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 jump boundary condition) and the $P_1$ approximation, and the computed temperature profiles are compared with those of the exact solution to examine their accuracy and validity. The Rosseland approximations are accurate for deep glass melts but they have serious error for relatively shallow ones. The $P_1$ approximation gives accurate results not only for shallow glass melts but also for deep ones, and thus it is shown that this approximation can be used as a substitute of the Rosseland approximations and the exact solution scheme.
Finally, the effect of the radiation on the stability limit of the uni-cell back flow and the longitudinal roll cells is investigated using the $P_1$ approximation. The upper free surface is subject to radiative cooling and the rigid bottom is insulated. A uni-cell back flow occurs when the mass flow decreases and steady cooling speed increases. In the meanwhile, steady two-dimensional roll cells appear when the Rayleigh number is greater than the critical value. When $\tau_1$ is about unity, the radiative emission and transmission is so effective that the vertical about unity, the radiative emission and transmission is so effective that the vertical temperature variation in the glass layer is minimal. This makes the flow very stable with regard to the longitudinal roll cells. Thus, with optical thickness $\tau_1$ ≒ 1.0, the uni-cell back flow may occur prior to the longitudinal roll cells. Either for optically thin and thick case, the flow is stabilized with decreasing the Prandtl number, increasing the dimensionless mass flow rate, increasing the optical depth and increasing the environmental temperature. The dependence on the mass flow rate is opposite to the previous case of non-radiative cooling, and it is the result of enhanced radiative heat transfer at elevated temperature. If the emissivity of the bottom wall is greater than 0.6, the emissivity has little effect on the stability of the flow. Decreasing the glass layer depth makes the flow more stable. The suppression criterion of back flow is exactly same as that of non-radiative case, i.e., $Ra_o/M ＜ 24$. The critical Rayleigh number and the associated wavenumber of the radiative cooling problem are larger than those of the non-radiative case. Even for optically thick cases, treatment of radiative transfer by the Rosseland approximation results in smaller critical Rayleigh number than reality by tens of percent. Therefore, elaborate analysis with radiation provides more room in the design of forehearth and shallow forming and refining processes.