The Kelvin-Helmholtz instability of a cylindrical flow with a shear layer

Abstract

A linear model analysis of the cylindrical shear flow leads to an eigenvalue problem based on a differential equation. A detailed analysis of the solution is presented. From this we extract the following physical principles, (i) Unstable modes only exist within a hemicycle in complex phase space, (ii) Each mode has an associated 'comoving point', i.e. there is always some fluid that comoves with the pattern. At the comoving point, neutral modes are oscillating or wave-like and unstable modes are over-reflected. (iii) Short wavelengths with large azimuthal numbers or long-wavelength pinch modes can be stabilized by the presence of a shear layer both for ordinary modes and reflecting modes, (iv) Reflecting modes have a band structure in wave-number versus growth rate space. For light jets, the growth rates are very small, (v) A shear layer suppresses pinching modes more effectively.