The upper branch stability of the Blasius boundary layer, including non-parallel flow effects

Abstract

The stability of the Blasius boundary layer is studied theoretically, with the aim of fixing the character of the upper branch of the neutral stability curve(s) and its dependence on non-parallel flow effects. Unlike most previous studies this work has a rational basis since, throughout, we consider the linear stability structure for asymptotically large Reynolds numbers ( Re ). The structure is five-zoned and quite complicated, more so than the structure (discussed in Smith (1979 a )) governing the lower branch stability properties, but nevertheless it lends itself to the systematic determination of the neutral frequency and of the influence of non-parallelism. The four leading terms in the asymptotic expansion of the neutral frequency are determined and then the non-parallel flow effects are considered. The latter are shown to be of relative order Re -3/10 in general, much larger than the relative order Re -1/2 suggested by the parallel flow approximations used extensively in the literature. The cause of this discrepancy lies partly in the relatively large wavelength of the Tollmien-Schlichting modes but, more especially, in a ‘transmission feature’, associated with the stability structure and brought about by the major determining role played by the small curvature of the boundary layer profile at the critical layer. This transmission feature enables even quite small effects in the disturbance velocity field to produce a much more profound effect in the neutral stability criteria. The results of this study are not inconsistent overall with previous numerical work but they do tend to suggest that linear non-parallel flow stability theory may well explain most of the related experimental observations, even near the critical Reynolds number.