Kutta–Joukowski theorem
   HOME

TheInfoList



OR:

The Kutta–Joukowski theorem is a fundamental theorem in
aerodynamics Aerodynamics, from grc, ἀήρ ''aero'' (air) + grc, δυναμική (dynamics), is the study of the motion of air, particularly when affected by a solid object, such as an airplane wing. It involves topics covered in the field of fluid dy ...
used for the calculation of
lift Lift or LIFT may refer to: Physical devices * Elevator, or lift, a device used for raising and lowering people or goods ** Paternoster lift, a type of lift using a continuous chain of cars which do not stop ** Patient lift, or Hoyer lift, mobil ...
of an airfoil (and any two-dimensional body including circular cylinders) translating in a uniform fluid at a constant speed large enough so that the flow seen in the body-fixed frame is steady and unseparated. The theorem relates the lift generated by an airfoil to the speed of the airfoil through the fluid, the
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
of the fluid and the circulation around the airfoil. The circulation is defined as the
line integral In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; ''contour integral'' is used as well, al ...
around a closed loop enclosing the airfoil of the component of the velocity of the fluid
tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
to the loop. It is named after
Martin Kutta Martin Wilhelm Kutta (; 3 November 1867 – 25 December 1944) was a German mathematician. Kutta was born in Pitschen, Upper Silesia (today Byczyna, Poland). He attended the University of Breslau from 1885 to 1890, and continued his studies in Mu ...
and Nikolai Zhukovsky (or Joukowski) who first developed its key ideas in the early 20th century. Kutta–Joukowski theorem is an inviscid theory, but it is a good approximation for real
viscous The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quantifies the inte ...
flow in typical aerodynamic applications. Kutta–Joukowski theorem relates lift to circulation much like the
Magnus effect The Magnus effect is an observable phenomenon commonly associated with a spinning object moving through a fluid. The path of the spinning object is deflected in a manner not present when the object is not spinning. The deflection can be expl ...
relates side force (called Magnus force) to rotation. However, the circulation here is not induced by rotation of the airfoil. The fluid flow in the presence of the airfoil can be considered to be the superposition of a translational flow and a rotating flow. This rotating flow is induced by the effects of
camber Camber may refer to a variety of curvatures and angles: * Camber angle, the angle made by the wheels of a vehicle * Camber beam, an upward curvature of a joist to compensate for load deflection due in buildings * Camber thrust in bike technology * ...
, angle of attack and the sharp
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 meets.Crane, Dale: ''Dictionary of Aeronautical Terms, third edition'', page 521. Aviation Supplies & Academics, 199 ...
of the airfoil. It should not be confused with a vortex like a
tornado A tornado is a violently rotating column of air that is in contact with both the surface of the Earth and a cumulonimbus cloud or, in rare cases, the base of a cumulus cloud. It is often referred to as a twister, whirlwind or cyclone, altho ...
encircling the airfoil. At a large distance from the airfoil, the rotating flow may be regarded as induced by a line vortex (with the rotating line perpendicular to the two-dimensional plane). In the derivation of the Kutta–Joukowski theorem the airfoil is usually mapped onto a circular cylinder. In many textbooks, the theorem is proved for a circular cylinder and the Joukowski airfoil, but it holds true for general airfoils.


Lift force formula

The theorem applies to two-dimensional flow around a fixed airfoil (or any shape of infinite
span Span may refer to: Science, technology and engineering * Span (unit), the width of a human hand * Span (engineering), a section between two intermediate supports * Wingspan, the distance between the wingtips of a bird or aircraft * Sorbitan ester ...
). The lift per unit span L'\,of the airfoil is given by where \rho_\infty\, and V_\infty\, are the fluid density and the fluid velocity far upstream of the airfoil, and \Gamma\, is the circulation defined as the
line integral In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; ''contour integral'' is used as well, al ...
:\Gamma= \oint_ V \cdot d\mathbf=\oint_ V\cos\theta\; ds\, around a closed contour C enclosing the airfoil and followed in the negative (clockwise) direction. As explained below, this path must be in a region of
potential flow In fluid dynamics, potential flow (or ideal 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 app ...
and not in the
boundary layer In physics and fluid mechanics, a boundary layer is the thin layer of fluid in the immediate vicinity of a bounding surface formed by the fluid flowing along the surface. The fluid's interaction with the wall induces a no-slip boundary cond ...
of the cylinder. The integrand V\cos\theta\, is the component of the local fluid velocity in the direction tangent to the curve C\, and ds\, is an infinitesimal length on the curve, C\,. Equation is a form of the ''Kutta–Joukowski theorem.'' Kuethe and Schetzer state the Kutta–Joukowski theorem as follows: :''The force per unit length acting on a right cylinder of any cross section whatsoever is equal to \rho_\infty V_\infty \Gamma and is perpendicular to the direction of V_\infty.''


