wake of a cylinder with free ends

The wake of a finite cylinder with free ends and an aspect ratio of 21.4 is simulated in three-dimensions and analysed theoretically. Close to the primary-instability threshold, the flow is shown to settle on a limit cycle with a uniform frequency throughout the flow-field. About 20% above the primary-instability threshold, a secondary instability sets in and the limit cycle becomes unstable. The new attractor of the flow can be identified as a limit T-2-torus characterized by two incommensurate frequencies. One of them is shown to evolve continuously from the primary-instability frequency, the other one, about 17 times smaller near the secondary-instability threshold, generates a slow modulation of the oscillations in the wake. The limit cycle and the limit torus are described in terms of their Fourier expansion and the spatial distribution of the most relevant Fourier components is investigated. The theoretical analysis and numerical results given shed some light on the mechanisms underlying a number of known but not satisfactorily explained three-dimensional effects in wakes of finite cylinders such as the ambiguity in the dominant Strouhal frequency, the existence of zones with different frequencies spanwise in the wake, the discreteness of coexisting frequencies observed in the wake as well as the spatial uniformity of the beating period. They moreover explain the Reynolds number variation of these effects and identify the recirculation around the cylinder ends as basically responsible for the onset of the secondary instability. The results are compared to the case of a cylinder with aspect ratio of 10.7 to determine the basic trends in aspect ratio dependence. It is shown that qualitatively the same behaviour is obtained, but that the secondary-instability threshold is shifted significantly upward to about twice the primary-instability threshold. Simulations of the wake of a finite NACA wing with incidence show that the form of the cross-section plays a minor role.