Interaction between longitudinal convection rolls and transverse waves in unstably stratified plane Poiseuille flow

Abstract

Nonlinear interaction between a Tollmien–Schlichting wave and longitudinal rolls resulting from Rayleigh–Bénard instability has been investigated in an unstably stratified plane Poiseuille flow of infinite extent. Cubic order amplitude equations for the interacting modes are derived on a weakly nonlinear basis in the neighborhood of a crossover point at which both modes become unstable on a linear basis simultaneously. A bifurcation analysis based on use of the actual numerical coefficients obtained from the governing equations and evaluated for various values of the Prandtl number is performed, and the results, such as the effect of longitudinal rolls on the subcritical instability characteristic of a Tollmien–Schlichting wave, are discussed.