Circulation and the Kutta condition

A lift-producing airfoil either has camber or operates at a positive angle of attack, the angle between the chord line and the fluid flow far upstream of the airfoil. Moreover, the airfoil must have a sharp trailing edge. Any real fluid is viscous, which implies that the fluid velocity vanishes on the airfoil. Prandtl showed that for large Reynolds number, defined as \mathord = \frac\,, and small angle of attack, the flow around a thin airfoil is composed of a narrow viscous region called the
boundary layer In physics and fluid mechanics, a boundary layer is the thin layer of fluid in the immediate vicinity of a bounding surface formed by the fluid flowing along the surface. The fluid's interaction with the wall induces a no-slip boundary cond ...
near the body and an
inviscid flow In fluid dynamics, inviscid flow is the flow of an inviscid (zero-viscosity) fluid, also known as a superfluid. The Reynolds number of inviscid flow approaches infinity as the viscosity approaches zero. When viscous forces are neglected, suc ...
region outside. In applying the Kutta-Joukowski theorem, the loop must be chosen outside this boundary layer. (For example, the circulation calculated using the loop corresponding to the surface of the airfoil would be zero for a viscous fluid.) The sharp trailing edge requirement corresponds physically to a flow in which the fluid moving along the lower and upper surfaces of the airfoil meet smoothly, with no fluid moving around the trailing edge of the airfoil. This is known as the
Kutta condition The Kutta condition is a principle in steady-flow fluid dynamics, especially aerodynamics, that is applicable to solid bodies with sharp corners, such as the trailing edges of airfoils. It is named for German mathematician and aerodynamicist Mart ...
. Kutta and Joukowski showed that for computing the pressure and lift of a thin airfoil for flow at large Reynolds number and small angle of attack, the flow can be assumed inviscid in the entire region outside the airfoil provided the Kutta condition is imposed. This is known as the
potential flow In fluid dynamics, potential flow (or ideal 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 app ...
theory and works remarkably well in practice.


Derivation

Two derivations are presented below. The first is a
heuristic A heuristic (; ), or heuristic technique, is any approach to problem solving or self-discovery that employs a practical method that is not guaranteed to be optimal, perfect, or rational, but is nevertheless sufficient for reaching an immediate ...
argument, based on physical insight. The second is a formal and technical one, requiring basic
vector analysis Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subjec ...
and complex analysis.


Heuristic argument

For a heuristic argument, consider a thin airfoil of chord c and infinite span, moving through air of density \rho. Let the airfoil be inclined to the oncoming flow to produce an air speed V on one side of the airfoil, and an air speed V + v on the other side. The circulation is then :\Gamma = Vc - (V + v)c = -v c.\, The difference in pressure \Delta P between the two sides of the airfoil can be found by applying
Bernoulli's equation In fluid dynamics, Bernoulli's principle states that an increase in the speed of a fluid occurs simultaneously with a decrease in static pressure or a decrease in the fluid's potential energy. The principle is named after the Swiss mathematic ...
: :\begin \frac (V)^2 + (P + \Delta P) &= \frac (V + v)^2 + P,\, \\ \frac (V)^2 + \Delta P &= \frac (V^2 + 2 V v + v^2),\, \\ \Delta P &= \rho V v \qquad \text \fracv^2),\, \end so the downward force on the air, per unit span, is :L' = c \Delta P = \rho V v c = -\rho V\Gamma\, and the upward force (lift) on the airfoil is \rho V\Gamma.\, A differential version of this theorem applies on each element of the plate and is the basis of
thin-airfoil theory An airfoil (American English) or aerofoil (British English) is the cross-sectional shape of an object whose motion through a gas is capable of generating significant lift, such as a wing, a sail, or the blades of propeller, rotor, or turbine. ...
.


