Question

Resistance to uniform turbulent flow in an open channel system is often expressed in terms of the resistance co-efficient using
A. Manning’s equation
B. Chezy’s equation
C. Darcy -Weisbach equation
D. Bernoulli’s equation
E. Francis Formula
Choose the correct answer from the options given below:

• A. A, B and D only
• B. A, B and C only
• C. B, D and E only
• D. C, D and E only

Hint:

Manning's \( n \), Chezy's \( C \), and Darcy-Weisbach's \( f \) are the three standard resistance coefficients used in open channel hydraulics.
They can be related to each other mathematically:
\[ C = \frac{R^{1/6}}{n} = \sqrt{\frac{8g}{f}} \]

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
In open channel hydraulics, uniform turbulent flow occurs when the gravity forces driving the flow are balanced by the frictional resistance forces exerted by the channel boundary.
Frictional resistance is modeled using equations that incorporate specific resistance coefficients.

Step 2: Detailed Explanation:
Let us analyze each equation and its association with flow resistance:
- A. Manning’s equation: This is an empirical formula widely used to calculate open channel flow velocity.
It uses Manning's roughness coefficient (\( n \)) to represent channel resistance:
\[ V = \frac{1}{n} R^{2/3} S^{1/2} \]
Thus, statement A is correct.
- B. Chezy’s equation: This is one of the oldest formulations for open channel flow, relating velocity to hydraulic radius and slope using the Chezy resistance coefficient (\( C \)):
\[ V = C \sqrt{R S} \]
Thus, statement B is correct.
- C. Darcy-Weisbach equation: This equation is derived from fluid mechanics principles and can be adapted to open channels by relating the friction factor (\( f \)) to the slope and hydraulic radius:
\[ V = \sqrt{\frac{8g}{f}} \sqrt{R S} \]
where \( f \) is the Darcy-Weisbach resistance coefficient.
Thus, statement C is correct.
- D. Bernoulli’s equation: This is an energy conservation equation along a streamline and does not express flow resistance in terms of a resistance coefficient.
- E. Francis Formula: This is an empirical formula used to calculate discharge over rectangular weirs, which is unrelated to uniform flow resistance.
Therefore, the equations that express flow resistance in terms of a resistance coefficient are A, B, and C.

Step 3: Final Answer:
The resistance to uniform turbulent flow is expressed using equations A, B, and C only.