Boundary Layer Theory: Displacement & Momentum Thickness Explained

[mrajkumar]

Hi,
could you tell the physical meaning of the displacement and momentum thickness of a boundary layer.And why the stream line diverges away from the body in the boundary layer to conserve mass?

 

[rcgldr]

update - I wan't sure about what you meant by momentum thickness, but thanks to boneh3ad's next post, it's explained there.

Link to flat plate article:

flat plate.htm

 

[boneh3ad]

The convexity or concavity of the surface is irrelevant for the question at hand, and by stating that this is a question about boundary layers automatically means this is a viscous flow.

The displacement thickness ($\delta_1$ or $\delta^*$) is a measure of the effect of the boundary layer on the flow of mass in a fluid. Essentially, you can solve the flow inviscidly and come up with a certain overall mass flow rate. Then solve the same flow accounting for the boundary layer, and the mass flow will be slightly smaller. The displacement thickness is the distance the surface would have to be displaced in order that the mass flow in the inviscid case would be the same as that in the viscous case with the original wall position. In essence, it is the distance the wall must move in order to get the same outer flow (inviscid) answer without solving for the boundary layer.

$$\delta^* = \int\limits_0^{\infty}\left(1 - \dfrac{u}{U_{\infty}}\right)dy.$$

The momentum thickness ($\delta_2$ or $\theta$) is similar, only it deals with the flow of momentum rather than mass. It shows up in the momentum integral boundary layer equation (as does $\delta^*$).

$$\theta = \int\limits_0^{\infty}\dfrac{u}{U_{\infty}}\left(1 - \dfrac{u}{U_{\infty}}\right)dy$$