Span efficiency factor ($e$)

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.