On Critical-Layer and Diffusion-Layer Nonlinearity in the Three-Dimensional Stage of Boundary-Layer Transition

This paper considers nonlinear interactions in the three-dimensional stage of transition to turbulence, taking an accelerating boundary layer as a prototype flow. Attention is focused on transition via subharmonic resonance in the upper-branch scaling regime. It is shown that the (weakly) nonlinear instability of the flow is described by a seven-zoned structure, cf. the five-zoned structure for the linear problem. The dominant nonlinear interactions take place both in a critical layer and in `diffusion layers'. The nonlinearly generated mean flow in turn interacts with the wall to attain a maximum magnitude near the wall. It is emphasized that both the nonlinear mechanism and the flow structure are generic for three-dimensional disturbances. And there is some similarity with the work in the context of wave/vortex interaction. Numerical solutions of the amplitude equations indicate that if the oblique modes initially have a small amplitude, they first experience a rapid growth caused by parametric resonance. Following this the cubic interactions of the oblique modes inhibit the growth and lead to a wavelength shortening. However, if the initial amplitudes of the oblique modes are sufficiently large, the parametric resonance can be completely bypassed. Numerical solutions also suggest that oblique modes with unequal initial amplitudes evolve to an equal-amplitude state.