Abstract

The motion of four point vortices with zero net circulation in a potential flow contained within a two-dimensional, singly periodic domain (i.e., a periodic strip) is determined under the assumption of a spatial symmetry that is preserved by the dynamics. This symmetry is inspired by the patterns observed in two-pair (2P) vortex wakes, in which four neighboring vortices appear as two pairs with a glide-reflective symmetry: the arrangement of each pair is related to the other by a reflection about the wake centerline and a half-period translation along the wake centerline. Under the assumed constraints, the problem can be reduced to an integrable Hamiltonian system. Vortex motions are classified using a bifurcation analysis of the phase space topology as determined by level curves of the Hamiltonian. Unlike the well-known von Karman point vortex model, in which a singly periodic system of two point vortices with glide-reflective symmetry is always in relative equilibrium, this four-point-vortex system exhibits a rich variety of relative motions for almost all possible initial conditions. Five distinct classes of relative vortex motion are identified, encompassing a total of 12 different types of motion, suggesting that experimental wakes with four vortices formed per shedding cycle may exhibit behaviors not yet explored in the literature. A finite number of initial conditions do correspond to relative equilibria, in which case the vortex configuration propagates downstream with invariant size and shape. Some of these relative equilibria are neutrally stable to perturbations that preserve the system constraints, while others are unstable, leading to large deviations from the equilibrium configuration. (C) 2015 AIP Publishing LLC.