This paper considers the dynamics of a rigid body interacting with point vortices in a perfect fluid. The fluid velocity is obtained using the classical complex variables theory and conformal transformations. The equations of motion of the solid-fluid system are formulated in terms of the solid variables and the position of the point vortices only. These equations are applied to study the dynamic interaction of an elliptic cylinder with vortex pairs for its relevance to understanding the swimming of fish in an ambient vorticity field. One finds two families of relative equilibria: moving Föppl equilibria and equilibria along the ellipse's axis of symmetry (the axis perpendicular to the direction of motion). The two families of relative equilibria are similar to those present in the classical problem of flow past a fixed body but their stability differs significantly from the classical ones.
We consider the passive locomotion of rigid bodies in inviscid point-vortex wakes. This work is motivated by a common belief that live and inanimate objects may extract energy from unsteady flows for locomotory advantages. Studies on energy extraction from unsteady flows focus primarily on energy efficiency. Besides efficiency, a fundamental aspect of energy extraction for locomotion purposes is stability of motion. Here, we propose idealized wake models using periodically generated point vortices to emulate shedding of vortices from an un-modeled moving or stationary object. We assess the stability of these point-vortex wakes and find that they are stable for a range of periods, unlike the von Kármán street model which is mainly unstable. We then investigate the dynamics of a rigid body submerged in such wakes. In particular, we calculate periodic trajectories where the rigid body “swims” passively against the flow by extracting energy from the ambient vortices. All the periodic trajectories we find are unstable. The largest instabilities reported are for elliptic bodies where rotational effects play a role in destabilizing their motion. Within the context of this model, we conclude that passive locomotion of rigid bodies in inviscid wakes is unstable. Questions as to whether passive stability can be achieved when accounting for fluid viscosity and body elasticity remain open.
We investigate the behavior of an infinite array of (reverse) von Kármán 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 Kármán 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 Kármán 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.
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