| A Family of Vortex Rings and
a Variational Application on Potential Flows Around Three-dimensional Bodies
Celso Pupo Pesce Universidade de Sao Paulo
Abstract The variational formulation and solution of general three-dimensional potential flows, firstly derived for the wave-body interaction problem in the presence of free-surface, Aranha & Pesce, 1989, gave rise to the construction of a special family of trial functions. This family is composed by circular-sectorial vortex rings, here named a-rings, i.e., rings that are postioned on the border of a circular sector with aperture angle a. An explicit formula for the velocity potential describing the a-rings family is here derived. A particular case is the well known circular vortex ring. The formula is given in terms of an uniformly valid series involving trigonometric and hypergeometric functions. Results concerning the complete circular ring are compared to the well known solution given, in closed form, in terms of Bessel functions, validating the present formula. Convergence is discussed. Graphical examples are shown for various rings of different sectorial angles. As an elementary application, the steady potential flow around three-dimensional bodies in unbounded fluid is formulated and solved under variational approach. The variational method is fully validated through the sphere problem and for a family of spheroids. Examples, concerning either translatory or rotatory motion around a transversal axis, are presented for the spheroid family. Finally an advancing finite cylinder ilustrates the ability of the a-rings family to represent potential flows around three-dimensional bodies with sharp edges. |