Complete sets of circular, elliptic and bipolar harmonic vortices on a plane

TL;DR

This paper introduces harmonic solutions to the steady Euler equations for incompressible fluids in 2D, describing various vortex geometries using harmonic potentials in different coordinate systems, and providing explicit streamlines and vortex representations.

Contribution

It presents a novel class of harmonic vortex solutions in circular, elliptic, and bipolar coordinates, linking vortex geometry with harmonic functions and streamlines.

Findings

Analytic expressions for streamlines of harmonic vortices

Representation of Rankine vortices using harmonic functions

Insight into vortex sheet discontinuities at boundaries

Abstract

A class of harmonic solutions to the steady Euler equations for incompressible fluids is presented in two dimensions in circular, elliptic and bipolar coordinates. Since the velocity field is solenoidal in this case, it can be written as the curl of a vector potential, which will then satisfy Poisson's equation with vorticity as a source term. In regions with zero vorticity, Poisson's equation reduces to Laplace's equation, and this allows for the construction of harmonic potentials inside and outside circles and ellipses, depending on the coordinate system. The vector potential is normal to the coordinate plane, and is proportional to the scalar harmonic functions on the plane, thereby guaranteeing that the velocity field is also harmonic and is located on the coordinate plane. The components of the velocity field normal to either a circle or an ellipse are continuous, but the tangential components are discontinuous, so that, in effect, a vortex sheet is introduced at these boundaries. This discontinuity is a measure of the vorticity, normal to the plane and distributed harmonically along the perimeter of the respective circles or ellipses. An analytic expression for the streamlines is obtained which makes visualisation of vortices of various geometries and harmonicities possible. This approach also permits a reformulation of the notion of multipolarity of vortices in the traditional sense of a multipolar expansion of the Green's function for Poisson's equation. As an example of the applicability of this formulation to known vortical structures, Rankine vortices of different geometries are expressed in terms of harmonic functions.