Kutta–Joukowski theorem

The Kutta–Joukowski theorem is a fundamental theorem of aerodynamics, that can be used for the calculation of the lift of an airfoil, or of any two-dimensional bodies 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 of the fluid, and the circulation. The circulation is defined as the line integral, around a closed loop enclosing the airfoil, of the component of the velocity of the fluid tangent to the loop. It is named after the German Martin Wilhelm Kutta and the Russian Nikolai Zhukovsky (or Joukowski) who first developed its key ideas in the early 20th century. Kutta–Joukowski theorem is an inviscid theory which for pressure and lift is however a good approximation to real viscous flow for typical aerodynamic applications.

Kutta–Joukowski theorem relates lift to circulation much like the Magnus effect relates side force (called Magnus force) to rotation. However, circulation here is not induced by rotation of the airfoil but by some intrinsic mechanism described below. Due to this circulation, the flow of air in response to the presence of the airfoil can be treated as the superposition of a translational flow and a rotating flow. This rotating flow is induced by joint effect of camber, angle of attack and sharp trailing edge of the airfoil and should not be confused with a vortex like a tornado encircling the cylinder or the wing of an airplane in flight. Seen from a distance large enough to the airfoil, this rotating flow may be regarded as induced by a line vortex (with the rotating line perpendicular to the twodimensional plane). In the derivation of the Kutta–Joukowski theorem the airfoil is usually mapped into a circular cylinder. This theorem is proved in many text books only for circular cylinder and Joukowski airfoil, but it holds true for general airfoils.

Lift force formula

The theorem refers to two-dimensional flow around an airfoil (or a cylinder of infinite span) and determines the lift generated by one unit of span. When the circulation $\Gamma\,$ is known, the lift $L\,$ per unit span (or $L'\,$) of the airfoil can be calculated using the following equation:

$L^\prime = -\rho_\infty V_\infty\Gamma,\,$

(1)

where $\rho_\infty\,$ and $V_\infty\,$ are the fluid density and the fluid velocity far upstream of the airfoil which is now regarded fix on a body fixed frame, and $\Gamma\,$ is the (anticlockwise positive) circulation defined as the line integral,

$\Gamma= \oint_{C} V \cdot d\mathbf{s}=\oint_{C} V\cos\theta\; ds\,$

around a closed contour $C$ enclosing the cylinder or airfoil and followed in the positive (anticlockwise) direction. This path must be in a region of potential flow and not in the boundary layer 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 (1) 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 $V_\infty.$

In using the Kutta–Joukowski theorem, caution should be paid on circulation $\Gamma$.

Circulation and Kutta condition

A lift producing airfoil either has camber or is translating in a uniform fluid at an angle of attack $\alpha>0\,$ (angle between the chord line and the direction of translation). Moreover, it must have a sharp trailing edge. To design a lift producing airfoil we certainly have borrowed much from bird wings since bird wings have section or airfoils with sharp trailing edge, with camber and moving in air at some angle of attack.

Any real flow is viscous and the fluid velocity vanishes on the airfoil so we should have a vanishing circulation if we treat the fluid as viscous and if the loop is chosen as the contour of the airfoil. Moreover, fluids moving along the lower and upper surfaces of the airfoil should meet at the sharp trailing edge since viscous dissipation prevents the fluid to turn round the sharp edge. This is known as the Kutta condition for real flow. Prandtl discovered that when the Reynolds number, defined as $Re=\frac{\rho V_{\infty}c_A}{\mu}\,$, is large enough and the angle of attack is small enough, then the flow around a thin enough airfoil is composed of a narrow viscous region called boundary layer near the body and an inviscid flow region outside.

Kutta and Joukowski discovered that for computing the pressure and lift of a thin enough airfoil for flow with large enough Reynolds number and at small enough angle of attack, the flow can be assumed inviscid in the entire region, provided the Kutta condition is imposed. This is known as the potential flow theory that works remarkably well in practice. Imposing the Kutta condition at the inviscid case is equivalent to giving a circulation.

In summary, a bird-wing like airfoil naturally produces lift and the flow in flight condition meets the Kutta condition. When using the potential flow theory (assumed inviscid and irrotational for calculating the pressure and lift, and the Prandtl boundary layer approximation may be then used to compute the frictional drag), requiring Kutta condition to be satisfied at flight condition yields a circulation required by applying the Kutta–Joukowski theorem that gives a lift force very close to that of the real flight.

Derivation

Two derivations are presented below. The first is a heuristic argument, based on physical insight. The second is a formal and technical one, requiring basic vector analysis and complex analysis.

Heuristic argument

For a rather 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:

$\frac {\rho}{2}(V)^2 + (P + \Delta P) = \frac {\rho}{2}(V + v)^2 + P,\,$

$\frac {\rho}{2}(V)^2 + \Delta P = \frac {\rho}{2}(V^2 + 2 V v + v^2),\,$

$\Delta P = \rho V v \qquad \text{(ignoring } \frac{\rho}{2}v^2),\,$

so the lift force per unit span is

$L' = c \Delta P = \rho V v c =-\rho V\Gamma.\,$

A differential version of this theorem applies on each element of the plate and is the basis of thin-airfoil theory.

