# Is there a Bernoullis equation for the rotational flux

## Kutta-Joukowski theorem - Krasna Łąka

The Kutta-Joukowski theorem is a fundamental theorem in aerodynamics that is used for calculating the lift of an airfoil and of all two-dimensional bodies, including circular cylinders, that move in a uniform fluid at a constant speed large enough that the flow that flows through it is seen in the solid frame, is uniform and inseparable. The phrase 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, which encloses the flow profile of the velocity component of the fluid tangent to the loop. It is named after Martin Kutta and Nikolai Zhukovsky (or Joukowski) who developed their key ideas in the early 20th century. The Kutta-Joukowski theorem is a non-viscous theory, but it is a good approximation for real viscous flow in typical aerodynamic applications.

The Kutta-Joukowski theorem relates the lift to the circulation, similar to how the Magnus effect relates the lateral force (called Magnus force) to the rotation. However, the circulation is not induced here by the rotation of the airfoil. The fluid flow in the presence of the airfoil can be viewed as the superposition of a translational flow and a rotating flow. This rotating flow is induced by the effects of the camber, angle of attack and the sharp trailing edge of the airfoil. It should not be confused with a vortex like a tornado that surrounds the airfoil. At a great distance from the airfoil, the rotating flow can be viewed as being induced by a line vortex (the rotating line being perpendicular to the two-dimensional plane). When deriving the Kutta-Joukowski theorem, the wing profile is usually mapped onto a circular cylinder. In many textbooks the theorem is proven for a circular cylinder and the Joukowski airfoil, but it applies to general airfoils.

### Buoyancy formula

The theorem applies to two-dimensional flow around a fixed flow profile (or any shape of infinite span). The lift per unit span the flow profile is given by (1)

in which and are the fluid density and fluid velocity far upstream of the wing and is the circulation, defined as the line integral around a closed contour which includes the airfoil and followed in the negative direction (clockwise). As explained below, this path must be in an area of ​​the potential flow and not in the boundary layer of the cylinder. The integrand is the component of the local fluid velocity in the direction tangential to the curve and is an infinitesimal length on the curve, . equation (1) is a form of the Kutta-Joukowski theorem.

Kuethe and Schetzer state the Kutta-Joukowski theorem as follows:

The force per unit of length acting on a right cylinder of any cross-section is the same and is perpendicular to the direction of ### Circulation and the Kutta State

A lift-generating airfoil either has a camber or operates at a positive angle of attack between the chord line and the fluid flow far upstream of the airfoil. In addition, the airfoil must have a sharp trailing edge.

All real liquid is viscous, which means that the liquid velocity disappears on the airfoil. Prandtl showed that for large Reynolds numbers, defined as and a small angle of attack. The flow around a thin airfoil consists of a narrow viscous area called the boundary layer near the body and an inviscid flow region outside. When applying the Kutta-Joukowski theorem, the loop outside of this boundary layer must be chosen. (For example, the circulation calculated using the loop corresponding to the surface of the airfoil would be zero for a viscous liquid.)

The requirement for a sharp trailing edge physically corresponds to a flow in which the liquid moves along the lower and the upper surfaces of the airfoil meet smoothly without liquid moving around the trailing edge of the airfoil. This is called the kutta condition.

known. Kutta and Joukowski showed that for the calculation of the pressure and the lift of a thin flow profile for the flow at a large Reynolds number and a small angle of attack, the flow can be assumed to be non-viscous in the entire region outside the wing, provided the Kutta condition is imposed. This is known as the potential flow theory and works remarkably well in practice.

### Derivation

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

### requires. Heuristic argument

For a heuristic argument, consider a thin airfoil with a chord and infinite span, moves through air with a density . Let the airfoil tilt towards the oncoming flow, an air speed on one side of the airfoil and an air speed on the other hand. The circulation is then The pressure difference between the two sides of the airfoil can by applying Bernoulli's equation:  so that the buoyancy force per unit span