Stability of unstably stratified shear flow in a channel under non-Boussinesq conditions

Abstract

We investigate the linear stability of unstably stratified Poiseuille flow between two horizontal parallel plates under non-Boussinesq conditions. It is shown that Squire's transformation can be used to reduce the three-dimensional stability problem to an equivalent two-dimensional one. The eigenvalue problem, consisting of the generalized Orr-Sommerfeld equations, is solved numerically using an integral Chebyshev pseudo-spectral method. The influence of the non-Boussinesq effects on stability is studied. The dependence of the critical Rayleigh number on the Reynolds number and temperature difference parameter is obtained. As in the Boussinesq case, results show that the most unstable mode is that of longitudinal rolls. However, in contrast to the Boussinesq case, the rolls are highly distorted for large temperature differences. In addition, the critical Rayleigh number increases with the increase of the temperature difference and is independent of the Reynolds number.