Problem $2$: $(40$ points total$)$

The airfoil section of the wing of the British Spitfire of World War II fame is an NACA $2213$ at the wing root, tapering to an NACA $2205$ at the wing tip. The root chord is $8\mathrm{.}33$ ft $.$ The measured profile drag coefficient of the NACA $2213$ airfoil is $0\mathrm{.}006$ at a Reynolds number of $\backslash (9\backslash$times $10^\{6\}\backslash ).$ Consider the Spitfire cruising at an altitude of $\backslash (18,000\backslash$mathrm$\{$ft$\}\backslash ).$

$($a$)$ At what velocity is it flying for the root chord Reynolds number to be $\backslash (9\backslash$times $10^\{6\}\backslash )?$

$($b$)$ At this velocity and altitude, assuming completely turbulent flow, estimate the skin$-$friction drag coefficient for the NACA $2213$ airfoil, and compare this with the total profile drag coefficient. Calculate the percentage of the profile drag coefficient that is due to pressure drag.

$($c$)$ A more reasonable calculation of the skin friction coefficient would be to assume an initially laminar boundary layer starting at the leading edge, and then transitioning to a turbulent boundary layer at some point downstream. Calculate the skin$-$friction coefficient with assuming a critical Reynolds number of $\backslash (10^\{6\}\backslash )$ for transition.

$($Note: Assume that $\backslash (\backslash$mu $\backslash )$ varies as the square root of temperature.$)$