Page 91 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

76 BASICS OF FLUID MECHANICS AND INTRODUCTION TO CFD

z—-kR=(Z-—R)*9(Z), o(R) #0,

a form which avoids the sharp cuspidal point and which, in the vicinity
of infinity, has the expansion

k(k—1) R?
z=Z4+
2 Z£
In this case the image of acircumference I passing through — R and

R will be the union of two circular arcs, symmetrical versus Og and
passing through —kR and kR.
Finally, if one considers the image of acircumference I), passing only
through Z = R and centered on the OX axis, this image will be tangent
to the previous symmetrical contour at KR where it has also a sharp point
with the angle of semitangents equal to 2km. Such an image is known
as a Karman-—Trefftz profile. An application on a dirigible balloon of

Karman—Trefftz type is given in chapter 6, 3.3.
Writing the Joukovski type transformation under the form

dz _(,_B\(,,8
dz Z)\ TZ}?

von Mises has considered the generalization

= (1-8) (1B)... (1-H) kc am ek

Again a circumference passing through Z = R is transformed onto a
(wing) profile of von Mises type, with a sharp point at a certain zg and
where the jump of each semitangent is kz.
We remark that if the Joukovski type profiles depend on two param-
eters (like the coordinates of the Ty center), the Karman — Trefftz type

profiles depend on three parameters (with the additional k) while the
von Mises type profiles depend on n+ 1 parameters.
JE. Carafoli has introduced the transformations of the type z = Z +
Ee + (Z—byP with pa positive integer (the order of the pole b). For small

a one obtains quasi—Joukovski profiles.
Caius Jacob has considered a class of profiles defined by the conformal
mappings expressed in terms of rational functions [70].
Recently, I. Taposu has emphasized a special class of profiles (“dolphin
profiles”) whose use in practice could improve the classical concepts of
aerodynamics [139].
In different laboratories around the world one deals with classes of

profiles (Naca, Géttingen, ONERA, RAE, Tzagy, etc.) which are given,
in general, “by points” and, seldom, by their analytical form.