PROBLEM $5\mathrm{.}1$

For boundary layer flow over the wing of a commercial aircraft is modeled as a straight flat plate with an approximate boundary layer velocity profile of

$u(y)=V_{}sin(\frac{y}{2}),0y$

The air stream velocity is $350f\frac{t}{sec}$ at $40,000ft,$ and the location of interest is $x=10ft.$ Assume standard air properties apply.

Boundary condition verification:

$($a$)$ Verify that this velocity profile satisfies the no$-$slip condition. $)=(0$

$($b$)$ Verify this velocity profile matches the free stream velocity at the edge of the boundary layer. at $y=)$

$($c$)$ Verify the smooth transition at the edge of the boundary layer. at $($:$y=\}$

Turbulent or Laminar?

$($a$)$ Determine the Reynolds number.

$R{e}_x=\frac{V_{}x}{?}$

$u$

where

$u$ is the kinematic viscosity. $($see Tables from standard atmospheric air$)$

$($b$)$ Is it laminar or turbulent? $($See lecture slides on boundary layer$)$

$3.$ Boundary layer characteristics: $($No need to evaluate integrals by hand. Evaluate the final expression using a computer software and provide proof, i$.$e$.,$ printout, screenshot, etc.$)$

$($a$)$ Determine the boundary layer thickness, $.($See lecture slides for laminar or turbulent incompressible flow.$)$

$($b$)$ Use the following equation to determine the ratio of the displacement thickness to the boundary layer thickness.

${}^{**}={\displaystyle \underset{0}{\overset{}{}}}(1-\frac{u}{V_{}})dy,\frac{{}^{**}}{}=$

$($c$)$ Use the following equation to determine the ratio of the momentum thickness to the boundary layer thickness.

$={\displaystyle \underset{0}{\overset{{}_0}{}}}(-\frac{u}{V_{}})(1-\frac{u}{V_{}})dy,\frac{}{}=$