fluid_mechanics

10.6 Rapidly Varied Flow 555
component of weight and the shear forces in the direction of flow produces a change in the fluid
momentum that requires a change in velocity and, from continuity considerations, a change in
depth. Whether the depth increases or decreases depends on various parameters of the flow, with
a variety of surface profile configurations [flow depth as a function of distance, y y 1x2] possible
(Refs. 5, 9).
10.6 Rapidly Varied Flow
In many cases the
flow depth may
change significantly
in a short distance.
V10.8 Erosion in a
channel
In many open channels, flow depth changes occur over a relatively short distance so that dydx 1.
Such rapidly varied flow conditions are often quite complex and difficult to analyze in a precise
fashion. Fortunately, many useful approximate results can be obtained by using a simple onedimensional
model along with appropriate experimentally determined coefficients when necessary.
In this section we discuss several of these flows.
Some rapidly varied flows occur in constant area channels for reasons that are not immediately
obvious. The hydraulic jump is one such case. As is indicated in Fig. 10.12, the flow may change
from a relatively shallow, high-speed condition into a relatively deep, low-speed condition within
a horizontal distance of just a few channel depths. Other rapidly varied flows may be due to a
sudden change in the channel geometry such as the flow in an expansion or contraction section of
a channel as is indicated in Fig. 10.13.
In such situations the flow field is often two- or three-dimensional in character. There may be
regions of flow separation, flow reversal, or unsteady oscillations of the free surface. For the purpose
of some analyses, these complexities can be neglected and a simplified analysis can be undertaken.
In other cases, however, it is the complex details of the flow that are the most important property of
the flow; any analysis must include their effects. The scouring of a river bottom in the neighborhood
of a bridge pier, as is indicated in Fig. 10.14, is such an example. A one- or two-dimensional model
of this flow would not be sufficient to describe the complex structure of the flow that is responsible
for the erosion near the foot of the bridge pier.
Many open-channel flow-measuring devices are based on principles associated with rapidly
varied flows. Among these devices are broad-crested weirs, sharp-crested weirs, critical flow flumes,
and sluice gates. The operation of such devices is discussed in the following sections.
F I G U R E 10.12
Hydraulic jump.
V10.9 Bridge pier
scouring
Flow
F I G U R E 10.13 Rapidly varied
flow may occur in a channel transition section.