Figure 2.18. Roll waves or slug flow.
There is no simple criterion for determining the size of roll waves, since their size depends upon
the magnitude of the discharge, the type of flow (laminar or turbulent), the roughness and slope
of the channel, the length of the channel, and the nature and frequency of the initial disturbances
which cause the waves to form. However, a necessary condition required to generate instability
of the free surface and induce the formation of roll waves in turbulent flows when Chezy's
equation is applicable is:
which can be expressed in alternate form for a wide channel as
for turbulent flow with a rough boundary in which yo is the normal depth, S is the slope of the
channel, and C is the Chezy discharge coefficient.
When the flow in a wide channel is turbulent with a smooth boundary, roll waves can form if
Fr ≥ 1.5
S ≥ 2.25
and when the flow in a wide channel is laminar, roll waves can form if
Fr ≥ 0.5
These conditions indicate that, for turbulent flow in a wide channel with a rough boundary, roll
waves can occur when the flow velocity is greater than twice the celerity of a wave that is, the
Froude number is greater than 2, or when the slope is four times as great as the slope required
for critical depth. They can also form for turbulent flow in a wide channel with a smooth boundary
if the velocity of flow is greater than 1.5 times the celerity of a wave, or the slope is 2.25 times the
slope required for critical depth. By way of contrast, roll waves can form in laminar flow in a wide
channel if the velocity is half the celerity of a gravity wave; in other words, the flow may never
pass through critical flow (Fr = 1.0).