Quasi-three dimensional analysis of global instabilities: onset of vortex shedding behind a wavy cylinder

Abstract

In this paper the global-stability theory is extended to account for weak spanwise-flow variations using a quasi-three-dimensional framework. The analysis considers the onset of vortex shedding behind a circular cylinder with a spanwise-varying diameter. The quasi-three-dimensional approach models the fully three-dimensional flow structure as a series of two-dimensional eigenvalue problems representing the sectional-flow behaviour. The sectional results are coupled together using the Ginzburg–Landau equation, which models the diffusive coupling and provides the global response. The onset of global instability (and thus vortex shedding) is linked to both the sectional growth rates (characterized by the maximum-diameter location) and the spanwise extent of the zone of instability. Unsteady numerical simulations are used to guide the global-stability analysis and to assess the fidelity of the predictions. Results from the stability analysis are shown to be in good agreement with the numerical simulations, which are in close agreement with experiments.