The
horseshoe vortex model is a simplified representation of the
vortexA vortex is a spinning, often turbulent,flow of fluid. Any spiral motion with closed streamlines is vortex flow. The motion of the fluid swirling rapidly around a center is called a vortex...
system of a
wingA wing is a surface used to produce lift for flight through the air or another gaseous or fluid medium. The wing shape is usually an airfoil. The word originally referred only to the foremost limbs of birds, but has been extended to include the wings of insects , bats, pterosaurs, and aircraft.A...
. In this model the wing
vorticityVorticity is a concept used in fluid dynamics. In the simplest sense, vorticity is the tendency for elements of the fluid to "spin."More formally, vorticity can be related to the amount of "circulation" or "rotation" in a fluid.The average vorticity in a small region of fluid flow is equal to the...
is modelled by a bound vortex of constant
circulationCirculation may refer to:*Circulatory system, a biological organ system whose primary function is to move substances to and from cells*Circulation , the path integral of the fluid velocity around a closed curve...
, travelling with the wing, and two
trailing vorticesWingtip vortices are tubes of circulating air which are left behind a wing as it generates lift. One wingtip vortex trails from the tip of each wing. The cores of vortices spin at very high speed and are regions of very low pressure. To first approximation, these low-pressure regions form with...
, therefore having a shape vaguely reminiscent of a horseshoe. (The
starting vortexThe starting vortex is a vortex which forms in the air adjacent to the trailing edge of an airfoil as it is accelerated from rest in a fluid. It leaves the airfoil , and remains stationary in the flow...
created as the wing begins to move through the fluid is considered to have been dissipated by the action of
viscosityViscosity is a measure of the resistance of a fluid which is being deformed by either shear stress or extensional stress. In everyday terms , viscosity is "thickness." Thus, water is "thin," having a lower viscosity, while honey is "thick," having a higher viscosity...
, as are the
trailing vorticesWingtip vortices are tubes of circulating air which are left behind a wing as it generates lift. One wingtip vortex trails from the tip of each wing. The cores of vortices spin at very high speed and are regions of very low pressure. To first approximation, these low-pressure regions form with...
well behind the aircraft.)
The
trailing vorticesWingtip vortices are tubes of circulating air which are left behind a wing as it generates lift. One wingtip vortex trails from the tip of each wing. The cores of vortices spin at very high speed and are regions of very low pressure. To first approximation, these low-pressure regions form with...
are responsible for the component of the
downwashthumb|right|300px|The effect of downwash from a hovering helicopter is clearly visible on the surface of water below.The term downwash has two meanings within the field of aerodynamics....
which creates induced drag.
The horseshoe vortex model is unrealistic in implying a constant
circulationIn fluid dynamics, circulation is the line integral around a closed curve of the fluid velocity. Circulation is normally denoted . If is the fluid velocity and is a unit vector along the closed curve :...
(and hence by the
Kutta–Joukowski theoremThe Kutta–Joukowski theorem is a fundamental theorem of aerodynamics. It is named after the German Martin Wilhelm Kutta and the Russian Nikolai Zhukovsky who first developed its key ideas in the early 20th century. The theorem relates the lift generated by a right cylinder to the speed of the...
constant lift) at all sections on the
wingspanThe wingspan of an airplane or a bird, is the distance from the left wingtip to the right wingtip. For example, the Boeing 777 has a wingspan of about ; and a Wandering Albatross caught in 1965 had a wingspan of , the official record for a living bird.The term wingspan, more technically extent,...
.
Discussion
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The
horseshoe vortex model is a simplified representation of the
vortexA vortex is a spinning, often turbulent,flow of fluid. Any spiral motion with closed streamlines is vortex flow. The motion of the fluid swirling rapidly around a center is called a vortex...
system of a
wingA wing is a surface used to produce lift for flight through the air or another gaseous or fluid medium. The wing shape is usually an airfoil. The word originally referred only to the foremost limbs of birds, but has been extended to include the wings of insects , bats, pterosaurs, and aircraft.A...
