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Cross-flow vortices and their secondary instability in Falkner-Skan-Cooke boundary layers.
Linear eigenvalue calculations and direct numerical simulations (DNS) of disturb
ance growth in Falkner-Skan-Cooke (FSC) boundary layers have been performed.
The growth rates of the small amplitude disturbances obtained from the DNS calcu
lations show substantial differences compared to linear local theory, i.e. non-p
arallel effects are large. This is in agreement with results obtained using the
parabolic stability equations (PSE), also presented in this paper.
Using single frequency and random disturbances in the DNS we find that the most
amplified waves have characteristics in agreement with linear theory.
With higher amplitude initial disturbances in the DNS calculations, saturated cr
oss flow vortices are obtained. In these vortices strong shear layers appear.
When a small random disturbance is added to a saturated cross-flow vortex, a low
frequency secondary instability is found located at the bottom shear layer of t
he cross flow vortex and a high frequency instability is found at the upper shea
r layer of the cross flow vortex. The growth rates of the secondary instabilitie
s are found from detailed analysis of simulations of single frequency disturbanc
es. The low frequency disturbance is amplified throughout the domain, but with a
lower growth rate than the high frequency disturbance, which is amplified only
once the cross-flow vortices have reached a level where nonlinear effects have c
aused saturation. The high frequency disturbance has a growth rate that is consi
derably higher than the growth rates for the primary instabilities.