  Experiments have shown that micron-sized distributed surface roughness can
significantly promote transition in a three-dimensional boundary layer
dominated by crossflow insta- bility. This sensitive effect has not yet been
fully explained physically and mathematically. Unlike past researches focusing
on the receptivity of the boundary layer to surface roughness, or on the local
stability of the modified mean flow, this paper seeks possible inherent
mechanisms by investigating the effects of distributed surface roughness on
crossflow instability through resonant interactions with eigenmodes. A key
observation is that the perturbation induced by roughness with specific
wavenumbers can interact with two eigenmodes (travelling and stationary
vortices) through triadic resonance, or interact with one eigenmode (stationary
vortices) through Bragg scattering. Unlike the usual triadic resonance of
neutral, or nearly neutral, eigenmodes, the present triadic resonance can take
place among modes with O(1) growth rates, provided that these are equal; unlike
the usual Bragg scattering involving neutral waves, crossflow stationary
vortices can also be unstable. For these amplifying waves, the generalized
triadic resonance and Bragg scattering are put forward, and the resulting
corrections to the growth rates are derived by a multiple-scale method. The
analysis is extended to the case where up to four crossflow vortices interact
with each other in the presence of suitable roughness components. The numerical
results for Falkner-Skan-Cooke boundary layers show that roughness with a small
height (a few percent of the local boundary-layer thickness) can change growth
rates substantially (by a more-or-less O(1) amount)....
