# The velocity profile inside the boundary layer for flow over a flat plate is given as $\frac{u}{{{u_\infty }}} = \sin \left( {\frac{\pi }{2}\frac{y}{ 0 votes 127 views in General closed The velocity profile inside the boundary layer for flow over a flat plate is given as \(\frac{u}{{{u_\infty }}} = \sin \left( {\frac{\pi }{2}\frac{y}{\delta }} \right),$Where U is the free stream velocity, δ is the local boundary layer thickness. If δ* is the local displacement thickness, the value of  $\frac{{{\delta ^*}}}{\delta }$ is

1. $\frac{2}{\pi }$
2. $1 - \frac{2}{\pi }$
3. $1 + \frac{2}{\pi }$
4. 0

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Correct Answer - Option 2 : $1 - \frac{2}{\pi }$

Concept:

Displacement thickness is given by

${{\rm{\delta }}^{\rm{*}}} =\int\limits_0^{\rm{\delta }} \left( {1 - \frac{{\rm{u}}}{{{{\rm{U}}_\infty }{\rm{\;}}}}} \right){\rm{dy}}$

Where,

u – velocity of the fluid

U - Free stream velocity

Calculations:

Given:

$\frac{u}{{{u_\infty }}} = \sin \left( {\frac{{xy}}{{2\delta }}} \right)$

$Displacement\;Thickness\;{\delta ^*} = \mathop \smallint \limits_0^\delta \left( {1 - \frac{u}{{{u_\infty }}}} \right)dy$

${\delta ^*} = \mathop \smallint \limits_0^\delta \left( {1 - \sin \frac{{\pi y}}{{2\delta }}} \right)dy$

${\delta ^*} = \left[ {y + \frac{{\cos \frac{{\pi y}}{{2\delta }}}}{{\frac{\pi }{{2\delta }}}}} \right]_0^\delta$

${\delta ^*} = \delta + \frac{{2\delta }}{\pi }\left( 0 \right) - 0 - \frac{{2\delta }}{\pi }$

${\delta ^*} = \delta - \frac{{2\delta }}{\pi } \Rightarrow \frac{{{\delta ^*}}}{\delta } = 1 - \frac{2}{\pi }$