The spread of vorticity in the wake behind a cylinder

Abstract

This paper is an attempt to investigate the effect on the configuration of vortices in the wake behind a cylinder of an allowance for the thickness of the vortices. The vortices themselves are assumed to be initially rectilinear and of equal circular section, and we assume also that they arrange themselves in an “unsymmetrical double row.” We therefore find a relationship between the “stability ratio” of the double row—that is, the ratio of the distance between the rows to the distance between consecutive vortices on the same row, in the stable configuration—and the diameters of the vortices. The problem in its initial stages can no longer be treated as one in two dimensions, for the “self-induction” of a vortex only enters when we deal with a three-dimensional disturbance, and it is the self-induction that produces the differ­ence between this and the original treatment of the subject. By the “self-induction” of a vortex we mean the effect of the vortex on itself. The isolated rectilinear vortex is treated separately and the results obtained from it are extended to meet the case of the double row of rectilinear vortices. The three-dimensional stability of the Benard-Karman street has already been discussed, but the present treatment introduces various simplifications which, while not altering the general nature of the problem, make the expressions more amenable to treatment and yield results that appear to have been masked by the complexities of the algebra in the previous investigation. I would like to express my thanks to Dr. H. Jeffreys for many helpful criticisms which have had the effect of altering entirely certain sections of this paper.