Boundary Layer Theory Questions and Answers

Answer: D

The assumption is \(\delta \ll L\).

Prandtl's boundary layer equations come from an order-of-magnitude analysis of the Navier-Stokes equations in a thin layer next to the wall. Because the layer is thin compared with the body length, gradients across it (in y) are much larger than gradients along it (in x). This lets the streamwise diffusion term \(\partial^{2}u/\partial x^{2}\) be dropped and shows that the pressure is nearly constant across the layer, imposed by the outer flow.

If \(\delta\) were comparable to or larger than L, none of these simplifications would hold, which rules out the other three options.

Answer: C

The new thickness is \(2\delta_{1}\).

For a laminar flat-plate boundary layer (Blasius):

\[ \begin{aligned} \delta &\approx \frac{5x}{\sqrt{Re_{x}}} = 5\sqrt{\frac{\nu x}{U_{\infty}}} \\ &\propto \sqrt{\nu} \end{aligned} \]

At the same x and U, multiplying the kinematic viscosity by 4 multiplies the thickness by \(\sqrt{4} = 2\).

\(4\delta_{1}\) assumes thickness proportional to viscosity, \(\delta_{1}/2\) has the dependence inverted, and \(\delta_{1}\) ignores viscosity altogether. A more viscous fluid diffuses momentum further from the wall, so the layer thickens.

Answer: A

The profile is \(u/U = 2(y/\delta) - (y/\delta)^{2}\).

Apply the three boundary conditions: no slip at the wall, free-stream velocity at the edge, and a smooth join with the free stream (zero slope) at the edge.

\[ \begin{aligned} u(0) = 0 &\Rightarrow a = 0 \\ \left.\tfrac{du}{dy}\right|_{\delta} = 0 &\Rightarrow b + 2c\delta = 0 \\ u(\delta) = U &\Rightarrow b\delta + c\delta^{2} = U \end{aligned} \]

These give \(b = 2U/\delta\) and \(c = -U/\delta^{2}\). The '+' option gives u = 3U at the edge. The 1.5 and 0.5 coefficients belong to a cubic profile and do not satisfy the zero-slope condition for a quadratic.

Answer: B

The displacement thickness is 2 mm.

With \(\eta = y/\delta\) and the profile \(u/U = 2\eta - \eta^{2}\):

\[ \begin{aligned} \delta^{*} &= \delta\int_{0}^{1}\left(1 - \frac{u}{U}\right)d\eta \\ &= \delta\int_{0}^{1}(1 - 2\eta + \eta^{2})\,d\eta \\ &= \delta\left(1 - 1 + \tfrac{1}{3}\right) = \frac{6}{3} = 2\ \text{mm} \end{aligned} \]

Displacement thickness measures the outward shift of the streamlines caused by the slowed fluid, so it must be positive and smaller than \(\delta\); the negative options are impossible. 2.25 mm would come from a different (cubic) profile.