The effect of compressibility on the speed of propagation of a vortex ring

Abstract

A circular vortex filament of radius R , cross sectional area π a 2 and circulation Г propagates steadily in an inviscid, calorically perfect gas. The flow outside the filament is assumed to be irrotational and isentropic. If it is further assumed that a/R ≪ 1, the cross section is approximately circular and the speed of propagation of the filament is shown to depend on the distribution of circulatory velocity v 0 and entropy s 0 within the core. If s 0 is constant and equal to its value in the isentropic exterior of the filament, the vortex ring is slowed down by compressibility effects, whatever the distribution of circulatory velocity. If the circulatory velocity corresponds to rigid rotation in the core cross section, the speed, U , of propagation is given by U = Γ/4π R [ln 8R/ a - ¼ - 5/12 M 2 + O ( M 4 )], where M is the Mach number Γ /2π ac ∞ and c ∞ is the sound speed far from the vortex ring. Numerical results for finite M are also given in this case. These results enable the cut-off theory of filament motion to be extended to compressible fluids.