Criteria for instability of helical disturbances in inviscid, swirling flows

This work considers the linear inviscid instability of columnar vortices with axial flow in unbounded domains subjected to 3D perturbations. The base flow parameters have a general dependence on the radial distance from the swirl axis. Following Howard and Gupta's approach, we develop two stability conditions in terms of an infinite set of helical disturbances via a normal modes expansion. We develop a generalization of Fj\o rtoft's necessary criterion which states that a wave-like disturbance may be unstable if the base shear velocity has an inflection point in the binormal direction of the helix which is also a vorticity maximum. A necessary condition for instability is that\[(W'-W'_0)d(\kappa\dot{\gamma}')/dr<0\]must be satisfied somewhere for any real constant $W'_0$ where $\kappa$ is the curvature of the helix, $W'$ is the binormal base velocity, and $\dot{\gamma}'$ is the binormal base shear rate. The second condition leads to a generalization of Rayleigh's criterion for centrifugal instability for helical disturbances. We find that a necessary and sufficient condition for instability is that\[Vd\Gamma'/dr<0\]be satisfied somewhere, where $V$ is the base azimuthal velocity and $\Gamma'$ is the base circulation due to the flux of vorticity tangent to the helical vortex tube.