Lifting-line theory

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.

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.

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.