Question

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?

• A. ( \frac{1}{2} \)
• B. ( \frac{1}{4} \)
• C. ( \frac{1}{6} \)
• D. ( \frac{1}{3} \)

Hint:

For linear velocity profiles, boundary layer thickness ratios often simplify neatly using basic integrals.

Answer 1

The Correct Option is C

For a linear velocity profile in a boundary layer, the velocity varies as:
\[ u(y) = U_{\infty} \left( \frac{y}{\delta} \right) \] where \( \delta \) is the boundary layer thickness.
The momentum thickness is defined as:
\[ \theta = \int_0^{\delta} \frac{u}{U_\infty} \left(1 - \frac{u}{U_\infty}\right) dy \] Substitute the linear profile \( \frac{u}{U_\infty} = \frac{y}{\delta} \):
\[ \theta = \int_0^{\delta} \left(\frac{y}{\delta}\right)\left(1 - \frac{y}{\delta}\right) dy \] Expand the integrand:
\[ \theta = \int_0^{\delta} \left(\frac{y}{\delta} - \frac{y^2}{\delta^2}\right) dy \] Compute the integrals:
\[ \int_0^{\delta} \frac{y}{\delta}\, dy = \frac{1}{\delta}\left[\frac{y^2}{2}\right]_0^\delta = \frac{\delta}{2} \] \[ \int_0^{\delta} \frac{y^2}{\delta^2}\, dy = \frac{1}{\delta^2}\left[\frac{y^3}{3}\right]_0^\delta = \frac{\delta}{3} \] So,
\[ \theta = \frac{\delta}{2} - \frac{\delta}{3} = \frac{\delta}{6} \] Thus the ratio of momentum thickness to boundary layer thickness is:
\[ \frac{\theta}{\delta} = \frac{1}{6} \] Hence, the correct answer is (C).