= Lift-induced drag
{title2=$C_{D_i}=C_L^2/(\pi e\,AR)$}
{wiki}
<Lift-induced drag> is the component of aerodynamic drag associated with the three-dimensional flow that supports a finite wing’s lift. The trailing vorticity produces <downwash>, changing the local relative-flow direction. A <force> approximately perpendicular to that local flow consequently has a component opposing the freestream motion. NASA’s primary account identifies the counter-rotating <wingtip vortices> and the downstream <force> component produced by their induced flow.<Induced Drag Coefficient by NASA Glenn Research Center>[Air Pressure of a Wing and Induced Drag Coefficient]{ref}
For a small induced angle $\alpha_i$, a <force> $L'$ perpendicular to the local flow has freestream components $L=L'\cos\alpha_i$ and $D_i=L'\sin\alpha_i$. Hence $D_i/L=\tan\alpha_i\approx\alpha_i$. This elementary projection explains why changing the flow direction can change drag even without adding direct viscous rubbing to that model.
A useful finite-wing model is
$$
\boxed{C_{D_i}=\frac{C_L^2}{\pi e\,AR},\qquad AR=\frac{b^2}{S},}
$$
where $b$ is <wingspan>, $S$ the wing reference area and $e$ the <span efficiency factor>. Use the same reference area in $C_L=L/(q_\infty S)$ and $C_{D_i}=D_i/(q_\infty S)$, with $q_\infty=\rho U^2/2$. NASA gives this coefficient relation and aspect-ratio definition.<Induced Drag Coefficient by NASA Glenn Research Center>[Drag Coefficient, displayed Cdi and AR equations]{ref} It is a model of the induced contribution, not a statement that total drag has only this dependence in every flow regime.
An original fixed-lift calculation eliminates the coefficients:
$$
\boxed{D_i=\frac{L^2}{\pi e q_\infty b^2}.}
$$
Take $L=6000\,\mathrm N$, $b=10\,\mathrm m$, $S=12.5\,\mathrm{m^2}$, $e=0.8$, $\rho=1.2\,\mathrm{kg/m^3}$ and $U=30\,\mathrm{m/s}$. Then $AR=8$, $q_\infty=540\,\mathrm{Pa}$, $C_L\approx0.8889$, and $D_i\approx265.3\,\mathrm N$. The corresponding induced-power contribution is $D_iU\approx7.96\,\mathrm{kW}$. Doubling the speed while holding lift, span, density and efficiency fixed reduces this modeled drag to about $66.3\,\mathrm N$.
What is held fixed matters. At fixed $C_L$ instead of fixed lift, induced drag grows with dynamic <pressure>. The familiar inverse-square-speed conclusion above therefore cannot be transferred to a test that holds angle of attack or coefficient fixed while changing speed. Changes in efficiency, separation or compressibility can also invalidate the assumed comparison.
<Lifting-line theory> relates spanwise loading, trailing vorticity and induced velocities. The ideal planar minimum for fixed lift and span has <elliptic span loading>; this is a statement about the load distribution, not merely the outline of the wing. Total drag measurements also contain profile and other contributions, so fitting a total-drag curve does not automatically isolate $D_i$.
A proposed wind-tunnel study measures <force> components and freestream conditions for several lift settings, while separately accounting for support loads, wall effects and profile drag. A downstream velocity survey tests the predicted downwash structure. Force balance alone observes total aerodynamic loads; the split into induced and other drag requires the additional flow information or a justified model. Report those assumptions before inferring an efficiency factor from the measured drag.
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