The Kutta–Joukowski theorem is a fundamental theorem in aerodynamics used for the calculation of lift of an airfoil (and any two-dimensional body including circular cylinders) translating in a uniform fluid at a constant speed so large that the flow seen in the body-fixed frame is steady and unseparated.
[1] It is named after Martin Kutta 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 flow in typical aerodynamic applications.
This rotating flow is induced by the effects of camber, angle of attack and the sharp trailing edge of the airfoil.
is the circulation defined as the line integral around a closed contour
As explained below, this path must be in a region of potential flow and not in the boundary layer of the cylinder.
is the component of the local fluid velocity in the direction tangent to the curve
Kuethe and Schetzer state the Kutta–Joukowski theorem as follows:[5] 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.
Prandtl showed that for large Reynolds number, defined as
, and small angle of attack, the flow around a thin airfoil is composed of a narrow viscous region called the boundary layer near the body and an inviscid flow 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.
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.
For a heuristic argument, consider a thin airfoil of chord
Let the airfoil be inclined to the oncoming flow to produce an air speed
between the two sides of the airfoil can be found by applying Bernoulli's equation: so the downward force on the air, per unit span, is and the upward force (lift) on the airfoil is
A differential version of this theorem applies on each element of the plate and is the basis of thin-airfoil theory.
First of all, the force exerted on each unit length of a cylinder of arbitrary cross section is calculated.
is the unit vector normal to the cylinder, and ds is the arc element of the borderline of the cross section.
Then the components of the above force are: 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.
and do some manipulation: Surface segments ds are related to changes dz along them by: Plugging this back into the integral, the result is: 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.
where the apostrophe denotes differentiation with respect to the complex variable z.
and the desired expression for the force is obtained: which is called the Blasius theorem.
From complex analysis it is known that a holomorphic function can be presented as a Laurent series.
will look thus: The function does not contain higher order terms, since the velocity stays finite at infinity.
As a result: Take the square of the series: Plugging this back into the Blasius–Chaplygin formula, and performing the integration using the residue theorem: And so the Kutta–Joukowski formula is: 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.
When the flow is rotational, more complicated theories should be used to derive the lift forces.