Two-dimensional local instability: Complete eigenvalue spectrum

Abstract

The study of "simple" flows such as Poiseuille flow have been for a long time studied and their simplicity has permitted us to highlight different instability mechanisms. More recently, several authors have shown, through transient growth studies and by the identification of the nonnormal character of the operator of Orr-Sommerfeld, that these two flows could have subcritical dynamics. However, the assumption of one-dimensionality of the basic flow limits the comparison with the experiment where the basic flow is not exactly one-dimensional. Although different studies have been realized on the stability of a two-dimensional basic flow, all were interested only in the most unstable mode. In this present paper, the stability of the laminar flow in a rectangular duct of an arbitrary aspect ratio is investigated numerically with for objective the computation of the complete spectrum. This study will highlight strong similarities between the 1D and 2D spectra but, in spite of a powerful numerical method, will show the numerical limitations observed to obtain a spectrum converged in a sufficiently large field in ω. © 2006 Springer.