At a discharge of 10 m³/s the depth of uniform flow in a rectangular channel 3 m wide is 2.2 m. The water flows round a 90° bend of inner radius 5.0 m. Assuming no energy loss at the bend, calculate the depth of water at the inner and outer radii of the bend.

Question

Accepted Answer

See Figure 10.17.

Since there is no energy loss between the entry to the bend and points within the bend, [latex]E_{0} = E_{A} = E_{B} = E [/latex]; that is,

[latex]y_{0} + \frac{V^{2}_{0} }{2g} = y_{A} + \frac{ u^{2}_{ heta ,A} }{2g} = y_{B} +\frac{ u^{2}_{ heta ,B} }{2g}[/latex] (i)

Hence, if [latex] u_{ heta ,A}[/latex] and [latex] u_{ heta ,B}[/latex] can be determined, [latex]y_{A}[/latex] and [latex]y_{B} [/latex] can be calculated.

[latex]y_{0}[/latex] = 2.2 m; [latex]V_{0} = \frac{10}{3 × 2.2} [/latex] = 1.515 m∕s; [latex]\frac{V^{2}_{0} }{2g}[/latex] = 0.117 m

[latex]\Rightarrow E_{0}[/latex] = 2.2 + 0.117 = 2.317 m

[latex] u_{ heta ,r} = \frac{K}{r}[/latex] and Q = [latex]\int_{r_{A} }^{r_{B}} u_{ heta ,r} y_{r} dr[/latex] (ii)

[latex]y_{r} = E − \frac{V^{2}_{ heta ,r} }{2g} = E − \frac{K^{2} }{2gr^{2}} [/latex]

And substituting in Equation (ii),

[latex]\Rightarrow Q = K \int_{r_{A} }^{r_{B}} \frac{1}{r} \left(E − \frac{K^{2} }{2gr^{2}} ight) dr = K \int_{r_{A} }^{r_{B}} \left( \frac{E}{r} − \frac{K^{2} }{2gr^{3}} ight) dr[/latex]

[latex]Q = K \left[E \ln \left(\frac{r_{B}}{r_{A}} ight) + \frac{K^{2} }{4g} \left(\frac{1}{r^{2}_{B} } - \frac{1}{r^{2}_{A} } ight) ight] [/latex]

[latex]10 = K \left[2.317 \ln \left(\frac{8}{5} ight) + \frac{K^{2} }{4g} \left(\frac{1}{8^{2} } - \frac{1}{5^{2} } ight) ight] [/latex]

By trial, K = 9.71.

[latex] \Rightarrow u_{ heta ,A} = \frac{9.71}{5}[/latex] = 1.942; [latex]\frac{ u ^{2}_{ heta ,A} }{2g}[/latex] = 0.192 m

[latex] u_{ heta ,B} = \frac{9.71}{5}[/latex]= 1.2137; [latex]\frac{ u ^{2}_{ heta ,B} }{2g}[/latex]= 0.075 m

[latex]\Rightarrow y_{A}[/latex] = 2.317 − 0.192 = 2.125 m
[latex]y_{B}[/latex] = 2.317 − 0.075 = 2.242 m

Question 10.7: At a discharge of 10 m³/s the depth of uniform flow in a rec...

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