Polygonal N-vortex arrays: A Stuart model

A class of exact planar solutions of the Euler equations representing stationary N-polygonal arrays of vortices are found. The solutions are parametrized by two parameters N and Ωmax. N denotes the number of vorticity extrema surrounding the origin; Ωmax denotes the extremal value of this vorticity. Except for a point vortex at the origin, the solutions have everywhere-smooth vorticity distributions and are generalizations of the classic exact solution of Stuart [J. Fluid Mech. 29, 417 (1967)] for an infinite row of smooth vortices. In the limit |Ωmax|→∞, the solutions reduce to the pure point vortex problem considered by Morikawa and Swenson [Phys. Fluids 14, 1058 (1971)]. The new solutions can be understood as “smoothed-out” counterparts to this point vortex problem.