Area-varying waves on curved vortex tubes with application to vortex breakdown

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Abstract
Equations which modify those derived by Widnall & Bliss (1971) and Moore & Saffman (1972) are presented in which jet-like flow along the axis of a vortex tube interacts with the motion of the tube. The equations describe two major effects. The first is the propagation of axial waves along the vortex tube which is similar to the flow of shallow water. A local decrease in cross-section area of the vortex tube produces higher swirling velocity and lower pressure. The resulting axial pressure gradient causes a propagating wave of area and axial velocity in order to move fluid into the region of smaller area. The second effect is instability to helical disturbances when the jet-like axial velocity is high enough to overcome the stabilizing effect of the swirling motion. An elementary nonlinear theory of vortex breakdown is presented which has an analogy with the formation of bores in shallow-water theory. A numerical example shows the growth of a helical disturbance behind a vortex breakdown front.