Kutta condition
Encyclopedia
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 edge
s of airfoil
s. It is named for German
mathematician
and aerodynamicist Martin Wilhelm Kutta
.
Kuethe and Schetzer state the Kutta condition as follows:
In fluid flow around a body with a sharp corner the Kutta condition refers to the flow pattern in which fluid approaches the corner from both directions, meets at the corner, and then flows away from the body. None of the fluid flows around the corner and remains attached to the body.
The Kutta condition is significant when using the Kutta–Joukowski theorem
to calculate the lift generated by an airfoil. The value of circulation of the flow around the airfoil must be that value which would cause the Kutta condition to exist.
through a fluid it generates no lift. There are two stagnation point
s on the body - one at the front and the other at the back. If the oval cylinder moves with a non-zero angle of attack through the fluid there are still two stagnation points on the body - one on the underside of the cylinder, near the front edge; and the other on the topside of the cylinder, near the back edge. The circulation around this smooth cylinder is zero and no lift is generated, despite the positive angle of attack.
If an airfoil with a sharp trailing edge begins to move with a positive angle of attack through air, the two stagnation points are initially located on the underside near the leading edge and on the topside near the trailing edge, just as with the cylinder. As the air passing the underside of the airfoil reaches the trailing edge it must flow around the trailing edge and along the topside of the airfoil toward the stagnation point on the topside of the airfoil. Vortex
flow occurs at the trailing edge and, because the radius of the sharp trailing edge is zero, the speed of the air around the trailing edge should be infinitely fast! Real fluids cannot move at infinite speed but they can move very fast. The very fast airspeed around the trailing edge causes strong viscous
forces to act on the air adjacent to the trailing edge of the airfoil and the result is that a strong vortex accumulates on the topside of the airfoil, near the trailing edge. As the airfoil begins to move it carries this vortex, known as the starting vortex
, along with it. Pioneering aerodynamicists were able to photograph starting vortexes in liquids to confirm their existence.
The vorticity in the starting vortex is matched by the vorticity in the bound vortex in the airfoil, in accordance with Kelvin's circulation theorem
. As the vorticity in the starting vortex progressively increases the vorticity in the bound vortex also progressively increases and causes the flow over the topside of the airfoil to increase in speed. The stagnation point on the topside of the airfoil moves progressively towards the trailing edge. After the airfoil has moved only a short distance through the air the stagnation point on the topside reaches the trailing edge and the starting vortex is cast off the airfoil and is left behind, spinning in the air where the airfoil left it. The starting vortex quickly dissipates due to viscous forces.
As the airfoil continues on its way, there is a stagnation point at the trailing edge. The flow over the topside conforms to the upper surface of the airfoil. The flow over both the topside and the underside join up at the trailing edge and leave the airfoil travelling parallel to one another. This is known as the Kutta condition.
When an airfoil is moving with a positive angle of attack, the starting vortex has been cast off and the Kutta condition has become established, there is a finite circulation of the air around the airfoil. The airfoil is generating lift, and the magnitude of the lift is given by the Kutta–Joukowski theorem
.
One of the consequences of the Kutta condition is that the airflow over the topside of the airfoil travels much faster than the airflow under the underside. A parcel of air which approaches the airfoil along the stagnation streamline is cleaved in two at the stagnation point, one half traveling over the topside and the other half traveling along the underside. The flow over the topside is so much faster than the flow along the underside that these two halves never meet again. They do not even re-join in the wake long after the airfoil has passed. This is sometimes known as "cleavage". There is a popular fallacy called the equal transit-time fallacy that claims the two halves rejoin at the trailing edge of the airfoil. This fallacy is in conflict with the phenomenon of cleavage that has been understood since Martin Kutta's discovery.
Whenever the speed or angle of attack of an airfoil changes there is a weak starting vortex which begins to form, either above or below the trailing edge. This weak starting vortex causes the Kutta condition to be re-established for the new speed or angle of attack. As a result, the circulation around the airfoil changes and so too does the lift in response to the changed speed or angle of attack.
The Kutta condition gives some insight into why airfoils always have sharp trailing edges, even though this is undesirable from structural and manufacturing viewpoints. An aircraft with a wing with a smoothly rounded trailing edge would generate little or no lift.
while neglecting viscous effects in the underlying conservation of momentum equation. It is important in the practical calculation of lift
on a wing
.
The equations of conservation of mass
and conservation of momentum applied to an inviscid fluid flow, such as a potential flow
, around a solid body result in an infinite number of valid solutions. One way to choose the correct solution would be to apply the viscous equations, in the form of the Navier–Stokes equations. However, these normally do not result in a closed-form solution. The Kutta condition is an alternative method of incorporating some aspects of viscous effects, while neglecting others, such as skin friction and some other boundary layer
effects.
The condition can be expressed in a number of ways. One is that there cannot be an infinite change in velocity at the trailing edge. Although an inviscid fluid (a theoretical concept that does not normally exist in the everyday world) can have abrupt changes in velocity, in reality viscosity smooths out sharp velocity changes. If the trailing edge has a non-zero angle, the flow velocity there must be zero. At a cusp
ed trailing edge, however, the velocity can be non-zero although it must still be identical above and below the airfoil. Another formulation is that the pressure must be continuous at the trailing edge.
