A family of vortex rings and a variational application to potential flows around three-dimensional bodies

Abstract

A variational formulation and solution of general three-dimensional potential flows gave rise to the construction of a special family of ‘trial functions’. This family is composed by circular-sector vortex rings, here named α-rings, i.e., rings that are positioned on the border of a circular sector with aperture angle α. An explicit formula for the velocity potential describing the α-rings family is here derived. A particular case is the well-known circular vortex-ring. The formula is given in terms of a 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 sector angles. As an elementary application, the steady potential flow around three-dimensional bodies in unbounded fluid is formulated and solved under the 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.