Spatially convective global modes in a boundary layer

Abstract

The linear stability of a weakly nonparallel flow, the case of a flat plate boundary layer, is revisited by a linear global stability approach where the two spatial directions are taken as inhomogeneous, leading to a fully nonparallel stability method. The resulting discrete eigenvalues obtained by the fully nonparallel approach seem to be in agreement with classical Tollmien–Schlichting waves. Then the different modes are compared with classical linear stability approach and weakly nonparallel method based on linear parabolized stability equations (PSEs). It is illustrated that the nonparallel correction provided by the linear global stability approach is well matched by linear PSE. Furthermore, physical interpretation of these spatio-temporal global modes is given where a real pulsation, which has more physical interest, is considered. In particular the use of a Gaster transformation and the pseudospectrum illustrate the local and global properties of these Tollmien–Schlichting modes. Finally, the contribution of different components of global modes (normal and streamwise) in the transient amplifying behavior associated with the convectively unstable boundary layer is analyzed and compared with a classical steepest descent method. Then, a discussion of an equivalent of the continuous branch is given.