# Horseshoe vortex

Horseshoe vortex

The horseshoe vortex model is a simplified representation of the vortex system of a wing. In this model the wing vorticity is modelled by a bound vortex of constant circulation, travelling with the wing, and two trailing vortices, therefore having a shape vaguely reminiscent of a horseshoe. [Millikan, Clark B., "Aerodynamics of the Airplane", Figure 1.35] [ McCormick, Barnes W., "Aerodynamics, Aeronautics, and Flight Mechanics", Chapter 3] (The starting vortex created as the wing begins to move through the fluid is considered to have been dissipated by the action of viscosity, as are the trailing vortices well behind the aircraft.)

The trailing vortices are responsible for the component of the downwash which creates induced drag. [ McCormick, Barnes W., "Aerodynamics, Aeronautics, and Flight Mechanics", Chapter 4]

The horseshoe vortex model is unrealistic in implying a constant vorticity (and hence by the Kutta–Joukowski theorem constant lift) at all points on the wingspan. In a more realistic model (due to Ludwig Prandtl) 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. [ McCormick, Barnes W., "Aerodynamics, Aeronautics, and Flight Mechanics", Figure 4.21] However, by using the horseshoe vortex model with a reduced effective wingspan but same midplane circulation, the flows induced far from the aircraft can be adequately modelled.

References

* Clancy, L.J. (1975), "Aerodynamics", Section 8.10, Pitman Publishing Limited, London ISBN 0 273 01120 0
* McCormick, Barnes W., (1979), "Aerodynamics, Aeronautics, and Flight Mechanics", John Wiley & Sons, Inc. New York ISBN 0-471-03032-5
* Millikan, Clark B., (1941), "Aerodynamics of the Airplane", Section 1-6 John Wiley and Sons, Inc., New York
* Piercy, N.A.V. (1944), "Elementary Aerodynamics", Article 213, The English Universities Press Ltd., London.
* Von Mises, Richard, (1959), "Theory of Flight", Chapter IX - section 4, Dover Publications, Inc., New York ISBN 0-486-60541-8

Notes

* Helmholtz's theorems
* Kutta condition
* Kutta–Joukowski theorem
* Prandtl's lifting-line model

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