Derive Momentum thickness and energy thickness for the given velocity profile

u/U=2(y/δ)- (y/δ)^2

(10 Marks) May-2018

Subject Fluid Mechanics 2

Topic Boundary layer theory

Difficulty Medium

A) Momentum thickness, $\theta$ is,

$\theta = \int_o^\delta \frac{u}{U} (1-\frac{u}{U}) dy$

velocity profile

$\frac{u}{U} = 2 (\frac{y}{ \delta}) - (\frac{y}{\delta})^2 $

$\therefore \theta = \int_o^\delta \{2 (\frac{y}{\delta}) - (\frac{y}{\delta})^2\}$ $\{ 1- 2(\frac{y}{\delta}) - (\frac{y}{\delta})^2 \}$ dy

= $\int_o^\delta [\frac{2y}{\delta} - \frac{y^2}{\delta^2}] [ 1 - \frac{2y}{\delta} + \frac{y^2}{\delta^2}] $ dy

= $\int_o^\delta [\frac{2y}{\delta} - \frac{4y^2}{\delta^2} + \frac{2y^3}{\delta^3} - \frac{y^2}{\delta^2} + \frac{2y^3}{\delta^3} - \frac{y^4}{\delta^4}]$ dy

$= \int_o^\delta (\frac{2y}{\delta} - \frac{5y^2}{\delta^2} + \frac{4 y^3}{\delta^2} - \frac{y^4}{\delta^4}) dy$

= $ [ \frac{2y^2}{2 \delta } - \frac{5y^3}{3 \delta ^2} + \frac{4y^4}{4 \delta ^3} - \frac{y^4}{5 \delta^4}]^\delta$

$= ( \frac{\delta^2}{\delta} - \frac{5\delta^3}{3\delta^2} + \frac{\delta^4}{\delta^3} - \frac{\delta^5}{5\delta^4})$

= $\delta - \frac{5 \delta}{3} = 0 - \frac{\delta}{5} = \frac{2 \delta}{15}$

B) Energy thickness $\delta **$

$\delta** = \int^\delta_o \frac{u}{u} ( 1 - \frac{u^2}{u^2}) dy = \int^\delta_o (\frac{2y}{\delta} - \frac{y^2}{\delta^2}) ( 1 - (\frac{2y}{\delta} - \frac{y^2}{\delta^2})^2) dy$

= = $\int^\delta_o (\frac{2y}{\delta} - \frac{y^2}{\delta^2}) (1 - [4y^2/\delta^2 + y^4/\delta^4 - 4y^3/\delta^3]) dy$

= $ \int^\delta_o (\frac{2y}{\delta} - y^2/\delta^2) ( 1 - 4y^2/ \delta^2 - y^4/ \delta^4 + 4y^3/ \delta^3) dy$

= $ \int^\delta_o ( 2y/ \delta - 8y^2/ \delta^3 - 2y^5/ \delta^5 + 8y^4/ \delta^4 - y^2/ \delta^2 + 4y^4/ \delta^4 + y^6/ \delta^6 - 4y^5/ \delta^5 ) dy$

= $\int^ \delta_o (2y/ \delta - y^2/ \delta^2 - 8y^3/ \delta^3 + 12y^4/ \delta^4 - 6y^5/ \delta^5 + y^6/ \delta^6 ) dy$

= $ [ \frac{2y^2}{2 \delta} - \frac{y^3}{3 \delta^2} - \frac{8y^4}{ \delta^3} + \frac{12y^5}{5 \delta^4} - \frac{6y^6}{6 \delta^5} + y^7/ 7 \delta^6 ] \delta_o$

$= \frac{\delta^2}{\delta} - \frac{\delta^3}{3\delta^2} - \frac{2\delta^4}{\delta^3} + \frac{12\delta^5}{5 \delta^4} - \frac{\delta^6}{\delta^5} + \frac{\delta^7}{7 \delta^6}$

= $\delta - \delta /3 - 2\delta + 12\delta - \delta + \delta /7 = \frac{22\delta}{105}$

(Now, always put $\delta = 1$ in above equation to get $\frac{22}{105}$, which is then multiplied by $\delta$ i.e. $\frac{22}{105} \times \delta = \frac{22}{105} \delta$ as ans)