A CRITICAL-LAYER ANALYSIS OF THE RESONANT TRIAD IN BOUNDARY-LAYER-TRANSITION - NONLINEAR-INTERACTIONS

A systematic theory is developed to study the nonlinear spatial evolut ion of the resonant triad in Blasius boundary layers. This triad consi sts of a plane wave at the fundamental frequency and a pair of symmetr ical, oblique waves at the subharmonic frequency. A low-frequency asym ptotic 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 initial ly causes double-exponential growth of the oblique waves. The plane wa ve, however, continues to follow the linear theory, even when the obli que waves' amplitude attains the same order of magnitude as that of th e plane wave. However, when the amplitude of the oblique waves exceeds that of the plane wave by a certain level, a nonlinear stage comes in to effect in which the self-interaction of the oblique waves becomes i mportant. The self-interaction causes rapid growth of the phase of the oblique waves, which causes a change of the sign of the parametric-re sonance 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 no nlinear regime, the net outcome is that the self-interaction of obliqu e waves causes the parametric resonance stage to be followed by an 'os cillatory' saturation stage.