The spillway in Fig. 8–13 is to be built such that the estimated average flow over its crest will be Q = 3000 m³/s. Determine the required flow over the crest of a model built to a scale of 1/25.

Question

Accepted Answer

Here the weight of the water is the most significant force influencing the flow, and so Froude number similitude must be achieved. This equality will provide the necessary relationship between the velocities and the size of the spillway. Thus,

[latex] \left(\frac{V}{\sqrt{g L}} ight)_m=\left(\frac{V}{\sqrt{g L}} ight)_p [/latex]

or

[latex] \frac{V_m}{V_p}=\left(\frac{L_m}{L_p} ight)^{1 / 2}=\left(\frac{1}{25} ight)^{1 / 2} [/latex] (1)

We can express the velocity ratio in terms of the flow ratio since Q = VA, where A is the product of the depth of the water passing over the crest, [latex] L_h [/latex], and the width of the spillway, [latex] L_w [/latex]. Thus,

[latex] \frac{V_m}{V_p}=\frac{Q_m A_p}{Q_p A_m}=\frac{Q_m\left(L_h ight)_p\left(L_w ight)_p}{Q_p\left(L_h ight)_m\left(L_w ight)_m}=\frac{Q_m}{Q_p}\left(\frac{25}{1} ight)^2 [/latex]

Substituting this into Eq. 1, we get

[latex] \frac{Q_m}{Q_p}=\left(\frac{1}{25} ight)^{5 / 2} [/latex]

Thus,

[latex] \begin{aligned} Q_m & =Q_p\left(\frac{1}{25} ight)^{5 / 2} \ & =\left(3000 m ^3 / s ight)\left(\frac{1}{25} ight)^{5 / 2}=0.960 m ^3 / s \end{aligned} [/latex]

Question 8.7: The spillway in Fig. 8–13 is to be built such that the estim...

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