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On the mechanisms of icicle evolution

Published online by Cambridge University Press:  18 March 2010

JEROME A. NEUFELD ,
RAYMOND E. GOLDSTEIN  and
M. GRAE WORSTER
JEROME A. NEUFELD*
Affiliation:
Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, CMS, Wilberforce Road, Cambridge CB3 0WA, UK
RAYMOND E. GOLDSTEIN
Affiliation:
Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, CMS, Wilberforce Road, Cambridge CB3 0WA, UK
M. GRAE WORSTER
Affiliation:
Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, CMS, Wilberforce Road, Cambridge CB3 0WA, UK
*
†Email address for correspondence: [email protected]
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Abstract

We present a study of a cylinder of ice melting in warm air in order to quantify the heat-transfer mechanisms controlling the evolution of its shape, which are inherent in a range of phenomena involving phase change and fluid flow. Motivated by the initial melting at the top of a flat-topped cylinder of ice, we analyse laminar, natural convection above a cooled, finite, horizontal plate (or below a heated, finite, horizontal plate) and show that, to a very good approximation, the partial-differential, boundary-layer equations can be separated with self-similar vertical profiles scaled by the boundary-layer thickness. We find that the horizontal evolution of the boundary-layer thickness is governed by equations describing a steady, viscous gravity current fed by diffusive entrainment, and therefore describe such flows as diffusive gravity currents. We first use the predictions of our model to examine previous experimental results in two dimensions. Our experimental results relating to the melting of ice in air are then compared with predictions based on our analysis of the axisymmetric thermal boundary layer. This comparison confirms the vertical thermal structure and shows that melting is governed in roughly equal measure by heat transfer from the air, the latent heat of condensation of water vapour, and the net radiative heat transfer from the surroundings to the ice.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 647: Dedicated to Professor S. H. Davis on his 70th birthday , 25 March 2010 , pp. 287 - 308
Copyright
Copyright © Cambridge University Press 2010

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References

REFERENCES

Aihara, T., Yamada, Y. & Endo, S. 1972 Free convection along the downward-facing surface of a heated horizontal plate. Intl J. Heat Mass Trans. 15, 2535–2549.CrossRefGoogle Scholar
Clifton, J. V. & Chapman, A. J. 1969 Natural-convection on a finite-size horizontal plate. Intl J. Heat Mass Trans. 12, 1573–1584.CrossRefGoogle Scholar
Dayan, A., Kushnir, R. & Ullmann, A. 2002 Laminar free convection underneath a hot horizontal infinite flat strip. Intl J. Heat Mass Trans. 45, 4021–4031.CrossRefGoogle Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Fujii, T., Honda, H. & Morioka, I. 1973 A theoretical study of natural convection heat transfer from downward-facing horizontal surfaces with uniform heat flux. Intl J. Heat Mass Trans. 16, 611–627.Google Scholar
Gill, W. N., Zeh, D. W. & Del Casal, E. 1965 Free convection on a horizontal plate. Zeit. Ang. Math. Phys. 16, 539–541.CrossRefGoogle Scholar
Hatfield, D. W. & Edwards, D. K. 1981 Edge and aspect ratio effects on natural convection from the horizontal heated plate facing downwards. Intl J. Heat Mass Trans. 24 (6), 1019–1024.CrossRefGoogle Scholar
Higuera, F. J. 1993 Natural convection below a downward facing horizontal plate. Eur. J. Mech. B. Fluids 12 (3), 289–311.Google Scholar
Higuera, F. J. & Weidman, P. D. 1995 Natural-convection beneath a downward facing heated plate in a porous-medium. Eur. J. Mech. B. Fluids 14 (1), 29–40.Google Scholar
Kuiken, H. K. 1968 An asymptotic solution for large Prandtl number free convection. J. Engng Math. 2 (2), 355–371.CrossRefGoogle Scholar
Ogawa, N. & Furukawa, Y. 2002 Surface instability of icicles. Phys. Rev. E 66, 041202.CrossRefGoogle ScholarPubMed
Schlichting, H. & Gersten, K. 2000 Boundary Layer Theory. Springer.CrossRefGoogle Scholar
Short, M. B., Baygents, J. C. & Goldstein, R. E. 2006 A free-boundary theory for the shape of the ideal dripping icicle. Phy. Fluids 18, 083101.CrossRefGoogle Scholar
Singh, S. N. & Birkebak, R. C. 1969 Laminar free convection from a horizontal infinite strip facing downwards. Zeit. Ang. Math. Phys. 20 (4), 454–461.CrossRefGoogle Scholar
Stewartson, K. 1958 On the free convection from a horizontal plate. Zeit. Ang. Math. Phys. 9 (3), 276–282.CrossRefGoogle Scholar
Tsilingiris, P. T. 2008 Thermophysical and transport properties of humid air at temperature range between 0 and 100°C. Energy Convers. Manage. 49, 1098–1110.CrossRefGoogle Scholar
Ueno, K. 2003 Pattern formation in crystal growth under parabolic shear flow. Phys. Rev. E 68, 021603.CrossRefGoogle ScholarPubMed
Ueno, K. 2004 Pattern formation in crystal growth under parabolic shear flow. Part 2. Phys. Rev. E 69, 051604.CrossRefGoogle Scholar
Wagner, C. 1956 Discussion of integral methods in natural convection flows by S. Levy. J. Appl. Mech. 78, 320–321.CrossRefGoogle Scholar
Warren, S. G. 1984 Optical constants of ice from the ultraviolet to the microwave. Appl. Optics 23 (8), 1206–1225.CrossRefGoogle Scholar
Wood, S. E. & Battino, R. 1990 Thermodynamics of Chemical Systems. Cambridge University Press.CrossRefGoogle Scholar
Worster, M. G. 2000 Solidification of fluids. In Perspectives in Fluid Dynamics: A Collective Introduction to Current Research (ed. Batchelor, G. K., Moffatt, H. K. & Worster, M. G.), pp. 393–446. Cambridge University Press.Google Scholar

Neufeld et al. supplementary material

Movie 1. A sequence of high resolution images, taken every 15 seconds, of a cylinder of ice melting in air with far-field temperature T0 = 21.7 oC and relative humidity RH = 43.0%. The cylindrical top of the ice, analysed in the accompanying paper, has an initial radius of R = 2.7 cm. The movie has been speeded up by a factor of 200.

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