On the nonlinear evolution of a pair of oblique Tollmien–Schlichting waves in boundary layers

Abstract

This paper is concerned with the nonlinear interaction and development of a pair of oblique Tollmien–Schlichting waves which travel with equal but opposite angles to the free stream in a boundary layer. Our approach is based on high-Reynolds-number asymptotic methods. The so-called ‘upper-branch’ scaling is adopted so that there exists a well-defined critical layer, i.e. a thin region surrounding the level at which the basic flow velocity equals the phase velocity of the waves. We show that following the initial linear growth, the disturbance evolves through several distinct nonlinear stages. In the first of these, nonlinearity only affects the phase angle of the amplitude of the disturbance, causing rapid wavelength shortening, while the modulus of the amplitude still grows exponentially as in the linear regime. The second stage starts when the wavelength shortening produces a back reaction on the development of the modulus. The phase angle and the modulus then evolve on different spatial scales, and are governed by two coupled nonlinear equations. The solution to these equations develops a singularity at a finite distance downstream. As a result, the disturbance enters the third stage in which it evolves over a faster spatial scale, and the critical layer becomes both non-equilibrium and viscous in nature, in contrast to the two previous stages, where the critical layer is in equilibrium and purely viscosity dominated. In this stage, the development is governed by an amplitude equation with the same nonlinear term as that derived by Wu, Lee & Cowley (1993) for the interaction between a pair of Rayleigh waves. The solution develops a new singularity, leading to the fourth stage where the flow is governed by the fully nonlinear three-dimensional inviscid triple-deck equations. It is suggested that the stages of evolution revealed here may characterize the so-called ‘oblique breakdown’ in a boundary layer. A discussion of the extension of the analysis to include the resonant-triad interaction is given.