Problem $6(15$ pts$.)$: A low subsonic closed$-$loop wind tunnel is to be fitted with a flow straightener in the settling chamber, immediately upstream from the contraction, to reduce swirl and feed a homogeneous inlet velocity profile into the test section. The honeycomb $($figure a$)$ consists of a grid of $N$ square section ducts $($figure b$)$ of side $a=5mm$ and length $L=4cm,$ completely filling the cross$-$section of the wind tunnel at the installation location in the plenum. The total area is $A=0\mathrm{.}5m^2$ and the average incoming axial velocity is $U=10\frac{m}{s}$ at full power.

Figure a$)$ Flow straightener $($Honeycomb$)$ consisting of $N$ individual cells. b$)$ Detail of one cell: a duct of square cross$-$section. The boundary layer $($BL$)$ develops on all four internal walls.

The boundary layers developing on the interior walls of each duct, to be taken as one$-$sided flat plates, are incompressible and can be considered two$-$dimensional and fully laminar. Both the blockage due to plates thickness and three$-$dimensional effects deriving from finite spanwise size shall be neglected. The air density is $=1\mathrm{.}225k\frac{g}{m^2}$ and the kinematic viscosity is $v=1\mathrm{.}4531{10}^{-5}\frac{m^2}{s}.$

Assume the velocity near the walls can be approximated by $u$~~$U_{}(\frac{2y}{}-\frac{y^2}{{}^2})$ and use the results derived in the lectures for this velocity distribution.

a$)$ Estimate the boundary layer displacement thickness ${}^{**}.$

b$)$ Explain why the cross$-$sectional area of each cell is effectively reduced.

c$)$ Estimate the increase in the flow speed by using the boundary layer displacement thickness.

d$)$ Find the drag force experienced by each individual cell $D_1$ and by the whole grid $D.$

e$)$ What is the additional power that will be required to drive the flow through the wind tunnel once the flow straightener has been installed?