Linear and nonlinear growth of secondary instabilities of crossflow vortices studied by PSE

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 stages of both the stationary crossflow vortices and their high-frequency secondary instabilities are modelled by nonlinear parabolized stability equations (PSE). The PSE computations provide the input data for the secondary instability code and its results in turn are used for initialization of the secondary instabilities within the PSE analysis. According to our knowledge, it is the first time that such an approach with bi–directional coupling of both numerical methods has been used for the analysis of a crossflow–dominated transition scenario. 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. The latter could be used as a criterion for imminent laminar–turbulent transition within the saturation region of the stationary crossflow modes which was lacking in any nonlinear PSE analysis of crossflow–dominated transition until now. It is well known that the regions of highest secondary instability amplitude spatially coincide with regions of high shear that develop in the time–averaged mean flow due to the action of the finite–amplitude stationary crossflow vortices. Here, something similar is found for the regions of highest rms–amplitudes of the nonlinearly generated first temporal higher harmonic of the finite-amplitude secondary instability mode which coincide with regions of highest gradients in the secondary–mode rms–amplitudes. This might point to the existence of a tertiary instability but needs further investigation.