Joukowsky transform

The Kutta - Zhukovsky transformation, often only Zhukovsky transformation or called by other transcription Joukowski transform is a mathematical method, finds the application in fluid mechanics and electrostatics. It is the simplest transformation that delivers applied to a circuit as a result of airfoils. It is named after Martin Wilhelm Kutta, and Nikolai Zhukovsky Jegorowitsch.

Definition

The Kutta - Zhukovsky transformation can be represented by complex numbers, there is a conformal mapping. So corresponds to a function with the equation

To create contours with curved wing center line, also geometric calculations are still necessary since the starting point of the transformation must be not the center, but a by x and y shifted dot inside the circle here.

Application

Together with the circle transformed to the image of the streamlines around the circle, the velocity and pressure distribution, which you can easily choose because of the symmetry so that they satisfy the flow equation. The historical and educational significance of the procedure depends on the fact that also satisfies the result of the transformation of the flow equation, and one can thus calculate the dynamic lift immediately. Thus, a comparison between theoretical and experimental research wing was possible.

History

Kutta used the transformation of airfoils, which consisted of infinitely thin circular arc segments. Zhukovsky showed that one can also calculate profiles of finite thickness and curved middle contour with this method. However, such a calculated profiles do not have serious drawbacks, such as flow separation and increased vortex formation, which is why later complicated transformation equations were used. Today it is numerical methods for the simulation of the flow one, has two advantages: on the one hand, one can choose the profile course free, even in three dimensions, on the other hand it does not rely on simplified flow equations and fields.