Numerical study of three-dimensional mixed convection due to buoyancy and centrifugal force in an oxide melt for Czochralski growth

Mixed convection due to buoyant and centrifugal forces in the crucible of a Czochralski apparatus has a significant effect on the quality and stability of growing oxide single crystals. Thus, the present investigation is concerned with the time-dependent and three-dimensional simulation of the flow and heat transfer in an oxide melt with a rotating crystal for a fundamental understanding of mixed convection and its effect on the surface waves. Based on an efficient block-structured finite-volume Navier-Stokes solver, quasi-direct numerical simulations (quasi-DNS) were carried out. A cylindrical crucible filled with a high Prandtl number (Pr=10.45) model fluid was considered for different combinations of Rayleigh numbers (Ra=4.0×10 ⁸ and 7.6×10 ⁸) and Reynolds numbers (Re=235-461). It was found that the surface wave pattern is strongly dependent on the structure of the three-dimensional and time-dependent fluid flow in the crucible. For all cases predicted, the waves exhibit regularity and the flow structure is fully symmetric in the vertical midplane as long as the centrifugal force has a significant effect on the flow below the disc. Furthermore, a downward buoyant jet was found to form at the disc center in all cases when the upper flow column due to the centrifugal force becomes restricted to the top. The transient flow development can be characterized by Gr/Re ². For Gr/Re ²=235 the flow field is always periodic whereas it attains a non-periodic state through a number of quasi-periodic modes for Gr/Re ² larger than 334. Increasing values of Gr/Re ² lead to an earlier onset of the non-periodicity. Except during the initial period, the hot fluid is pushed below the disc primarily by the radial buoyant convection. This hot fluid interacts with the cold column of the downward buoyant flow controlling the temperature at the crystal/melt interface. The shape, the rotation rate of the waves, and the transient nature of the flow field were found to be in agreement with experimental observations.