Question

Match the Gradually Varied Flow (GVF) profiles on a mild slope (\(M_1, M_2, M_3\)) with the nature of their water surface slope (\(dy/dx\)) relative to the flow direction.

Profile & Nature of Slope (\(dy/dx\)) 
P. \(M_1\) Profile & 1. Positive (Rising Curve) 
Q. \(M_2\) Profile & 2. Negative (Drawdown Curve) 
R. \(M_3\) Profile & 3. Positive (Rising Curve) 

Select the correct classification: 

• A. \(M_1\) is Rising, \(M_2\) is Rising, \(M_3\) is Falling
• B. \(M_1\) is Rising, \(M_2\) is Falling, \(M_3\) is Rising
• C. \(M_1\) is Falling, \(M_2\) is Rising, \(M_3\) is Falling
• D. \(M_1\) is Rising, \(M_2\) is Falling, \(M_3\) is Falling

Hint:

For any GVF profile, remember this rule: The water surface always tends towards the normal depth line (\(y_n\)). If the flow is subcritical (\(Fr<1\), as in M1 and M2), this happens from upstream. If the flow is supercritical (\(Fr>1\), as in M3), this happens from downstream control.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks to determine the shape of the water surface profile (rising or falling) for the three possible Gradually Varied Flow (GVF) profiles on a mild slope: M1, M2, and M3. A rising profile has a positive slope (\(dy/dx>0\)), and a falling profile has a negative slope (\(dy/dx<0\)).
Step 2: Key Formula or Approach:
The governing equation for Gradually Varied Flow is: \[ \frac{dy}{dx} = \frac{S_0 - S_f}{1 - Fr^2} \] where:

• \(\frac{dy}{dx}\) is the slope of the water surface.
• \(S_0\) is the channel bed slope (positive for a mild slope).
• \(S_f\) is the friction slope.
• \(Fr\) is the Froude number.

For a mild slope (M), the normal depth (\(y_n\)) is greater than the critical depth (\(y_c\)), i.e., \(y_n>y_c\).
Step 3: Detailed Explanation:
We analyze the sign of the numerator (\(S_0 - S_f\)) and the denominator (\(1 - Fr^2\)) for each profile zone.
- Relationship between depth and slopes/Froude number: - If flow depth \(y>y_n\), then \(S_f<S_0\). - If flow depth \(y<y_n\), then \(S_f>S_0\). - If flow depth \(y>y_c\), then flow is subcritical, \(Fr<1\). - If flow depth \(y<y_c\), then flow is supercritical, \(Fr>1\). M1 Profile: The flow depth \(y\) is in Zone 1, so \(y>y_n>y_c\).

• \(y>y_n \implies S_0 - S_f>0\) (Numerator is positive).
• \(y>y_c \implies Fr<1 \implies 1 - Fr^2>0\) (Denominator is positive).
• \(\frac{dy}{dx} = \frac{(+)}{(+)} = +\). The profile is Rising. This is a backwater curve.

M2 Profile: The flow depth \(y\) is in Zone 2, so \(y_n>y>y_c\).

• \(y<y_n \implies S_0 - S_f<0\) (Numerator is negative).
• \(y>y_c \implies Fr<1 \implies 1 - Fr^2>0\) (Denominator is positive).
• \(\frac{dy}{dx} = \frac{(-)}{(+)} = -\). The profile is Falling. This is a drawdown curve.

M3 Profile: The flow depth \(y\) is in Zone 3, so \(y_n>y_c>y\).

• \(y<y_n \implies S_0 - S_f<0\) (Numerator is negative).
• \(y<y_c \implies Fr>1 \implies 1 - Fr^2<0\) (Denominator is negative).
• \(\frac{dy}{dx} = \frac{(-)}{(-)} = +\). The profile is Rising.

Step 4: Final Answer:
Summarizing the results:

• \(M_1\) is a Rising curve.
• \(M_2\) is a Falling curve.
• \(M_3\) is a Rising curve.

This matches option (B).