Question

Consider the flow past a curved wall as shown in the figure. Which of the following statement(s) is/are correct? 

• A. \( P \) is the separation point.
• B. Between \( T \) and \( U \), the pressure gradient in the streamwise direction at the wall is positive.
• C. \( U \) is the stagnation point.
• D. Between \( T \) and \( U \), the streamwise-velocity gradient in the normal direction at the wall is negative.

Hint:

For boundary layer analysis:
1. Pressure gradients can be identified by observing flow acceleration or deceleration along the wall.
2. Velocity gradients in the normal direction indicate how velocity changes perpendicular to the wall.
3. Stagnation points are locations where the velocity of the fluid is zero.

Answer 1

The Correct Option is B

Step 1: Analyze the flow and the terms.
The flow past a curved wall involves boundary layer characteristics, including separation, stagnation, pressure gradients, and velocity gradients. We analyze each statement: 1. Statement (A): \( P \) is the separation point.
The separation point is characterized by a zero wall shear stress and a reversed flow. However, the diagram does not explicitly indicate \( P \) as the separation point.
Statement (A) is incorrect.
2. Statement (B): Between \( T \) and \( U \), the pressure gradient in the streamwise direction at the wall is positive.
Between \( T \) and \( U \), the curvature of the wall indicates a deceleration of flow, resulting in a positive pressure gradient in the streamwise direction.
Statement (B) is correct.
3. Statement (C): \( U \) is the stagnation point. At a stagnation point, the flow velocity becomes zero. The diagram does not suggest \( U \) is a stagnation point, as flow velocity at \( U \) is not zero.
Statement (C) is incorrect.
4. Statement (D): Between \( T \) and \( U \), the streamwise-velocity gradient in the normal direction at the wall is negative.
The streamwise-velocity gradient in the normal direction (\( \partial u / \partial y \)) represents the rate of change of velocity perpendicular to the wall. Between \( T \) and \( U \), the deceleration of flow results in a negative gradient.
Statement (D) is correct.
Step 2: Verify the options.
Option (A): Incorrect, as \( P \) is not clearly defined as the separation point.
Option (B): Correct, as there is a positive pressure gradient between \( T \) and \( U \).
Option (C): Incorrect, as \( U \) is not the stagnation point.
Option (D): Correct, as the streamwise-velocity gradient in the normal direction is negative between \( T \) and \( U \).
Conclusion: The correct statements are (B) and (D).