A mathematical model of 2P and 2C vortex wakes

We present a mathematical model of the vortex wake modes that appear behind neighboring and/or oscillating, flapping, and swimming bodies in which there are four vortices generated in an anti-symmetric pattern during each shedding cycle. The two-dimensional potential flow model consists of four point vortices with strengths ± Γ in a spatially periodic domain. The relative vortex positions are restricted by a discrete symmetry that is motivated by the spatial symmetry observed in experimental wakes. The strength restriction and the imposed symmetry result in the model system being an integrable Hamiltonian dynamical system. We find that the point vortex motion can be one of four distinct types based on the values of linear impulse and Hamiltonian. Two of these types correspond to 2P wakes and consist of two oppositely signed, counter-rotating vortex pairs. One of these types corresponds to 2C wakes and consists of two like-signed, co-rotating vortex pairs. The fourth type is an exchanging mode in which the two vortices near the wake centerline translate faster than the outer two vortices. Scaled comparisons of the model with both a 2P and a 2C experimental wake show good representation of the experimentally observed vortex dynamics and lead to estimates of the experimental vortex strengths.