Nonlinear Evolution of Two-Dimensional Tollmien–Schlichting Waves Triggered by a Linearly Damping Eigenmode

抄録

Numerical simulation has been carried out on the evolution of two-dimensional disturbances in plane Poiseuille flow. In a linearly stable region, we introduced a single anti-symmetric eigenmode belonging to the Orr–Sommerfeld spectra, and examined whether or not Tollmien–Schlichting waves were excited beyond a threshold of the subcritical instability. It is shown that a wall mode may easily trigger the nonlinear growth beyond the threshold. An initial threshold amplitude, above which the subcritical instability eventually sets in, depends on the initially introduced eigenmode: both the fifth and the eighth eigenmodes have the initial thresholds about ten times larger than that of the first eigenmode. Except for the initially introduced one, for t≤1, all the eigencomponents embedded in the fundamental Fourier mode are found to exhibit transient growth of ∝t². It is due to the nonlinear self-interaction of the initial disturbance.