Global asymptotic analysis of the Kelvin–Helmholtz instability of supersonic shear layers

Abstract

A global asymptotic analysis of the traveling wave Kelvin–Helmholtz instability of a supersonic finite-width velocity shear layer, having a linear velocity profile, is carried out. The structure of the resulting traveling wave solutions, comprising a composite Wilhelm–Kramers–Brillouin–Jeffreys and boundary-layer solution that satisfies outgoing spatially damping radiative wave boundary conditions, agrees closely with previously computed numerical results. The condition for the occurrence of the traveling wave instability is derived and the absence of this mode in compressible vortex sheets is explained. The energy density and energy flux density of both the unstable standing and traveling wave modes is considered in the context of the radiative boundary conditions.