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Long waves on inclined films at high Reynolds number

Published online by Cambridge University Press:  26 April 2006

Th. Prokopiou ,
M. Cheng  and
H.-C. Chang
Th. Prokopiou
Affiliation:
Department of Chemical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA
M. Cheng
Affiliation:
Department of Chemical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA
H.-C. Chang
Affiliation:
Department of Chemical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA
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Abstract

At large Reynolds number (Re > 10), waves on inclined films grow rapidly downstream in both amplitude and wavelength to the extent that linear stability theory cannot adequately describe their velocity–wavenumber relationship. The wavelength increases indefinitely until solitary waves are formed very far downstream. In a recent experiment of Brauner & Maron (1982), this evolution to long waves is observed to occur by successive wavelength doubling. In this analysis, we develop a second-order integral boundary-layer approximation for long waves at intermediate Re of O(ε−1), where ε is the dimensionless wavenumber scaled with respect to the film thickness. (A second-order theory is needed because it introduces important dissipation terms which allow periodic and solitary waveforms to exist when surface tension is negligible.) After showing that this model can adequately describe infinitesimal waves at inception, we verify the existence of solitary waves and long-wavelength periodic waves near the critical Reynolds number with a weakly nonlinear analysis. These finite-amplitude waves are then numerically continued into the more important high-Re and strongly nonlinear regions. It is shown that the solitary wave speed approaches 1.67 times the Nusselt velocity, and the thickness of the substrate film approaches 0.47 times the Nusselt film thickness at large Re. These results are favourably compared to experimental data of Chu & Dukler (1974, 1975). We also confirm the period-doubling scenario of Brauner & Maron by showing that short finite-amplitude monochromatic waves are unstable to subharmonic instability.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 222 , January 1991 , pp. 665 - 691
Copyright
© 1991 Cambridge University Press

