Waves on the erodible bed of an open channel

In the first part of this work the stability of the erodible bed of a stream with a free surface is studied within the framework of classical hydraulics, in which the velocity variation with depth is reduced to a single mean velocity and the bed friction is related in a general way to the local depth and mean velocity. Only two-dimensional motions can be studied in this way. In considering bed friction and a difference in phase between deposition and the mean velocity gradient along the channel, this work combines aspects of earlier studies of Exner and Kennedy. In the absence of a phase difference between erosion and mean velocity, the analysis proceeds without linearization. When the development from an equili- brium flow down a uniform slope is considered, the bed waves formed under subcritical flows are found to move downstream, while those under super-critical flows upstream; in both régimes bed waves are damped. In each case the side of a wave facing in the direction of motion is the steeper. When a phase shift is introduced as well, the analysis is carried forward only after linearization. The primary effect of bed slope and friction is a shift in the ranges of phase angle for which growth can take place and a corresponding alteration in the wavelengths for maximum growth. Friction also reduces bed-wave celerity. Consideration is given to the physical processes represented by the artifice of a phase difference between erosion and mean velocity gradient. The second section of this investigation concerns two-dimensional potential flow over a wavy stream bed. This problem is considered from a point of view different from that adopted previously by Kennedy; a modified criterion is proposed for the maximum Froude number at which bed waves will form. It is in better agreement with measured data. In the third part of this paper, the potential analysis is extended to include class of three-dimensional motions. The conditions are found for the formation of dunes (bed waves 180° out of phase with the surface waves above) and anti-dunes (the two in phase). The criterion separating dunes and antidunes for two dimensions is found to give a lower limit on the Froude number for antidunes of the more general three-dimensional class. In the antidune regime the streamwise perturbation to the velocity can change sign between surface and stream bed; the limit for this is determined. The erosion equation relating changes in bed level to local stream speed is generalized to include sediment convection in two dimensions. With this equation, three distinct regimes of bed-wave motion are found: at low Froude numbers, dunes moving downstream; at higher Froude numbers, antidunes moving upstream; and, finally, antidunes moving downstream. The criterion separating the last two r6gimes is just that mentioned above as the limit of flows whose velocity perturbation has the same sign from surface to stream bed. It is argued that this fundamental change in the flow marks also the end of the region of growth of small bed waves. It is found that three-dimensional dunes and upstream-moving antidunes can exist beyond the two-dimensional limit, but that the latter applies for all bed waves such that the ratio of stream depth to wavelength (d/λ) is small. This explains why the modified criterion for two dimensions provides for small d/λ an envelope for data obtained from observations of a wide variety of bed forms, but fails to do so for larger d/λ. The celerity of bed waves beneath three-dimensional flows is discussed. It is suggested that a class of waves occurring in natural streams will move more slowly than would two-dimensional waves in similar conditions. The work concludes with a comparison of several methods of modelling erosive flows.