Question 5.4: A long channel has a compound cross section (Fig. 5-11) and ......
A long channel has a compound cross section (Fig. 5-11) and a free overfall at the downstream end. The channel discharge is 2.5 m³ /s, and the Manning n for the main channel and for the floodplain are 0.013 and 0.0144, respectively. Discuss and sketch the water surface profiles for the following four channel bottom slopes, S_{0} = 0.0094, 0.0049, 0.0029 and 0.001.
Given:
Q = 0.5 m³ /s;
n_{m} = 0.013;
n_{f} = 0.0144;
Channel cross section, as shown in Fig. 5-11.
Determine:
Water-surface profiles for S_{o} = 0.0094, 0.0049, 0.0029, and 0.001.
The specific energy and Froude number versus the flow depth for the channel cross section and for the specified discharges are plotted in Fig. 5-12. There are three critical depths: y_{c_{1}}= 0.86 m, y_{c_{2}}= 1.002 m and y_{c_{3}}= 1.12 m.
\quadLet us consider each of the channel bottom slopes one by one.
S_{o} = 0.0094
For this bottom slope, y_{n}= 0.75 m (determined by solving the Manning equation by trial and error). As shown in Fig. 5-12, y_{n} < y_{c_{1}} and the Froude number is greater than 1. Therefore, the flow is supercritical, control is at the upstream end, and the free overfall at the downstream end does not affect the flow. Once the flow depth approaches the normal depth, it changes only slightly at the downstream end. The water-surface profile is shown in Fig. 5-13a.
S_{o} = 0.0049
For this bottom slope y_{n}= 0.97 m. Thus, y_{c_{1}} < y_{n} < y_{c_{2}} and F_{rc} < 1. Therefore, the flow is subcritical (Fig. 5-12) and the control is at the downstream end. The water-surface profile may be computed starting at the downstream end with a depth equal to y_{c_{1}}. The water-surface profile is shown schematically in Fig. 5-13b.
S_{o} = 0.0029
For this slope, y_{n}= 1.07 m; i.e., y_{c_{2}} < y_{n} < y_{c_{3}}. Figure 5-12 shows that the Froude number for this normal depth is greater than 1. Therefore, the flow is supercritical; control is at the upstream end and the free overfall at the downstream end does not affect the flow in the channel. The water-surface profile is shown in Fig. 5-13c.
S_{o} = 0.001
For this slope, y_{n} = 1.2 m; i.e., y_{n} > y_{c_{3}} and the flow is subcritical, as indicated by Fig. 5-12. The water-surface profile may be computed by starting at the downstream end with depth equal to y_{c_{3}}. The flow depth varies from normal depth, y_{n} at some upstream point to the critical depth, y_{c_{3}} near the downstream end. The flow varies rapidly from one critical depth, y_{c_{3}} to the other, y_{c_{1}} at the downstream end. The flow profile is shown in Fig. 5-13d.
