Subharmonic instabilities of Tollmien-Schliehting waves in two-dimensionalPoiseuille flow

The stability of; the: upper branch of shear traveling waves in two-dimensi onal Poiseuille flow, when the total flux through the channel is held const ant, 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 rue imposed. We mainly cons ider channels long enough to contain M=4 and M=8 basic wavelengths. In thes e: cases, subharmonic bifurcations are found to be dominant except in a sma ll region of parameters. From this type of bifurcation, we show that if the wave number is decreased, the periodic train of finite amplitude waves evo lves continuously towards the stable localized wave packets obtained in lon g channels by other authors and whose existence has been associated to the vicinity of an inverted Hopf bifurcation. Depending on the basic wave numbe r of the periodic train destabilized, different types of solutions for a gi ven length of the channel can be obtained. Furthermore, for moderate Reynol ds numbers, configurations of linearly stable wave trains exist, provided t hat their basic wave number is alpha approximate to 1.5. [S1063-651X(99)152 08-5].