Question

For an incompressible boundary layer flow over a flat plate (figure shown), the momentum thickness is expressed as

• A. $\displaystyle \int_{0}^{\infty}\frac{u}{U_\infty}\,dy$
• B. $\displaystyle \int_{0}^{\infty}\left(1-\frac{u}{U_\infty}\right)\!dy$
• C. $\displaystyle \int_{0}^{\infty}\frac{u}{U_\infty}\!\left(1-\frac{u}{U_\infty}\right)\!dy$
• D. $\displaystyle \int_{0}^{\infty}\left(1-\frac{u^2}{U_\infty^{2}}\right)\!dy$

Hint:

Remember the trio: $\delta^*: 1-\frac{u}{U_\infty}$; \ $\theta: \frac{u}{U_\infty}\Big(1-\frac{u}{U_\infty}\Big)$; \ energy thickness: $\frac{u}{U_\infty}\Big(1-\frac{u^2}{U_\infty^2}\Big)$.

Answer 1

The Correct Option is C

Step 1: Recall boundary-layer integral thickness definitions.
Let $U_\infty$ be the free-stream speed, $u(y)$ the local mean velocity. - Displacement thickness: $\displaystyle \delta^*=\int_0^\infty\!\left(1-\frac{u}{U_\infty}\right)dy.$ - Momentum thickness: $\displaystyle \theta=\int_0^\infty\!\frac{u}{U_\infty}\left(1-\frac{u}{U_\infty}\right)dy.$ - Energy thickness: $\displaystyle \int_0^\infty\!\frac{u}{U_\infty}\left(1-\frac{u^2}{U_\infty^2}\right)dy.$
Step 2: Identify the correct expression.
The momentum thickness $\theta$ accounts for the loss of momentum flux due to the boundary layer and is exactly \[ \theta=\int_{0}^{\infty}\frac{u}{U_\infty}\left(1-\frac{u}{U_\infty}\right)dy, \] which corresponds to option (C). Final Answer: \[ \boxed{\text{(C) }\displaystyle \int_{0}^{\infty}\frac{u}{U_\infty}\left(1-\frac{u}{U_\infty}\right)dy} \]