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References

Alekseenko, S. V., Nakoryakov, V. Ye. & Pokusaev, B. G., 1985 Wave formation on a vertical falling liquid film. AIChE J. 31, 1446–1460.Google Scholar
Aluko, M. & Chang, H.-C. 1984 PEFLOQ: An algorithm for the bifurcation analysis of periodic solutions. Comput. Chem. Engng 8, 355–365.Google Scholar
Armbruster, D., Guckenheimer, J. & Holmes, P., 1988 Heteroclinic cycles and modulated travelling waves in systems with 0(2) symmetry. Physica 29D, 257–282.Google Scholar
Batchelor, G. K.: 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Benjamin, T. B.: 1957 Wave formation in laminar flow down an inclined plane. J. Fluid Mech. 2, 554–574.Google Scholar
Benjamin, T. B.: 1984 Impulse, force and variational principles. IMA J. Appl. Maths 32, 3–68.Google Scholar
Benney, D. J.: 1966 Long waves on liquid films. J. Maths Phys. 45, 150–155.Google Scholar
Bertschy, J. R., Chin, R. W. & Abernathy, F. H., 1983 High-strain-rate free-surface Boundary-layer flows. J. Fluid Mech. 126, 443–461.Google Scholar
Brauner, N.: 1987 Roll wave celerity and average film thickness in turbulent wavy film flow. Chem. Engng Sci. 42, 265–273.Google Scholar
Brauner, N. & Maron, D. M., 1982 Characteristics of inclined thin films, waviness and the associated mass transfer. Intl J. Heat Mass Transfer 25, 99–110.Google Scholar
Brauner, N. & Maron, D. M., 1983 Modelling of wavy flow in inclined thin films. Chem. Engng Sci. 38, 775–788.Google Scholar
Brock, R. R.: 1970 Periodic permanent roll waves. J. Hydraul. Div. ASCE 12, (HY), 2565–2580.Google Scholar
Carr, J.: 1981 Applications of Center Manifold Theory. Springer.
Chang, H.-C.: 1987 Evolution of nonlinear waves on vertically falling films — a normal form analysis. Chem. Engng Sci. 42, 515–533.Google Scholar
Chang, H.-C.: 1989 Onset of nonlinear waves on falling films. Phys. Fluids A 1, 1314.Google Scholar
Choi, I.: 1977 Contributions à 1′étude des mechanisms physiques de la generation des ôndes de capillarite-gravité à une interface air—eau. Thesis, Université D'Aix Marseille.
Chu, K. J. & Dukler, A. E., 1974 Statistical characteristics of thin, wavy films: II. Studies of the substrate and its wave structure. AICHE J. 20, 695–706.Google Scholar
Chu, K. J. & Dukler, A. E., 1975 Statistical characteristics of thin wavy films: III. Structure of the large waves and their resistance to gas flows. AIChE J. 21, 583–595.Google Scholar
Doedel, E. J. & Kernevez, J. P., 1986 Software for continuation problems in ordinary differential equations with applications. Caltech. Applied Maths Rep.Google Scholar
Dressler, R. F.: 1949 Mathematical solution of the problem of roll-waves in inclined open channels. Commun. Pure Appl. Maths. 2, 149–194.Google Scholar
Gjevik, B.: 1970 Occurrence of finite-amplitude surface waves on falling liquid films. Phys. Fluids 13, 1919–1925.Google Scholar
Guckenheimer, J. & Holmes, P., 1983 Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer.
Hanratty, T. J.: 1983 In Waves on Fluid Interfaces (ed. R. E. Meyer), pp. 221–259. Academic.
Hwang, S.-H. & Chang, H-C. 1987 Turbulent and inertial roll waves in inclined film flow. Phys. Fluids 30, 1259–1268.Google Scholar
Janssen, P. A. E.: 1986 The period-doubling of gravity-capillary waves. J. Fluid Mech. 172, 531–546.Google Scholar
Von Kármán, T. 1921 Z. angew. Math. Meth. I, 233.
Lee, J.: 1969 Kapitza's method of film flow description. Chem. Engng Sci. 24, 1309–1320.Google Scholar
Lin, S. P.: 1969 Finite amplitude stability of a parallel flow with a free surface. J. Fluid Mech. 36, 113–126.Google Scholar
Lin, S. P.: 1974 Finite-amplitude side-band instability of a viscous film. J. Fluid Mech. 63, 417–429.Google Scholar
Lin, S. P.: 1983 Waves on Fluid Interfaces (ed. R. E. Meyer), pp. 261–289. Academic.
Nakaya, C.: 1975 Long waves on a thin fluid layer flowing down an inclined plane. Phys. Fluids 18, 1407–1420.Google Scholar
Nakaya, C.: 1989 Waves on a viscous fluid film down a vertical wall. Phys. Fluids A 1, 1143–1154.Google Scholar
Needham, D. J. & Merkin, J. H., 1984 On roll waves down an inclined channel. Proc. R. Soc. Lond. A 394, 259–278.Google Scholar
Needham, D. J. & Merkin, J. H., 1986 On infinite period bifurcations with an application to roll waves. Acta Mech. 60, 1–16.Google Scholar
Pttmir, A., Manneville, P. & Pomeau, Y., 1983 On solitary waves running down an inclined plane. J. Fluid Mech. 135, 27–50.Google Scholar
Shkadov, V. Y.: 1967 Wave conditions in the flow of thin layer of a viscous liquid under the action of gravity. Izv. Akad. Nauk. SSSR Mekh. Zhidk. i Gaza 1, 43–50.Google Scholar
Shkadov, V. Y.: 1968 Theory of wave flows of a thin layer of viscous liquid. Izv. Akad. Nauk. SSSr Mekh. Zhid. i Gaza 2, 20.Google Scholar
Stoker, J. J.: 1957 Water Waves. Interscience.
Yih, C-S.: 1963 Stability of liquid flow down an inclined plane. Phys. Fluids 6, 321–330.Google Scholar
Zufibia, J. A.: 1987 Symmetry breaking in periodic and solitary gravity-capillary waves on water of finite depth. J. Fluid Mech. 184, 183–206.Google Scholar