fluid_mechanics

568 Chapter 10 ■ Open-Channel Flow
SOLUTION
From Eq. 10.35 we have
Q bq baC d 12gy 1
F I G U R E E10.9
250
Free outflow
Submerged outflow
or
20 ft 11.0 ft2 C d 22132.2 fts 2 216.0 ft2
Q 393C d cfs
The value of C d is obtained from Fig. 10.25 along the vertical line
y 1a 6 ft1 ft 6. For y 3 6 ft 1i.e., y 3a 6 y 1a2 we
obtain C d 0, indicating that there is no flow when there is no
head difference across the gate. The value of C d increases as y 3a
decreases, reaching a maximum of C d 0.56 when y 3a 3.2.
Thus, with y 3 3.2a 3.2 ft
Q 393 10.562 cfs 220 cfs
The flowrate for 3.2 ft y 3 6 ft is obtained from Eq. 1 and
the C d values of Fig. 10.24 with the results as indicated in Fig.
E10.9.
COMMENT For y 3 6 3.2 ft the flowrate is independent of
y 3 , and the outflow is a free 1not submerged2 outflow. For such
cases the inertia of the water flowing under the gate is sufficient
to produce free outflow even with y 3 7 a.
(1)
Q, cfs
200
150
100
50
Q
y 1 = 6 ft
a = 1 ft
y 3
0
0 1 2 3 4 5 6
y 3 , ft
10.7 Chapter Summary and Study Guide
open-channel flow
Froude number
critical flow
subcritical flow
supercritical flow
wave speed
specific energy
specific energy diagram
uniform depth flow
wetted perimeter
hydraulic radius
Chezy equation
Manning equation
Manning coefficient
rapidly varied flow
hydraulic jump
sharp-crested weir
weir head
broad-crested weir
underflow gate
This chapter discussed various aspects of flows in an open channel. A typical open-channel flow
is driven by the component of gravity in the direction of flow. The character of such flows can
be a strong function of the Froude number, which is ratio of the fluid speed to the free-surface
wave speed. The specific energy diagram is used to provide insight into the flow processes
involved in open-channel flow.
Uniform depth channel flow is achieved by a balance between the potential energy lost by
the fluid as it coasts downhill and the energy dissipated by viscous effects. Alternately, it represents
a balance between weight and friction forces. The relationship among the flowrate, the slope
of the channel, the geometry of the channel, and the roughness of the channel surfaces is given
by the Manning equation. Values of the Manning coefficient used in the Manning equation are
dependent on the surface material roughness.
The hydraulic jump is an example of nonuniform depth open-channel flow. If the Froude
number of a flow is greater than one, the flow is supercritical, and a hydraulic jump may occur.
The momentum and mass equations are used to obtain the relationship between the upstream
Froude number and the depth ratio across the jump. The energy dissipated in the jump and the
head loss can then be determined by use of the energy equation.
The use of weirs to measure the flowrate in an open channel is discussed. The relationships
between the flowrate and the weir head are given for both sharp-crested and broad-crested weirs.
The following checklist provides a study guide for this chapter. When your study of the
entire chapter and end-of-chapter exercises has been completed you should be able to
write out meanings of the terms listed here in the margin and understand each of the related
concepts. These terms are particularly important and are set in italic, bold, and color type
in the text.
determine the Froude number for a given flow and explain the concepts of subcritical, critical,
and supercritical flows.
plot and interpret the specific energy diagram for a given flow.