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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 ...
An elliptical spanwise lift distribution cannot be achieved by an untwisted wing with an elliptical planform because there is a logarithmic term in the lift distribution that becomes important near the wing tips. [4] Elliptical wing planforms are more difficult to manufacture. [5]
The acronym is a reference to early German Aerospace Engineer Ludwig Prandtl, whose theory of the bell-shaped lift distribution deeply influenced Bowers. [ 2 ] The Prandtl-D1 and the Prandtl-D3 models are preserved in the National Air and Space Museum and the California Science Center , respectively.
Typically, the elliptical spanwise distribution of lift produces the minimum induced drag [15] for a planar wing of a given span. A small number of aircraft have a planform approaching the elliptical — the most famous examples being the World War II Spitfire [ 13 ] and Thunderbolt .
Given the distribution of bound vorticity and the vorticity in the wake, the Biot–Savart law (a vector-calculus relation) can be used to calculate the velocity perturbation anywhere in the field, caused by the lift on the wing. Approximate theories for the lift distribution and lift-induced drag of three-dimensional wings are based on such ...
The distribution of forces on a wing in flight are both complex and varying. This image shows the forces for two typical airfoils, a symmetrical design on the left, and an asymmetrical design more typical of low-speed designs on the right. This diagram shows only the lift components; the similar drag considerations are not illustrated.
A horseshoe vortex caused by a (purely theoretical) uniform lift distribution over an aircraft’s wing. The starting vortex is also shown. Any spanwise change in lift distribution sheds a trailing vortex, according to the lifting-line theory. The starting vortex is also shown.
With an initial estimate of the pressure distribution on the wing, the structural designers can start designing the load-bearing parts of the wings, fin and tailplane and other lifting surfaces. Additionally, while the VLM cannot compute the viscous drag, the induced drag stemming from the production of lift can be estimated.