Question

The velocity profile inside the boundary layer for flow over a plate is given as${\frac{u}{U}}_{\infty }=sin\left(\frac{\pi }{2}\frac{y}{\delta }\right)$ where $U_{\infty }$ is the free stream velocity and $\delta$ is the local boundary layer thickness. If $\delta \ast$ is the local displacement thickness, the value of $\frac{{\delta }^{\ast }}{\delta }$ is

• A. $\frac{2}{\pi }$
• B. $1-\frac{2}{\pi }$
• C. $1+\frac{2}{\pi }$
  
• D. 0

Answer 1

The correct option is B $1-\frac{2}{\pi }$

$Given,\frac{u}{U_{\infty }}=sin\left(\frac{\pi }{2}\frac{y}{\delta }\right)$

$where\delta =BoundarylayerthicknessLocaldisplacementthickness,$

${\delta }^{\ast }=\underset{0}{\overset{\delta }{\int }}(1-\frac{u}{U_{\infty }})dy=\underset{0}{\overset{\delta }{\int }}[1-sin\left(\frac{\pi y}{2\delta }\right)]dy$

$={[y+\frac{2\delta }{\pi }\times cos\left(\frac{\pi y}{2\delta }\right)]}_0^{\delta }=[\delta +0-0-\frac{2\delta }{\pi }]$

${\delta }^{\ast }=\delta (1-\frac{2}{\pi })$

$or\frac{{\delta }^{\ast }}{\delta }=1-\frac{2}{\pi }$