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

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

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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 }\end{aligned}

As $\left(\frac{\partial u}{\partial y}\right)_{y=0}$ is positive.

Hence flow will not separate or flow will remain attached with the surface.

$2^{nd}$ Velocity Profile

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

Differentiating w.r.t. $y$, $$\frac{\partial u}{\partial y}=4 U \left(\frac{y}{\delta}\right) \times \frac{1}{\delta}-3U \left(\frac{y}{\delta}\right)^{2} \times \frac{1}{\delta}$$

At $y=0,$

\begin{aligned}\left(\frac{\partial u}{\partial y}\right)_{y=0} &=4 U \times \left(\frac{0}{\delta}\right) \frac{1}{\delta}-3U \times \left(\frac{0}{\delta}\right)^{2} \frac{1}{\delta}\\ &=0 \end{aligned}

As $\left(\frac{\partial u}{\partial y}\right)_{N=0}=0$, the flow is on the verge of separation.

$3^{rd}$ Velocity Proflle

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

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

at $y=0,$

\begin{aligned}\left(\frac{\partial u}{\partial y}\right)_{y=0} &=-\frac{2 U}{\delta}+2 U\left(\frac{\partial}{\delta}\right) \times \frac{1}{\delta} \\ &=-\frac{2 U}{\delta}\end{aligned}

As $\left(\frac{\partial u}{\partial y}\right)_{y=0}$ is negative the flow has separated.