fluid_mechanics

10.3 Energy Considerations 541
V10.5 Bicycle
through a puddle
of a river 1such as a submerged log2 may cause the surface of the river to dip below the level it
would have had if the log were not there, or it may cause the surface level to rise above its
undisturbed level. Which situation will happen depends on the value of Fr. Similarly, for supercritical
flows it is possible to produce steplike discontinuities in the fluid depth 1called a hydraulic jump;
see Section 10.6.12. For subcritical flows, however, changes in depth must be smooth and
continuous. Certain open-channel flows, such as the broad-crested weir 1Section 10.6.32, depend
on the existence of critical flow conditions for their operation.
As strange as it may seem, there exist many similarities between the open-channel flow of
a liquid and the compressible flow of a gas. The governing dimensionless parameter in each case
is the fluid velocity, V, divided by a wave speed, the surface wave speed for open-channel flow or
sound wave speed for compressible flow. Many of the differences between subcritical 1Fr 6 12
and supercritical 1Fr 7 12 open-channel flows have analogs in subsonic 1Ma 6 12 and supersonic
1Ma 7 12 compressible gas flow, where Ma is the Mach number. Some of these similarities are
discussed in this chapter and in Chapter 11.
10.3 Energy Considerations
The slope of the
bottom of most
open channels is
very small; the
bottom is nearly
horizontal.
A typical segment of an open-channel flow is shown in Fig. 10.6. The slope of the channel bottom
1or bottom slope2, S 0 1z 1 z 2 2/, is assumed constant over the segment shown. The fluid depths
and velocities are y 1 , y 2 , V 1 , and V 2 as indicated. Note that the fluid depth is measured in the vertical
direction and the distance x is horizontal. For most open-channel flows the value of S 0 is very small
1the bottom is nearly horizontal2. For example, the Mississippi River drops a distance of 1470 ft in
its 2350-mi length to give an average value of S 0 0.000118. In such circumstances the values of
x and y are often taken as the distance along the channel bottom and the depth normal to the bottom,
with negligibly small differences introduced by the two coordinate schemes.
With the assumption of a uniform velocity profile across any section of the channel, the onedimensional
energy equation for this flow 1Eq. 5.842 becomes
where is the head loss due to viscous effects between sections 112 and 122 and z 1 z 2 S 0 /.
p 1
g V 2 1
2g z 1 p 2
g V 2 2
2g z 2 h L
(10.5)
h L
Since the pressure is essentially hydrostatic at any cross section, we find that p 1g y 1 and
p 2g y 2 so that Eq. 10.5 becomes
y 1 V 2 1
2g S 0/ y 2 V 2 2
2g h L
(10.6)
One of the difficulties of analyzing open-channel flow, similar to that discussed in Chapter 8 for pipe
flow, is associated with the determination of the head loss in terms of other physical parameters.
Without getting into such details at present, we write the head loss in terms of the slope of the
energy line, S f h L/ 1often termed the friction slope2, as indicated in Fig. 10.6. Recall from
1 S f
y 1
V 1
2
___
2g
V1
F I G U R E 10.6
Energy line
S (1)
0
1
z 1
(2)
h L
2
___ V 2
2g
y 2
z 2
V 2
Typical open-channel geometry.
Slope = S f
Slope = S 0
Horizontal datum
x