1. Example. A smooth (n=0.012) rectangular channel with b = 6 ft and So=0.002 supports a steady flow of 160 ft3/s. The depth increases to 8 ft as the water approaches a low-head dam. Starting at that location, determine the depth profile over a distance of 3500 ft back upstream. What determines the depth at which the stream stabililizes upstream?
Solution. We can set up a spreadsheet, considering small increments in y and using the preceding equation to solve for the distance Dl required for that Dy to occur.
y (ft)
V (ft/s) P A Rh E DE
Vavg
Rh,avg
Sf,avg DL L
8.00
3.33
22.00
48.00
2.18
8.173
0.0
7.85
3.40
21.70
47.10
2.17
8.029
-0.143
3.37
-82.4
7.70
3.46
21.40
46.20
2.16
7.886
3.43
-82.8
-165.2
3.65
7.31
13.30
21.90
1.65
4.479
-0.086
7.16
1.66
-277.3
3.50
7.62
13.00
21.00
1.62
4.401
-0.077
7.46
1.63
-652.7

3. Why does the water depth change so slowly at the upstream end of the reach?
We get a hint about why this occurs from the values shown in the table for the velocity and friction slope. Moving upstream from the location of known depth, the stream gets shallower, so the velocity and therefore Sf increase. Eventually, Sf gets close to So, meaning that the net force on the water approaches zero. At this point, the water velocity, and therefore its depth, approach constant values. A little thought shows that this represents uniform flow conditions. To confirm, we an compute the uniform (normal) depth:

4. As we inferred, the approach of the depth to a constant value reflects an approach of the stream to uniform flow conditions.