Aircraft Drag Polar

Aircraft parabolic drag polar

$C_D = C_{D0} + K\,C_L^2 \label{eq:Drag:Polar:Parabolic:A}$ $K = \frac{1}{\pi\, \mathrm{A\!R} \, e} \label{eq:K:Drag:Polar:Parabolic}$

$C_{D0}$, parasite drag coefficient at zero lift.
$C_{D\mathrm{i}} = C_L^2/\big(\pi \mathrm{A\!R} \, e\big)$ , drag coefficient due to lift (induced drag).
$e$, Oswald efficiency factor, includes all effects from airplane.
$\mathrm{A\!R}_\mathrm{e}=\mathrm{A\!R}\,e$ , effective aspect ratio.
$b_\mathrm{e}=b\sqrt{e}$, effective wing span.

Parabolic polar:

Actual polar:

Aircraft parabolic and actual drag polar.

Generalized parabolic polar

$C_D = C_{D\mspace{2mu}\mathrm{min}} + K\,\left( C_L - C_{L\mspace{2mu}\mathrm{ideal}} \right)^2 \label{eq:Drag:Polar:Parabolic:B}$

$C_{L\mspace{2mu}\mathrm{ideal}} = C_L @ C_{D\mspace{2mu} \mathrm{min}}$, ideal lift coefficient, i.e. $C_L$ at the angle of attack of minimum $C_D$.

Equation (\ref{eq:Drag:Polar:Parabolic:B}) is unnecessarily too complicated. An acceptable approximation is given by the simple parabolic drag polar (\ref{eq:Drag:Polar:Parabolic:A}).

Validity of aircraft parabolic drag polar for normal operative conditions.

Equivalent parassite area

$f = C_{D0} \, S \label{eq:f:Parassite:Equivalent:Area}$ $f = C_{f\mspace{2mu}\mathrm{e}} \, S_\mathrm{wet} \label{eq:f:Parassite:Equivalent:Area:Swet}$ $f = C_{f\mspace{2mu}\mathrm{e}} \, S_\mathrm{wet} \quad \Rightarrow \quad C_{D0} = \frac{f}{S} \quad \Rightarrow \quad C_{D0} = C_{f\mathrm{e}} \, \frac{S_\mathrm{wet}}{S} \label{eq:CD0:f:Swet}$ $C_{f\mspace{2mu}\mathrm{e}} = 1.5 \cdot C_{f\mspace{2mu}\mathrm{turb}} \label{eq:Cfe:Cfturb}$

$C_{f\mspace{2mu}\mathrm{turb}}$, friction coefficient of a flat plate aligned with the flow, calculated in turbulent regime at the aircraft flight Reynolds number $\mathrm{Re} = \rho V \bar{c}/\mu$.

Oswald efficiency factor

$e = f \left( M, \mathrm{A\!R}, \Lambda_\mathrm{le} \right) \label{eq:e:Drag:Polar}$

(various calculation methods in literature)

Flight Mechanics for Pilots

Teaching material for student pilots of the Italian Air Force Academy

Aircraft Drag Polar

Aircraft parabolic drag polar

Generalized parabolic polar

Equivalent parassite area

Oswald efficiency factor