Question

A sharp flat plate of length \( L \) and infinite width is immersed parallel to a fluid stream having velocity \( u_\infty \).
At a point on the plate, far away from the leading edge and not near the trailing edge, the boundary layer thickness, the displacement thickness, and the momentum thickness are denoted as \( \delta \), \( \delta^* \), and \( \theta \), respectively.
Which one of the following options correctly represents the relation between these thicknesses?

• A. \( \delta>\delta^*>\theta \)
• B. \( \delta>\theta>\delta^* \)
• C. \( \delta^*>\theta>\delta \)
• D. \( \theta>\delta^*>\delta \)

Hint:

In boundary layer theory, the boundary layer thickness is always the largest, followed by the displacement thickness, and the smallest is the momentum thickness.

Answer 1

The Correct Option is A

In boundary layer theory, the boundary layer thickness \( \delta \) represents the distance from the leading edge of the plate to the point where the flow velocity reaches approximately 99% of the free-stream velocity \( u_\infty \).
The displacement thickness \( \delta^* \) represents the distance by which the outer flow is displaced due to the presence of the boundary layer.
The momentum thickness \( \theta \) accounts for the reduction in momentum due to the velocity gradient within the boundary layer.
For typical boundary layer profiles, the relationship between these thicknesses is as follows:
- The boundary layer thickness \( \delta \) is the largest since it defines the overall extent of the region affected by the boundary layer.
- The displacement thickness \( \delta^* \) is smaller than \( \delta \), but still larger than the momentum thickness \( \theta \).
- The momentum thickness \( \theta \) is the smallest of these three, as it represents the reduction in momentum in the boundary layer.
Therefore, the correct relation is \( \delta>\delta^*>\theta \).
Thus, the correct answer is (A) \( \delta>\delta^*>\theta \).