A critical-layer analysis of the resonant triad in boundary-layer transition: nonlinear interactions

Abstract

A systematic theory is developed to study the nonlinear spatial evolution of the resonant triad in Blasius boundary layers. This triad consists of a plane wave at the fundamental frequency and a pair of symmetrical, oblique waves at the subharmonic frequency. A low-frequency asymptotic scaling leads to a distinct critical layer wherein nonlinearity first becomes important, and the critical layer's nonlinear, viscous dynamics determine the development of the triad.The plane wave initially causes double-exponential growth of the oblique waves. The plane wave, however, continues to follow the linear theory, even when the oblique waves’ amplitude attains the same order of magnitude as that of the plane wave. However, when the amplitude of the oblique waves exceeds that of the plane wave by a certain level, a nonlinear stage comes into effect in which the self-interaction of the oblique waves becomes important. The self-interaction causes rapid growth of the phase of the oblique waves, which causes a change of the sign of the parametric-resonance term in the oblique-waves amplitude equation. Ultimately this effect causes the growth rate of the oblique waves to oscillate around their linear growth rate. Since the latter is usually small in the nonlinear regime, the net outcome is that the self-interaction of oblique waves causes the parametric resonance stage to be followed by an ‘oscillatory’ saturation stage.