On the linear evolution of disturbances in plane Poiseuille flow
The linear evolution of disturbances due to a ribbon vibrating at frequency $\omega_0$ in plane Poiseuille flow is computed by solving the associated initial boundary value problem in the Fourier-Laplace plane, followed by inversion. A novel algorithm for identifying the temporal modes of the Orr-Sommerfeld equation (OSE) in the complex wavenumber plane, which are required in the inversion, is presented. Unlike in many prior studies, the performance of the Laplace integral first, not only avoids complicated causality arguments and confusion, in locating upstream and downstream modes, that is prevalent in literature but also yields a spatio-temporally uniform solution. It also reveals that the solution consists of a time-periodic part at $\omega_0$ , associated with the relevant spatial mode (the Tollmein-Schlichting wave) and a transient wavepacket, associated mainly with the saddle points of the OSE and is computed by the method of steepest descents, which can also include contributions from the spatial pole. Which of these parts dominates depends on the Reynolds number and {\omega}0. A secondary stability analysis of this dominant part is seen to explain the disturbance growth observed in the seminal experiments of Nishioka, Iida & Ichikawa (J. Fluid Mech., vol.72 , 1975, p.731) and Nishioka, Iida & Kanbayashi (NASA TM-75885, 1981). Threshold amplitudes for instability at a subcritical Reynolds number Re = 5000 are obtained from the time-averaged three dimensional disturbances, by combining the secondary base states and the growing Floquet modes. The observed minima of the threshold amplitude curves in the experiments are explained in terms of the instabilities of these two base states. Computations, for another subcritical (4000) and a supercritical (6000) Reynolds number, are also validated with the experimental data.