BL, displacement and momentum thickness

Boundary layer thickness ($\delta$)

It is the distance from the wall, measured normal to it, at which the velocity reaches 99% of the free-stream velocity $U$ ($u = 0.99U$).

Application: It marks out the region where viscous effects matter. It is used to fix the size of the region to be analysed, to place probes and decide the wall clearance in wind tunnels, and as the upper limit of the integrals that give $\delta^*$ and $\theta$.

Displacement thickness ($\delta^*$)

It is the distance by which the external flow is shifted away from the wall because of the velocity deficit in the boundary layer:

$$\delta^* = \int_0^{\delta}\left(1 - \frac uU\right)dy$$

Application: In wind tunnels, diffusers and ducts the walls are moved outward (or the area is enlarged) by $\delta^*$ to keep the core velocity constant. It also gives the pressure change in a duct entrance, and the effective contour of an aerofoil or body for the outer potential flow.

Momentum thickness ($\theta$)

It is the thickness of a fluid layer that, moving at velocity $U$, carries the momentum lost because of the boundary layer:

$$\theta = \int_0^{\delta}\frac uU\left(1 - \frac uU\right)dy$$

Application: It is used in the momentum integral equation to find the wall shear stress and the drag force on a surface: $\tau_0 = \rho U^2\,d\theta/dx$, and $F_D = \rho U^2\theta\,b$ for a plate of width $b$ ($\theta$ taken at the trailing edge).