Degeneracies of the temporal Orr-Sommerfeld eigenmodes in plane Poiseuille flow

Abstract

Degenerating stable temporal Orr-Sommerfeld eigenmodes are studied for plane Poiseuille flow. The discrete spectrum of the eigenmodes is shown to possess infinitely many degeneracies, each appearing at a certain combination of k (the modulus of resultant wavenumber) and αR (the streamwise wavenumber time the Reynolds number). The eigenmodes are found to degenerate in a specific manner which confines the streamwise phase velocities of the degeneracies to be around $\frac{2}{3}$ of the centreline velocity. The responses of the degeneracies are investigated through the initial-value problem. The responses of the first four symmetric and the first two antisymmetric degeneracies are evaluated numerically for arbitrary initial disturbances expanded in terms of Chebyshev polynomials. The first symmetric and the first antisymmetric degeneracies exhibit temporal growth of the amplitudes in the wavenumber space. The maximum amplitudes are at most 7 times larger than the corresponding initial amplitudes. The amplitudes of the responses of the other four degeneracies decay rapidly owing to their higher damping rates. The time for which the degeneracy-response is in the growing phase is shown to be stretched with increasing Reynolds number. The degeneracies can therefore be active for longer periods of time at larger Reynolds numbers.