 Lift-induced drag

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.^[ref]

For a small induced angle ${\alpha }_i$, a force $L^{\prime }$ perpendicular to the local flow has freestream components $L=L^{\prime }\cos {\alpha }_i$ and $D_i=L^{\prime }\sin {\alpha }_i$. 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 iswhere $b$ 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.^[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:Take $L=6000\mathrm{N}$, $b=10\mathrm{m}$, $S=12.5{\mathrm{m}}^2$, $e=0.8$, $\rho =1.2kg/{\mathrm{m}}^3$ and $U=30m/s$. Then $AR=8$, $q_{\infty }=540\mathrm{Pa}$, $C_L\approx 0.8889$, and $D_i\approx 265.3\mathrm{N}$. The corresponding induced-power contribution is $D_{i}U\approx 7.96\mathrm{kW}$. 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.