NUMERICAL SIMULATION OF FLOWS IN CZOCHRALSKI CRYSTAL GROWTH AND TAYLOR VORTICES

Abstract

The global method of generalized differential quadrature (GDQ) is applied to simulate the flows in Czochralski crystal growth configuration and Taylor vortex flow. In the first part of present study, the single-domain GDQ solver is first validated by its application to the benchmark problem of flows in Czochralski crystal growth suggested by Wheeler (1990). It is demonstrated that the GDQ method can obtain accurate numerical solutions using just a few points, needing only small computational resources. The multi-domain generalized differential quadrature (GDQ) method is also employed in this investigation to simulate the same cases as in the single-domain study. The effect of interface treatment on the numerical solution is studied through four types of interface approximations. The performance of the four interface approximations is validated by the benchmark problem suggested by Wheeler. It is demonstrated in this study that the multi-domain GDQ approach is an efficient method which can obtain accurate numerical solutions by using very few grid points, and the overlapped interface approximation provides the most accurate numerical results. In the second part of this study, the generalised differential quadrature (GDQ) method is applied to simulate the Taylor vortex flow. As a validation of the present numerical method, the two-four cells flow transition locus is first generated by the GDQ method. The results by the GDQ method are quite comparable with both the experimental results by Benjamin (1978b) and the finite-difference results by Ball & Farouk ( 1988). It is demonstrated in this study that the two-four cell transition locus is not singular and the locus presented by Benjamin is the limit locus of the two-four cell transition. There may exist a series of transition locus above this limit locus and could be realised with different critical acceleration rate. The effect of acceleration rate on two-cell vortex formation was investigated using a time-dependent solver. Meanwhile, under the similar scheme as that used in time-dependent solver to increase the Reynolds number from zero continuously, the time-independent solver was also used in this study. It was found that the realization of the two-cell primary mode is not arbitrary but process dependent. For time-dependent solver, there exists a critical acceleration rate. Below this critical acceleration rate, the two-cell primary mode could be realized, and above it, only the four-cell flow will appear. It was also demonstrated in this study that the time-independent solver can mimic the time-variant evolution problem reliably and accurately if the proper conditions to increase the Reynolds number are provided.