Journal of Scientific Computing

, Volume 12, Issue 1, pp 75–97 | Cite as

A Legendre-Spectral Element Method for Flow and Heat Transfer About an Accelerating Droplet

  • Hoa D. Nguyen
  • Seungho Paik
  • Rod W. Douglass


The problem of flow and heat transfer associated with a spherical droplet accelerated from rest under gravitational force is studied; using a Legendre-spectral element method in conjunction with a mixed time integration procedure to advance the solution in time. An influence matrix technique which exploits the superposition principle is adapted to resolve the lack of vorticity boundary conditions and to decouple the equations from the interfacial couplings. The computed flow and temperature fields, the drag coefficient, the Nusselt number, and the interfacial velocity and vorticity are presented for a drop moving vertically in a quiescent gas of infinite extent to illustrate the evolution of the flow and temperature fields. Comparison of the predicted drag coefficient and the Nusselt number against previous numerical and experimental results indicate good agreement.

Spectral element accelerating droplet heat transfer influence matrix Nusselt number drag 


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  1. Abdel-Alim, A. H., and Hamielec, A. E. (1975). A theoretical and experimental investigation of the effect of internal circulation on the drag of spherical doplets falling at terminal velocity in liquid media, Ind. Eng. Chem., Fund. 14, 308–312.Google Scholar
  2. Canuto, C., Hussaini, M. H., Quarteroni, A., and Zang, T. A. (1988). Spectral methods in Fluid Dynamics, Springer-Verlag, New York.Google Scholar
  3. Chao, B. T. (1962). Motion of spherical has bubbles in a viscous liquid at large Reynolds numbers, Phys. Fluids 5, 69–79.Google Scholar
  4. Chisnell, R. F. (1987). The unsteady motion of a drop moving vertically under gravity, J. Fluid Mech. 176, 443–464.Google Scholar
  5. Clift, R., Grace, J. R., and Weber, M. E. (1978). Bubbles, Drops, and Particles, Academic Press, New York.Google Scholar
  6. Elzinga, E. R., and Banchero, J. T. (1961). AIChE J. 7, 394.Google Scholar
  7. Fox, L., and Parker, I. (1968). Chebyshev Polynomials in Numerical Analysis, Oxford University Press, London.Google Scholar
  8. Froessling, N. (1938). Gerlands Beitr. Geophysics 52, 170.Google Scholar
  9. Gottlieb, D., and Orszag, S. A. (1977). Numerical Analysis of Spectral methods: Theory and Applications, SIAM, Philadelphia.Google Scholar
  10. Hadamard, J. (1911). Mouvement permanent lent d'une sphère liquide et visqueuse dans un liquide visqueux, C. R. Acad. Sci. 152, 1735–1738.Google Scholar
  11. LeClair, B. P., Hamielec, A. E., Pruppacher, H. R., and Hall, W. D. (1972). A theoretical and experimental study of the internal circulation in water drops falling at terminal velocity in air, J. Atmos. Sci. 29, 728–740.Google Scholar
  12. Lin, C. L., and Lee, S. C. (1973). Transient state analysis of separated flow around a sphere, Comput. and Fluids 1, 235–250.Google Scholar
  13. Nguyen, H. D., Paik, S., and Chung, J. N. (1992). A combined Galerkin/collocation spectral method for transient solution of flow past a spherical droplet, in Sohal, M. S., and Rabas, T. J., (eds.), Two-Phase Flow in Energy Exchange Systems, The American Society of Mechanical Engineers, New York, p. 87.Google Scholar
  14. Nguyen, H. D., Paik, S., and Chung, J. N. (1993a). Unsteady conjugate heat transfer associated with a translating spherical droplet: a direct numerical simulation. Num. Heat Transfer., Part A 24, 161–180.Google Scholar
  15. Nguyen, H. D., Paik, S., and Chung, J. N. (1993b). Unsteady mixed convection heat transfer from a solid sphere: the conjugate problem, Int. J. Heat Mass Transfer 36, 4443–4453.Google Scholar
  16. Oliver, D. L. R;, and Chung, J. N. (1985). Steady flows inside and around a fluid sphere at low Reynolds numbers, J. Fluid Mech. 154, 215–230.Google Scholar
  17. Oliver, D. L. R., and Chung, J. N. (1986). Conjugate unsteady heat transfer from a spherical droplet at low Reynolds numbers, Int. J. heat Mass Transfer 29, 879–887.Google Scholar
  18. Oliver, D. L. R., and Chung, J. N. (1987). Flow about a fluid sphere at low to moderate Reynolds numbers, J. Fluid Mech. 177, 18.Google Scholar
  19. Oliver, D. L. R., and Chung, J. N. (1990). Unsteady conjugate heat transfer from a translating fluid sphereat moderate Reynolds numbers, Int. J. Heat Mass Transfer 33, 401–408.Google Scholar
  20. Patera, A. T. (1984). A spectral element method for fluid dynamics: laminar flow in a channel expansion, J. Comput. Phys. 54, 468–488.Google Scholar
  21. Rivkind, V. Ya., Ryskin, G. M., and Fishbein, G. A. (1976). Appl. Math. Mech. 40, 687.Google Scholar
  22. Rybczynski, W. (1911). Uber die fortschreitende bewegung einer flussigen kugel in einem zaben medium, Bull. Int. Acad. Pol. Sci. Lett., Cl. Sci. Math. Nat., Ser. A, 40.Google Scholar
  23. Taylor, T. D., and Acrivos, A. (1964). On the deformation and drag of a falling viscous drop at low Reynolds number, J. Fluid Mech. 18, 466.Google Scholar

Copyright information

© Plenum Publishing Corporation 1997

Authors and Affiliations

  • Hoa D. Nguyen
    • 1
  • Seungho Paik
  • Rod W. Douglass
    • 1
  1. 1.Idaho National Engineering LaboratoryIdaho Falls

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