For the following velocity profiles, determine whether the flow has separated or on the verge of separation or will be attached with the surface.

1. $\frac{u}{U}=\frac{3}{2}(\frac{y}{\delta})-\frac{1}{2}(\frac{y}{\delta})^2$

2. $\frac{u}{U}=2(\frac{y}{\delta})^2-(\frac{y}{\delta})^3$

3. $\frac{u}{U}=-2(\frac{y}{\delta})+(\frac{y}{\delta})^2$

Solution :

Given :

$1^{st}$ Velocity Profile

$$\begin{aligned} \frac{u}{U} &=\frac{3}{2}\left(\frac{y}{\delta}\right)-\frac{1}{2}\left(\frac{y}{\delta}\right)^{3} \\ \quad u &=\frac{3 U}{2}\left(\frac{y}{\delta}\right)-\frac{U}{2}\left(\frac{y}{\delta }\right)^{3} \end{aligned}$$

Differentiating w.r.t. $y$, $$ \frac{\partial u}{\partial y}=\frac{3 U}{2} \times \frac{1}{\delta }-\frac{U}{2} \times 3\left(\frac{y}{\delta }\right)^{2} \times \frac{1}{\delta } $$

At $y=0$,

$$\begin{aligned}\left(\frac{\partial u}{\partial y}\right)_{y=0} &=\frac{3 U}{2\delta }-\frac{3 U}{2}\left(\frac{0}{\delta }\right)^{2} \times \frac{1}{\delta }\\ &=\frac{3 U}{2\delta …