Petrov‐Galerkin methods for natural convection in directional solidification of binary alloys

Abstract

SUMMARYA Petrov-Galerkin finite element method is presented for calculation of the steady, axisymmetric thermosolutal convection and interface morphology in a model for vertical Bridgman crystal growth of nondilute binary alloys. The Petrov-Galerkin method is based on the formulation for biquadratic elements developed by Heinrich and Zienkiewicz and is introduced into the calculation of the velocity, temperature and concentration fields. The algebraic system is solved simultaneously for the field variables and interface shape by Newton's method. The results of the Petrov-Galerkin method are compared critically with those of Galerkin's method using the same finite element grids. Significant improvements in accuracy are found with the Petrov-Galerkin method only when the mesh is refined and when the formulation of the residual equations is modified to account for the mixed boundary conditions that arise at the solidification interface. Calculations for alloys with stable and unstable solute gradients show the occurrence of classical flow transitions and morphological instabilities in the solidification system.

A computer simulation of crystal growth by the traveling-solvent method (TSM): pseudo-steady-state calculations

“…Unexpectedly, near the centerline the growth front breaks down. This breakdown may be caused by numerical errors or related to the convection-induced morphological stability [35,36]. Nevertheless, since the solution still converges, the results are also presented here for comparison.…”

Spectral simulations of oscillatory convection at low Prandtl number

“…For instance, the stability analysis in a cavity of aspect ratio A = 4 predicts Grc,osc at 13 722 (Pr = 0, R-F), 14767 and 16598 (Pr = 0015, R-F-C and R-F-A respectively, ' Table I). 28 Ben Hadid and R o u x ,~~ using a finite difference method, obtained 20000 < Grc, osc 6 < Grc, osc < 33 500 obtained a steady solution for Gr = 16700 and an oscillatory one for Gr = 79200, but no precise investigation concerning the definite nature of the solution has been done in the intermediate range of Gr.…”

Heat Transfer Analysis and Design for Bulk Crystal Growth: Perspectives on the Bridgman Method

“…Large-scale, computer-aided analysis of coupled heat transfer and melt convection during the vertical Bridgman growth of single crystals was pioneered by Brown and coworkers [113][114][115][116][117][118][119]. In particular, Adornato and Brown [114] studied the effect of different furnace profiles and ampoule materials, and compared their predictions to the growth experiments conducted by Wang and Witt [14].…”