• # question_answer If the velocity distribution in a turbulent boundary layer is given by $\frac{u}{{{u}_{\infty }}}={{\left( \frac{y}{\delta } \right)}^{1/9}}$ then the ratio of displacement thickness to nominal boundary layer thickness will be: A) 1.0                               B) 0.6C) 0.3                               D) 0.1

Displacement thickness, $\delta '=\int\limits_{0}^{\delta }{\left( 1-\frac{u}{{{U}_{x}}} \right)}\,\,dy$ $=\int\limits_{0}^{\delta }{\left[ 1-{{\left( \frac{y}{\delta } \right)}^{1/9}} \right]}\,\,dy$ $=\left| y-\frac{1}{{{\delta }^{1/9}}}.{{y}^{10/9}}\times \frac{9}{10} \right|_{0}^{\delta }=\delta =\frac{9}{10}\,\delta =\frac{\delta }{10}$ $\frac{\delta '}{\delta }=\frac{1}{10}=0.1$