. In this model the wing
vorticityVorticity is a concept used in fluid dynamics. In the simplest sense, vorticity is the tendency for elements of the fluid to "spin."More formally, vorticity can be related to the amount of "circulation" or "rotation" in a fluid.The average vorticity in a small region of fluid flow is equal to the...
is modelled by a bound vortex of constant
circulationCirculation may refer to:*Circulatory system, a biological organ system whose primary function is to move substances to and from cells*Circulation , the path integral of the fluid velocity around a closed curve...
, travelling with the wing, and two
trailing vorticesWingtip vortices are tubes of circulating air which are left behind a wing as it generates lift. One wingtip vortex trails from the tip of each wing. The cores of vortices spin at very high speed and are regions of very low pressure. To first approximation, these low-pressure regions form with...
, therefore having a shape vaguely reminiscent of a horseshoe. (The
starting vortexThe starting vortex is a vortex which forms in the air adjacent to the trailing edge of an airfoil as it is accelerated from rest in a fluid. It leaves the airfoil , and remains stationary in the flow...
created as the wing begins to move through the fluid is considered to have been dissipated by the action of
viscosityViscosity is a measure of the resistance of a fluid which is being deformed by either shear stress or extensional stress. In everyday terms , viscosity is "thickness." Thus, water is "thin," having a lower viscosity, while honey is "thick," having a higher viscosity...
, as are the
trailing vorticesWingtip vortices are tubes of circulating air which are left behind a wing as it generates lift. One wingtip vortex trails from the tip of each wing. The cores of vortices spin at very high speed and are regions of very low pressure. To first approximation, these low-pressure regions form with...
well behind the aircraft.)
The
trailing vorticesWingtip vortices are tubes of circulating air which are left behind a wing as it generates lift. One wingtip vortex trails from the tip of each wing. The cores of vortices spin at very high speed and are regions of very low pressure. To first approximation, these low-pressure regions form with...
are responsible for the component of the
downwashthumb|right|300px|The effect of downwash from a hovering helicopter is clearly visible on the surface of water below.The term downwash has two meanings within the field of aerodynamics....
which creates induced drag.
The horseshoe vortex model is unrealistic in implying a constant
circulationIn fluid dynamics, circulation is the line integral around a closed curve of the fluid velocity. Circulation is normally denoted . If is the fluid velocity and is a unit vector along the closed curve :...
(and hence by the
Kutta–Joukowski theoremThe Kutta–Joukowski theorem is a fundamental theorem of aerodynamics. It is named after the German Martin Wilhelm Kutta and the Russian Nikolai Zhukovsky who first developed its key ideas in the early 20th century. The theorem relates the lift generated by a right cylinder to the speed of the...
constant lift) at all sections on the
wingspanThe wingspan of an airplane or a bird, is the distance from the left wingtip to the right wingtip. For example, the Boeing 777 has a wingspan of about ; and a Wandering Albatross caught in 1965 had a wingspan of , the official record for a living bird.The term wingspan, more technically extent,...
. In a more realistic model (due to
Ludwig PrandtlLudwig Prandtl was a German scientist. He was a pioneer in the development of rigorous systematic mathematical analyses which he used to underlay the science of aerodynamics, which have come to form the basis of the applied science of aeronautical engineering...
) the vortex strength reduces along the
wingspanThe wingspan of an airplane or a bird, is the distance from the left wingtip to the right wingtip. For example, the Boeing 777 has a wingspan of about ; and a Wandering Albatross caught in 1965 had a wingspan of , the official record for a living bird.The term wingspan, more technically extent,...
, and the loss in vortex strength is shed as a vortex-sheet from the trailing edge, rather than just at the wing-tips. 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.
See also
- Helmholtz's theorems
In fluid mechanics, Helmholtz's theorems describe the three-dimensional motion of fluid in the vicinity of vortex filaments. These theorems apply to inviscid flows and flows where the influence of viscous forces is small and can be ignored....
- Kutta condition
The Kutta condition is a principle in steady flow fluid dynamics, especially aerodynamics, that is applicable to solid bodies which have sharp corners such as the trailing edges of airfoils...
- Kutta–Joukowski theorem
The Kutta–Joukowski theorem is a fundamental theorem of aerodynamics. It is named after the German Martin Wilhelm Kutta and the Russian Nikolai Zhukovsky who first developed its key ideas in the early 20th century. The theorem relates the lift generated by a right cylinder to the speed of the...
- Prandtl's lifting-line model