Steady vortex flows past a circular cylinder are obtained numerically as solutions
of the partial differential equation Δψ = f(ψ), f(ψ)
= ω(1 − H(ω − α)), where H is the
Heaviside function. Only symmetric solutions are considered so the flow may be
thought of as that past a semicircular bump in a half-plane. The flow is transplanted
by the complex logarithm to a semi-infinite strip. This strip is truncated at a finite
height, a numerical boundary condition is used on the top, and the difference equations
resulting from the five-point discretization for the Laplacian on a uniform grid are
solved using Fourier methods and an iteration for the nonlinear equation. If the area
of the vortex region is prescribed the magnitude of the vorticity ω is adjusted in an
inner iteration to satisfy this area constraint.
Three types of solutions are discussed: vortices attached to the cylinder, vortex
patches standing off from the cylinder and strips of vorticity extending to infinity.
Three families of each type of solution have been found. Equilibrium positions for
point vortices, including the Föppl pair, are related to these families by continuation.