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On ice-induced instability in free-surface flows

Published online by Cambridge University Press:  19 April 2007

EVGENIY SHAPIRO  and
SERGEI TIMOSHIN
EVGENIY SHAPIRO
Affiliation:
Department of Aerospace Sciences, Cranfield University, Cranfield MK43 0AL, UK
SERGEI TIMOSHIN
Affiliation:
Department of Mathematics, University College London, Gower Street, London WC1 6BT, UK
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Abstract

The problem of stability of a water-coated ice layer is investigated for a free-surface flow of a thin water film down an inclined plane. An asymptotic (double-deck) theory is developed for a flow with large Reynolds and Froude numbers which is then used to investigate linear two-dimensional, three-dimensional and nonlinear two-dimensional stability characteristics. A new mode of upstream-propagating instability arising from the interaction of the ice surface with the flow is discovered and its properties are investigated. In the linear limit, closed-form expressions for the dispersion relation and neutral curves are obtained for the case of Pr = 1. For the general case, the linear stability problem is solved numerically and the applicability of the solution with Pr = 1 is analysed. Nonlinear double-deck equations are solved with a novel global-marching-type scheme and the effects of nonlinearity are investigated. An explanation of the physical mechanism leading to the upstream propagation of instability waves is provided.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 577 , 25 April 2007 , pp. 25 - 52
Copyright
Copyright © Cambridge University Press 2007

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References

REFERENCES

Abramowitz, M. & Stegun, I. A. 1972 Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover.Google Scholar
Alexiades, V. & Solomon, A. D 1993 Mathematical modeling of Melting and Freezing Processes. Hemisphere.Google Scholar
Batchelor, G. K., Moffatt, H. K. & Worster, M. G. 2000 Perspectives in Fluid Dynamics: A Collective Introduction to Current Reasearch. Cambridge University Press.Google Scholar
Bowles, R. I. 1995 Upstream influence and the form of standing hydraulic jumps in liquid-layer flows on favourable slopes. J. Fluid. Mech. 284, 63.CrossRefGoogle Scholar
Brattkus, K. & Davis, S. H. 1988 Flow-induced morphological instabilities: stagnation-point flows. J. Cryst. Growth 89, 423.CrossRefGoogle Scholar
Briggs, R. J. 1964 Electron-Stream Interaction with Plasmas. MIT Press.CrossRefGoogle Scholar
Cassel, K. W., Ruban, A. I. & Walker, J. D. A. 1995 An instability in supersonic boundary-layer flow over a compression ramp. J. Fluid Mech. 300, 265.CrossRefGoogle Scholar
Chung, T. J. 2002 Computational Fluid Dynamics. Cambridge University Press.CrossRefGoogle Scholar
Davis, S. H. 1990 Hydrodynamic interactions in directional solidification. J. Fluid. Mech. 212, 241.CrossRefGoogle Scholar
Forth, S. A. & Wheeler, A. A. 1989 Hydrodynamic and morphological stability of unidirectional solidification of a freezing binary alloy: a simple model. J. Fluid Mech. 202, 339.CrossRefGoogle Scholar
Gajjar, J. S. B. 1987 Fully developed free surface flows – liquid layer flow over a convex corner. Computers Fluids 15, 337–360.CrossRefGoogle Scholar
Gilpin, R. R. 1979 The morphology of ice structure in a pipe at or near transient Reynolds numbers. Heat Transfer – San Diego, AIChE Symposium Series 189, vol. 75, p. 89.Google Scholar
Gilpin, R. R. 1981 Ice formation in a pipe containing flows in the transition and turbulent regimes. J. Heat Transfer 103, 363.CrossRefGoogle Scholar
Glicksman, M. E., Coriell, S. R. & McFadden, G. B. 1986 Interaction of flows with the crystal-melt interface. Annu. Rev. Fluid Mech. 18, 307.CrossRefGoogle Scholar
Hooper, A. P. & Boyd, W. G. C. 1983 Shear-flow instability at the interface between two shearing fluids. J. Fluid Mech. 128, 507.CrossRefGoogle Scholar
Isaacson, E. & Keller, H. B. 1966 Analysis of Numerical Methods. Dover.Google Scholar
Kikuchi, Y., Shigemasa, Y., Oe, A. & Ogata, T. 1986 Steady-state freezing of liquids in laminar flow between two parallel plates. J. Nucl. Sci. Technol. 23, 979.CrossRefGoogle Scholar
Lee, J. S. 1993 Freezing problems in pipe flows. PhD thesis Iowa State University.Google Scholar
Rothmayer, A. P. & Tsao, J. C. 1998 A mechanism for ice roughness formation on an airfoil leading edge, contributing to glaze ice accretion. AIAA Paper 98–0485.Google Scholar
Rothmayer, A. P. & Tsao, J. C. 2000 Triple-deck simulation of surface glaze ice accretion. AIAA Paper 2000–0234.Google Scholar
Seki, N., Fukusako, S. & Younan, G. W. 1984 Ice-formation phenomena for water flow between two cooled parallel plates. J. Heat Transfer 106, 498.CrossRefGoogle Scholar
Shapiro, E. 2004 Instabilities in multiphase and icing flows. PhD thesis, University of London.Google Scholar
Shapiro, E. & Timoshin, S. 2006 Linear stability of ice growth under a gravity-driven water film. Phys. Fluids 18, 074106.CrossRefGoogle Scholar
Streitz, J. T. & Ettema, R. 2002 Observations from an aufeis windtunnel. Cold Regions Sci. Technol. 34,(2) 85.CrossRefGoogle Scholar
Sychev, V. V., Ruban, A. I., Sychev, V. V. & Korolev, G. L. 2000 Asymptotic Theory of Separated Flows. Cambridge University Press.Google Scholar
Timoshin, S. N. 1997 Instabilities in a high-reynolds-number boundary layer on a film-coated surface. J. Fluid Mech. 353, 163.CrossRefGoogle Scholar
Timoshin, S. N. & Hooper, A. P. 2000 Mode coalescence in two-fluid boundary-layer stability problem. Phys. Fluids 12, 1969.CrossRefGoogle Scholar
Vargas, M. & Reshotko, E. 1998 Physical mechanisms of glaze ice scallop formations on swept wings. AIAA Paper 98–0491.CrossRefGoogle Scholar
Yao, L. S. & Prusa, J. 1989 Melting and freezing. Adv. Heat Transfer 19, 1.CrossRefGoogle Scholar