Singularity formation in the strongly nonlinear wide-vortex/Tollmien–Schlichting-wave interaction equations

Abstract

A combined numerical/analytical study of the wide-vortex/wave interaction equations, describing boundary-layer instability, is presented. Depending on the obliqueness β of the wave input, different solution properties are obtained. For β = 1, oscillations in the wave amplitude lead to the evolution of a strongly three-dimensional mean flow, while for β = 2 the interaction is characterized by the development of a singularity in the wave pressure amplitude. This latter behaviour is modelled using an approximate form for the mean flow skin friction and the resulting amplitude equation is analysed using a combination of numerical and asymptotic techniques. A simple method is described for determining the singularity location for a given spanwise wavenumber, and the asymptotic behaviour of the pressure amplitude as the singularity is approached is deduced.