Froude model of a spillway

Criterion of similarity

Flow over a spillway is a free-surface flow dominated by gravity, so the Froude number is the same in the model and prototype. The scale ratio $L_r = L_p/L_m = 25$. The same fluid (water) and the same $g$ are used.

Ratios from Froude law

$$\begin{aligned} V_r &= \sqrt{L_r} = \sqrt{25} = 5 \\ T_r &= \frac{L_r}{V_r} = \sqrt{L_r} = 5 \\ Q_r &= V_r L_r^2 = L_r^{5/2} = 25^{2.5} = 3125 \\ F_r &= \frac{\rho_p}{\rho_m} L_r^3 = L_r^3 = 15\,625 \end{aligned}$$

(a) Model discharge

$$Q_m = \frac{Q_p}{Q_r} = \frac{1200}{3125} = 0.384\ \text{m}^3/\text{s}$$

(b) Velocity and time ratios

$V_p/V_m = 5$ and $T_p/T_m = 5$.

(c) Force on prototype gate

$$F_p = 40 \times 15\,625 = 625\,000\ \text{N} = 625\ \text{kN}$$

(d) Prototype time

$$T_p = 5 \times 2 = 10\ \text{minutes}$$

Comment

The Reynolds number is not matched in the model ($Re_r = V_r L_r = 125$), so viscous effects and surface tension are not scaled correctly. For a large model with a turbulent flow this error is small. This is incomplete similarity.

Answer: $Q_m = 0.384\ \text{m}^3/\text{s}$; $V_r = T_r = 5$; $F_p = 625\ \text{kN}$; $T_p = 10\ \text{min}$ (Froude law)