Initiation of longitudinal roll sell convection in a fully developed, steadily cooled flowing glass layer is investigated. This study is motivated by the flow of glass melt in the forehearth of a glass melting tank, although the results may be applied to more general situations.
The upper free surface is subject to convective and/or radiative cooling and the rigid bottom is insulated. This study is focused on the occurrence of longitudinal roll cells, aside from the uni-cell back flow. The SIMPLER algorithm with periodic boundary condition is used to directly simulate the flow field numerically in an unsteady manner.
As the first analysis, the Rosseland approximation is used to treat the radiative heat transfer in the flowing layer as conduction. The radiative cooling at the upper surface is also treated as a convective cooling using an effective heat transfer coefficient. Steady two-dimensional roll cells appear when Pr≥1. In this case, the conventional critical Rayleigh number based on the lower and upper surface temperature difference and the associated wavenumber increase with decreasing the Prandtl number, increasing the Biot number and decreasing M. When the Prandtl number is unity, 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. Besides, the uni-cell back flow does not occur if $Ra_o /M$ ＜ 24.
As a prerequisite for improved handling of radiative transfer in the glass melt, a simple but realistic heat transfer problem of steadily cooled flowing glass layer is analyzed using two Rosseland approximations (one with prescribed boundary temperature and the other with Deissler ju...