Secondary instability of cross-flow vortices in Falkner–Skan–Cooke boundary layers

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Abstract
Linear eigenvalue calculations and spatial direct numerical simulations (DNS) of disturbance growth in Falkner–Skan–Cooke (FSC) boundary layers have been performed. The growth rates of the small-amplitude disturbances obtained from the DNS calculations show differences compared to linear local theory, i.e. non-parallel effects are present. With higher amplitude initial disturbances in the DNS calculations, saturated cross-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 mode is found located at the bottom shear layer of the cross-flow vortex and a high-frequency secondary instability is found at the upper shear layer of the cross-flow vortex. The growth rates of the secondary instabilities are found from detailed analysis of simulations of single-frequency disturbances. 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 started to saturate. The high-frequency disturbance has a growth rate that is considerably higher than the growth rates for the primary instabilities, and it is conjectured that the onset of the high-frequency instability is well correlated with the start of transition.

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