On the spatio-temporal stability of primary and secondary crossflow vortices in a three-dimensional boundary layer

Abstract

To examine possible links between a global instability and laminar–turbulent breakdown in a three-dimensional boundary layer, the spatio-temporal stability of primary and secondary crossflow vortices has been investigated for the DLR swept-plate experiment. In the absence of any available procedure for the direct verification of pinching for three-dimensional wave packets the alternative saddle-point continuation method has been applied. This procedure is known to give reliable results only in a certain vicinity of the most unstable ray. Therefore, finding no absolute instability by this method does not prove that the flow is absolutely stable. Accordingly, our results obtained this way need to be confirmed experimentally or by numerical simulations. A geometric interpretation of the time-asymptotic saddle-point result explains certain convergence and continuation problems encountered in the numerical wave packet analysis. Similar to previous results, all our three-dimensional wave packets for primary crossflow vortices were found to be convectively unstable.

Due to prohibitive CPU time requirements the existing procedure for the verification of pinching for two-dimensional wave packets of secondary high-frequency instabilities could not be implemented. Again saddle-point continuation was used. Surprisingly, all two-dimensional wave packets of high-frequency secondary instabilities investigated were also found to be convectively unstable. This finding was corroborated by recent spatial direct numerical simulations of Wassermann & Kloker (2001) for a similar problem. This suggests that laminar–turbulent breakdown occurs after the high-frequency secondary instabilities enter the nonlinear stage, and spatial marching techniques, such as the parabolized stability equation method, should be applicable for the computation of these nonlinear states.