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. $\delta^*>\delta>\theta$
• D. $\theta>\delta^*>\delta$

Hint:

Remember the hierarchy: Boundary layer thickness $\delta$ (largest) > displacement thickness $\delta^*$ > momentum thickness $\theta$ (smallest).

Answer 1

The Correct Option is A

Step 1: Recall definitions.
- Boundary layer thickness $\delta$: distance from the plate where velocity reaches $\approx 99%$ of $u_\infty$. - Displacement thickness $\delta^*$: measures the reduction in mass flow due to the presence of the boundary layer. - Momentum thickness $\theta$: measures the reduction in momentum flux due to the boundary layer.

Step 2: Relationship between thicknesses.
For a flat plate laminar or turbulent boundary layer: \[ \delta>\delta^*>\theta. \] Reason: - $\delta$ is always the largest (physical thickness of velocity profile). - $\delta^*$ is smaller but accounts for lost mass flux. - $\theta$ is the smallest since it measures momentum deficit. Final Answer:
\[ \boxed{\delta>\delta^*>\theta} \]