On the stability of thermally driven shear flow heated from below

Abstract

The onset of convection in shear flow driven by lateral heating and also uniformly heated from below is investigated numerically by Galerkin's method. Stress-free as well as rigid, perfectly conducting boundaries are considered. The analysis is valid for small and moderate Prandtl numbers. The magnitude of the lateral basic temperature gradient may be expressed by a dimensionless Grashof number G, while the uniform heating from below is represented by a Rayleigh number Ra. Depending on the values of G, Ra and the Prandtl number Pr, a variety of interesting situations arise. In particular it is demonstrated that the form of the most unstable mode, i.e. whether it is a roll with axis aligned along the basic flow (a longitudinal roll) or one with axis normal to the basic flow (a transverse roll), depends on the value of the Prandtl number. For small values of G, the marginally stable disturbances are found to be steady, while for larger values of G, oscillatory instability occurs. For all values of G considered here (G [lsim ] 3000), computations of the energy balance for the marginally stable disturbances show that the main instability mechanism is of thermal origin, while the effect of shear may be important in selecting the preferred mode of disturbance.