fluid_mechanics

10.4 Uniform Depth Channel Flow 553
For many open
channels, the surface
roughness
varies across the
channel.
In many man-made channels and in most natural channels, the surface roughness 1and hence
the Manning coefficient2 varies along the wetted perimeter of the channel. A drainage ditch, for
example, may have a rocky bottom surface with concrete side walls to prevent erosion. Thus,
the effective n will be different for shallow depths than for deep depths of flow. Similarly, a river
channel may have one value of n appropriate for its normal channel and another very different
value of n during its flood stage when a portion of the flow occurs across fields or through
floodplain woods. An ice-covered channel usually has a different value of n for the ice than for
the remainder of the wetted perimeter 1Ref. 72. 1Strictly speaking, such ice-covered channels are
not “open” channels, although analysis of their flow is often based on open-channel flow
equations. This is acceptable, since the ice cover is often thin enough so that it represents a fixed
boundary in terms of the shear stress resistance, but it cannot support a significant pressure
differential as in pipe flow situations.2
A variety of methods has been used to determine an appropriate value of the effective
roughness of channels that contain subsections with different values of n. Which method gives the
most accurate, easy-to-use results is not firmly established, since the results are nearly the same
for each method 1Ref. 52. A reasonable approximation is to divide the channel cross section into N
subsections, each with its own wetted perimeter, P i , area, A i , and Manning coefficient, n i . The P i
values do not include the imaginary boundaries between the different subsections. The total flowrate
is assumed to be the sum of the flowrates through each section. This technique is illustrated by
Example 10.7.
E XAMPLE 10.6
Uniform Flow, Variable Roughness
GIVEN Water flows along the drainage canal having the properties
shown in Fig. E10.6a. The bottom slope is S 0 1 ft500 ft
0.002.
FIND Estimate the flowrate when the depth is y 0.8 ft
0.6 ft 1.4 ft.
SOLUTION
We divide the cross section into three subsections as is indicated
in Fig. E10.6a and write the flowrate as Q Q 1 Q 2 Q 3 ,
where for each section
The appropriate values of A i , P i , R hi , and n i are listed in
Table E10.6. Note that the imaginary portions of the perimeters
between sections 1denoted by the vertical dashed lines in Fig.
E10.6a2 are not included in the P i . That is, for section 122
and
■ TABLE E10.6
A i
Q i 1.49
n i
A i R 2 3 12
h i
S 0
A 2 2 ft 10.8 0.62 ft 2.8 ft 2
P 2 2 ft 210.8 ft2 3.6 ft
P i
R hi
i ( ft 2 ) (ft) (ft)
1 1.8 3.6 0.500 0.020
2 2.8 3.6 0.778 0.015
3 1.8 3.6 0.500 0.030
n i
3 ft
2 ft
3 ft
n 1 = 0.020 n 3 = 0.030
(1)
0.6 ft
n 2 =
0.015
(2)
F I G U R E E10.6a
so that
or
Thus, the total flowrate is
y
0.8 ft
(3)
R h2
A 2 2.8 ft2
0.778 ft
P 2 3.6 ft
Q Q 1 Q 2 Q 3 1.4910.0022 1 2
c 11.8 ft2 210.500 ft2 2 3
0.020
11.8 ft2 210.500 ft2 2 3
d
0.030
Q 16.8 ft 3 s
(Ans)
COMMENTS If the entire channel cross section were considered
as one flow area, then A A A 2 A 3 6.4 ft 2
1
and
P P or R h AP 6.4 ft 2 1 P 2 P 3 10.8 ft,
10.8 ft
0.593 ft. The flowrate is given by Eq. 10.20, which can be written
as
Q 1.49
n eff
12.8 ft2 210.778 ft2 2 3
0.015
AR 2 3
h S 1 2
0