For steady 2D incompressible laminar boundary layer flow over a flat plate with zero pressure gradient, assuming a parabolic velocity profile $\frac{u}{U_\infty} = 2\left(\frac{y}{\delta}\right) - \left(\frac{y}{\delta}\right)^2$, the momentum thickness $\theta$ expressed as a fraction of the boundary layer thickness $\delta$ is:

(A) $\theta = \frac{2}{15} \delta$

(B) $\theta = \frac{1}{6} \delta$

(C) $\theta = \frac{1}{3} \delta$

(D) $\theta = \frac{3}{10} \delta$

Answer Key:

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### 1. Strategy & Theoretical Principles By definition, the momentum thickness $\theta$ is: $$\theta = \int_0^\delta \frac{u}{U_\infty} \left(1 - \frac{u}{U_\infty}\right) \, dy$$ Let $\eta = \frac{y}{\delta} \implies dy = \delta \, d\eta$. For $\frac{u}{U_\infty} = 2\eta - \eta^2$: $$\frac{\theta}{\delta} = \int_0^1 (2\eta - \eta^2)(1 - 2\eta + \eta^2) \, d\eta$$ ### 2. Analytical Formulation & Integration Expand the product: $$(2\eta - \eta^2)(1 - 2\eta + \eta^2) = 2\eta - 4\eta^2 + 2\eta^3 - \eta^2 + 2\eta^3 - \eta^4 = 2\eta - 5\eta^2 + 4\eta^3 - \eta^4$$ Integrate term by term from 0 to 1: $$\frac{\theta}{\delta} = \left[ 2 \frac{\eta^2}{2} - 5 \frac{\eta^3}{3} + 4 \frac{\eta^4}{4} - \frac{\eta^5}{5} \right]_0^1 = 1 - \frac{5}{3} + 1 - \frac{1}{5} = 2 - \frac{5}{3} - \frac{1}{5}$$ ### 3. Step-by-Step Calculation Find the common denominator (15): $$\frac{\theta}{\delta} = \frac{30 - 25 - 3}{15} = \frac{2}{15}$$ Therefore: $$\theta = \frac{2}{15} \delta \approx 0.1333 \delta$$ ### 4. Distractor Forensics - **Option A ($2/15 \delta$):** Correct momentum thickness. - **Option B ($1/6 \delta$):** Corresponds to linear profile $\theta = \delta/6$. - **Option C ($1/3 \delta$):** Displacement thickness $\delta^* = \delta/3$ for this parabolic profile. - **Option D ($3/10 \delta$):** Sinusoidal profile momentum thickness ($0.137 \delta$) miscalculation. ### 5. Exam Trap & Standard Reference - **Trap Alert:** Distinguish displacement thickness $\delta^* = \int_0^1 (1 - \frac{u}{U}) d\eta = 1 - (1 - 1/3) = 1/3 \delta$ from momentum thickness $\theta = 2/15 \delta$. - **Standard Reference:** Schlichting, H. & Gersten, K., *Boundary-Layer Theory*, Chapter 8.

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