Lifting-line theory

The Prandtl lifting-line theory is a mathematical model that predicts lift distribution over a three-dimensional wing based on its geometry. It is also known as the Lanchester–Prandtl wing theory.

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 loses strength along the whole wingspan because it is shed as a vortex-sheet from the trailing edge, rather than just at the wing-tips.

On a three-dimensional, finite wing, lift over each wing segment (local lift per unit span, ${\displaystyle l}$ or ${\displaystyle {\tilde {L}}}$) does not correspond simply to what two-dimensional analysis predicts. Instead, this local amount of lift is strongly affected by the lift generated at neighboring wing sections.

As such, it is difficult to predict analytically the overall amount of lift that a wing of given geometry will generate. The lifting-line theory yields the lift distribution along the span-wise direction, ${\displaystyle {\tilde {L}}_{(y)}}$ based only on the wing geometry (span-wise distribution of chord, airfoil, and twist) and flow conditions (${\displaystyle \rho }$, ${\displaystyle V_{\infty }}$, ${\displaystyle \alpha _{\infty }}$).

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