The Kutta condition does not apply to unsteady flow. Experimental observations show that the stagnation point
(one of two points on the surface of an airfoil where the flow speed is zero) begins on the top surface of an airfoil (assuming positive effective angle of attack
) as flow accelerates from zero, and moves backwards as the flow accelerates. Once the initial transient effects have died out, the stagnation point is at the trailing edge as required by the Kutta condition.
Mathematically, the Kutta condition enforces a specific choice among the infinite allowed values of circulation.
Fluid dynamics
In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...
, especially aerodynamics
Aerodynamics
Aerodynamics is a branch of dynamics concerned with studying the motion of air, particularly when it interacts with a moving object. Aerodynamics is a subfield of fluid dynamics and gas dynamics, with much theory shared between them. Aerodynamics is often used synonymously with gas dynamics, with...
, that is applicable to solid bodies which have sharp corners such as the trailing edge
Trailing edge
The trailing edge of an aerodynamic surface such as a wing is its rear edge, where the airflow separated by the leading edge rejoins. Essential control surfaces are attached here to redirect the air flow and exert a controlling force by changing its momentum...
s of airfoil
Airfoil
An airfoil or aerofoil is the shape of a wing or blade or sail as seen in cross-section....
s. It is named for German
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
and aerodynamicist Martin Wilhelm Kutta
Martin Wilhelm Kutta
Martin Wilhelm Kutta was a German mathematician.Kutta was born in Pitschen, Upper Silesia . He attended the University of Breslau from 1885 to 1890, and continued his studies in Munich until 1894, where he became the assistant of Walther Franz Anton von Dyck. From 1898, he spent a year at the...
.
Kuethe and Schetzer state the Kutta condition as follows:
- A body with a sharp trailing edge which is moving through a fluid will create about itself a circulation of sufficient strength to hold the rear stagnation point at the trailing edge.
In fluid flow around a body with a sharp corner the Kutta condition refers to the flow pattern in which fluid approaches the corner from both directions, meets at the corner, and then flows away from the body. None of the fluid flows around the corner and remains attached to the body.
The Kutta condition is significant when using the Kutta–Joukowski theorem
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...
to calculate the lift generated by an airfoil. The value of circulation of the flow around the airfoil must be that value which would cause the Kutta condition to exist.
The Kutta condition applied to airfoils
When a smooth symmetric body, such as a cylinder with oval cross-section, moves with zero angle of attackAngle of attack
Angle of attack is a term used in fluid dynamics to describe the angle between a reference line on a lifting body and the vector representing the relative motion between the lifting body and the fluid through which it is moving...
through a fluid it generates no lift. There are two stagnation point
Stagnation point
In fluid dynamics, a stagnation point is a point in a flow field where the local velocity of the fluid is zero. Stagnation points exist at the surface of objects in the flow field, where the fluid is brought to rest by the object...
s on the body - one at the front and the other at the back. If the oval cylinder moves with a non-zero angle of attack through the fluid there are still two stagnation points on the body - one on the underside of the cylinder, near the front edge; and the other on the topside of the cylinder, near the back edge. The circulation around this smooth cylinder is zero and no lift is generated, despite the positive angle of attack.
If an airfoil with a sharp trailing edge begins to move with a positive angle of attack through air, the two stagnation points are initially located on the underside near the leading edge and on the topside near the trailing edge, just as with the cylinder. As the air passing the underside of the airfoil reaches the trailing edge it must flow around the trailing edge and along the topside of the airfoil toward the stagnation point on the topside of the airfoil. Vortex
Vortex
A 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...
flow occurs at the trailing edge and, because the radius of the sharp trailing edge is zero, the speed of the air around the trailing edge should be infinitely fast! Real fluids cannot move at infinite speed but they can move very fast. The very fast airspeed around the trailing edge causes strong viscous
Viscosity
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity...
forces to act on the air adjacent to the trailing edge of the airfoil and the result is that a strong vortex accumulates on the topside of the airfoil, near the trailing edge. As the airfoil begins to move it carries this vortex, known as the starting vortex
Starting vortex
The 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...
, along with it. Pioneering aerodynamicists were able to photograph starting vortexes in liquids to confirm their existence.
The vorticity in the starting vortex is matched by the vorticity in the bound vortex in the airfoil, in accordance with Kelvin's circulation theorem
Kelvin's circulation theorem
In fluid mechanics, Kelvin's circulation theorem states In an inviscid, barotropic flow with conservative body forces, the circulation around a closed curve moving with the fluid remains constant with time. The theorem was developed by William Thomson, 1st Baron Kelvin...
. As the vorticity in the starting vortex progressively increases the vorticity in the bound vortex also progressively increases and causes the flow over the topside of the airfoil to increase in speed. The stagnation point on the topside of the airfoil moves progressively towards the trailing edge. After the airfoil has moved only a short distance through the air the stagnation point on the topside reaches the trailing edge and the starting vortex is cast off the airfoil and is left behind, spinning in the air where the airfoil left it. The starting vortex quickly dissipates due to viscous forces.
