Generalized critical-layer analysis of fully coupled resonant-triad interactions in boundary layers

Abstract

The critical-layer analysis of the nonlinear resonant-triad interaction by Goldstein & Lee (1992) is extended to include viscous effects. A generalized scaling which is valid both for the quasi-equilibrium and non-equilibrium critical-layer analyses in zero- or non-zero-pressure-gradient boundary layers is obtained. A system of partial differential equations which governs the fully coupled non-equilibrium critical-layer dynamics is obtained and it is solved by using a numerical method. Amplitude equations and their viscous limits are also presented. The parametric-resonance growth rate of the non-equilibrium critical-layer solution with finite viscosity is larger than that of the viscous-limit quasi-equilibrium solution. The viscosity delays both the onset of the fully coupled interaction and the ultimate downstream location of the singularity. The difference between the non-equilibrium critical-layer solution and the corresponding quasi-equilibrium critical-layer solution becomes smaller, at least in the parametric resonance region, as the viscosity parameter becomes large. However, the non-equilibrium solution with finite viscosity always ends in a singularity at a finite downstream position unlike the viscous-limit solution.