  The fundamental resonance (FR) in the nonlinear phase of the boundary-layer
transition to turbulence appears when a dominant planar instability mode
reaches a finite amplitude, and the low-amplitude oblique traveling modes with
the same frequency as the dominant mode, together with the stationary streak
modes, undergo the strongest amplification among all the Fourier components.
This regime may be the most efficient means to trigger the natural transition
in hypersonic boundary layers. In this paper, we aim to reveal the intrinsic
mechanism of the FR in the weakly nonlinear framework based on the
large-Reynolds-number asymptotic technique. It is found that the FR is in
principle a triad resonance among a dominant planar fundamental mode, a streak
mode and an oblique mode. In the major part of the boundary layer, the
nonlinear interaction of the fundamental mode and the streak mode seeds for the
growth of the oblique mode, whereas the interaction of the oblique mode and the
fundamental mode drives the roll components (transverse and lateral velocity)
of the streak mode, which leads to a stronger amplification of the streamwise
component of the streak mode due to the lift-up mechanism. The main-layer
solution of the streamwise velocity, spanwise velocity and temperature of both
the streak and the oblique modes become singular as the wall is approached, and
so a viscous wall layer appears underneath. The wall layer produces an outflux
velocity to the main-layer solution, inclusion of which leads to an improved
asymptotic theory, whose accuracy is confirmed by comparing with the
calculations of the nonlinear parabolised stability equations (NPSE) at
moderate Reynolds numbers and the secondary instability analysis (SIA) at
sufficiently high Reynolds numbers.
