Long wavelength disturbances to non-planar parallel flow

Abstract

The instability of unbounded parallel inviscid flows which are neither plane nor axisymmetric is studied for disturbances with long wavelengths in the flow direction. The details of the variation of the flow velocity on any scale smaller than the wavelength are shown to have no effect on these disturbances and it is only the non-uniformity of the flow at infinity which is relevant. There is a class of disturbance which can only exist because of the non-uniformity, and it is governed by an equation similar to the Rayleigh equation for inviscid plane parallel flow. A number of the properties of the solution can be found and a large class of flows can be shown to be unstable. When the flow at infinity is linear in sectors it is easy to find simple solutions. Particular flows, which are either uniform in sectors with vortex sheets along the dividing radii or are continuous, but with a linear variation in each sector, are studied in detail and are shown to be unstable, even when the non-uniformity is confined to a narrow sector.