# Horseshoe vortex

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A horseshoe vortex caused by a (purely theoretical) uniform lift distribution over an aircraft’s wing
Any change in lift distribution sheds a new trailing vortex, according to the lifting-line theory
A realistic lift distribution causes the shedding of a complex vorticity pattern behind the aircraft.

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 wingtip vortices, therefore having a shape vaguely reminiscent of a horseshoe.[1][2] A starting vortex is shed as the wing begins to move through the fluid, which dissipates under the action of viscosity,[3] as do the trailing vortices far behind the aircraft.

The trailing wingtip vortices are responsible for the component of the downwash which creates induced drag.[4]

The horseshoe vortex model is unrealistic in that it implies a constant circulation (and hence, according to the Kutta–Joukowski theorem, constant lift) at all sections on the wingspan. In a more realistic model, the lifting-line theory, the vortex strength varies along the wingspan, and the loss in vortex strength is shed as a vortex-sheet all along the trailing edge, rather than as a single trail at the wing-tips.[5] Nevertheless, the simpler horseshoe vortex model used with a reduced effective wingspan but same midplane circulation provides an adequate model for the flows induced far from the aircraft.

The term horse-shoe vortex is also used in Wind Engineering to describe the vortex of strong winds that form around the base of a tall building. This effect is amplified by the presence of a low-rise building just upwind. This effect was studied at the UK Building Research Establishment between 1963 and 1973[6] and the cause of the effect is described in contemporary wind engineering text books.[7]

In hydrodynamics, a form of horseshoe vortex forms around bluff bodies in the flowing water, for instance around bridge piers.[8] They can cause scouring of bed materials from both upstream and downstream of the pier.[citation needed]

## References

• Anderson, John D. (2007), Fundamentals of Aerodynamics, Section 5.3 (4th ed.), McGraw-Hill, New York NY. ISBN 978-0-07-295046-5
• Clancy, L.J. (1975), Aerodynamics, Section 8.10, Pitman Publishing Limited, London ISBN 0-273-01120-0
• Cook, N.J. (1985), The designer's guide to wind loading of building structures, Part 1, Butterworths, London ISBN 0-408-00870-9
• 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
• Penwarden, A.D., Wise, A.F.E., (1975) Wind environment around buildings, HMSO, London ISBN 0-11-670533-7.
• 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

1. ^ Millikan, Clark B., Aerodynamics of the Airplane, Figure 1.35
2. ^ McCormick, Barnes W., Aerodynamics, Aeronautics, and Flight Mechanics, Chapter 3
3. ^ "Shed Vortex". NASA Glenn Research Center. Retrieved April 11, 2015.
4. ^ McCormick, Barnes W., Aerodynamics, Aeronautics, and Flight Mechanics, Chapter 4
5. ^ McCormick, Barnes W., Aerodynamics, Aeronautics, and Flight Mechanics, Figure 4.21
6. ^ Penwarden, AD. Wise, AFE. Wind environment around buildings, cover illustration
7. ^ Cook, NJ. The designer's guide to wind loading of building structures, Part 1, Figure 8.7
8. ^ Dargahi, Bijan (1989). "The turbulent flow field around a circular cylinder". Experiments in Fluids. Retrieved 2016-04-22.