Kutta–Joukowski theorem

The set of Kutta - Joukowski by other transcription also Kutta - Zhukovsky, Kutta Zhoukovski or English Kutta - Zhukovsky, describes in fluid mechanics, the proportionality of the dynamic lift for circulation

In which

. stand It is named after the German mathematician Martin Wilhelm Kutta and the Russian physicist and aviation pioneer Nikolai Zhukovsky Jegorowitsch.

Mathematically, the circulation, the result of the line integral. As soon as it is non-zero integral, a vortex is available.

The circulation here describes the measure of a rotating flow to a profile. This effect occurs for example at a flow around airfoil employed when the flow lines of the parallel flow and circulation flow superimposed. This causes a lift force F is on the upper side of the wing, which leads to the lifting of the wing.

Physical requirements

The Kutta - Joukowski formula is valid only under certain conditions on the flow field. It is the same as for the Blasius formula. That is, the flow must be two - dimensional stationary, incompressible, frictionless, irrotational and effectively. That is, in the direction of the third dimension, in the direction of the wing span, all variations are to be negligible. Forces in this direction therefore add up. Overall, they are proportional to the width. Because of the freedom of rotation extending the power lines from infinity to infinity in front of the body behind the body. Moreover, since true freedom from friction, the mechanical energy is conserved, and it may be the pressure distribution on the airfoil according to the Bernoulli equation can be determined. Summing the pressure forces initially leads to the first Blasius formula. From this the Kutta - Joukowski formula can be accurately derived with the aids function theory.

Mathematical properties and derivation

The computational advantages of the Kutta - Joukowski formula will be applied when formulating with complex functions to advantage. Then the level of the airfoil profile is the Gaussian number plane, and the local flow velocity is a holomorphic function of the variable. There exists a primitive function ( potential), so that

If we now proceed from a simple flow field (eg flow around a circular cylinder ) and it creates a new flow field by conformal mapping of the potential ( not the speed ) and subsequent differentiation with respect to, the circulation remains unchanged:

This follows ( heuristic ) the fact that the values ​​of at the conformal transformation is only moved from one point on the complex plane at a different point. Note that necessarily is a function of ambiguous when circulation does not disappear.

Because of the invariance can for example be the flow around a Joukowski profile directly from the circulation around a circular profile win. If you limit yourself with the transformations to those which do not alter the flow velocity at large distances from the airfoil ( specified speed of the aircraft ) as follows from the Kutta - Joukowski formula that all by such transformations apart resulting profiles have the same buoyancy.

For the derivation of the Kutta - Joukowski formula from the first Blasius formula the behavior of the flow velocity at large distances must be specified: In addition to holomorphy in the finite is as a function of continuous at the point. Then can be in a Laurent series development:

It is obvious. After the residue theorem also applies

It continues the series in the first Blasius formula and multiplied out. Again, only the term with the first negative power results in a contribution:

This is the Kutta Joukowski formula, both the vertical and the horizontal component of the force ( lift and drag ). From the prefactor follows that the power under the specified conditions (especially freedom from friction ) is always perpendicular to the inflow direction is (so-called d' Alembert's paradox).