Multiply-interacting Vortex Streets

Abstract

We investigate the behavior of an infinite array of (reverse) von K'arm'an streets. Our primary motivation is to model the wake dynamics in large fish schools. We ignore the fish and focus on the dynamic interaction of multiple wakes where each wake is modeled as a reverse von K'arm'an street. There exist configurations where the infinite array of vortex streets is in relative equilibrium, that is, the streets move together with the same translational velocity. We examine the topology of the streamline patterns in a frame moving with the same translational velocity as the streets which lends insight into fluid transport through the mid-wake region. Fluid is advected along different paths depending on the distance separating two adjacent streets. Generally, when the distance between the streets is large enough, each street behaves as a single von K'arm'an street and fluid moves globally between two adjacent streets. When the streets get closer to each other, the number of streets that enter into partnership in transporting fluid among themselves increases. This observation motivates a bifurcation analysis which links the distance between streets to the maximum number of streets transporting fluid among themselves. We also show that for short times, the analysis of streamline topologies for the infinite arrays of streets can be expected to set the pattern for the more realistic case of a finite array of truncated streets, which is not in an equilibrium state and its dynamic evolution eventually destroys the exact topological patterns identified in the infinite array case. The problem of fluid transport between adjacent streets may be relevant for understanding the transport of oxygen and nutrients to inner fish in large schools as well as understanding flow barriers to passive locomotion.