Discussion
The velocity profile in an incompressible, laminar boundary layer is shown in the figure below. U is the free-stream velocity, u(y) is the stream-wise velocity component. The area of the black shaded region in the figure below represents the
Answer: C
The shaded area represents the displacement thickness.
The plot is drawn in the non-dimensional velocity \(u/U\) against y. The black region lies between the profile and the line \(u/U = 1\), so its area is
\[ \delta^{*} = \int_{0}^{\infty} (1 - u/U)\,dy \]
which is the definition of displacement thickness: the distance by which the wall would have to be moved out, in an inviscid flow, to carry the same mass-flow deficit that the boundary layer causes.
Boundary layer thickness is a height (where u reaches about 0.99U), not an area. Momentum thickness is \(\int \frac{u}{U}(1 - \frac{u}{U})dy\), which carries an extra factor \(u/U\) and is not the shaded region. The shape factor is the ratio of displacement to momentum thickness, a pure number.
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