fluid_mechanics

10.4 Uniform Depth Channel Flow 551
Q, cfs
1200
1000
800
600
400
200
Finished concrete
Brickwork Asphalt
Rubble masonry
concrete case,
10.2 fts
Fr
132.2 fts 2 5 ft2 0.804
1 2
The flow is subcritical.
(Ans)
COMMENT The same results would be obtained for the
channel if its size were given in meters. We would use the same
value of n but set k 1 for this SI units situation.
0
0 0.005 0.01 0.015 0.02 0.025 0.03
n
F I G U R E E10.3b
In some instances a trial-and-error or iteration method must be used to solve for the dependent
variable. This is often encountered when the flowrate, channel slope, and channel material are
given, and the flow depth is to be determined as illustrated in the following examples.
E XAMPLE 10.4
Uniform Flow, Determine Flow Depth
GIVEN Water flows in the channel shown in Fig. E10.3a at a
rate of Q 10.0 m 3 s. The canal lining is weedy.
FIND
Determine the depth of the flow.
SOLUTION
In this instance neither the flow area nor the hydraulic radius are
known, although they can be written in terms of the depth, y.
Since the flowrate is given in m 3 /s, we will solve this problem using
SI units. Hence, the bottom width is (12 ft) (1 m/3.281 ft) 3.66 m
and the area is
where A and y are in square meters and meters, respectively. Also,
the wetted perimeter is
so that
y
A y a
tan 40° b 3.66y 1.19y2 3.66y
y
P 3.66 2 a b 3.11y 3.66
sin 40°
R h A P 1.19y2 3.66y
3.11y 3.66
where R h and y are in meters. Thus, with n 0.030 1from Table
10.12, Eq. 10.20 can be written as
Q 10 k n AR2 3
h S 1 2
0
1.0
0.030 11.19y2 3.66y2 a 1.19y2 3.66y
23
3.11y 3.66 b
10.00142 1 2
which can be rearranged into the form
11.19y 2 3.66y2 5 51513.11y 3.662 2 0
where y is in meters. The solution of Eq. 1 can be easily obtained
by use of a simple rootfinding numerical technique or by trial-
(1)
and-error methods. The only physically meaningful root of Eq. 1
1i.e., a positive, real number2 gives the solution for the normal
flow depth at this flowrate as
y 1.50 m
(Ans)
COMMENT By repeating the calculations for various
flowrates, the results shown in Fig. E10.4 are obtained. Note that
the water depth is not linearly related to the flowrate. That is, if
the flowrate is doubled, the depth is not doubled.
y, m
3.0
2.5
2.0
1.5
1.0
0.5
(10, 1.50)
0
0 5 10 15 20 25 30
F I G U R E E10.4
Q, m 3 /s