Application of Higher Order BDF Discretization of the Boussinesq Equation and the Heat Transport Equation

  • G. Bärwolff
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 21)

Abstract

During the growth of crystals there were observed crystal defects under some conditions of the growth device. As a result of experiments a transition from the twodimensional flow regime of a crystal melt in axisymmetric zone melting devices to an unsteady threedimensional behavior of the velocity and temperature field was found. This behavior leads to striations as undesirable crystal defects. For the investigation of this symmetry break a mathematical model of the crystal melt was formulated for

i) the theoretical description of the experimentally observed behavior and

ii) the identification of critical parameters of the growth device, i.e. the evaluation of bifurcation points

To describe and to avoid such a behavior it is necessary to solve the unsteady three-dimensional Boussinesq equation coupled with the heat transport equation efficiently

To improve first and second Euler, leapfrog and Adams-Bashforth methods higher order explicit and BDF (Backward Differenciation Formulas) methods are applied and constructed for time dependend calculations and a Newton method is discussed for the resulting nonlinear equation systems for implicit integration methods and the steady state solution

Keywords

Convection Lution 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bärwolff, G., König, F. and Seifert, G., (1997) Thermal buoyancy convection in vertical zone melting configurations, ZAMM 77 (1997) 10, 757–766CrossRefGoogle Scholar
  2. 2.
    Bernert, K., (1990) Differenzenverfahren zur Lösung der Navier-StokesGleichungen über orthogonalen Netzen, Wissenschaftliche Schriftenreihe der TU Chemnitz, Heft 10, ChemnitzGoogle Scholar
  3. 3.
    Chorin, A.J., (1968) Numerical Solution of Navier-Stokes Equation, Mathematics of computation, Vol. 22 p, 745–760MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Butler, T.D., (1974) Recent Advances in Computational Fluid Dynamics, in: Computing Methods in Applied Sciences and Engineering, Part 2. Lecture Notes in Computer Science No. 11, Springer Berlin, Heidelberg, New YorkGoogle Scholar
  5. 5.
    Roache, P.D., (1972) Computational Fluid Dynamics, Hermosa publishers, AlbuquerqueGoogle Scholar
  6. 6.
    Emmrich, E. (2000) Error Analysis for Second Order BDF Discretization of the Incompressible Navier-Stokes Problem, Proc. of the 4th Summer Conference on Numerical Modelling in Continuum Mechanics, Prague, August 2000Google Scholar
  7. 7.
    Golub, G.H. and Ortega, J.M., (1992) Scientific Computing and Differential Equations, Academic Press, IncMATHGoogle Scholar
  8. 8.
    Jiang, B., (1998) The Least-Squares Finite Element Method, Springer Berlin, Heidelberg, New YorkGoogle Scholar
  9. 9.
    Basu, B., Enger, S., Breuer, M. and Durst, F., (2000) Three-dimensional simulation of flow and thermal field in a Czochralski melt using block-structured finite-volume method, Journ. of Crystal Growth 219(2000), 123–143CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • G. Bärwolff
    • 1
  1. 1.TU BerlinInst. f. Mathematik, SekrBerlinGermany

Personalised recommendations