Problem

From flow experiments with a smooth flat plate we conclude that the velocity profile in a laminar bound$-$

ary layer can be represented as a sine function:

$\frac{u}{u_0}=sin(\frac{}{2}),$ for $,01$

with $=\frac{y}{}.u_0$ is the speed of the external flow parallel to the plate.

$($a$)$ Derive the equations for momentum thickness $,$ boundary layer thickness $,$ and wall shear stress

${}_w$ as functions of the coordinate $x$ along the plate.

$(8$ of $40$ points$)$

$($b$)$ Find the function for the displacement thickness ${}_1(x)$

$(2$ of $40$ points$)$

$($c$)$ Create a plot with curves for the thicknesses $,{}_1,$ and $$ as a function of $x$ for the following flow

conditions: flow velocity $u_0=0\mathrm{.}8\frac{m}{s},$ density $=998\mathrm{.}5986k\frac{g}{m^3},$ kinematic viscosity

$u=1\mathrm{.}0542.$

${10}^{-6}\frac{m^2}{s}$ and $0mx1\mathrm{.}0m.$

Add curves for the results obtained by Blasius.

$(2$ of $40$ points$)$

$($d$)$ Is the assumption of laminar flow in the boundary layer justified for the whole plate given the flow

data above? Reason your answer.

$(2$ of $40$ points$)$

$($e$)$ Create a plot with a curve for the wall shear stress ${}_w$ over $x$ for the same flow conditions. Add a

curve for the result obtained by Blasius.

$(2$ of $40$ points$)$

$($f$)$ Derive the equation for the local friction coefficient $C_{f_x}$ for the velocity profile $(1)$ and compare it to

Blasius' exact result