Onset of nonlinearity in a two-dimensional thin shear layer

Abstract

This study numerically investigates an early stage of nonlinear interaction for the better understanding of the onset of nonlinear behaviors. Two-dimensional shear flow is chosen as a canonical flow. When two disturbances of different wavenumbers satisfying no resonance condition are initially given, new components appear one after another while the original disturbances grow. The vorticity budget analysis shows that the beat of the two exciting modes plays an essential role in forming the sum and difference nonlinear components, namely the secondary modes. For the nonlinear interaction mechanism, the high vorticity around the center of the shear layer is locally transported in the transverse direction at specific streamwise sections where the amplitude of the vertical velocity fluctuation becomes relatively larger compared to other sections. The distance between these specific sections corresponds to the wavelength of the beat. The vertically dispersed vorticity will then be convected in the horizontal directions by the mean flow. As a result, several regions of concentrated vorticity appear which eventually develop into vortices. The amplification mechanism is found to be the same for both the primary and secondary modes, though the secondary modes arise from the additional perturbation deriving from the initial perturbation.