Dynamic interaction of point vortices and a two-dimensional cylinder

Abstract

In this paper we consider the system of an arbitrary two-dimensional cylinder interacting with point vortices in a perfect fluid. We present the equations of motion and discuss their integrability. Simulations show that the system of an elliptic cylinder ͑with nonzero eccentricity͒ and a single point vortex already exhibits chaotic features and the equations of motion are nonintegrable. We suggest a Hamiltonian form of the equations. The problem we study here, namely, the equations of motion, the Hamiltonian structure for the interacting system of a cylinder of arbitrary cross-section shape, with zero circulation around it, and N vortices, has been addressed by Shashikanth ͓Regular Chaotic Dyn. 10, 1 ͑2005͔͒. We slightly generalize the work by Shashikanth by allowing for nonzero circulation around the cylinder and offer a different approach than that by Shashikanth by using classical complex variable theory.

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

“…Note that the governing equations were simultaneously derived in Borisov et al (2007) following a different approach with emphasis on the Hamiltonian structure of the system. Also note that the interactions of multiple rigid bodies in potential flow was considered in the recent works of Wang (2004), Kanso et al (2005), Nair and Kanso (2007) and Crowdy et al (2007).…”