Zhukovskii function

The rational function

$$w=\lambda(z)=\frac12\left(z+\frac1z\right)$$

of the complex variable $z$. It is important for its applications in fluid mechanics, which were discovered by N.E. Zhukovskii (see [1], [2]), particularly in constructing and studying the Zhukovskii profile (Zhukovskii wing). Suppose that a circle $K$ is given in the $z$-plane passing through the points $z=\pm1$ (Fig. a), together with a circle $K'$ touching $K$ on the outside at $z=1$, with centre $\alpha$ and radius $\rho$. Under the mapping $w=\lambda(z)$, the image of $K'$ is a closed curve $L'$ with a cusp at the point $w=1$, touching an arc of the circle $L$ (the image of $K$) at that point; this image is represented in Fig. band is the Zhukovskii profile.

Figure: z099280a

Figure: z099280b

The function $w=\lambda(\rho t+\alpha)$ maps the exterior of the unit circle in the $t$-plane to the exterior of $L'$. To obtain a Zhukovskii profile of a more general shape and disposition, the generalized Zhukovskii function is applied (see [3], [4], [5]):

$$w=\frac12(a-b)z+\frac12(a+b)\frac1z,\quad a>b>0.$$

References

[1]	N.E. Zhukovskii, "Collected works" , 2. Hydrodynamics , Moscow-Leningrad (1949) (In Russian)
[2]	N.E. Zhukovskii, "Collected works" , 6. The theoretical foundations of flying , Moscow-Leningrad (1950) (In Russian)
[3]	A.I. Markushevich, "Theory of functions of a complex variable" , 1–2 , Chelsea (1977) (Translated from Russian)
[4]	L.J. [L.I. Sedov] Sedov, "Two-dimensional problems in hydrodynamics and aerodynamics" , Acad. Press (1965) (Translated from Russian)
[5]	N.E. Kochin, I.A. Kibel', N.V. Roze, "Theoretical hydrodynamics" , 1 , Interscience (1964) (Translated from Russian)

Comments

The Zhukovskii profiles (or Zhukovskii aerofoils) suffer the drawback that, as mentioned above, they have a cusp at the trailing edge. This implies that if one had to build wings with such a profile, one should obtain a very thin, and hence fragile, rear part of the wing. For this reason more general profiles, having a singularity with distinct tangents at the trailing edge, have been introduced (von Kármán–Trefftz profiles). Another generalization of the Zhukovskii profile goes in the direction of enlarging the number of parameters (von Mises profiles).

The Zhukovskii aerofoils are usually called the Kutta–Zhukovskii aerofoils in the Western literature. "Zhukovskii" is often spelled "Joukowski" in the Western literature.

References

[a1]	G. Birkhoff, "Hydrodynamics, a study in logic, fact and similitude" , Princeton Univ. Press (1960) Zbl 0095.20303
[a2]	J. Lighthill, "An informal introduction to theoretical fluid mechanics" , Clarendon Press (1986)
[a3]	R. von Mises, "Theory of flight" , Dover, reprint (1959)