Interactions of three components and subcritical self-sustained amplification of disturbances in plane Poiseuille flow

A low-dimensional nonlinear model for the normal velocity (v) and normal vorticity (eta) disturbance development in plane Poiseuille flow is studied. The study is restricted to the interactions of a pair of oblique components of the form e(/(alpha x+/-beta z)) and the component of the form e(/2 beta z) where alpha and beta are streamwise and spanwise wave numbers, respectively. The disturbances considered are also assumed to be highly elongated in the streamwise direction. Owing to the non-normal properties of the basic equations, the eta disturbance is first transiently amplified. Then, if the Reynolds number (R) and the initial disturbance are sufficiently large, the nonlinear interactions lead to a self-sustained process of disturbance amplification at subcritical R. For large R (R greater than or similar to 5000). the threshold disturbance amplitude scales like R-3. The, results also strongly indicate that the nonlinear feedback from eta to v is crucial for the establishment of the self-sustained process. (C) 1999 American Institute of Physics. [S1070-6631(99)00703-5].