Instability of a plane axisymmetric flow with a shear discontinuity in the velocity profile

Abstract

It is proposed to investigate the stability of a plane axisymmetric flow with an angular velocity profile Ω(r) such that the angular velocity is constant when r rO + L but varies monotonically from Ω1 to Ω2 near the point rO, the thickness of the transition zone being small L ≪ rO, whereas the change in velocity is not small ¦Ω2 − Ω1¦ ∼ Ω2, Ω1. Obviously, as L → O short-wave disturbances with respect to the azimuthal coordinate ϕ (kϕ=m/rO ≫ 1/rO) will be unstable with a growth rate-close to the Kelvin—Helmholtz growth rate. In the case L=O (i.e., for a profile with a shear-discontinuity) we find the instability growth rate γO and show that where the thickness of the discontinuity L is finite (but small) the growth rate does not differ from γO up to terms proportional to kϕL ≪ 1 and 1/m ≪ 1. Using this example it is possible to investigate the effect of rotation on the flow stability. It is important to note that stabilization (or destabilization) of the flow in question by rotation occurs only for three-dimensional or axisymmetric perturbations.