An object with square cross$-$section $($and side length d$)$ is centrally placed in a channel with width $4$d$.$ The

flow is two$-$dimensional $($nothing varies in the out$-$of$-$page direction$).$ The upstream flow $($at section $1)$ has uniform

velocity V$1$ and uniform pressure p$1,$ and we assume that the flow downstream $($at section $2)$ has parallel streamlines

and has a uniform pressure p$2.$ At section $2,$ assume that there is a central wake region of height w with zero velocity

and a uniform velocity V$2$ in the region of the channel above and below $($see Figure$).$ You may make any other suitable

assumptions as needed.

$($a$)(8$ points$)$ Obtain the drag coefficient CD $=$ FD$/(0\mathrm{.}5\backslash$rho V $2$

$1$ A$)$ for the object as a function of w$.$ Here FD denotes

the drag force exerted on the body, A denotes the frontal area of the object, equal to d $\backslash$times $1,$ assuming unit depth

in out$-$of$-$plane direction. What is CD when w $=2$d$?$

$($b$)(2$ points$)$ What is CD when w $=0?$ How do you explain the deviation of this solution from the true value of

CD $2?$