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:**

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

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

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

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

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

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