Lifting-line theory
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. 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.[6][7]
Introduction
![](http://upload.wikimedia.org/wikipedia/commons/thumb/a/a0/1915ca_laminar_fluegel%28cropped%29.jpg/220px-1915ca_laminar_fluegel%28cropped%29.jpg)
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.
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An unrealistic lift distribution that neglects three-dimensional effects
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The observed lift distribution on a (finite) trapezoidal wing
Principle and derivation
Lifting line theory supposes wings that are
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The lift distribution over a wing can be modeled with the concept ofcirculation
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A vortex is shed downstream for every span-wise change in lift
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 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).[9]
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,
Suppose the freestream flow attacks the airfoil at position y at angle α(y) (relative to the liftless angle for the airfoil at position y — thus a uniform flow across a wing may still have varying α(y)). By the small-angle approximation, the effective angle of attack at y of the combined freestream and vortex system is α(y)+w(y)⁄V. Combining the above formulae,
(1) |
All the quantities in this equation except V and Γ are geometric properties of the wing, and so an engineer can (in principle) solve for Γ(y) given a fixed V. As in the derivation of
Once the velocity V, circulation Γ, and fluid density ρ are known, the lift generated by the wing is assumed to be the net lift produced by each airfoil with the prescribed circulation... ...and the drag is likewise the total across airfoils: From these quantities and the aspect ratio AR, the span efficiency factor may be computed.[15][16][11]
Effects of control inputs
Control surface deflection changes the shape each airfoil slice, which can produce a different angle-of-no-lift for that airfoil, as well as a different angle-of-attack response. These do not require substantial modification to the theory, only changing ∂αC(y,0) and α(y) in (1). However, a body with rapidly moving wings, such as a rolling aircraft or flapping bird, experiences a vertical flow across the wing due to the wing's change in orientation, which appears as a missing term in the theory.
Rolling wings
When the aircraft is rolling at rate p about the fuselage, an airfoil at (signed) position y experiences a vertical airflow at rate py, which correspondingly adds py⁄V to the effective angle of attack. Thus (1) becomes: which correspondingly modifies both the lift and the induced drag.[17] This "drag force" comprises the main production of thrust for flapping wings.[17]
Elliptical wings
The
Useful approximations
A useful approximation for the 3D lift coefficient for elliptical circulation distribution[citation needed] isNote that this equation becomes the
Limitations
The lifting line theory does not take into account
See also
- Horseshoe vortex
- Kutta condition
- Thin airfoil theory
- Vortex lattice method
Notes
- ISBN 0-07-237335-0.
- ISBN 0-7506-5111-3.
- ISBN 0-486-43485-0.
- ^ Lanchester, Frederick W. (1907). Constable (ed.). Aerodynamics.
- ^ Prandtl, Ludwig (1918). Königliche Gesellschaft der Wissenschaften zu Göttingen (ed.). Tragflügeltheorie.
- Abbott, Ira H., and Von Doenhoff, Albert E., Theory of Wing Sections, Section 1.4.
- ^ Clancy, L. J., Aerodynamics, Section 8.11.
- ISBN 978-81-85618-24-1.
- ^ Batchelor 1993, p. 585-586.
- ^ Acheson, D. J. (1990). Elementary Fluid Dynamics. Oxford Applied Mathematics and Computing Science. Oxford: Clarendon Press (published 2009). pp. 134–136, 138.
- ^ a b Auld, Douglass; Srinivas (1995). "3-D Lifting Line Theory". Aerodynamics for Students. University of Sydney.
- ^ Batchelor 1993, p. 586-587.
- ISBN 978-1-62410-092-5, retrieved 2020-12-02
- ^ doi:10.2514/1.262.
- ^ Abbott, Ira H., and Von Doenhoff, Albert E., Theory of Wing Sections, Section 1.3
- ^ Clancy, L.J., Aerodynamics, Equation 5.7
- ^ .
- ^ Scott, Jeff (10 August 2003). "Question #136: Lift Coefficient & Thin Airfoil Theory". Ask a Rocket Scientist: Aerodynamics. Aerospaceweb.org.
References
- ISBN 0-273-01120-0
- Abbott, Ira H., and Von Doenhoff, Albert E. (1959), Theory of Wing Sections, Dover Publications Inc., New York. Standard Book Number 486-60586-8