GATE 2021 AE – Question 52
For boundary-layer profile $u/U=(y/\delta)^2$, the shape factor $\delta^*/\theta$ is
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Correct answer: 5
Explanation
With $u/U=(y/\delta)^2$ and $\eta=y/\delta$:
**Displacement thickness:**
$$\delta^*=\int_0^\delta\left(1-\frac uU\right)dy=\delta\int_0^1(1-\eta^2)\,d\eta=\delta\left(1-\frac13\right)=\frac{2\delta}{3}.$$
**Momentum thickness:**
$$\theta=\int_0^\delta\frac uU\left(1-\frac uU\right)dy=\delta\int_0^1\eta^2(1-\eta^2)\,d\eta=\delta\left(\frac13-\frac15\right)=\frac{2\delta}{15}.$$
**Shape factor:**
$$H=\frac{\delta^*}{\theta}=\frac{2\delta/3}{2\delta/15}=\frac{15}{3}=\mathbf{5}.$$