Journal of Fluid Mechanics



Some steady vortex flows past a circular cylinder


ALAN ELCRAT a1, BENGT FORNBERG a2, MARK HORN a3 and KENNETH MILLER a1
a1 Department of Mathematics, Wichita State University, Wichita, KS 67260, USA
a2 Department of Applied Mathematics, University of Colorado, Boulder, CO 80309, USA
a3 Intergraph, Huntsville, AL 35894, USA

Abstract

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.

(Received September 15 1998)
(Revised May 15 1999)



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