Formal derivation


Lift forces for more complex situations

The lift predicted by the Kutta-Joukowski theorem within the framework of inviscid potential flow theory is quite accurate, even for real viscous flow, provided the flow is steady and unseparated. In deriving the Kutta–Joukowski theorem, the assumption of irrotational flow was used. When there are free vortices outside of the body, as may be the case for a large number of unsteady flows, the flow is rotational. When the flow is rotational, more complicated theories should be used to derive the lift forces. Below are several important examples. ; Impulsively started flow at small angle of attack : For an impulsively started flow such as obtained by suddenly accelerating an airfoil or setting an angle of attack, there is a
vortex sheet A vortex sheet is a term used in fluid mechanics for a surface across which there is a discontinuity in fluid velocity, such as in slippage of one layer of fluid over another. While the tangential components of the flow velocity are discontinuous ...
continuously shed at the trailing edge and the lift force is unsteady or time-dependent. For small angle of attack starting flow, the vortex sheet follows a planar path, and the curve of the
lift coefficient In fluid dynamics, the lift coefficient () is a dimensionless quantity that relates the lift generated by a lifting body to the fluid density around the body, the fluid velocity and an associated reference area. A lifting body is a foil or a com ...
as function of time is given by the Wagner function. In this case the initial lift is one half of the final lift given by the Kutta–Joukowski formula. The lift attains 90% of its steady state value when the
wing A wing is a type of fin that produces lift while moving through air or some other fluid. Accordingly, wings have streamlined cross-sections that are subject to aerodynamic forces and act as airfoils. A wing's aerodynamic efficiency is e ...
has traveled a distance of about seven chord lengths. ; Impulsively started flow at large angle of attack : When the angle of attack is high enough, the trailing edge vortex sheet is initially in a spiral shape and the lift is singular (infinitely large) at the initial time. The lift drops for a very short time period before the usually assumed monotonically increasing lift curve is reached. ; Starting flow at large angle of attack for wings with sharp leading edges : If, as for a flat plate, the leading edge is also sharp, then vortices also shed at the leading edge and the role of leading edge vortices is two-fold: 1) they are lift increasing when they are still close to the leading edge, so that they elevate the Wagner lift curve, and 2) they are detrimental to lift when they are convected to the trailing edge, inducing a new trailing edge vortex spiral moving in the lift decreasing direction. For this type of flow a vortex force line (VFL) map can be used to understand the effect of the different vortices in a variety of situations (including more situations than starting flow) and may be used to improve vortex control to enhance or reduce the lift. The vortex force line map is a two dimensional map on which vortex force lines are displayed. For a vortex at any point in the flow, its lift contribution is proportional to its speed, its circulation and the cosine of the angle between the streamline and the vortex force line. Hence the vortex force line map clearly shows whether a given vortex is lift producing or lift detrimental. ; Lagally theorem : When a (mass) source is fixed outside the body, a force correction due to this source can be expressed as the product of the strength of outside source and the induced velocity at this source by all the causes except this source. This is known as the Lagally theorem. For two-dimensional inviscid flow, the classical Kutta Joukowski theorem predicts a zero drag. When, however, there is vortex outside the body, there is a vortex induced drag, in a form similar to the induced lift. ; Generalized Lagally theorem : For free vortices and other bodies outside one body without bound vorticity and without vortex production, a generalized Lagally theorem holds, with which the forces are expressed as the products of strength of inner singularities (image vortices, sources and doublets inside each body) and the induced velocity at these singularities by all causes except those inside this body. The contribution due to each inner singularity sums up to give the total force. The motion of outside singularities also contributes to forces, and the force component due to this contribution is proportional to the speed of the singularity. ; Individual force of each body for multiple-body rotational flow : When in addition to multiple free vortices and multiple bodies, there are bound vortices and vortex production on the body surface, the generalized Lagally theorem still holds, but a force due to vortex production exists. This vortex production force is proportional to the vortex production rate and the distance between the vortex pair in production. With this approach, an explicit and algebraic force formula, taking into account of all causes (inner singularities, outside vortices and bodies, motion of all singularities and bodies, and vortex production) holds individually for each body with the role of other bodies represented by additional singularities. Hence a force decomposition according to bodies is possible. ; General three-dimensional viscous flow : For general three-dimensional, viscous and unsteady flow, force formulas are expressed in integral forms. The volume integration of certain flow quantities, such as vorticity moments, is related to forces. Various forms of integral approach are now available for unbounded domain and for artificially truncated domain. The Kutta Joukowski theorem can be recovered from these approaches when applied to a two-dimensional airfoil and when the flow is steady and unseparated. ; Lifting line theory for wings, wing-tip vortices and induced drag : A wing has a finite span, and the circulation at any section of the wing varies with the spanwise direction. This variation is compensated by the release of streamwise vortices, called
trailing vortices Wingtip vortices are circular patterns of rotating air left behind a wing as it generates lift.Clancy, L.J., ''Aerodynamics'', section 5.14 One wingtip vortex trails from the tip of each wing. Wingtip vortices are sometimes named ''trailing' ...
, due to conservation of vorticity or Kelvin Theorem of Circulation Conservation. These streamwise vortices merge to two counter-rotating strong spirals separated by distance close to the wingspan and their cores may be visible if relative humidity is high. Treating the trailing vortices as a series of semi-infinite straight line vortices leads to the well-known lifting line theory. By this theory, the wing has a lift force smaller than that predicted by a purely two-dimensional theory using the Kutta–Joukowski theorem. This is due to the upstream effects of the trailing vortices' added downwash on the angle of attack of the wing. This reduces the wing's effective angle of attack, decreasing the amount of lift produced at a given angle of attack and requiring a higher angle of attack to recover this lost lift. At this new higher angle of attack, drag has also increased. Induced drag effectively reduces the slope of the lift curve of a 2-D airfoil and increases the angle of attack of C_ (while also decreasing the value of C_).


See also

*
Horseshoe vortex The horseshoe vortex model is a simplified representation of the vortex system present in the flow of air around a wing. This vortex system is modelled by the ''bound vortex'' (bound to the wing) and two '' trailing vortices'', therefore having ...


References


Bibliography

* Milne-Thomson, L.M. (1973) ''Theoretical Aerodynamics'', Dover Publications Inc, New York {{DEFAULTSORT:Kutta-Joukowski theorem Aerodynamics Fluid dynamics Physics theorems Aircraft wing design