Question

What is the momentum thickness for the boundary layer with velocity distribution $\frac{u}{U}=\frac{y}{\delta }$?

• A. $\delta /6$
• B. $\delta /2$
• C. $3\delta /2$
• D. $2\delta$

Answer 1

The correct option is A $\delta /6$

The momentum thickness is given by
$\theta ={\int }_0^{\delta }\frac{u}{U}(1-\frac{u}{U})dy={\int }_0^{\delta }\frac{y}{\delta }(1-\frac{y}{\delta })dy={\int }_0^{\delta }(\frac{y}{\delta }-\frac{y^2}{{\delta }^2})dy=\frac{1}{{\delta }^2}{\int }_0^{\delta }(\delta y-y^2)dy\frac{1}{{\delta }^2}{[\frac{\delta y^2}{2}-\frac{y^3}{3}]}_0^{\delta }=\frac{1}{{\delta }^2}[\frac{{\delta }^3}{2}-\frac{y^3}{3}]=\delta \left[\frac{3-2}{6}\right]=\frac{\delta }{6}$