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 lift distribution over a wing can be modeled with the concept of circulationA vortex is shed downstream for every span-wise change in liftThe shed vortex can be modeled as a vertical velocity distributionThe upwash and downwash induced by the shed vortex can be computed at each neighbor segment. 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, l {displaystyle l} or L ~ {displaystyle { ilde {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. 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, L ~ ( y ) {displaystyle { ilde {L}}_{(y)}} based only on the wing geometry (span-wise distribution of chord, airfoil, and twist) and flow conditions ( ρ {displaystyle ho } , V ∞ {displaystyle V_{infty }} , α ∞ {displaystyle alpha _{infty }} ). The lifting-line theory applies the concept of circulation and the Kutta–Joukowski theorem, so that instead of the lift distribution function, the unknown effectively becomes the distribution of circulation over the span, Γ ( y ) {displaystyle Gamma _{(y)}} . Modeling the (unknown and sought-after) local lift with the (also unknown) local circulation allows us to account for the influence of one section over its neighbors. In this view, any span-wise change in lift is equivalent to a span-wise change of circulation. According to Helmholtz's theorems, a vortex filament cannot begin or terminate in the air. Any span-wise change in lift can be modeled as the shedding of a vortex filament down the flow, behind the wing. This shed vortex, whose strength is the derivative of the (unknown) local wing circulation distribution, d ⁡ Γ d ⁡ y {displaystyle {operatorname {d} Gamma over operatorname {d} y}} , influences the flow left and right of the wing section.