Discussion
A typical boundary layer over a flat plate has a linear velocity profile with zero velocity at the wall and freestream velocity, \(U_{\infty}\), at the outer edge of the boundary layer. What is the ratio of the momentum thickness to the thickness of the boundary layer?
Answer: C
The ratio is \(\frac{1}{6}\).
With \(u/U_{\infty} = \eta = y/\delta\), the momentum thickness is
\[ \begin{aligned} \theta &= \delta\int_{0}^{1} \eta(1 - \eta)\,d\eta \\ &= \delta\left(\frac{1}{2} - \frac{1}{3}\right) = \frac{\delta}{6} \end{aligned} \]
\(1/2\) is the displacement thickness ratio for the same profile, \(\int_{0}^{1}(1 - \eta)\,d\eta\), and \(1/3\) and \(1/4\) do not come from either integral.
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