Lifting-line theory
The Lanchester-Prandtl lifting-line theory[1] is a mathematical model in aerodynamics that predicts lift distribution over a three-dimensional wing from the wing's geometry.
[2] The theory was expressed independently[3] by Frederick W. Lanchester in 1907,[4] and by Ludwig Prandtl in 1918–1919[5] after working with Albert Betz and Max Munk.
[6][7] It is difficult to predict analytically the overall amount of lift that a wing of given geometry will generate.
One might expect that understanding the full wing simply involves adding up the independently calculated forces from each airfoil segment.
Lifting-line theory corrects some of the errors in the naive two-dimensional approach by including some interactions between the wing slices.Lifting line theory supposes wings that are long and thin with negligible fuselage, akin to a thin bar (the eponymous "lifting line") of span 2s driven through the fluid.
From the Kutta–Joukowski theorem, the lift L(y) on a 2-dimensional segment of the wing at distance y from the fuselage is proportional to the circulation Γ(y) about the bar at y.
When the aircraft is stationary on the ground, these circulations are all equal, but when the craft is in motion, they vary with y.
By Helmholtz's theorems, the generation of spatially-varying circulation must correspond to shedding an equal-strength vortex filament downstream from the wing.
[8] In the lifting line theory, the resulting vortex line is presumed to remain bound to the wing, so that it changes the effective vertical angle of the incoming freestream air.
The vertical motion induced by a vortex line of strength γ on air a distance r away is γ⁄4πr, so that the entire vortex system induces a freestream vertical motion at position y of
This flow changes the effective angle of attack at y; if the circulation response of the airfoils comprising the wing are understood over a range of attack angles, then one can develop an integral equation to determine Γ(y).
[9] Formally, there is some angle of orientation such that the airfoil at position y develops no lift.
Suppose the freestream flow attacks the airfoil at position y at angle α(y) (relative to the liftless angle for the airfoil at position y — thus a uniform flow across a wing may still have varying α(y)).
By the small-angle approximation, the effective angle of attack at y of the combined freestream and vortex system is α(y)+w(y)⁄V.
Combining the above formulae, All the quantities in this equation except V and Γ are geometric properties of the wing, and so an engineer can (in principle) solve for Γ(y) given a fixed V. As in the derivation of thin-airfoil theory, a common approach is to expand Γ as a Fourier series along the wing, and then keep only the first few terms.
[12][13][14] Once the velocity V, circulation Γ, and fluid density ρ are known, the lift generated by the wing is assumed to be the net lift produced by each airfoil with the prescribed circulation...
From these quantities and the aspect ratio AR, the span efficiency factor
These do not require substantial modification to the theory, only changing ∂αC(y,0) and α(y) in (1).
When the aircraft is rolling at rate p about the fuselage, an airfoil at (signed) position y experiences a vertical airflow at rate py, which correspondingly adds py⁄V to the effective angle of attack.
[17] This "drag force" comprises the main production of thrust for flapping wings.
[17] The efficiency e is theoretically optimized in an elliptical wing with no twist, in which
which yields the equation for the elliptic induced drag coefficient:
According to lifting-line theory, any wing planform can achieve the same efficiency through twist (a position-varying increase in pitch) relative to the fuselage.
[14] A useful approximation for the 3D lift coefficient for elliptical circulation distribution[citation needed] is
The theory also presupposes that flow around the wings is in equilibrium, and does not address bodies that are quickly accelerated relative to the freestream air.
