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## 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.

## 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.