In this exercise, you will have to develop a numerical solver to solve for the self$-$similar

boundary layer profile when there are favorable and adverse pressure gradients. To do this, you can

follow a similar numerical approach as was done previously in problem $1,$ but your governing equations

will have to properly account for the accelerating $($or decelerating$)$ outer velocity.

Recall the similarity equation used in the Falkner Skan solution is

$()$

$2$

$10$

f ff f

$\backslash$alpha $\backslash$beta $$

$$

$++=$

$$

$,$

where

$()$

df u

f d Ux

$\backslash$eta

$==$

$,()$

y

x

$\backslash$eta $=\backslash$xi

$,$ and U x$()$ is the velocity outside of the boundary layer. For planar

flows over wedge shapes, $\backslash$alpha $=1,$ and $\backslash$beta $=2$

$\backslash$pi

$\backslash$theta $,$ where $\backslash$theta is the turning angle $($or the half wedge angle$).$

a$)$ For the case of planar stagnation flow $($a turning angle of $90$ degrees$),$ determine a suitable

scaling function $()$

x

$\backslash$xi

$,$ and plot the self similar boundary layer profile.

b$)$ For the case of flow around a wedge with a half angle of $10$ degrees, plot the corresponding selfsimilar velocity profile and compare it to the Blasius solution for flow over a flat plate. Both the

Blasius and wedge flow velocity profiles should be on the same graph.

c$)$ Now consider a case where the flow expands around a $10$ degree corner, meaning the turning

angle is negative $10$ degrees. Plot the self$-$similar velocity profile and compare it to the Blasius

profile on the same graph.

d$)$ Determine the turning angle at which the shear stress at the wall becomes zero and the

boundary layer separates from the surface.

Problem #$3.$ In this exercise, you will have to develop a numerical solver to solve for the self$-$similar boundary layer profile when there are favorable and adverse pressure gradients. To do this, you can follow a similar numerical approach as was done previously in problem $1,$ but your governing equations will have to properly account for the accelerating $($or decelerating$)$ outer velocity.

Recall the similarity equation used in the Falkner Skan solution is

$f^{\text{'}\text{'}\text{'}}+f{f}^{\text{'}\text{'}}+[1-(f^{\text{'}}{)}^2]=0$

where $f^{\text{'}}=\frac{df}{d}=\frac{u}{U(x)},=\frac{y}{(x)},$ and $U(x)$ is the velocity outside of the boundary layer. For planar flows over wedge shapes, $=1,$ and $=\frac{2}{},$ where $$ is the turning angle $($or the half wedge angle$).$

a$)$ For the case of planar stagnation flow $($a turning angle of $90$ degrees$),$ determine a suitable scaling function $(x),$ and plot the self similar boundary layer profile.

b$)$ For the case of flow around a wedge with a half angle of $10$ degrees, plot the corresponding selfsimilar velocity profile and compare it to the Blasius solution for flow over a flat plate. Both the Blasius and wedge flow velocity profiles should be on the same graph.

c$)$ Now consider a case where the flow expands around a $10$ degree corner, meaning the turning angle is negative $10$ degrees. Plot the self$-$similar velocity profile and compare it to the Blasius profile on the same graph.

d$)$ Determine the turning angle at which the shear stress at the wall becomes zero and the boundary layer separates from the surface.