Vortex Patches

A vortex patch is a bounded region of uniform vorticity in two-dimensional, incompressible, inviscid fluid flow. The streamfunction satisfies the Poisson equation with the vorticity acting as a source term. The standard formulation is to write the streamfunction as a convolution of the vorticity with the two-dimensional free-space Greens function. A simple application of Greens theorem converts the area integral to a boundary integral. Numerical methods must then account for the singular nature of the boundary integral, and high accuracy is difficult when filamentation takes place, that is, when long, very thin filaments of vorticity erupt from the main boundary. A new boundary integral is derived based on a different viewpoint. A particular solution is readily known which represents solid body rotation. To the particular solution must be added a homogeneous solution, and the combination must satisfy the boundary conditions. A standard boundary integral can be used to solve the Laplace equation with Dirichlet boundary conditions. This approach leads to a boundary integral without singularity and easily approximated by the trapezoidal rule that ensures spectral accuracy. Results indicate that high accuracy is possible with even modest resolution. The method is used to explore the mathematical properties of filamentation.