Section 4: Soil and Water Conservation Engineering
Fluid Mechanics
Subtopic: Flow in open channels
Question Statement:
Which of the following statements regarding gradually varied flow (GVF) surface profiles in open channel hydraulics are correct?
Options:
(A) An $M_1$ profile occurs on a mild slope ($S_0 < S_c$) when the actual depth $y$ is greater than both normal depth $y_n$ and critical depth $y_c$ ($y > y_n > y_c$), representing a backwater curve
(B) An $M_2$ profile is a drawdown curve ($y_n > y > y_c$) occurring upstream of a sudden free overfall on a mild channel
(C) An $S_1$ curve occurs on a steep slope ($y > y_c > y_n$) behind a high check dam or weir, producing a subcritical backwater profile
(D) Flow profiles across the critical depth line $y = y_c$ are always perfectly smooth and horizontally asymptotic
Answer Key:
A, B, C
Step-by-Step Solution & Explanation
- Statements A, B, and C are correct:
1. $M_1$ occurs when $y > y_n > y_c$, $\frac{dy}{dx} > 0$ (backwater curve behind a dam on mild slope).
2. $M_2$ occurs when $y_n > y > y_c$, $\frac{dy}{dx} < 0$ (drawdown profile approaching a free drop).
3. $S_1$ profile occurs when a structure forces flow depth above $y_c$ on a steep slope.
- Statement D is incorrect: At critical depth $y = y_c$, $Fr = 1$ and the GVF differential equation $\frac{dy}{dx} = \frac{S_0 - S_f}{1 - Fr^2}$ indicates an infinite slope ($\,\frac{dy}{dx} \to \infty$), signifying physical breakdown into hydraulic jumps or standing waves.