BL, displacement thickness and ratio, 1/7 profile

The ratio of displacement thickness to boundary layer thickness for the 1/7 power-law profile is $\delta^*/\delta = 1/8 = 0.125$.

Boundary layer

When a real fluid flows past a solid surface, the fluid layer touching the surface sticks to it (no-slip). Velocity rises from zero at the wall to the free-stream velocity $U$ over a thin region near the surface. This thin region, where viscous shear is important, is the boundary layer. Outside it the flow behaves as an ideal (inviscid) fluid.

Boundary layer thickness ($\delta$)

It is the distance from the wall at which the velocity reaches 99% of the free-stream velocity: $u = 0.99\,U$ at $y = \delta$.

Displacement thickness ($\delta^*$)

It is the distance by which the external streamlines are shifted away from the wall because of the slow-moving fluid in the boundary layer. Equivalently, it is the thickness of a layer of free-stream flow that has the same discharge as the loss of discharge caused by the boundary layer.

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

Ratio for $u/U = (y/\delta)^{1/7}$

$$\begin{aligned} \delta^* &= \int_0^{\delta}\left[1 - \left(\frac y\delta\right)^{1/7}\right]dy = \delta - \frac{\delta}{1 + 1/7}\\ &= \delta - \frac{7}{8}\delta = \frac{\delta}{8} \end{aligned}$$ $$\frac{\delta^*}{\delta} = \frac18 = 0.125$$

Answer: $\delta^*/\delta = 1/8$, so the displacement thickness is 12.5% of the boundary layer thickness.