Boundary Layer Theory Questions and Answers

Answer: C

The axial pressure gradient is equal to zero.

Outside the boundary layer the free stream is uniform, so by Bernoulli's equation \(dp/dx = -\rho U\,dU/dx = 0\). Boundary-layer theory shows that the pressure does not vary across the thin layer (\(\partial p/\partial y \approx 0\)), so the layer feels the same zero gradient. This is the Blasius flat-plate problem.

A positive (adverse) or negative (favourable) gradient arises only when the outer velocity changes along the surface, as over a curved body. Pressure gradient and velocity gradient have different dimensions, so they cannot simply be equal.

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 \(\delta\).

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.