fluid_mechanics

566 Chapter 10 ■ Open-Channel Flow
(b)
or
Similarly, for the triangular weir, Eq. 10.32 gives
where H and Q are in meters and m 3 s and C wt is obtained from
Fig. 10.21. For example, with H 0.20 m, we find C wt 0.60,
or Q 2.36 10.60210.202 52 0.0253 m 3 s. The triangular weir
results are also plotted in Fig. E10.8.
(c) For the broad-crested weir, Eqs. 10.28 and 10.29 give
Thus, with P w 1 m
or
Q C wt 8 15 tan au 2 b 12g H5 2
C wt 8 15 tan145°2 2219.81 m s 2 2 H 5 2
Q 2.36C wt H 5 2
Q C wb b1g a 2 3 b 32
H 3 2
1.125 a 1 H 12
P w
b b1g a 2 32
2 HP w 3 b H 3 2
Q 1.125 a 1 H
2 H b 12
12 m2 29.81 ms 2 a 2 3 b 32
H 3 2
Q 3.84 a 1 H
2 H b 1/2
H 3/2
(3)
(2)
where, again, H and Q are in meters and m 3 s. This result is also
plotted in Fig. E10.8.
COMMENTS Although it appears as though any of the three
weirs would work well for the upper portion of the flowrate range,
neither the rectangular nor the broad-crested weir would be very
accurate for small flowrates near Q Q min because of the small
head, H, at these conditions. The triangular weir, however, would
allow reasonably large values of H at the lowest flowrates. The
corresponding heads with Q Q min 0.02 m 3 s for rectangular,
triangular, and broad-crested weirs are 0.0312, 0.182, and 0.0375 m,
respectively.
In addition, as discussed in this section, for proper operation
the broad-crested weir geometry is restricted to 0.08 HL w
0.50, where L w is the weir block length. From Eq. 3 with Q max
0.60 m 3 s, we obtain H max 0.349. Thus, we must have L w
H max 0.5 0.698 m to maintain proper critical flow conditions at
the largest flowrate in the channel. However, with Q Q min
0.02 m 3 s, we obtain H min 0.0375 m. Thus, we must have L w
H min 0.08 0.469 m to ensure that frictional effects are not important.
Clearly, these two constraints on the geometry of the weir
block, L w , are incompatible.
A broad-crested weir will not function properly under the
wide range of flowrates considered in this example. The sharpcrested
triangular weir would be the best of the three types considered,
provided the channel can handle the H max 0.719-m
head.
(Photograph courtesy
of Pend Oreille Public
Utility District.)
10.6.4 Underflow Gates
A variety of underflow gate structures is available for flowrate control at the crest of an overflow
spillway (as shown by the figure in the margin), or at the entrance of an irrigation canal or river
from a lake. Three types are illustrated in Fig. 10.24. Each has certain advantages and
disadvantages in terms of costs of construction, ease of use, and the like, although the basic
fluid mechanics involved are the same in all instances.
The flow under a gate is said to be free outflow when the fluid issues as a jet of supercritical
flow with a free surface open to the atmosphere as shown in Fig. 10.24. In such cases it is customary
to write this flowrate as the product of the distance, a, between the channel bottom and the bottom
of the gate times the convenient reference velocity 12gy 1 2 12 . That is,
q C d a12gy 1
(10.35)
where q is the flowrate per unit width. The discharge coefficient, C d , is a function of the contraction
coefficient, C c y 2a, and the depth ratio y 1a. Typical values of the discharge coefficient for free
V10.15 Spillway
gate
y 1
y 2
a
(a) (b) (c)
F I G U R E 10.24 Three variations of underflow gates: (a) vertical gate, (b) radial gate,
(c) drum gate.