The observed nonlinear saturation of crossflow vortices in the DLR swept-plate transition
experiment, followed by the onset of high-frequency signals, motivated us to
compute nonlinear equilibrium solutions for this flow and investigate their instability
to high-frequency disturbances. The equilibrium solutions are independent of
receptivity, i.e. the way crossflow vortices are generated, and thus provide a unique
characterization of the nonlinear flow prior to turbulence. Comparisons of these
equilibrium solutions with experimental measurements exhibit strong similarities. Additional
comparisons with results from the nonlinear parabolized stability equations
(PSE) and spatial direct numerical simulations (DNS) reveal that the equilibrium
solutions become unstable to steady, spatial oscillations with very long wavelengths
following a bifurcation close to the leading edge. Such spatially oscillating solutions
have been observed also in critical layer theory computations. The nature of the
spatial behaviour is herein clarified and shown to be analogous to that encountered
in temporal direct numerical simulations. We then employ Floquet theory to systematically
study the dependence of the secondary, high-frequency instabilities on
the saturation amplitude of the equilibrium solutions. With increasing amplitude, the
most amplified instability mode can be clearly traced to spanwise inflectional shear
layers that occur in the wake-like portions of the equilibrium solutions (Malik et al.
1994 call it ‘mode I’ instability). Both the frequency range and the eigenfunctions
resemble recent experimental measurements of Kawakami et al. (1999). However,
the lack of an explosive growth leads us to believe that additional self-sustaining
processes are active at transition, including the possibility of an absolute instability
of the high-frequency disturbances.