Kutta-Joukowski's theorem

The Kutta-Joukowski theorem, after another transcription, also called Kutta-Schukowski , Kutta-Zhoukovski or English Kutta-Zhukovsky , describes the proportionality of the dynamic lift to the circulation in fluid mechanics

${\displaystyle F_{\mathrm {A} }'=-\rho \cdot v_{\infty }\cdot \Gamma }$

in which

${\displaystyle F_{\mathrm {A} }'}$for the buoyancy force per span

${\displaystyle \rho }$for the density of the medium flowing around

${\displaystyle v_{\infty }}$ for the undisturbed approach velocity

${\displaystyle \Gamma }$ for circulation

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

Mathematically, the circulation is the result of the line integral . As soon as this integral is different from zero, there is a vortex. ${\displaystyle \Gamma }$ ${\displaystyle \Gamma =\oint vds}$

The circulation describes the extent of a flow rotating around a profile. This effect occurs, for example, on a raised wing around which the air flows, when the streamlines of the parallel flow and the circulation flow overlap. This has the effect that a lift force forms on the upper side of the wing , which leads to the lifting of the wing. ${\displaystyle F_{\mathrm {A} }}$

Physical requirements

The Kutta-Joukowski formula only applies to the flow field under certain conditions. They are the same as for the Blasius formulas . That means, the flow must be stationary , incompressible , frictionless , rotation-free and effectively 2-dimensional. I.e. in the direction of the third dimension, in the case of the wing, the direction of the span, all variations should be negligible. Forces in this direction therefore add up. Overall, they are proportional to the width . Because of the freedom of rotation, the streamlines run from infinity in front of the body to infinity behind the body. Since there is also freedom from friction, the mechanical energy is preserved and the pressure distribution on the wing can be determined using Bernoulli's equation . Adding up the compressive forces leads to the first Blasius formula . From this, the Kutta-Joukowski formula can be derived exactly with the aid of function theory . ${\displaystyle b}$

Mathematical properties and derivation

The computational advantages of the Kutta-Joukowski formula only come into play when it is formulated with complex functions. Then the plane of the wing profile is the Gaussian number plane and the local flow velocity is a holomorphic function of the variables . There is an antiderivative (potential) such that ${\displaystyle v^{*}=v_{x}-\mathrm {i} v_{y}}$ ${\displaystyle z=x+\mathrm {i} y}$ ${\displaystyle \Phi (z)}$

${\displaystyle v^{*}={\frac {\mathrm {d} \Phi }{\mathrm {d} z}}}$

If you now start from a simple flow field (e.g. flow around a circular cylinder) and create a new flow field by conforming the potential (not the velocity) and then deriving it, the circulation remains unchanged: ${\displaystyle z}$

${\displaystyle \Gamma =\oint v^{*}(z)\mathrm {d} z\qquad {\text{invariant unter konformen Transformationen von }}\Phi }$

This follows (heuristically) from the fact that the values ​​of are only shifted from one point on the numerical level to another point during the conformal transformation. It should be noted that if the circulation does not disappear , there is necessarily an ambiguous function. ${\displaystyle \Phi }$ ${\displaystyle \Phi }$

Because of the invariance, z. B. gain the circulation around a Joukowski profile directly from the circulation around a circular profile. If the transformations are limited to those that do not change the flow speed at large distances from the wing (given speed of the aircraft), it follows from the Kutta-Joukowski formula that all profiles resulting from such transformations have the same lift.

To derive the Kutta-Joukowski formula from the 1st Blasius formula, the behavior of the flow velocity at large distances must be specified: In addition to the holomorphism in the finite, let as a function of continuous in the point . Then you can develop into a Laurent series : ${\displaystyle v^{*}}$ ${\displaystyle w=1/z}$ ${\displaystyle w=0}$ ${\displaystyle v^{*}}$

${\displaystyle v^{*}(z)=A_{0}+{\frac {A_{1}}{z}}+{\frac {A_{2}}{z^{2}}}+\cdots }$

It is obvious . According to the residual theorem also applies ${\displaystyle A_{0}=v_{\infty }}$

${\displaystyle A_{1}={\frac {\Gamma }{2\pi \mathrm {i} }}}$

You insert the series in the 1st Blasius formula and multiply from. Again, only the term with the first negative power gives a contribution:

${\displaystyle F^{*}=\mathrm {i} {\frac {\rho }{2}}\oint (v^{*})^{2}\mathrm {d} z=\mathrm {i} {\frac {\rho }{2}}\oint {\frac {2A_{0}A_{1}}{z}}\mathrm {d} z=\mathrm {i} \rho v_{\infty }\Gamma }$

This is the Kutta-Joukowski formula for both the vertical and horizontal components of force (lift and drag). The pre-factor shows that the force is always perpendicular to the direction of flow under the stated conditions (especially freedom from friction) (so-called d'Alembert's paradox ). ${\displaystyle \mathrm {i} \!\,}$

Individual evidence

1. ↑ Schlichting / Truckenbrodt Sections 2.5 and 6.2
2. ↑ Schlichting / Truckenbrodt, Section 6.212

literature

• Hermann Schlichting , Erich Truckenbrodt : Aerodynamics of the Airplane, Volume 1, Springer 1967