Question

For a laminar boundary layer developing over a flat plate, the displacement thickness is defined as: $\delta^* = \int_0^\delta \left(1 - \frac{u}{\text{U}_\infty}\right) dy$. Physically, $\delta^*$ represents

• A. Momentum loss in the boundary layer
• B. Reduction in mass flow due to the presence of the boundary layer
• C. Energy lost in the boundary layer
• D. Vorticity in the flow

Hint:

Remember the physical definitions of boundary layer thicknesses:
Displacement thickness ($\delta^*$) $\rightarrow$ Mass flow rate deficit.
Momentum thickness ($\theta$) $\rightarrow$ Momentum flow rate deficit.
Energy thickness ($\delta^{**}$) $\rightarrow$ Kinetic energy flow rate deficit.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks for the physical significance of the boundary layer parameter called "displacement thickness" ($\delta^*$).

Step 2: Key Formula or Approach:
The velocity of fluid inside the boundary layer is less than the free-stream velocity ($u < U_\infty$) due to viscous shear forces. This deficit in velocity causes a reduction in the mass flow rate near the plate.

Step 3: Detailed Explanation:

• Let us calculate the mass flow rate reduction through a section of height $h$ (where $h > \delta$) due to boundary layer formation:
\[ \Delta \dot{m} = \int_0^h \rho U_\infty dy - \int_0^h \rho u dy = \int_0^h \rho (U_\infty - u) dy \]

• Since $u = U_\infty$ for $y > \delta$, we can restrict the limit of integration to the boundary layer thickness $\delta$:
\[ \Delta \dot{m} = \int_0^\delta \rho (U_\infty - u) dy = \rho U_\infty \int_0^\delta \left(1 - \frac{u}{U_\infty}\right) dy \]

• If we represent this mass flow rate deficit as an equivalent layer of thickness $\delta^*$ of free-stream fluid carrying the same mass deficit:
\[ \Delta \dot{m} = \rho U_\infty \delta^* \]

• Equating these two expressions:
\[ \delta^* = \int_0^\delta \left(1 - \frac{u}{U_\infty}\right) dy \]

• Physically, this is the distance by which the solid boundary would have to be displaced outward in an inviscid, frictionless flow to maintain the same mass flow rate as the actual viscous flow.

Step 4: Final Answer:
Physically, $\delta^*$ represents the reduction in mass flow due to the presence of the boundary layer.