"Lifting-Line" theory // Lanchester Lanchester–Prandtl theory // "3D Wing" theory
It is well known that on a three three-dimensional, finite wing, the lift over each wing segment , (the local lift per unit span) does not correspond merely to that predicted by two-dimensional analysis with airfoils. Instead, this local amount of lift is strongly affected by the lift generated at . the neighboring wing sections.
An unrealistic lift distribution, 3D effects are neglected
An observed lift distribution, over a (finite) trapezoidal wing
The Lifting-Line theory implements the following idea. The loss in vortex strength along the wingspan is attributed mainly to shedding of a vortex down the flow, behind the wing, rather than at the wing tips. , The Lifting-Line theory yields the lift distribution along the span wise direction of the span-wise 3D wing , based only on the wing geometry (span wise distribution of chord, airfoil, and twist) (span-wise and flow conditions (density, velocity, and angle of attack). The Lifting-Line theory makes use of the concept of circulation over the wing span. Instead of lift distribution function, the unknown effectively becomes circulation distribution. distribution Then Γ( y ) is related to L( y ) with the Kutta–Joukowski theorem, L ( y ) = − ρ ⋅ U ⋅ Γ( y ) .
According to the Helmholtz theorems, a vortex filament cannot begin or terminate in the air. theorems, Change in circulation represents shedding of a vortex filament down the flow, behind the wing. This shed vortex influences the flow left and right of the wing section. This shed vortex induces an upwash on the outboard and downwash on the inboard. inboard The shed vortex can be modeled as a vertical velocity distribution distribution. If the change in lift distribution is known at given lift section, it is possible to predict how that section influences the lift over its neighbors: the vertical induced velocity (upwash or downwash) induced downwash can be quantified...