Lift-induced drag ($C_{D_i}=C_L^2/(\pi eAR)$)

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.

Downwash is a downward component of fluid velocity, commonly the induced flow associated with a lifting wing and its trailing vortices. For positive upward lift, this flow changes the direction of the local relative wind and contributes to lift-induced drag. NASA's primary explanation connects the trailing vortex system, reduced effective incidence and rearward component of the aerodynamic force.^[ref]

Choose streamwise velocity $U>0$ and let $w>0$ denote downward induced speed at a section. The induced angle and effective angle of attack areAn ideal local lifting force normal to that relative flow is tilted rearward. Its vertical and downstream projections obey $D_i/L=\tan {\alpha }_i=w/U$; hence $D_i\approx L{\alpha }_i$ for small induced angles. This projection isolates the induced contribution in the stated model; it does not include skin friction or every source of total drag.

For an original uniform-downwash example, take $U=50m/s$, $w=2m/s$ and total lift $L=10000\mathrm{N}$. Then ${\alpha }_i\approx 2.291^{\circ }$ and $D_i=400\mathrm{N}$ in the projection model. The power required to overcome this contribution is $U{D}_i=20\mathrm{kW}$. At geometric incidence $6^{\circ }$, the effective incidence is about $3.709^{\circ }$.

A real finite wing generally has span-dependent loading and induced speed. In small-angle lifting-line theory,Using an unweighted arithmetic mean of measured $w$ need not reproduce that integral. For elliptic span loading, the ideal lifting-line downwash is constant across the span and ${\alpha }_i\simeq C_L/(\pi AR)$, consistent with $C_{D_i}=C_L^2/(\pi AR)$. These statements concern the ideal planar model rather than arbitrary wakes or separated flow.

A proposed observation measures flow direction and speed at a specified plane around a loaded wing, using calibrated directional probes or timed tracer images. Repeat with zero and reversed lift, and compare the sign and spatial distribution of the induced flow. Record the freestream reference and wing incidence as well as the probe positions. Wake-plane dependence of a downwash measurement matters: a velocity measured well downstream cannot simply be inserted as the wing-section induced speed without a model connecting the two planes. Visible tracer deflection demonstrates flow direction only to the extent that the tracer follows the fluid; it is not by itself a quantitative lift or drag measurement.

Wake-plane dependence of a downwash measurement means that a measured downward velocity depends on where the observation plane lies relative to the lifting wing and its evolving wake. The induced velocity entering a wing-section force calculation is not generally equal to a downstream probe reading. Classical lifting-line theory explicitly adjusts local incidence using the induced flow at the lifting line, under assumptions such as steady flow, incompressible flow and a nearly planar high-aspect-ratio wing.^[ref]

An original straight-filament model isolates one reason for the difference. Consider a vortex filament of circulation magnitude $\Gamma$ starting at streamwise coordinate zero and extending downstream forever. At a point with streamwise coordinate $x$ and perpendicular distance $r>0$, its induced azimuthal speed isThe integral follows by differentiating $(s-x)/[r\sqrt{(s-x{)}^2+r^2}]$. At $x=0$, only one half of the corresponding infinite straight filament contributes; far downstream, the local filament extends far in both axial directions relative to $r$.

For two straight trailing filaments at transverse positions $y=\pm a$ with opposite circulation signs chosen to induce downward flow between them, their midpoint contributions add:With stipulated $\Gamma =20{\mathrm{m}}^2/s$ and $a=5\mathrm{m}$, this trailing-filament contribution is $0.6366m/s$ at $x=0$, $1.0868m/s$ at $x=5\mathrm{m}$, and approaches $1.2732m/s$. This is an idealized geometry calculation, not a measured wake or a universal factor-of-two correction for every wing.

The model omits finite vortex cores, the bound-vortex contribution away from the lifting line, the complete distributed vortex sheet, wake deformation and wake contraction in a downwash survey. Each can change the spatial transfer. In particular, a wing-plane value and a far-wake value must be connected using the same load and wake model before using one to infer the other.

In the small-angle lifting-line model, the wing-plane induced-drag calculation weights downwash by the circulation or local lift. Its source derivation integrates the local induced contribution along the wing.^[ref] Substituting an arbitrary downstream speed into that expression can misestimate drag even if the speed measurement itself is accurate.

A proposed survey measures velocity components at several downstream stations with the same probe calibration and freestream reference, covering both vortex cores and the intervening region. Record the wing loading, probe orientation and cross-plane coordinates, and track how vortex separation and circulation estimates change. A centerline reading alone does not recover the whole momentum field. Matching an axial trend tests the proposed transfer model only over the surveyed region; it does not establish that a rolled-up real wake remains a pair of straight mathematical filaments.

