Lifting-line theory

Lifting-line theory or Lanchester-Prandtl wing theory was published by Ludwig Prandtl in 1918–1919 after working with Albert Betz and Max Munk on the problem of a useful mathematical tool for examining lift from "real world" wings.

In this model, the vortex strength reduces along the wingspan, and the loss in vortex strength is shed as a vortex-sheet from the trailing edge, rather than just at the wing-tips. [Abbott, Ira H., and Von Doenhoff, Albert E., "Theory of Wing Sections", Section 1.4] [Clancy, L.J., "Aerodynamics", Section 8.11]

The theory states that::$C\_L\; =\; C\_\{L\_a\}\; (\; frac\{AR\}\{AR+2\}\; )\; alpha$where
*$C\_L$ is the lift coefficient,
*$C\_\{L\_a\}$ is the 2D lift coefficient slope (see Thin Airfoil Theory),
*$AR$ is the Aspect Ratio, and
*$alpha$ is the angle of attack in radians.

The theoretical maximum for $C\_\{L\_a\}$ is 2$pi$. Note that this equation becomes the Thin Airfoil equation if "AR" goes to infinity. [ [http://www.aerospaceweb.org/question/aerodynamics/q0136.shtml Aerospace Web's explanation of line lifting theory] ]

Lifting-line theory also states an equation for induced drag:. [Abbott, Ira H., and Von Doenhoff, Albert E., "Theory of Wing Sections", Section 1.3] [Clancy, L.J., "Aerodynamics", Equation 5.7] :$C\_\{D\_i\}\; =\; frac$C_L}^2}{pi AR} where
*$C\_\{D\_i\}$ is the drag coefficient for induced drag,
*$C\_L$ is the lift coefficient, and
*$AR$ is the Aspect Ratio.

See also

* Horseshoe vortex
* Thin Airfoil Theory

References

* Clancy, L.J. (1975), "Aerodynamics", Pitman Publishing Limited, London. 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

Notes