Three-dimensional Numerical Simulation of Unsteady Marangoni Convection in the CZ Method using GSMAC-FEM
Haruhiko Kohno, Takahiko Tanahashi

doi:10.3970/cmes.2001.002.155
 Source CMES: Computer Modeling in Engineering & Sciences, Vol. 2, No. 2, pp. 155-170, 2001 Download Full length paper in PDF format. Size = 686,630 bytes Keywords Marangoni convection, Czochralski method vertical magnetic field, GSMAC-FEM Abstract Three-dimensional (3D) unsteady numerical simulations are carried out by means of the finite element method (FEM) with the generalized simplified marker and cell (GSMAC) method in silicon melt with a non-deformable free surface with Prandtl number Pr =$1.8534\times 10^{-2}$, Marangoni number Ma =$0.0 - 6.2067\times 10^{2}$, Grashof number Gr =$7.1104\times 10^{6}$, and the aspect ratio As = 1.0 in the Czochralski (CZ) method. The flow state becomes unstable earlier by increasing the absolute value of the thermal coefficient of surface tension in the range of$\sigma _{T}=0.0 - 1.5\times 10^{-5}$N/mK. Although the velocity distribution in the circumferential direction is isotropy in any direction first, its magnitude becomes periodic and has the wavelength equal to 1/8 of the circumference. Then the wavelength doubles, and the flow pattern becomes finally asymmetrical. Moreover, the oscillation of the velocity distribution is observed just under the single crystal, and the amplitude is found to depend on the value of$\sigma _{T}$. After imposing the vertical magnetic field more than 0.05T to the melt from 50s, the flow pattern becomes restored to symmetry. But the instability remains under the single crystal and it indicates that the influence of Marangoni convection can not be neglected in the crystal growing process.