The Motion of Solid Bodies in Potential Flow with Circulation: a Geometric Outlook

The motion of a circular body in 2D potential flow is studied using symplectic reduction. The equations of motion are obtained starting from a kinetic-energy type system on a space of embeddings and reducing by the particle relabelling symmetry group and the special Euclidian group. In the process, we give a geometric interpretation for the Kutta-Joukowski lift force in terms of the curvature of a connection on the original phase space. INTRODUCTION It has been known since the pioneering work of Kirchhoff, Stokes, and Lamb that the motion of a rigid body in a potential flow has a very succinct description with the ambient fluid manifesting itself only through the appearance of added masses and added moments of inertia. If the circulation around the body is non-zero, or if isolated point vortices are present in the fluid, additional effects have to be taken into account. In the former case, the body experiences an additional lift force, proportional to its velocity and the circulation. The resulting dynamics was first studied by Chaplygin and Lamb (see [1] and the references therein). In the case of point vortices, these effects are sufficiently subtle for the equations of motion of the system to have been derived only recently (see [2]). The Kirchhoff equations for a rigid body in a potential flow were studied from a geometric point of view in [3, 4]. In this paper, we show that this formalism can be extended to the case where the circulation around the body is not necessarily zero. ∗Address all correspondence to this author. †Permanent address: Department of Mathematical Physics and Astronomy, Ghent University, Krijgslaan 281, B-9000 Ghent, Belgium. The motion of a rigid body in a perfect fluid, even with circulation, can be viewed as a prime example of geometric reduction theory (see [5,6]). From this point of view, the body-fluid system first is defined as a dynamical system on an infinite-dimensional configuration space Q, consisting of two parts: one accounting for the position of the body, and the other consisting of maps taking the fluid labels to their respective positions in material space at a certain instant. The degrees of freedom of the system on Q can then be reduced to a finite number by realizing that two distinct symmetry groups act on Q, and dividing out by these group actions. First, there is the group of volume preserving diffeomorphisms, which acts on the label space of the fluid and simply permutes the labels of the fluid particles. Secondly, the whole system (consisting of solid and fluid) is invariant under global translations and rotations. Dividing out by these symmetry groups naturally leads to the Kirchhoff equations with an additional lift force proportional to the circulation (equation 25 below). We limit ourselves to the case of a rigid body moving in a potential flow with circulation but no external vorticity, as this case is not overly complicated but already exhibits many of the interesting features present in geometric reduction theory. A significant feature of our analysis is that the lift force experienced by the body turns out to be nothing but the velocity vector of the body contracted with a certain curvature tensor (see equation 20). In this way, we provide an alternative geometric description of what is known in the classical literature on fluid dynamics as the Kutta-Joukowski theorem. The layout of the paper is as follows. After describing the Proceedings of DSCC2008 2008 ASME Dynamic Systems and Control Conference October 20-22, 2008, Ann Arbor, Michigan, USA 1 Copyright © 2008 by ASME DSCC2008-2292 problem setting and recalling some well-known facts from potential flow theory and rigid body dynamics, we describe the geometric approach to perfect fluid dynamics. We then apply this theory to the case at hand, i.e. the dynamics of a rigid circular body in a potential flow with circulation. The bulk of the paper is devoted to reducing the dynamics with respect to the symmetry groups described above. In the process, we use a certain connection on the unreduced phase space and calculate its curvature. In the final sections, the equations of motion are derived and we discuss the physical significance of this curvature. The paper ends with an outlook on possible generalizations of this approach. PROBLEM SETTING We consider a rigid body of cylindrical shape moving in an inviscid, incompressible fluid. The body – considered to be uniform and neutrally-buoyant (the body weight is balanced by the force of buoyancy) – may be represented by a disc in R2 and for the sake of convenience we assume that the fluid fills the complement of the body in R2. This assumption can easily be relaxed, for example to the case where the fluid moves in a bounded container, or on a two-dimensional surface different from R2 (such as the 2-sphere). The reference configuration of the fluid will be denoted by F0, and that of the body by B0. The space taken by the fluid at a generic time t will be denoted by F . Note however that as time progresses, the position of the body changes and hence so does its complement F . Rigid body kinematics. Introduce an orthonormal inertial frame {e1,2,3} where {e1,e2} span the plane of motion and e3 is the unit normal to this plane. The configuration of the submerged rigid body can then be described by a rotation β about e3 and a translation r= xoe1+yoe2 of a point O (often chosen to coincide with the mass center) in the {e1,e2} directions (see figure ). The angular and translational velocities expressed relative to the inertial frame are of the form β̇e3 and v= vx e1+vy e2 where vx = ẋo, vy = ẏo (the dot denotes derivative with respect to time t). It is convenient for the following development to introduce a moving frame {b1,2,3} attached to the body. The point transformation from the body to the inertial frame can be represented as x= RX+ r, R= ( cosβ −sinβ sinβ cosβ )