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. 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 around the airfoil. 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 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. Kutta–Joukowski theorem relates lift to circulation much like the Magnus effect 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, angle of attack and the sharp trailing edge of the airfoil. It should not be confused with a vortex like a tornado 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). 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 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 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 is a form of the Kutta–Joukowski theorem. Kuethe and Schetzer state the Kutta–Joukowski theorem as follows:

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, 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. This is known as the Kutta condition. 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 theory and works remarkably well in practice.

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 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 The difference in pressure \Delta P 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.

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.