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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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