Subharmonic instabilities of Tollmien-Schlichting waves in two-dimensional Poiseuille flow

The stability of the upper branch of shear traveling waves in two-dimensional Poiseuille flow, when the total flux through the channel is held constant, is considered. Taking into account the length of the periodic channel, perturbations of the same wave number (superharmonic), and different wave number (subharmonic) of the uniform wave trains are imposed. We mainly consider channels long enough to contain M=4 and M=8 basic wavelengths. In these cases, subharmonic bifurcations are found to be dominant except in a small region of parameters. From this type of bifurcation, we show that if the wave number is decreased, the periodic train of finite amplitude waves evolves continuously towards the stable localized wave packets obtained in long channels by other authors and whose existence has been associated to the vicinity of an inverted Hopf bifurcation. Depending on the basic wave number of the periodic train destabilized, different types of solutions for a given length of the channel can be obtained. Furthermore, for moderate Reynolds numbers, configurations of linearly stable wave trains exist, provided that their basic wave number is alpha approximately 1.5.