Discussion
A sharp flat plate of length L and infinite width is immersed parallel to a fluid stream having velocity \(u_{\infty}\). At a point on the plate, far away from the leading edge and not near the trailing edge, the boundary layer thickness, the displacement thickness, and the momentum thickness are denoted as \(\delta\), \(\delta^{*}\), and \(\theta\), respectively.
Which one of the following options correctly represents the relation between these thicknesses?
Answer: A
The order is \(\delta \gt \delta^{*} \gt \theta\).
The boundary layer thickness \(\delta\) is where the velocity reaches 99% of the free stream. The displacement thickness \(\delta^{*} = \int_{0}^{\delta}(1 - u/U)\,dy\) and the momentum thickness \(\theta = \int_{0}^{\delta}(u/U)(1 - u/U)\,dy\) are integrals of deficits that are always smaller than 1 and shrink toward the edge, so both are less than \(\delta\). The extra factor \(u/U \le 1\) makes \(\theta\) smaller than \(\delta^{*}\).
For the laminar Blasius profile, for example, \(\delta^{*} \approx \delta/2.9\) and \(\theta \approx \delta/7.5\), and their ratio (the shape factor) is about 2.6. The other orders all put \(\theta\) above \(\delta^{*}\) or a thickness above \(\delta\).
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