A wingtip vortex is a concentrated rotating part of the wake behind a finite lifting wing. Spanwise variation of the wing’s circulation produces trailing vorticity, which can roll up into a strong counter-rotating pair. The two rotating wakes induce downward motion between them and upward motion farther outside. NASA’s account connects the tip-region pressure-driven flow with the pair and its induced change of local flow direction.^[ref]

A useful idealization treats each concentrated wake as a straight line vortex. Outside its core, circular symmetry and the definition of circulation giveFor stipulated $|\Gamma |=20{\mathrm{m}}^2/s$ and $r=0.50\mathrm{m}$, the ideal tangential speed is about $6.37m/s$. The singularity as $r\to 0$ is a defect of the line model: a real vortex has a finite core and a nonuniform vorticity distribution.

Consider an original cross-flow model with cores separated horizontally by $d$, strengths $-\Gamma$ and $+\Gamma$, and orientations chosen to give downwash between them. Each ideal vortex induces downward speed $|\Gamma |/(2\pi d)$ at the other core. At $d=10\mathrm{m}$ and $|\Gamma |=20{\mathrm{m}}^2/s$, this pair-interaction speed is about $0.318m/s$. At the midpoint, adding their two velocities gives $2|\Gamma |/(\pi d)\approx 1.273m/s$ downward. These are distinct locations and distinct speeds.

The wake is not generally two permanently straight tubes with fixed strength. A distributed trailing sheet rolls up, the cores move under induced velocities, and viscosity, turbulence and ambient motion alter the field. A wing’s actual span loading and operating state determine the wake, so the illustrative circulation and core size cannot be assigned to an aircraft merely from its visible span.

The connection to lift-induced drag is the complete induced-flow and load system. A short line sketched at each wingtip is not an extra independent friction force to add to an already complete drag calculation. The flow carries momentum and kinetic energy while changing the force direction experienced by the finite wing.

A proposed laboratory wake survey places a finite wing at a controlled load in a wind tunnel and measures transverse velocity at a downstream plane using calibrated directional probes or timed tracer images. Reverse the sign of lift to check the expected reversal of rotation, and compare measured circulation on several enclosing paths outside the cores. Tracers must follow the flow adequately, while finite sampling, tunnel walls and the probe’s own disturbance limit the inference. A visible tracer pattern alone does not supply the full three-dimensional velocity or pressure field.

The span efficiency factor $e$ compares a wing’s lift-induced drag with a reference induced-drag level at the same lift, span and dynamic pressure. In the conventional finite-wing coefficient model,It describes the induced contribution under stated geometry and flow assumptions. NASA’s primary account gives this relation and identifies the elliptic-load ideal in its lifting-line discussion.^[ref]

For an ideal planar wing in classical lifting-line theory, elliptic span loading minimizes induced drag for fixed lift and span, corresponding to $e=1$ in this normalization. Deviations of the loading can increase induced drag and lower $e$. The load distribution depends on twist, section properties and operating conditions as well as planform. An outline alone therefore does not specify a universal efficiency value. Nonplanar systems and different reference conventions require their own interpretation before applying the planar bound.

For an original coefficient example, choose $C_L=0.80$, $AR=8$ and $e=0.80$. Then $C_{D_i}\approx 0.031831$. With $q_{\infty }=500\mathrm{Pa}$ and $S=20{\mathrm{m}}^2$, lift is $8000\mathrm{N}$ and induced drag is about $318.3\mathrm{N}$. At $e=1$ with those other quantities fixed, induced drag would be about $254.6\mathrm{N}$. Thus the chosen $e=0.8$ corresponds to a 25-percent induced-drag increase over that reference, not a statement that twenty percent of all input power is lost.

A drag-polar fit illustrates why efficiency definitions can differ. Suppose a model haswhere $k_v$ is an additional lift-dependent profile-drag coefficient and $e_i$ is the induced efficiency. Fitting the entire quadratic term as $C_L^2/(\pi AR{e}_{\mathrm{fit}})$ givesFor stipulated $e_i=0.90$, $AR=8$ and $k_v=0.010$, the fitted value is about $0.734$. The algebra does not identify $e_{\mathrm{fit}}$ with the induced-only factor. It shows how a total-drag fit can absorb another modeled contribution even when the induced flow itself has not changed.

The inversion is also sensitive to a small extracted drag. At fixed $C_L$ and $AR$, a small fractional error in $C_{D_i}$ gives approximately the opposite fractional error in $e$. If induced drag is obtained by subtracting a large estimated profile contribution from total drag, uncertainty in that subtraction can dominate. Different reference areas or lift conditions also change the comparison.

A proposed test combines measured lift and drag with a separately justified profile-drag estimate or a calibrated wake-based induced-drag inference. Repeat over the intended coefficient range and report the geometry, area convention, flow regime and uncertainty. A single fitted number is useful only alongside the model that separates induced drag from the rest of the measured aerodynamic force.

