Hamiltonian Structure and Dynamics of a Neutrally Buoyant Rigid Sphere Interacting with Thin Vortex Rings

Abstract

In a previous paper, we presented a (noncanonical) Hamiltonian model for the dynamic interaction of a neutrally INTRODUCTIONThe objective of this paper is essentially twofold. First, we want to present the equations of motion and the Hamiltonian structure of the system consisting of a neutrally buoyant rigid sphere interacting dynamically with N arbitrarily-shaped and arbitrarily-oriented vortex rings, modeled as N closed curves in R 3 , as in Figure 1. These equations and Hamiltonian structure will be derived as a special case of the model described in [1]. The simple geometry of the sphere allows an explicit representation of the image velocity field of the rings and we will follow the work of [2] for this. Second, with a view to studying dynamic orbits of such a system, we focus on the case of an axisymmetric configuration in which the rings are all circles (in parallel planes) with centers along a common line passing through the center of the sphere, as in Figure 2. For this axisymmetric case, the system equations become ordinary differential equations and are thus easily integrated numerically.Motivations for constructing models like these come from locomotion problems in a fluid environment-both in nature and engineering-such as, for example, the swimming of neutrally buoyant fish and the energy-efficient design of small, autonomous underwater vehicles where coherent vortical structures in the vicinity of the moving body play an important role. Other potential applications, more in line with the theme of this conference, include transport phenomena of small particles in a fluid environment or of particles in microgravity environments. The models incorporate nonlinear effects and, within an inviscid framework, fully couple the solid-fluid dynamics. It is of course necessary to develop these models more, in particular to include the effects of fluid viscosity and, perhaps, turbulence. Nevertheless, as a start, these non-trivial, lowdimensional models provide a platform for applying the powerful theoretical tools of dynamical systems and nonlinear control to study the complex phenomena of solid-fluid interactions in locomotion problems.

Geometric-control formulation and averaging analysis of the unsteady aerodynamics of a wing with oscillatory controls

“…Most of these efforts are nicely summarized in Arnold's (2013) and Kambe's (2009) books. With a similar spirit, there are several efforts made to construct a Hamiltonian structure for the dynamics of point vortices interacting with a rigid body in an ideal flow in the two-dimensional case of zero circulation around the body (Shashikanth et al 2002;Shashikanth 2005), arbitrary circulation around the body (Borisov, Mamaev & Ramodanov 2003, 2007, the three-dimensional case (Shashikanth et al 2008(Shashikanth et al , 2010Dritschel & Boatto 2015) and the case of unsteady (time-varying) point vortices (Hussein et al 2018).…”