Page 90 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

Dynamics of Inviscid Fluids 75

whose focuses are located at the points A(1,0) and A‘(—1,0). In the

case when r = 1 the image in the plane (Z) will be the segment [—1,1]
run in both senses (on the “upper border” and then, in the opposite
sense, on the “lower border”). Obviously, in this case, the considered
transformation would map both the outside and inside of the unit disk
|Z| <1, onto the whole plane (z) with a cut along the segment [—1, 1]
(in accord with the existence of two inverse transformations Z = z+

Vz* —1, where, to fix the ideas, the positive determination of the root
at z = x > 1 is considered).
If is a circumference passing by A and A’, its image will be only a
circular arc joining A and A’ and crossing the center C of , an arc which
is run in both senses. Let’s now consider acircumference I’; passing only
through the singular point A (and not through A’). Its image will be

a closed curve with a sharp cuspidal point at A where the tangent is
the same with that to the arc ACA’ which is also “the skeleton” of this
contour.
This image contour is called the Joukovski (wing) profile, and the ini-
tial considered transformation is of Joukovski or Kutta—Joukovski type.
iY
Obviously to a fluid flow around I, of 3 velocity at far field, it

could associate a fluid flow of Vo velocity at infinity, past the considered
Joukovski profile, the incidences in both flows being the same.
The Joukovski profiles are technically hard to make and more, they are
not very realistic for practical purposes. That is why their importance

is mainly theoretical.
The above Joukovski type transformation could be generalized by
considering
1 R?
z= 5 (z + =)

or even zg = Z+ Rr the last transformation having the advantage of
equal velocities at far field in the associated flows. We remark that the
last form could be rewritten as

z-2R (Z—R)\?’

z+2R \Z+R)’

and it transforms the outside of |Z| = R onto the whole plane (z) with
a cut along the segment [—2R,2R]. A direct generalization would be

z—kR Z—R\*
_ l<k<2

z+kR (Fa) pe SRSA,

which points out that