Discussion
The velocity profile inside the boundary layer for flow over a flat plate is given as \(\frac{u}{U_{\infty}} = \sin\left(\frac{\pi y}{2 \delta}\right)\), where \(U_{\infty}\) is the free stream velocity and \(\delta\) is the local boundary layer thickness. If \(\delta^{*}\) is the local displacement thickness, the value of \(\frac{\delta^{*}}{\delta}\) is
Answer: B
The ratio is \(1 - \frac{2}{\pi}\).
The displacement thickness is the integral of the velocity deficit across the boundary layer. With \(\eta = y/\delta\):
\[ \begin{aligned} \frac{\delta^{*}}{\delta} &= \int_{0}^{1}\left[1 - \sin\left(\tfrac{\pi\eta}{2}\right)\right]d\eta \\ &= 1 - \frac{2}{\pi} \approx 0.363 \end{aligned} \]
since \(\int_{0}^{1}\sin(\pi\eta/2)\,d\eta = 2/\pi\). \(2/\pi\) is that integral alone, the flow carried rather than the deficit; \(1 + 2/\pi\) has the sign wrong, and 0 would mean no boundary layer.
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