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 …