The continuous spectrum for a boundary layer in a streamwise pressure gradient

Solutions of the Orr–Sommerfeld equation belonging to the continuous spectrum are presented for boundary layers developing in the presence of a streamwise pressure gradient. Although the continuous spectrum has received considerable attention in the Blasius case, in most engineering applications transition to turbulence occurs in a region where there is a pressure gradient. This investigation, so far as we know, is the first to examine what effect this has on the eigenfunctions. Our results show that when there is a pressure gradient the magnitude of the eigenfunctions near the edge of the boundary layer can be much larger than it is for Blasius flow. This is particularly true when the pressure gradient is adverse, but such is the case even when it is favorable. We also investigate the effect of Reynolds number and frequency on the penetration depth; the latter term refers to one of the properties of these modes that distinguishes them from Tollmien–Schlichting waves, namely, that their magnitude is largest near the edge of the boundary layer, but much smaller inside.