$($a$)$ Derive an expression for the displacement thickness $$ for flow over a flat plate subject to a zero freestream

pressure gradient.

$(8$ marks$)$

$($b$)$ A small open circuit wind tunnel, a schematic of which is shown in the figure below, is being designed for

calibration of hot wire anemometers. The air is at $20C.$ The test section of the wind tunnel is $30cm$ in

diameter and $30cm$ in length. The flow through the test section must be as uniform as possible. The wind

tunnel speed ranges from $1$ to $8\frac{m}{s},$ and the design is to be optimised for an air speed of $V=4\frac{m}{s}$ through the

test section.

$($i$)$ For the case of nearly uniform flow at $4\frac{m}{s}$ at the test section inlet, by how much will the centreline air

speed change by the end of the test section?

$(12$ marks$)$

$($ii$)$ Recommend a design that will lead to a more uniform test section flow.

$(5$ marks$)$

The following equations can be used without proof:

$\frac{{}_{lyn}^{**}}{x}=\frac{1\mathrm{.}721}{R{e}_x^{\frac{1}{2}}},\frac{{}_{tyb}^{**}}{x}=\frac{0\mathrm{.}02}{R{e}_x^{\frac{1}{7}}}$