Camber Line Design $(45\backslash \%)$

Using thin$-$airfoil theory, design an airfoil camber line that meets the following th

$\backslash$begin$\{$itemize$\}$

$\backslash$item The lift coe$$cient is $c$\_$l $=0\mathrm{.}5$$

$\backslash$item There is no leading$-$edge suction peak

$\backslash$item The airfoil has $10\backslash \%$ static stability, SM $=10\backslash \%$

$\backslash$end$\{$itemize$\}$

$\backslash$begin$\{$enumerate$\}[$label$=(\backslash$alph$*)]$

$\backslash$item aBased on the requirements, determine the thin$-$airfoil theory constants $B$\_1$$ and $B$\_2$$$.$

$\backslash$item Assume that the camber line is given by

$$

$\backslash$frac$\{$z$\_$c$\}\{$c$\}=$ a $\backslash$frac$\{$x$\}\{$c$\}(1+$b $\backslash$frac$\{$x$\}\{$c$\})$

$$

Determine the constants $a$ and $b$$,$ and the angle of attack $$\backslash$alpha$$,$ such that the design requirements are met. Provide a computer$-$generated plot of the camber line.

$\backslash$item Plot the di$$erence in pressure coe$$cient versus $$\backslash$frac$\{$x$\}\{$c$\}$$$,$

$$

$\backslash$Delta c$\_$p $\backslash$equiv c$\_\{$p$,$lower$\}\backslash$Bigg$(\backslash$frac$\{$x$\}\{$c$\}\backslash$Bigg$)-$ c$\_\{$p$,$upper$\}\backslash$Bigg$(\backslash$frac$\{$x$\}\{$c$\}\backslash$Bigg$)$

$$

and verify numerically, using point values of $$\backslash$Delta c$\_$p$ and trapezoidal integration $($trapz$)$ that

$$

c$\_$l $=\backslash$frac$\{1\}\{$c$\}\backslash$int$\_0^$c $\backslash$Delta c$\_$p$($x$)$ dx

$$

$\backslash$item Verify your answer by running mfoil, starting with the NACA $$0006$$ airfoil, and modifying the camber according to your results in this problem. How well does this airfoil satisfy the given requirements? Include a plot of the pressure coe$$cient$.$

$\backslash$end$\{$enumerate$\}$