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Wave propagation and boundary instability in erodible-bed channels

Published online by Cambridge University Press:  28 March 2006

Mario H. Gradowczyk
Mario H. Gradowczyk
Affiliation:
Department of Civil Engineering, Massachusetts Institute of Technology Formerly: Instituto de Cálculo & Departmento de Meteorología, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires.
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Abstract

Wave propagation in one-dimensional erodible-bed channels is discussed by using the shallow-water approximation for the fluid and a continuity equation for the bed. In addition to gravity waves, a third wave, which gives the velocity of propagation of a bed disturbance, is found. An appropriate dimensional analysis yields the quasi-steady approximation for the complete shallow-water equations.

The well-known linear stability analysis of free-surface flows is extended to include the erodibility of the bed. The critical Froude number Fc above which the free-surface of the fluid may become unstable is obtained. It is shown that erodibility increases the stability of the free surface, in qualitative agreement with previous experiments if qb > qs, qb and qs being respectively the contact-bed discharge and suspended-material discharge. The stability theory is also used to discuss coupled beds and surface waves. From it, five different configurations have been obtained: a sinusoidal wave pattern moving downstream, a transition zone and antidunes moving upstream, moving downstream and stationary. These bed forms are in agreement with experimental results; hence shallow-water theory seems to give a reasonable explanation of the boundary instability.

It is shown that the quasi-steady approximation and Kennedy's (1963) stability analysis will be in agreement if (kh)2 [Lt ] 1, where k is the wave number, and h is the depth of the water. When the phase shift δ is introduced in the quasi-steady approximation, the five bed patterns derived from the full equations are found again.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 33 , Issue 1 , 12 July 1968 , pp. 93 - 112
Copyright
© 1968 Cambridge University Press

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References

Anderson, A. G. 1953. The characteristics of sediment waves formed by flow in open channels. Proc. Third Mid-Western Conf. on Fluid Mech. University of Minnesota, pp. 379–95.Google Scholar
Chabert, J. & Chauvin, J. L. 1963 Formation des dunes et des rides dans les modèles fluviaux Bull. Centre Rech. Essais de Chatou, 4, 31–51.Google Scholar
Coenish, V. 1934 Ocean Waves and Kindred Geophysical Phenomena. Cambridge University Press.
Dressler, R. F. 1949 Mathematical solution of the problem of roll-waves in inclined open channels Communs Pure Appl. Math. 2, 149–94.Google Scholar
Dressler, R. F. & Pohle, F. V. 1953 Resistance effects on hydraulic instability Communs Pure Appl. Math. 6, 93–6.Google Scholar
Exner, F. 1925 Ueber die Wechselwirkung zwischen Wasser und Geschiebe in Fluessen. Sitzungeberichte der Oesterreichischen Akad. der Wiss. Abt. II 134, pp. 165–203.Google Scholar
Forchheimer, P. 1930 Hydraulik. Leipzig: Teubner.
Frank, E. 1946 On the zeros of polynomials with complex coefficients Bull. Am. Math. Soc. 52, 144–57.Google Scholar
Friedrichs, K. D. 1948 On the derivation of the shallow-water theory Communs Pure Appl. Math. 1, 81–5.Google Scholar
Gradowczyk, M. H. & Folguera, H. C. 1965 Analysis of scour in open channels by means of mathematical models La Houille Blanche, 20, 761–9.Google Scholar
Gradowczyk, M. H., Maggiolo, O. J. & Raggi, R. 1967 A bed wave moving downstream from obstacles Proc. XII Congress of the I.A.H.R. Fort Collins 3, 314–21.Google Scholar
Guy, H. P., Simons, D. B. & Richardson, E. V. 1966 Summary of alluvial channel data from flume experiments, 1956–1961, U.S. Geol. Survey, Prof. Paper 462–1, 96 pp.Google Scholar
Jeffreys, H. 1925 The flow of water in an inclined channel of rectangular section. Phil. Mag. 649, 793–807.Google Scholar
Kennedy, J. F. 1963 The mechanics of dunes and antidunes in erodible-bed channels J. Fluid Mech. 16, 521–44.Google Scholar
Knoroz, V. Z. 1959 The effect of the channel macro-roughness on its hydraulic resistance. Izvestiia Vsesouzhogo Neuchno-Issledovatel 'skago Instituta Gidrotekhniki (Trudy Laboratori i Plantin; Gidrouzlov. 62, 75–96) Translated by U.S. Geol Survey, Denver, Colorado 1961.
Lighthill, M. J. & Whitman, G. B. 1955 On kinematic waves. I. Flood movement in long rivers. Proc. Roy. Soc A 229, 281–316.Google Scholar
Liu, K. H. 1957 Mechanics of sediment ripple formation, Proc Am. Soc. Civ. Engrs. 83, no. HY 2.Google Scholar
Maggiolo, O. J. & Borghi, J. 1965 Sobre la evolucion en el tiempo del processo de socavación Revista de Ingenieria, 59, 149–60.Google Scholar
MEYER-PETER, E. & Muller, R. 1948 Formula for bed load transport Congress I.A.H.R., Stockholm, 3, 39–65.Google Scholar
Nordin, C. F. 1963 A preliminary study of sediment transport parameters. U.S. Geol. Survey, Prof. Paper 462-C 21 pp.Google Scholar
Reynolds, A. J. 1965 Waves on the erodible bed of an open channel J. Fluid Mech. 22, 113–33.Google Scholar
Rouse, H. 1938 Fluid Mechanics for Hydraulic Engineers. New York: McGraw-Hill.
Shields, A. 1936 Anwendung der Aenlichkeitsmechanik und der Turbulenzforschung auf die Geschiebebewegung, Mitt. der Preuss Versuchsanst. f. Wasserbau und Schiffbau, Heft 26, Berlin.Google Scholar
Simons, D. B. & Richardson, E. V. 1961 Forms of bed roughness in alluvial channels J. Am. Soc. Civil Engrs, Hyd. Div. 87, 87–105.Google Scholar
Thomas, H. A. 1940 Propagation of waves in steep prismatic conduits. Proc. Hydraulic Conference, University of Iowa, Studies in Engineering, Bull. 20.Google Scholar
U. S. WATERWAYS Exp. STATION, 1935. Studies of river bed materials and their movement, with special reference to the lower Mississippi River, Paper 17, January 1935.
Vanoni, V. A. & Brooks, X. H. 1957 Laboratory studies of the roughness and suspended load of alluvial streams. California Institute of Technology, Sedimentation Laboratory M.R.D., Sediment Series, no. 11, 121 pp.Google Scholar