Numerical simulation of the stability of steadily cooled flowing layer

Initiation of longitudinal roll cell convection in a fully developed, steadily cooled flowing layer has been investigated. The upper free surface is subject to convective cooling and the rigid bottom is insulated. The critical Rayleigh number and the associated wavenumber are obtained as functions of the Prandtl number, the dimensionless mass flow rate of main flow and the Biot number at the upper surface. Linear stability theory is not valid in this case. The SIMPLER algorithm with periodic boundary condition is used to directly simulate the flow field numerically in an unsteady manner. In the region of steady bifurcation, the critical Rayleigh number is significantly greater than the results using the linear stability theory. However, when the Prandtl number is greater than 10, the linear stability theory is asymptotically valid, and the critical Rayleigh number and the associated wavenumber are very close to the results from the linear stability theory. Oscillatory motion, or Hopf bifurcation, occurs when the Prandtl number is less than 0.1.