Advances in Computational Mathematics

, Volume 6, Issue 1, pp 295–308 | Cite as

Free surface Nvier-Stokes flows with simultaneous heat transfer and solidification/melting

  • K. A. Pericleous
  • M. Cross
  • G. Moran
  • P. Chow
  • K. S. Chan


A mathematical model to analyse some key aspects of the metal cast process is described involving the filling of the mould by liquid metal and simultaneously, undergoing both cooling and solidification (re-melting) phase change. A computational solution procedure based upon a finite volume discretisation approach, on both structured and unstructured meshes, is described. The overall flow solution procedure is based on the pressure correction algorithm SIMPLE suitably adapted to: (a) solve for the free surface with minimal smearing by the SEA algorithm, and (b) solve for the solidification/melting phase change using an enthalpy conservation algorithm developed by Voller, but with its root in the work of Crank many years ago.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    C. Bailey, P. Chow, M. Cross, Y. Fryer and K. Pericleous, Multiphysics modelling of metals casting processes, Proc. Roy. Soc. London Ser. A 452 (1996) 459–486.Google Scholar
  2. [2]
    J. Campbell,Castings (Butterworth-Heinemann, Oxford, 1991).Google Scholar
  3. [3]
    CFDS-FLOW3D (AEA Technology, Harwell, Oxon, UK).Google Scholar
  4. [4]
    K. S. Chan, Free surface flow and application to the filling and solidification of liquid metals into vessels of arbitrary shape, Ph.D. thesis, University of Greenwich, London (1994).Google Scholar
  5. [5]
    P. M. Y. Chow, Control volume unstructured mesh procedure for convection-diffusion solidification process, Ph.D. thesis, University of Greenwich, London (1993).Google Scholar
  6. [6]
    J. Crank,Free and Moving Boundary Problems (Clarendon Press, Oxford, 1984).Google Scholar
  7. [7]
    M. Cross and J. Campbell, eds.,Modeling of Casting, Welding and Advanced Solidification Processes VII (Materials, Metals and Minerals Society, Warrendale, PA, 1995).Google Scholar
  8. [8]
    S. E. Hibbert, N. C. Markatos and V. R. Voller, Computer simulations of moving interface, convective, phase change process, Internat. J. Heat Mass Transfer 31 (1988) 1785–1795.CrossRefGoogle Scholar
  9. [9]
    B. P. Leonard, Universal limiter for transient interpolation modelling of the advective transport equations, Tech. Memo 100916, NASA Lewis, Cleveland, OH (1988).Google Scholar
  10. [10]
    A. Lazaridis, A numerical solution of the multi-dimensional solidification (or melting) problems, Internat. J. Heat Mass Transfer 13 (1970) 1459–1477.CrossRefGoogle Scholar
  11. [11]
    J. C. Martin and W. J. Moyce, An experimental study of the collapse of liquid column on a rigid horizontal plane, Philos. Trans. Roy. Soc. London Ser. A 244 (1952) 312.Google Scholar
  12. [12]
    T. T. Maxwell, Numerical modelling of free surface flows, CFDU, Ph.D. thesis, Imperial College, London (1977).Google Scholar
  13. [13]
    S. V. Patankar and D. B. Spalding, A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows, Internat. J. Heat Mass Transfer 15 (1972) 1787.CrossRefGoogle Scholar
  14. [14]
    K. A. Pericieous, K. S. Chan and M. Cross, Free surface flow and heat transfer in cavities: the SEA algorithm, Numerical Heat Transfer, Part B 27 (1995) 487.Google Scholar
  15. [15]
    PHOENICS (CHAM Limited, Wimbledon, London, UK).Google Scholar
  16. [16]
    PHYSICA (University of Greenwich, London, UK).Google Scholar
  17. [17]
    W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, eds.,Numerical Recipes in Fortran, The Art of Scientific Computing (Cambrigde University Press, 1992) chapter 19.Google Scholar
  18. [18]
    C. M. Rhie and W. L. Chow, A numerical study of the turbulent flow past an isolated aerofoil with trailing edge separation, AIAA J. 21 (1982) 1525–1532.Google Scholar
  19. [19]
    P. Sampson and R. D. Gibson, Solidification of a liquid metal flowing through a circular pipe: a prediction of nozzle blockage, Adv. Engrg. Software 3 (1981) 17–25.CrossRefGoogle Scholar
  20. [20]
    P. Sampson and R. D. Gibson, A mathematical model of nozzle blockage by freezing — II. Turbulent flow, Internat. J. Heat Mass Transfer 25 (1982) 119–126.CrossRefGoogle Scholar
  21. [21]
    D. B. Spalding, A method for computing steady and unsteady flows possessing discontinuities of density, CHAM Report 910/2 (1974).Google Scholar
  22. [22]
    B. van Leer, Towards the ultimate conservation difference scheme, IV. A new approach to numerical convection, J. Comput. Phys. 23 (1977) 276.CrossRefGoogle Scholar
  23. [23]
    V. R. Voller and M. Cross, Solidification processes, algorithms and codes, in:Mathematical Modelling of Materials Processing, eds. M. Cross et al. (Oxford University Press, Oxford, 1993) pp. 93–111.Google Scholar
  24. [24]
    V. R. Voller and C. Prakash, A fixed grid numerical modelling methodology for convection/diffusion mushy region phase change problems, Int. J. Heat Mass Transfer 30 (1987) 1709–1718.CrossRefGoogle Scholar
  25. [25]
    V. R. Voller, C. R. Swaininathan and B. G. Thomas, Fixed grid techniques for phase change problems — a review, Internat. J. Numer. Methods Engrg. 30 (1990) 875–898.Google Scholar

Copyright information

© J.C. Baltzer AG, Science Publishers 1996

Authors and Affiliations

  • K. A. Pericleous
    • 1
  • M. Cross
    • 1
  • G. Moran
    • 1
  • P. Chow
    • 1
  • K. S. Chan
    • 1
  1. 1.Centre for Numerical Modelling and Process AnalysisUniversity of GreenwichLondonUK

Personalised recommendations