The effect of axial pressure gradient on axisymmetrical and helical vortex waves

Abstract

The effect of axial pressure gradient in producing gradual changes in amplitude and wavelength of both axisymmetrical and helical waves on a vortex core is studied analytically using an idealized model of a vortex with uniform axial vorticity across the core in an unbounded space. Possible implications of the findings on the vortex breakdown phenomenon are also considered. In agreement with previous literature on the subject, the steady solution for a supercritical vortex flow in an adverse pressure gradient is found to become complex (and hence break down) at a certain critical point. The axisymmetric waves on the core obey a hyperbolic system of equations. An explicit solution of these equations for small-amplitude axisymmetric waves indicates that the wave amplitude becomes unbounded at the critical point in the linear theory. An implicit nonlinear solution of these equations shows the formation of ‘‘shocks’’ along the core considerably upstream of the critical point, which seem to correspond to formation of bubble-type vortex breakdowns. A solution is also obtained for evolution of long helical waves on a vortex core in the presence of an axial pressure gradient, and this solution is used to suggest an explanation for why the vortex breakdown at times adopts a bubble form while at other times it adopts a spiral form.