Elliptic span loading is a distribution of lift per unit span proportional to a half-ellipse, vanishing at both tips of a planar wing. For span $b$ and total lift $L$, writeThe normalization follows by setting $s=2y/b$ and using ${\int }_{-1}^1\sqrt{1-s^2}ds=\pi /2$. It specifies a load distribution, rather than the wing outline. Classical lifting-line theory identifies this loading as the minimum-induced-drag solution for fixed total lift and span in its ideal planar setting.^[ref]

The loading corresponds to circulation magnitude $\Gamma (y)={\Gamma }_0\sqrt{1-(2y/b{)}^2}$, with $L^{\prime }(y)=\rho U\Gamma (y)$ under the chosen positive-lift convention. Its ideal downwash at the lifting line is constant,Using dynamic pressure $q=\rho U^2/2$ recovers $D_i=L^2/(\pi q{b}^2)$ and the span efficiency factor $e=1$ in that normalization.

A Fourier series makes the minimum transparent. In the classical lifting-line expansion, put $y=(b/2)\cos \vartheta$ andAt fixed $C_L$ and $AR$, $A_1$ is fixed and every higher-order contribution to drag is nonnegative. The minimum has only $A_1$, yielding the half-ellipse. A higher-harmonic penalty in span loading quantifies departure from it: if $A_3=0.1A_1$ is the only additional term, $1/e=1+3(0.1{)}^2=1.03$, so $e\approx 0.971$ and induced drag rises three percent under the stated constraints.

For an original loading example, $L=6000\mathrm{N}$ and $b=10\mathrm{m}$ give a central load $763.94N/m$. At $y=b/4$ it is $0.8660$ of the central value. With $\rho =1.2kg/{\mathrm{m}}^3$ and $U=50m/s$, the model gives ${\Gamma }_0\approx 12.732{\mathrm{m}}^2/s$, $w\approx 0.6366m/s$ and $D_i\approx 76.39\mathrm{N}$.

A proposed test integrates measured surface pressure around several wing sections to estimate $L^{\prime }(y)$, then compares the normalized distribution and a wake survey with the model. An elliptic planform does not alone guarantee elliptic loading: section characteristics, incidence variation, deformation and operating conditions also enter. Nonplanar lifting systems, fixed bending-moment constraints and viscous or compressible effects change the optimization problem; the fixed-lift, fixed-span planar result should not be generalized beyond its assumptions.

A higher-harmonic penalty in span loading is the increase of ideal lift-induced drag caused by nonfundamental terms in a wing's circulation distribution at fixed total lift and span. In the classical planar lifting-line theory expansion, choose $y=(b/2)\cos \theta$ andHere $b$ is span, $U$ is freestream speed, and the coefficients $A_n$ are dimensionless. The primary force calculation uses this circulation expansion, the local lift law and sine orthogonality to obtain the lift and induced drag.^[ref]

Because $L^{\prime }(y)=\rho U\Gamma (y)$ and $|dy|=(b/2)\sin \theta d\theta$, the integral for total lift selects only $A_1$. The induced angle contains ${\sum }_{n}n{A}_n\sin (n\theta )/\sin \theta$. Integrating its product with the loading then givesThe orthogonality identity behind both reductions isA higher harmonic can thus have zero net lift contribution while still adding positive induced drag. Its effect on local loading has not disappeared merely because its span integral vanishes.

For nonzero $C_L$, division by the elliptic reference givesThe pure first harmonic gives elliptic span loading and $e=1$. At a fixed relative coefficient magnitude, a larger order carries a larger penalty. Changing a coefficient's sign can redistribute the loading without changing its own squared contribution in this ideal model.

For an original example, take $A_3=0.10A_1$, $A_5=0.05A_1$ and all other higher terms zero. The excess is $3(0.10{)}^2+5(0.05{)}^2=0.0425$, so $e=1/1.0425\approx 0.95923$. With $C_L=0.8$ and $AR=8$,At $q=500\mathrm{Pa}$ and $S=20{\mathrm{m}}^2$, the corresponding induced drags are about $254.65$ and $265.47\mathrm{N}$. The higher harmonics add $4.25\%$ to this induced contribution, rather than $4.25\%$ to every source of aircraft drag.

A truncation error in an inferred span-efficiency factor matters when fitting sparse load measurements. Omitting nonzero terms makes the ideal-model drag sum too small at a fixed known $A_1$, while fitting noisy high orders can inflate it because the terms are squared and weighted by order. If a justified bound is ${\sum }_{n\ge 2}n(A_n/A_1{)}^2\le \delta$, then $e\ge 1/(1+\delta )$. A fit residual alone does not supply that bound.

A proposed experiment reconstructs section lift or circulation from a spanwise pressure survey, fits a declared set of modes, and compares the predicted induced contribution with an independent wake-based estimate. Report sampling density, load uncertainty and truncation choice. Viscosity, compressibility, nonplanar geometry or constraints other than fixed lift and span change the physical optimization problem; the quadratic penalty belongs to the stated lifting-line model.