Sn. Timoshin et Ft. Smith, SINGULAR MODES IN RAYLEIGH INSTABILITY OF 3-DIMENSIONAL STREAMWISE-VORTEX FLOWS, Journal of Fluid Mechanics, 333, 1997, pp. 139-160
The upper-branch neutral modes of inviscid instability in a boundary-l
ayer flow with significant longitudinal vortices present are shown to
possess typically a logarithmically singular, non-inflectional, critic
al layer. This contrasts with previous linear and nonlinear suggestion
s implemented in vortex-wave interaction and secondary instability the
ories, which are re-examined. The analysis here is based first on pert
urbation techniques applied to a Rayleigh unstable planar motion suppl
emented by a vortex centred around the inflection level, followed by t
he extension to more general cases. Flows with order one and larger sp
anwise scales are considered. Multiple solutions, their limit properti
es and parametric continuations are illustrated with concrete examples
.