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Chapter 15

Q. 15.3

Water flows through a 4 ft wide rectangular irrigation channel that gradually contracts to a width of 3 ft as shown in Figure 15.18. If the water depth upstream of the contraction is 1 ft, and the flow velocity there is 2 ft/s, find the water depth and flow speed downstream of the contraction.

15.18

Step-by-Step

Verified Solution

We can solve this problem by applying Eq. 15.13 between a point upstream where the velocity, width, and depth are V1, w1, and y1, and a point downstream where the velocity, width, and depth are V2, w2, and y2 as shown in Figure 15.18. Writing Eq. 15.13 between these points gives

\frac{1}{2g}\left[\frac{V_1y_1w_1}{y_2w_2} \right]^2+y_2= \frac{V^2_1}{2g} +y_1

Multiplying by y^2 _2 and rearranging we obtain the cubic equation

y^3_2-y^2_2\left[\frac{V^2_1}{2g}+y_1\right]\frac{1}{2g} \left[\frac{V_1y_1w_1}{w_2} \right]^2=0

After inserting the data, we have y^3_2-y^2_2[1.062 ft] + 0.1104 ft = 0, which can be solved to obtain the three solutions: −0.286 ft, 0.412 ft, and 0.936 ft. The negative root can be immediately discarded on physical grounds. Next we calculate the downstream velocities and corresponding Froude numbers for each of the positive roots. For y2 = 0.412 ft we find:

V_2=\frac{V_1y_1w_1}{y_2w_2}=\frac{(2\ \mathrm{ft} /s)(1\ \mathrm{ft} )(4\ \mathrm{ft} )}{(0.412\ \mathrm{ft} )(3\ \mathrm{ft} )} =6.47\ \mathrm{ft} /s

and

Fr_2=\frac{V_2}{\sqrt{gy_1} }=\frac{6.47\ \mathrm{ft} /s}{\sqrt{(32.2\ \mathrm{ft} /s^2)(1\ \mathrm{ft} )}}=1.14

For y2 = 0.936 ft we find:

V_2=\frac{V_1y_1w_1}{y_2w_2}=\frac{(2\ \mathrm{ft} /s)(1\ \mathrm{ft} )(4\ \mathrm{ft} )}{(0.936\ \mathrm{ft} )(3\ \mathrm{ft} )} =2.85\ \mathrm{ft} /s

and

F_2=\frac{V_2}{\sqrt{gy_1} }=\frac{2.85\ \mathrm{ft} /s}{\sqrt{(32.2\ \mathrm{ft} /s^2)(1\ \mathrm{ft} )}}=0.50

Calculating the upstream Froude number, we find

Fr_1=\frac{V_1}{\sqrt{gy_1} }=\frac{2\ \mathrm{ft} /s}{\sqrt{(32.2\ \mathrm{ft} /s^2)(1\ \mathrm{ft} )}}=0.352

hence the flow is subcritical. Since the depth of a subcritical for flow must decrease in a contraction, the correct depth downstream of the contraction is 0.936 ft and the corresponding flow speed is 2.85 ft/s.