Long wavelength streamwise vortices caused by wall curvature or roughness

Long wavelength instabilities of boundary layers caused by centrifugal effects or wall roughness are investigated. The wall roughness is modelled by small amplitude surface waviness. The instability is described in the nonparallel regime where it develops on the same length scale as the unperturbed flow. It is shown that instabilities initiated by disturbances close to the leading edge initially deform rapidly into algebraically growing eigensolutions but then deform into exponentially growing disturbances. The disturbances ultimately develop in a quasi-parallel manner and then pass successively through the high Gortler number, or equivalent large roughness parameter, regimes first described by Denier et al. (Nasa Contractor ICASE Report 90-31, 1990; Philos Trans R Soc A 334:51-85, 1991). It is shown that the mode which develops downstream is the most rapidly growing one available and not the second most unstable mode as claimed in a recent paper.