Spectra of the Orr-Sommerfeld equation: The plane Poiseuille flow for the normal fluid revisited (Q2736740)

When solving the Orr-Sommerfeld equation (OSE) of hydrodynamic instability theory by numerical techniques, the occurrence of the so-called spurious modes is often observed. In the present paper, a spectral Chebyshev approach combined with a preconditioned complex matrix solver is used to solve the OSE, and the author reports (for plane Poiseuille flow) the occurrence of strange spectra which are so far not related to the spurious modes, and which have never been mentioned in the literature before. These strange spectra, being one pair of eigenvalues corresponding to the specific Reynolds number \(Re\) and the wavenumber \(a\), have almost the same real parts (phase speeds) but different imaginary parts (amplifying factors). The behavior of these strange spectra disappears as \(\text{Re} \to 400\). In the present paper, seven such pairs of eigenvalues are listed for Reynolds numbers ranging between 500 and 5750, and for wavenumbers ranging between 0.8 and 1.8.