On the mother bodies of steady polygonal uniform vortices. Part I: numerical experiments

Abstract

ABSTRACT The existence of an integral relation between self-induced velocity of a uniform, planar vortex and Schwarz function of its boundary opens the way to understand the kinematics of the vortex by analysing the internal singularities of that function. In general, they are branch cuts and form the so-called “mother body” of the vortex, because they generate the same external velocities of the vortex, by means of a relation identical to the Biot–Savart law for a vortex sheet. The jump of the Schwarz function across the cuts plays the role of the (complex) density of circulation. This paper investigates the singularities of polygonal vortices, which are highly nontrivial steady vortices widely present in Nature, and having fascinating properties, some of them still not well understood. By means of the equation of the dynamics of the Schwarz function specialised for steady vortices, a numerical tool based on elementary properties of the holomorphic functions is used for detecting the internal singularities and evaluating their strengths. In this way, it is shown that an nagonal vortex possesses n internal branch cuts. In a reference system having origin on the centre of vorticity of the vortex and real axis crossing one of its vertices, these cuts start from the origin and are directed along the n roots of the unity, so that they are aligned with the vertices. The positions of the branch points and the values assumed by the Schwarz function in these points are calculated by evaluating this function just outside the vortex boundary. Once the conditions on the branch points are defined, a power series representation of the Schwarz function is proposed, that is able to explain the behaviour of its real and imaginary parts in neighbourhoods of these points. Some conjectures about the external singularities are also discussed.