Debris Flows and Landslides - River Dynamics and Integrated River Management - page 229

The matrix of the viscous debris flow consists of water, silt, and clay. The concentration of silt and
clay is high, and, therefore, the matrix exhibits non-Newtonian features and yield stress. The development
of the intermittent wave pattern is related to the yield strength of the matrix and Froude number (Details
are given later).

Fig. 4.44 Viscous debris flow develops into a series of roll waves

4.3.3.3 Roll Waves
Wang and his collaborators studied experimentally and theoretically the mechanism of the development
of roll waves (Wang et al., 1990; Wang, 2002). The experiments were done in a flume of 26-m long and
50-cm wide, using clay suspensions as the flowing medium, which is a viscous-plastic fluid and behaves
similarly to a viscous debris flow. The roll waves generally developed as follows: a slight fluctuation in
velocity occurred as clay suspension flowed in the flume, then some ripples appeared on the surface. The
ripples grew into waves as they propagated downstream, and more ripples formed at the same time.
Sometimes the waves grew so large that their maximum discharges were more than double the incoming
discharge, and the residual mud stopped moving after the waves passed. The roll waves stopped growing
when they reached a certain amplitude. The growth process is shown in Fig. 4.45 and a fully developed
roll wave has the form shown in Fig. 4.46, in which the streamlines of the flow are seen by a viewer
moving with the wave.

Fig. 4.45 Wave height, ' h , increases with the distance, L , down the slope. Ripples may grow into waves if the
yield stress,

W , is large. The waves stopped growing when they reached a certain amplitude (after Wang, 2002)

The wave always propagates more rapidly than the flow between waves. A portion of the mud moves
upward like a fountain under the extrusion of the wave as it was caught by the wave. Then it divides into
two parts, one part flowing forward at a velocity 2 u (where u is the speed of the wave) and forming a
rolling front and the other part flowing at a velocity less than u that gradually lags behind the wave.
Wang (2002) studied the mechanism of the development of the roll waves and derived the following
equation by applying the Bingham model and the de Saint Venant equations:

ª º

d 1 W K

() ' u

B

' 1 Fr ) u (4.9)

« »

d t 2 U h d

() gh

« »

¬ ¼

m