Kutta-Zhukovsky transformation

Archetype and image of a Kutta-Schukowski transformation

Frictionless, incompressible flow about an airfoil profile , calculated with the Joukowski transformation. The shades of blue represent the pressure (the darker the higher).

The Kutta-Schukowski transformation , often just called Schukowski transformation or after another transcription called Joukowski transformation , is a mathematical process that is used in fluid mechanics and electrostatics . It is the simplest transformation that, when applied to a circle , yields aerofoil profiles. It is named after Martin Wilhelm Kutta and Nikolai Jegorowitsch Schukowski .

definition

The Kutta-Schukowski transformation can be represented with complex numbers ; it is a conformal mapping . So it corresponds to a function with the equation ${\displaystyle f\colon \mathbb {C} \setminus \{0\}\rightarrow \mathbb {C} }$

${\displaystyle \zeta =f(z)=z+{\frac {a^{2}}{z}}}$

with a real parameter ɑ. In order to create wing contours with a curved center line, geometric calculations are also necessary, since the starting point of the transformation does not have to be the center, but a point within the circle that has been shifted around and around . ${\displaystyle x}$ ${\displaystyle y}$

properties

With and the imaginary unit you get ${\displaystyle x,y\in \mathbb {R} ,\;z=x+\mathrm {i} y}$ ${\displaystyle \mathrm {i} }$

${\displaystyle \zeta ={\frac {x^{2}+y^{2}+a^{2}}{x^{2}+y^{2}}}x+\mathrm {i} {\frac {x^{2}+y^{2}-a^{2}}{x^{2}+y^{2}}}y}$

All real numbers and the complex ones on the circle around the origin with radius ɑ are mapped to real numbers:

${\displaystyle z\in \mathbb {R} \;{\textsf {oder}}\;|z|=a\;\rightarrow \;\zeta \in \mathbb {R} }$

A circle through the origin with a larger radius | z | > ɑ is mapped onto an ellipse .

Illustration of a circular disk on the plane with a slot

The function ζ = f (z) maps the outside (or inside) of a circle with radius ɑ in the z-plane to the ζ-plane with a slot. The reverse of this figure

${\displaystyle z_{1,2}(\zeta )={\frac {\zeta }{2}}\pm {\sqrt {{\frac {\zeta ^{2}}{4}}-a^{2}}}}$

is not unique for all points lying on the flanks of the slot, with the exception of the ends of the slot. The two values ​​z ₁ and z ₂ are reciprocal to each other (z ₂  = ɑ ² / z ₁ ) and the number to be used is the value of which is greater than or equal to ɑ (or less than or equal to ɑ). On the flanks is z _1,2  = ɑ e ^{± iφ} , | z _1,2 | = ɑ, ζ = 2ɑ  cos (φ) ∈ ℝ and z ₂ is complex conjugate to z ₁ . The slot ends themselves are at ζ = ± 2ɑ or z = ± ɑ. For all other points of the ζ-plane (ζ ∉ ℝ or | ζ | ≥ 2ɑ) the mapping z (ζ) is unique.

These properties are used in fracture mechanics when calculating the Griffith crack with the Airy stress function .

Singularity at z = ± a

Because of the positions z = ± ɑ, the mapping has a singularity. The point z = -ɑ is usually mapped into the interior of the profile and then does not appear. If the circle leads through z = ɑ in the z-plane, then the tangents to the branches of the curve, which arrive in the ζ-plane at point ζ = 2ɑ, are parallel. The rear edge angle is then 0 ° as in the pictures. ${\displaystyle {\tfrac {\partial \zeta }{\partial z}}=1-{\tfrac {a^{2}}{z^{2}}}}$

application

Together with the circle, one also transforms the image of the streamlines around the circle, the velocity and pressure distribution , which can be calculated analytically with the assumption of a potential flow around the circle. The historical and educational significance of the method is based on the fact that the result of the transformation of the flow equation is sufficient and one as the dynamic lift with the Blasius'schen formula or the set of Kutta Joukowski can be calculated. With the formula, discovered in 1902, a comparison between theoretical and experimental wing research became possible and the first lift-generating wing profiles could be developed.

history

Kutta used the transformation for wing profiles, which consisted of infinitely thin circular arc segments. Schukowski showed that this method can also be used to calculate profiles of finite thickness and a curved central contour. However, profiles calculated in this way still have serious disadvantages, such as flow separation and increased vortex formation , which is why more complicated transformation equations were used later. Today, numerical methods are used to simulate the flow, which has two advantages: On the one hand, you can freely choose the profile course, including three-dimensional, on the other hand, you are not dependent on simplified flow equations and fields.

Web links

• Geometric basics (PDF file; 29 kB)
• interactive Kutta-Schukowski transformation Java applet requires a Java-enabled browser
• Applet for ideal flows , enables the conformal mapping of a flow field interactively with the Kutta-Schukowski transformation.

Individual evidence

1. ^ Spurk (2010), p. 414.
2. ↑ ^{a b} Spurk (2010), p. 416.

literature

• JH Spurk: Fluid Mechanics . Springer Verlag, Heidelberg, Dordrecht, London, New York 2010, ISBN 978-3-642-13142-4 , pp. 414 ff ., doi : 10.1007 / 978-3-642-13143-1 ( limited preview in Google Book Search [accessed April 29, 2020]).