Section 4: Soil and Water Conservation Engineering
Fluid Mechanics
Subtopic: Flow through orifices, weirs and notches
Question Statement:
Regarding sharp-crested and broad-crested weirs used in agricultural runoff gauging, which of the following statements are CORRECT?
Options:
(A) The theoretical discharge over a rectangular sharp-crested weir is proportional to $H^{3/2}$
(B) The discharge over a $90^\circ$ triangular V-notch weir is proportional to $H^{5/2}$
(C) A Cipoletti weir has trapezoidal side slopes of $1\text{ horizontal to } 4\text{ vertical}$ to compensate for end contractions
(D) In a broad-crested weir with critical flow over the crest, the critical depth is $y_c = \frac{2}{3}H$, where $H$ is the total upstream head
Answer Key:
A, B, C, D
Step-by-Step Solution & Explanation
All four statements represent established hydraulic theorems:
1. Rectangular weir: $Q = \frac{2}{3} C_d L \sqrt{2g} H^{3/2} \propto H^{3/2}$. (Statement A True)
2. Triangular V-notch: $Q = \frac{8}{15} C_d \sqrt{2g} \tan(\theta/2) H^{5/2} \propto H^{5/2}$. (Statement B True)
3. Cipoletti weir sides slope outward at $1:4$ ($1\text{ H}:4\text{ V}$), compensating exactly for the reduction in flow caused by two end contractions in a rectangular notch. (Statement C True)
4. For a broad-crested rectangular weir, critical depth occurs on the crest where $E = y + \frac{v^2}{2g} = \frac{3}{2} y_c \implies y_c = \frac{2}{3} H$. (Statement D True)