Hamiltonian structure for a neutrally buoyant rigid body interacting with N vortex rings of arbitrary shape: the case of arbitrary smooth body shape

Abstract

We present a (noncanonical) Hamiltonian model for the interaction of a neutrally buoyant, arbitrarily shaped smooth rigid body with N thin closed vortex filaments of arbitrary shape in an infinite ideal fluid in Euclidean three-space. The rings are modeled without cores and, as geometrical objects, viewed as N smooth closed curves in space. The velocity field associated with each ring in the absence of the body is given by the Biot-Savart law with the infinite self-induced velocity assumed to be regularized in some appropriate way. In the presence of the moving rigid body, the velocity field of each ring is modified by the addition of potential fields associated with the image vorticity and with the irrotational flow induced by the motion of the body. The equations of motion for this dynamically coupled body-rings model are obtained using conservation of linear and angular momenta. These equations are shown to possess a Hamiltonian structure when written on an appropriately defined Poisson product manifold equipped with a Poisson bracket which is the sum of the Lie-Poisson bracket from rigid body mechanics and the canonical bracket on the phase space of the vortex filaments. The Hamiltonian function is the total kinetic energy of the system with the self-induced kinetic energy regularized. The Hamiltonian structure is independent of the shape of the body, (and hence) the explicit form of the image field, and the method of regularization, provided the self-induced velocity and kinetic energy are regularized in way that satisfies certain reasonable consistency conditions.

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

“…The problem setting and underlying assumptions are the same as in [1], however, for the sake of completeness, we describe these again. We consider a rigid smooth sphere that is immersed in an ideal (inviscid, incompressible) fluid.…”

“…The Lie-Poisson, or the momentum, equations [3] for the rings-sphere system are obtained from the Lie-Poisson equations in [1] by making the following observation. For the special geometry of the sphere and for the inviscid, free-slip boundary conditions in this problem, the angular velocity of the sphere Ω cannot affect and cannot be affected by the dynamics of the system.…”

“…Thus, the final coupled equations of motion of the dynamically interacting system of a rigid smooth sphere and N vortex rings of arbitrary shape, in a body-fixed frame, are given by (1) and (5):…”

“…Note that the equation for the angular momentum (impulse) A of the system (see [1]) is decoupled from the above equations and hence plays no role in the dynamics of the system.…”

“…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.…”