NON LINEAR INTERACTIONS AND TEMPERATURE OSCILLATIONS IN LOW-PRANDTL MELT IN CZOHRALSKI MODEL: VALIDATION OF COMPUTATIONAL SOLUTIONS FOR GRAVITY DRIVEN AND ROTATORY FLOWS

Non-linear interactions and temperature oscillations of gravity-driven fluid flow with rotation as typical example of non-linear coupling fluid system is studied as a benchmark of computer model of Czohralski growth. A focus of the paper is concentrated on the new results of multiparametric analysis of non-linear coupling fluid flows in Czohralski growth (hydrodynamic model) using the axisymmetric model on the basis of unsteady Navier-Stokes equations (Boussinesq approach).

Two types of numerical approaches: stream function-vorticity and velocity-pressure formulations and two types of numerical methods were used: "upwind" for qualitative pictures multiparametric simulation and second order approximation for quantitative analysis of flow and temperature field structure, convective instability and oscillations.

Parametric function of the characteristics in the melt can be written as follows:

A = (r, z, t, Rex, Rec, Gr, Pr, H/Rc, Rx/Rc, g, go)

here g indicates the boundary conditions and go - the initial values.

First part of the paper is devoted to the benchmark proposal [1] (H/Rx=1, Rx/Rc= 0.4, Pr=0.05) in the case of prescribed temperatures Tx and Tc on the seed and crucible wall and isolation of the crucible bottom and linear distribution on the melt surface. Results of tne calculations errors for different meshes and for different values of Rex, Rec and Gr are presented. Fig.l shows selected pictures for general gravity -driven and rotation regimes. Critical Grashof number for onset of temperature oscillations in the melt is also defined.

Second part of the paper related to the multiparametrical analysis of the general industrial ground-based regime with counter rotation, without rotation as well as zero gravity regime with rotation .Dependencies of oscillation amplitude on the Grashof number for the case of zero rotation as well as for the coupling nonlinear case are presented. It is shown for above mentioned ground-based case that temperature oscillations due to gravity-driven convection in Czohralski growth melt flow may be of two types: Rayleigh-Benard type due to cooling above of ctystal growth surface and side heating.

Experimental validation of computational solutions is discussed on the basis of comparisons with experimental data for temperature oscillations in the GaAs melt [3] and in the water [4]. Non-symmetry effects and their impact on the above mentioned results are also discussed.

This work was supported partly by joint NASA-RSA program (project No TM-11)

References

1. Wheeler A.A. Four Test Problems for the Numerical Simulation of Flow in Czochralski Crystal Growth. J. Crystal Growth,1990, v.102, p.691.
2. Buckle U., Schafer M. Benchmark results for the numerical simulation of flow in Czochralski crystal growth. / J. Crystal Growth 126,1993, p.682-694.
3. Kosushkin V.G., Polezhaev V.I., Zakharov B.G. Ground-based experiments and alternatives in GaAs cryatal growth. In.: Proceedings of the Microgravity Science and Applications Session. International Aerospace Congress, Moscow, ( Eds.) R.K. Crouch, V.I. Polezhaev,1995, p.141-146.
4. Jones A.D.W. T'he temperature field of a model Czochralski melt. J. Crystal Growth 69,1984, p.165- 172.
5. Polezhaev V.I., Etmakov M.K, Griaznov V.IL. et al. Computer Laboratory on Convective Processes in Microgravity: Concepts, Current Results and Perspective. 46 Internat́onal Astronautical Congress, IAF-95 J.3.11, October 2-6,1995, / Oslo, Notyvay.