Vortex equilibrium in flows past bluff bodies

The equilibrium conditions of a point vortex in the separated flow past a locally deformed wall is studied in the framework of the two-dimensional potential flow. Equilibrium locations are represented as fixed points of the vortex Hamiltonian contour line map. Their pattern is ascribable to the Poincaré–Birkhoff fixed-point theorem. An ‘equilibrium manifold’, representing the generalization of the Föppl curve for circular cylinders, is defined for arbitrary bodies. The property ∂ω/∂ψ ˜ =0 holds on it, with ψ ˜ being the stream function and ω the streamline slope of the pure potential flow. A ‘Kutta manifold’ is defined as the locus of vortices in flows that separate at a prescribed point (Kutta condition). The existence of standing vortices that satisfy the Kutta condition is discussed for symmetric bodies. On the basis of an asymptotic expansion of the equilibrium manifold, Kutta manifold and body geometry, it is shown that different classes of symmetric bodies exist which are ranked by the number of allowable standing vortices that satisfy the Kutta condition.