Linear and nonlinear growth of secondary instabilities of stationary crossflow vortices studied by parabolized stability equations

Abstract

Laminar–turbulent transition in (quasi–) three–dimensional boundary layers dominated by stationary crossflow vortices is studied for the setup of the DLR swept–flat plate experiment. The linear and subsequent nonlinear development of both the stationary crossflow vortices and their high–frequency secondary instabilities are modelled by nonlinear parabolized stability equations (PSE). In contrast to previous work, secondary instability theory is used only for initialization of secondary instabilities within the PSE analysis. Thereby, the nonlinear development of secondary instabilities including the generation of higher harmonics could be studied up to the stages where the feedback of the finite–amplitude secondary instability modes on the stationary crossflow vortices is no longer negligible and the skin–friction coefficient starts to deviate from that due to the mean flow distortion caused by the stationary crossflow modes alone. Moreover, the present approach allows rather efficient studies on the nonlinear development of secondary instabilities independent of their frequency.