Vortex equilibria using least-squares methods

Numerical methods and results for computing rotating or stationary equilibria of vortex patches and sheets, some in the presence of point vortices, are presented. The methods are based on those recently developed by Trefethen and colleagues for solving Laplace's equation in the complex plane by series and rational approximation. They share the common feature of finding the coefficients of the approximation by the fitting of boundary conditions using least-squares. Application of these methods to vortex patches requires their extension to the solution of Poisson's and Laplace's equation in two domains with matching conditions across the patch boundary. In the case of vortex sheets, the streamlines of the solution are computed along with the circulation density of the sheet. The use and accuracy of the methods is demonstrated by reproducing known results for equilibrium patches and vortex sheets, some having point vortices present. Several new numerical equilibrium solutions are also computed: a single straight sheet with two and four satellite point vortices respectively, and a three-sheeted structure, with the sheets emanating from a common point of rotation. New numerical solutions are also found for steady, doubly-connected vortex layers of uniform vorticity surrounding solid objects and such that the fluid velocity vanishes on the outer free boundary.