Flow Field of a Point Vortex Inside an Elliptical Boundary

Two-dimensional point vortices in Euler fluids are a common tool to model airfoils and geophysical flows. For instance, ocean vortices can interact with coastlines and other geometries that affect their motion. Such interactions have been studied around circular islands and in bays. While the flow external to an elliptical island is known and has applications to airfoil theory, the corresponding flow inside an elliptical boundary has not been studied. Here, we show an analytic solution for the flow field due to an ideal two-dimensional point vortex in an elliptical boundary, as well as the motion of the vortex. We use conformal mapping to find an image system that satisfies the boundary conditions of no normal flow through the ellipse walls. This results in flow fields similar to those within a circular boundary but with the streamlines stretched to fill the ellipse. Similar to the circular case, the point vortex traces out concentric ellipses as it moves around the model bay.