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On the evolution of roll waves

  • C. Kranenburg (a1)


An amplitude evolution equation is derived for roll waves that occur in a uniform open-channel flow down an incline. A periodic series of roll waves, which is a manifestation of the instability of high-velocity turbulent flow, is shown to be itself unstable to subharmonic disturbances. These disturbances develop into longer and higher roll waves. In this way roll waves tend to increase in size as time goes on, in agreement with experimental evidence.



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Alavian V. 1986 Behavior of density currents on an incline. J. Hydraul. Engng ASCE 112, 2742.
Benjamin, T. B. & Lighthill M. J. 1954 On cnoidal waves and bores Proc. R. Soc. Lond. A 224, 448460.
Cornish V. 1934 Ocean Waves and Kindred Geophysical Phenomena. Cambridge University Press.
Craik A. D. D. 1985 Wave Interactions and Fluid Flows. Cambridge University Press.
Crighton D. G. 1979 Model equations of nonlinear acoustics. Ann. Rev. Fluid Mech. 11, 1133.
Dressler R. F. 1949 Mathematical solution of the problem of roll-waves in inclined open channels. Commun. Pure Appl. Maths 2, 149194.
Jeffreys H. 1925 The flow of water in an inclined channel of rectangular section. Phil. Mag. 49, 793807.
Kaup D. J. 1980 The Estabrook–Wahlquist method with examples of application Physica D 1, 391411.
Kranenburg C. 1990 On the stability of gradually varying flow in wide open channels. J. Hydraul. Res. 28, 621628.
Liggett J. A. 1975 In Unsteady Flow in Open Channels (ed. K. Mahmood & V. Yevjevich), vol. 1. Water Resources Publications.
Mayer P. G. 1961 Roll waves and slug flows in inclined open channels. Trans. ASCE, 505535.
Merkin, J. H. & Needham D. J. 1986 An infinite period bifurcation arising in roll waves down an open inclined channel Proc. R. Soc. Lond. A 405, 103116.
Needham, D. J. & Merkin J. H. 1984 On roll waves down an open inclined channel Proc. R. Soc. Lond. A 394, 259278.
Nimmo, J. J. C. & Crighton D. G. 1982 Bäcklund transformations for nonlinear parabolic equations: the general results Proc. R. Soc. Lond. A 384, 381401.
Novik O. B. 1971 Model description of roll-waves. J. Appl. Math. Mech. 35, 938951.
Roache P. J. 1976 Computational Fluid Dynamics. Hermosa.
Whitham G. B. 1974 Linear and Nonlinear Waves. Wiley.
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On the evolution of roll waves

  • C. Kranenburg (a1)


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