As the airfoil continues on its way, there is a stagnation point at the trailing edge. The flow over the topside conforms to the upper surface of the airfoil. The flow over both the topside and the underside join up at the trailing edge and leave the airfoil travelling parallel to one another. This is known as the Kutta condition.
When an airfoil is moving with a positive angle of attack, the starting vortex has been cast off and the Kutta condition has become established, there is a finite circulation of the air around the airfoil. The airfoil is generating lift, and the magnitude of the lift is given by the Kutta–Joukowski theorem
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...
.
One of the consequences of the Kutta condition is that the airflow over the topside of the airfoil travels much faster than the airflow under the underside. A parcel of air which approaches the airfoil along the stagnation streamline is cleaved in two at the stagnation point, one half traveling over the topside and the other half traveling along the underside. The flow over the topside is so much faster than the flow along the underside that these two halves never meet again. They do not even re-join in the wake long after the airfoil has passed. This is sometimes known as "cleavage". There is a popular fallacy called the equal transit-time fallacy that claims the two halves rejoin at the trailing edge of the airfoil. This fallacy is in conflict with the phenomenon of cleavage that has been understood since Martin Kutta's discovery.
Whenever the speed or angle of attack of an airfoil changes there is a weak starting vortex which begins to form, either above or below the trailing edge. This weak starting vortex causes the Kutta condition to be re-established for the new speed or angle of attack. As a result, the circulation around the airfoil changes and so too does the lift in response to the changed speed or angle of attack.
The Kutta condition gives some insight into why airfoils always have sharp trailing edges, even though this is undesirable from structural and manufacturing viewpoints. An aircraft with a wing with a smoothly rounded trailing edge would generate little or no lift.
The Kutta condition in aerodynamics
The Kutta condition allows an aerodynamicist to incorporate a significant effect of viscosityViscosity
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity...
while neglecting viscous effects in the underlying conservation of momentum equation. It is important in the practical calculation of lift
Lift (force)
A fluid flowing past the surface of a body exerts a surface force on it. Lift is the component of this force that is perpendicular to the oncoming flow direction. It contrasts with the drag force, which is the component of the surface force parallel to the flow direction...
on a wing
Wing
A wing is an appendage with a surface that produces lift for flight or propulsion through the atmosphere, or through another gaseous or liquid fluid...
.
The equations of conservation of mass
Conservation of mass
The law of conservation of mass, also known as the principle of mass/matter conservation, states that the mass of an isolated system will remain constant over time...
and conservation of momentum applied to an inviscid fluid flow, such as a potential flow
Potential flow
In fluid dynamics, potential flow describes the velocity field as the gradient of a scalar function: the velocity potential. As a result, a potential flow is characterized by an irrotational velocity field, which is a valid approximation for several applications...
, around a solid body result in an infinite number of valid solutions. One way to choose the correct solution would be to apply the viscous equations, in the form of the Navier–Stokes equations. However, these normally do not result in a closed-form solution. The Kutta condition is an alternative method of incorporating some aspects of viscous effects, while neglecting others, such as skin friction and some other boundary layer
Boundary layer
In physics and fluid mechanics, a boundary layer is that layer of fluid in the immediate vicinity of a bounding surface where effects of viscosity of the fluid are considered in detail. In the Earth's atmosphere, the planetary boundary layer is the air layer near the ground affected by diurnal...
effects.
The condition can be expressed in a number of ways. One is that there cannot be an infinite change in velocity at the trailing edge. Although an inviscid fluid (a theoretical concept that does not normally exist in the everyday world) can have abrupt changes in velocity, in reality viscosity smooths out sharp velocity changes. If the trailing edge has a non-zero angle, the flow velocity there must be zero. At a cusp
Cusp
Cusp may refer to:*Beach cusps, a pointed and regular arc pattern of the shoreline at the beach*Behavioral cusp an important behavior change with far reaching consequences*Cusp catastrophe...
ed trailing edge, however, the velocity can be non-zero although it must still be identical above and below the airfoil. Another formulation is that the pressure must be continuous at the trailing edge.
The Kutta condition does not apply to unsteady flow. Experimental observations show that the stagnation point
Stagnation point
In fluid dynamics, a stagnation point is a point in a flow field where the local velocity of the fluid is zero. Stagnation points exist at the surface of objects in the flow field, where the fluid is brought to rest by the object...
(one of two points on the surface of an airfoil where the flow speed is zero) begins on the top surface of an airfoil (assuming positive effective angle of attack
Angle of attack
Angle of attack is a term used in fluid dynamics to describe the angle between a reference line on a lifting body and the vector representing the relative motion between the lifting body and the fluid through which it is moving...
) as flow accelerates from zero, and moves backwards as the flow accelerates. Once the initial transient effects have died out, the stagnation point is at the trailing edge as required by the Kutta condition.
Mathematically, the Kutta condition enforces a specific choice among the infinite allowed values of circulation.