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A bound vortex in an aerodynamic model is a vortex-filament representation attached to the modeled lifting body, rather than a wake element convected downstream. In lifting-line theory, a spanwise distribution of circulation $\Gamma (y)$ represents the wing loading. The word bound describes the representation's relation to the body; it does not identify a literal tube of fluid fixed inside the wing.

The local Kutta–Joukowski theorem connects the modeled circulation to lift per unit span:The MIT derivation uses that connection to couple the section law to the induced flow.^[ref] Here $c(y)$ is airfoil chord; the sign of $\Gamma$ is chosen consistently with positive lift.

An original example stipulates $\rho =1.2$ kg/m${}^3$, $U=40$ m/s and $\Gamma =5$ m${}^2$/s at one section. The predicted $L^{\prime }$ is 240 N/m. For chord 0.50 m, $q=960$ Pa and $c_l=240/(960\cdot 0.50)=0.50$. The same circulation at a different chord need not give the same sectional coefficient.

The finite-span model includes the wake. A horseshoe vortex joins a finite bound segment to trailing legs; continuously varying spanwise circulation generates a distributed trailing vortex sheet. Adjacent modeled segments with circulations 5 and 3 m${}^2$/s have a circulation difference of magnitude 2 m${}^2$/s, which determines the intervening wake-filament strength under the chosen orientation. Omitting those trailing contributions would omit the downwash that couples the wing sections.

Total lift follows from $L=\rho U\int \Gamma (y)dy$, but that single integral does not uniquely determine the distribution. Different span loadings can have equal total lift and different lift-induced drag; the originating force calculation keeps the circulation distribution when evaluating drag.^[ref]

A proposed apparatus comparison reconstructs section loading from surface pressures and independently measures the downstream velocity field. Agreement tests the lifting-line representation and wake prediction. A force reading alone cannot locate a unique physical vortex filament, and the model does not resolve the full viscous flow near the surface.

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.

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.

The induced angle of attack ${\alpha }_i$ is the change between the geometric flow direction and the local effective flow direction caused by the wing's induced velocity. In the usual small-angle lifting-line theory convention, downward downwash is positive andHere $U$ is the reference streamwise speed and $w$ the local downward speed. The originating lecture notes explicitly distinguish geometric, induced and effective incidence.^[ref]

If the local velocity has only these two components, its geometric inclination is $\arctan (w/U)$. The ratio $w/U$ is the small-angle approximation, with angles in radians. An original example uses $U=40$ m/s and $w=1.0$ m/s: ${\alpha }_i=\arctan (0.025)\simeq 1.432{}^{\circ }$. A geometric incidence of $5{}^{\circ }$ then gives effective incidence about $3.568{}^{\circ }$. This does not subtract the zero-lift angle of attack, which is a separate term in the section lift law.

For ideal elliptic span loading, the angle is constant across the span and approximatelyThe latter relation is the lift-induced drag contribution, with unit span efficiency factor.^[ref] For stipulated $C_L=0.60$ and $AR=8$, the approximation gives ${\alpha }_i\simeq 0.02387$ radians and $C_{D_i}\simeq 0.01432$.

A nonuniform wing generally has different local induced angles. Its total induced drag involves the loading-weighted integral $D_i\simeq \int L^{\prime }(y){\alpha }_i(y)dy$; an unweighted arithmetic mean of wake angles need not reproduce that integral. A wake velocity traverse and sectional surface pressure integration for airfoil lift test different pieces of this prediction.

The angle concerns induced flow at the modeled lifting line. A velocity measurement far downstream requires the corresponding wake model before being substituted into this formula. The small-angle, attached-flow approximation also does not account for all sources of drag or for strong crossflow.

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.

Lifting-line theory represents a finite aircraft wing by a spanwise bound vortex whose circulation varies with position. The trailing vortex filaments induce downwash, reducing the local effective angle of attack. In its classical form, the model assumes steady, incompressible potential flow, a comparatively large wing aspect ratio, small sweep and locally approximately two-dimensional section behavior.^[ref]

Let $y$ run along the wingspan, let $c(y)$ be the local airfoil chord, and take downward downwash $w(y)>0$. Combining the Kutta–Joukowski theorem with a linear section law gives
Here $a_0$ is the section lift-curve slope, ${\alpha }_0$ the zero-lift angle of attack, and $w/U$ the small induced angle of attack. Because the wake induced by the entire circulation distribution determines $w(y)$, these equations couple different sections. Solving independent two-dimensional airfoils and adding their lift misses that coupling.^[ref]

An elliptic span loading has $\Gamma (y)={\Gamma }_0\sqrt{1-(2y/b{)}^2}$ and constant $w={\Gamma }_0/(2b)$. Integration gives
This is the unit span efficiency factor case.^[ref] For stipulated $b=10\mathrm{m}$, $U=30m/s$, $\rho =1.2kg/{\mathrm{m}}^3$ and $L=6000\mathrm{N}$, it gives ${\Gamma }_0\simeq 21.22{\mathrm{m}}^2/s$, $w\simeq 1.061m/s$ and $D_i\simeq 212.2\mathrm{N}$.

A proposed test compares sectional surface pressure integration for airfoil lift with a traverse of the downstream velocity field. The two observations test the predicted loading and wake, rather than only one total force. Flow separation, strong sweep and compressibility require further modeling; the classical calculation does not predict all drag or aerodynamic stall.

The section lift-curve slope $a_0$ is the local derivative of a two-dimensional section lift coefficient with respect to its effective angle of attack, at stated flow conditions:The local linear approximation is $c_l\simeq a_0({\alpha }_{\mathrm{eff}}-{\alpha }_0)$, where ${\alpha }_0$ is the zero-lift angle of attack. Lifting-line theory uses this section relation while separately solving for the induced flow from the finite wing.^[ref]

The angle unit matters. In incompressible thin airfoil theory, the ideal slope is $2\pi$ per radian, or $2\pi (\pi /180)\simeq 0.1097$ per degree.^[ref] A tabulated slope of 0.10 per degree is about 5.73 per radian. Treating 0.10 as a per-radian value introduces a factor of about 57.3.

For an original section model with $a_0=6.0$ per radian and ${\alpha }_0=-2{}^{\circ }$, an effective incidence of $3{}^{\circ }$ givesThe relevant angular increment is five degrees, not three. With sectional dynamic pressure $q=500$ Pa and airfoil chord $c=0.30$ m, the resulting lift per unit span is $L^{\prime }=qc{c}_l\simeq 78.54$ N/m.

The finite-wing lift-curve slope is different because increased lift changes the downwash. For an ideal untwisted elliptic wing with uniform section slope, ${\alpha }_i=C_L/(\pi AR)$ givesTaking $a_0=2\pi$ and wing aspect ratio $AR=8$ yields $a\simeq 5.027$ per radian. This particular reduction is a consequence of the stated ideal loading, not a universal correction for arbitrary swept or separated wings.

A proposed wind tunnel measurement fits nearby section-force measurements against incidence while recording Reynolds number and Mach number. The fitted interval should remain within the approximately linear regime; aerodynamic stall can make one constant slope inadequate.

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.