The Lanchester–Prandtl lifting-line theory is a mathematical model in aerodynamics that predicts lift distribution over a three-dimensional wing from the wing's geometry. The theory was expressed independently by Frederick W. Lanchester in 1907, and by Ludwig Prandtl in 1918–1919 after working with Albert Betz and Max Munk. In this model, the vortex bound to the wing develops along the whole wingspan because it is shed as a vortex-sheet from the trailing edge, rather than just as a single vortex from the wing-tips.
Introduction
It is difficult to predict analytically the overall amount of lift that a wing of given geometry will generate. When analyzing a three-dimensional finite wing, a traditional approach slices the wing into cross-sections and analyzes each cross-section independently as a wing in a two-dimensional world. Each of these slices is called an airfoil, and it is easier to understand an airfoil than a complete three-dimensional wing.
One might expect that understanding the full wing simply involves adding up the independently calculated forces from each airfoil segment. However, this approximation is grossly incorrect: on a real wing, the lift from each infinitesimal wing section is strongly affected by the airflow over neighboring wing sections. Lifting-line theory corrects some of the errors in the naive two-dimensional approach by including some interactions between the wing slices.
Principle and derivation 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.
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 w ( y ) = ∫ − s s d Γ ( y ~ ) 4 π ( y − y ~ ) , {\displaystyle w(y)=\int _{-s}^{s}{\frac {d\Gamma ({\tilde {y}})}{4\pi (y-{\tilde {y}})}},} where the integral is understood in the sense of a Cauchy principal value. 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). Formally, there is some angle of orientation such that the airfoil at position y develops no lift. For airstreams of velocity V oriented at an angle α relative to the liftless angle, the airfoil will develop some circulation V⋅C(y,α); for small α, Taylor expansion approximates that circulation as V⋅∂C⁄∂α(y,0)⋅α. If the airfoil is ideal and has chord c(y), then theory predicts that C ( y , α ) = 2 π c ( y ) sin ( α ) , {\displaystyle C(y,\alpha )=2\pi c(y)\sin(\alpha ){\text{,}}} but real airfoils may be less efficient.
… excerpt ends here. Continue reading the full article.
