Asymptotic approach to the problem of boundary layer instability in transonic flow

Tollmien–Schlichting waves can be analyzed using the Prandtl equations involving selfinduced pressure. This circumstance was used as a starting point to examine the properties of the dispersion relation and the eigenmode spectrum, which includes modes with amplitudes increasing with time. The fact that the asymptotic equations for a nonclassical boundary layer (near the lower branch of the neutral curve) have unstable fluctuation solutions is well known in the case of subsonic and transonic flows. At the same time, similar solutions for supersonic external flows do not contain unstable modes. The bifurcation pattern of the behavior of dispersion curves in complex domains gives a mathematical explanation of the sharp change in the stability properties occurring in the transonic range.