Non-parallel stability analysis of three-dimensional boundary layers along an infinite attachment line

Abstract

Instability of a non-parallel similar-boundary-layer flow to small and wavy disturbances is governed by partial differential equations with respect to the non-dimensional vertical coordinate ζ and the local Reynolds number R1 based on chordwise velocity of external stream and a boundary-layer thickness. In the particular case of swept Hiemenz flow, the equations admit a series solution expanded in inverse powers of R12 and then are decomposed into an infinite sequence of ordinary differential systems with the leading one posing an eigenvalue problem to determine the first approximation to the complex dispersion relation. Numerical estimation of the series solution indicates a much lower critical Reynolds number of the so-called oblique-wave instability than the classical value Rc = 583 of the spanwise-traveling Tollmien–Schlichting instability. Extension of the formulation to general Falkner–Skan–Cooke boundary layers is proposed in the form of a double power series with respect to 1/R12 and a small parameter ε denoting the difference of the Falkner–Skan parameter m from the attachment-line value m = 1.