Formal derivation

Formal derivation of Kutta–Joukowski theorem

First of all, the force exerted on each unit length of a cylinder of arbitrary cross section is calculated. Let this force per unit length (from now on referred to simply as force) be $\mathbf{F}\,$. So then the total force is: $\mathbf{F}=-\oint_C p \mathbf{n}\, ds,$ where C denotes the borderline of the cylinder, $p$ is the static pressure of the fluid, $\mathbf{n}\,$ is the unit vector normal to the cylinder, and ds is the arc element of the borderline of the cross section. Now let $\phi$ be the angle between the normal vector and the vertical. Then the components of the above force are: $F_x= -\oint_C p \sin\phi\, ds \quad, \qquad F_y= \oint_C p \cos\phi\, ds.$ Now comes a crucial step: consider the used two-dimensional space as a complex plane. So every vector can be represented as a complex number, with its first component equal to the real part and its second component equal to the imaginary part of the complex number. Then, the force can be represented as: $F=F_x+iF_y=-\oint_Cp(\sin\phi-i\cos\phi)\,ds .$ The next step is to take the complex conjugate of the force $F$ and do some manipulation: $\bar{F}=-\oint_C p(\sin\phi+i\cos\phi)\,ds=-i\oint_C p(\cos\phi-i\sin\phi)\, ds=-i\oint_C p e^{-i\phi}\,ds.$ Surface segments ds are related to changes dz along them by: $dz=dx+idy=ds(\cos\phi+i\sin\phi)=ds\,e^{i\phi} \qquad \Rightarrow \qquad d\bar{z}=e^{-i\phi}ds.$ Plugging this back into the integral, the result is: $\bar{F}=-i\oint_C p \, d\bar{z}.$ Now the Bernoulli equation is used, in order to remove the pressure from the integral. Throughout the analysis it is assumed that there is no outer force field present. The mass density of the flow is $\rho.$ Then pressure $p$ is related to velocity $v=v_x+iv_y$ by: $p=p_0-\frac{\rho |v|^2}{2}.$ With this the force $F$ becomes: $\bar{F}=-ip_0\oint_C d\bar{z} +i \frac{\rho}{2} \oint_C |v|^2\, d\bar{z} = \frac{i\rho}{2}\oint_C |v|^2\,d\bar{z}.$ Only one step is left to do: introduce $w=f(z),$ the complex potential of the flow. This is related to the velocity components as $w'=v_x-iv_y=\bar{v},$ where the apostrophe denotes differentiation with respect to the complex variable z. The velocity is tangent to the borderline C, so this means that $v=\pm |v| e^{i\phi}.$ Therefore, $v^2d\bar{z}=|v|^2dz, \,$ and the desired expression for the force is obtained: $\bar{F}=\frac{i\rho}{2}\oint_C w'^2\,dz,$ which is called the Blasius–Chaplygin formula. To arrive at the Joukowski formula, this integral has to be evaluated. From complex analysis it is known that a holomorphic function can be presented as a Laurent series. From the physics of the problem it is deduced that the derivative of the complex potential $w$ will look thus: $w'(z)=a_0+\frac{a_1}{z}+\frac{a_2}{z^2}+\dots .$ The function does not contain higher order terms, since the velocity stays finite at infinity. So $a_0\,$ represents the derivative the complex potential at infinity: $a_0=v_{x\infty}-iv_{y\infty}\,$. The next task is to find out the meaning of $a_1\,$. Using the residue theorem on the above series: $a_1=\frac{1}{2\pi i} \oint_C w'\, dz.$ Now perform the above integration: $\oint_C w'(z)\,dz =\oint_C (v_x-iv_y)(dx+idy)= \oint_C (v_x\,dx+v_y\,dy)+i\oint_C(v_x\,dy-v_y\,dx)=\oint_C \mathbf{v}\,{ds} +i\oint_C(v_x\,dy-v_y\,dx).$ The first integral is recognized as the circulation denoted by $\Gamma.$ The second integral can be evalutated after some manipulation: $\oint_C(v_x\,dy-v_y\,dx)=\oint_C\left(\frac{\partial \psi}{\partial y}dy+\frac{\partial\psi}{\partial x}dx\right)=\oint_C d\psi=0.$ Here $\psi\,$ is the stream function. Since the C border of the cylinder is a streamline itself, the stream function does not change on it, and $d\psi=0 \,$. Hence the above integral is zero. As a result: $a_1=\frac{\Gamma}{2\pi i}.$ Take the square of the series: $w'^2(z)=a_0^2+\frac{a_0\Gamma}{\pi i z} +\dots.$ Plugging this back into the Blasius–Chaplygin formula, and performing the integration using the residue theorem: $\bar{F}=\frac{i\rho}{2}\left[2\pi i \frac{a_0\Gamma}{\pi i}\right]=i\rho a_0 \Gamma = i\rho \Gamma(v_{x\infty}-iv_{y\infty})=\rho\Gamma v_{y\infty}+ i\rho\Gamma v_{x\infty}=F_x-iF_y.$ And so the Kutta–Joukowski formula is: $F_x=\rho \Gamma v_{y\infty} \quad, \qquad F_y= -\rho \Gamma v_{x\infty}.$

Lift forces for more complex situations

The lift predicted by 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.

a) Kutta Joukowski Theorem for steady irrotational flow. 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.

b) 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 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 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 has traveled a distance equal to seven chord lengths.

c) 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.

d) 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,(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.

e) 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.

f) 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.

g) 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.

h) 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.

i) 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), due to conservation of vorticity or Kelvin Theorem of Circulation Conservation. These streamwise vortices merge to two counter-rotating strong spirals, called wing tipe vortices, separated by distance close to the wing span and may be visible if the sky is cloudy. 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. Most importantly, there is an induced drag. This induced drag is a pressure drag which has nothing to do with frictional drag.

See also

• Magnus Effect
• Horseshoe vortex
• Kutta condition
• Lift coefficient
• Wing

References

• Batchelor, G. K. (1967) An Introduction to Fluid Dynamics, Cambridge University Press
• Clancy, L.J. (1975), Aerodynamics, Pitman Publishing Limited, London ISBN 0-273-01120-0
• A.M. Kuethe and J.D. Schetzer (1959), Foundations of Aerodynamics, John Wiley & Sons, Inc., New York ISBN 0-471-50952-3