The Hamiltonian structure of a two-dimensional rigid circular cylinder interacting dynamically with N point vortices

Abstract

This paper studies the dynamical fluid plus rigid-body system consisting of a two-dimensional rigid cylinder of general cross-sectional shape interacting with N point vortices. We derive the equations of motion for this system and show that, in particular, if the vortex strengths sum to zero and the rigid-body has a circular shape, the equations are Hamiltonian with respect to a Poisson bracket structure that is the sum of the rigid body Lie-Poisson bracket on se(2)*, the dual of the Lie algebra of the Euclidean group on the plane, and the canonical Poisson bracket for the dynamics of N point vortices in an unbounded plane. We then use this Hamiltonian structure to study the linear and nonlinear stability of the moving Föppl equilibrium solutions using the energy-Casimir method.

Swimming in an inviscid fluid

“…When neglecting vorticity, the derived equations reduce to a known model for the locomotion of an articulated body in potential flow, [4]. The equations are also consistent with the models developed in [2,5,15] for the case of the dynamics of a rigid body in a fluid with point vortices.…”

“…The derived equations, when neglecting vorticity, reduce to the model developed in [4] for swimming in potential flow, and are also consistent with the models developed in [2,5,15] for a rigid body interacting dynamically with point vortices. The effects of cyclic shape changes and the presence of vorticity on the locomotion of a submerged body are discussed through examples.…”

“…This case was studied in [4] for a system of articulated rigid bodies using Hamilton's variational principle and a Lagrangian function equal to the kinetic energy of the solid-fluid system. Another approach would be to model the wake using discrete vortex structures as done in [2,5,15] for a rigid body interacting with point vortices. Indeed, equations (2) are consistent with those derived in [2,5,15] where P shape = Π shape = 0 and an additional set of equations governing the dynamic behavior of the point vortices is obtained.…”

“…Another approach would be to model the wake using discrete vortex structures as done in [2,5,15] for a rigid body interacting with point vortices. Indeed, equations (2) are consistent with those derived in [2,5,15] where P shape = Π shape = 0 and an additional set of equations governing the dynamic behavior of the point vortices is obtained. Finally, note that P w and Π w can be computed by coupling (2) to a numerical solver based on vortex shedding models such as those presented in [3,17].…”

Stability of a coupled body–vortex system

“…The dynamics of a rigid body interacting with point vortices was considered in Borisov and Mamaev (2003) and in Shashikanth et al (2002); Shashikanth (2005) for the case when the vortex strength sum to zero ( N k=1 Γ k = 0). The equations governing the motion of the rigid body were derived using Newtonian mechanics while the motion of the point vortices was described by a Kirchhoff-Routh function.…”