A plot of the cross sectional outline of a Joukowski wing is presented

Trefftz, E.

English

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N.A.C'A" Technical _emorandnm No. 336
GRAPHIC CONSTRUCTION OF JCUKOWSKI WINGS.*
By E. Trefftz.
In plotting the cross-section_=.l outline (or profile) of
a Joukowski wing, we proceed as follows (Fig. 6).
We first plot an xy system of coordinates with the ori-
gin 0 such that the x axis forms the angle @ with the hor-
izontal direction of the wing and mark on the x axis the
point L, for which x = - _, and on the y axis the point
F, for which y = f.
Wc now describe two circles and label them K_ and K 2.
The center l.{_ of the first circle is situated on the straight
line LF at a distance 26 from the ooint F (beyond the
section LF). The circle, moreover, passes through, the point
L The second circle likewise Gas.sos _nrouoh the point L
and its center X_ is likewise on LF, the position of _,[
on LF being determined by the following condition. If OV_
is the portion of the positive x axis cut off by the circle
K_ s,ud OV2 the portion cut off by the circle K2, then
OV_ x 0_ = t_ •
We now draw, from the point O, the two lines OA_ and
OA2, so as to form equal angles with the x axis, A_ being
the ooint of intersection of the first line with the circle
X_ and A_ the intersection of the second line with the circle
From "Zeitsehrift f_r Flugtechnik und )_otor!uftschiffahrt,"
L_a7 31, 1913, pp. 130 and 131.
84
https://ntrs.nasa.gov/search.jsp?R=19800006778 2020-03-21T19:28:51+00:00Z
N.A.C-A. Technical _{emorsndum No. 336
K2. Than the center P of the line AqA_ is the point sought
on the Joukowski win_ profile.
In plotting the preceding fibres, $4 noints were found
in this manner for each one, by shifting the first line from
the point L 15° each time and drawing the second line sym-
metrically with reference to the x axis.
In order to determine the pressure on each point of tl:e
profile, when the win_7 is exposed to a horizontal wind having
the velocity V, we must know the velocity q at which the
air flows by each point of the profile. The pressure on each
unit area of the win_ surface is then proportional to q2 .
We can now find the values of q in a very simple man-
ner. For this purpose, we draw a horizontal line through the
point L, If we designate by h the distance of the point
A_ (of the circle K_) from this horizontal line, we obtain,
for any desired point P of the figure, the corresponding
value of q in the following manner.
gram the distance between the points
dlc of which we had found the point
?(e take from the dia-
AI and Am, at the mid-
P, and also the distances
of the point A_
the circle K_
from the origin O, from the center !_ of
and from the horizontal line passing through
L. We then have
O___A_ 2_h
q : V A_. A_ l,i_ AI
The _thematical proof for the given constructions is sin-
pie. As already mentioned, the profile of a joukov:ski _.:_'ing
85
N.A.C.A. Technical :.{emorandum No. 336
can be constructed by describin C on the z plane, with the aid
of the formula z : .[ + 4_ ' the circle K, determined by
the camber and radii difference. This circle passes through
the point _ - 2"
The systems of coordinates are plotted both in the
plane and in the z plane in such _znner that the _ axis and
the x axis form the an_o-le _ with the horizontal wlna direc-
tion.
If we now describe, in the z plane, both circles, which
we obtain from the given circle K in the _ plane by ermploy-
ing the two conversion formulas
zI := 2_ and z_ = ---
then these are the same two circles we designated above by
Kl and K2.
The point A_ has the coordinate z_ and the point A_
has the coordinate Z2, hence the center of A_ A2
coordinate z = ½ (zl + z_) == _ + 4--_' as desired.
therefore an actual point on the Joukowski curve.
has the
P is
The following for_mlia holds good for the velocity q at
which the air flows by _very point on the Joukowski fibre.
f
From z : + it follows that
86
N.A.C.A. Technical 3Icmorandum No. 535
t_ _ I (2[ - t_h- z_ - z_
whence we obtain
_incc the absolute value of z_ - zz equals the distance
A_ As and the absolute value of z_ =-- the distance OA_.
For _ (_,N), we obtain, from formula S of the preceding
article, _ _ _Vh in which h is the distance of the point
A_ from the horizon±al line passing through L. In the cx-
pression there _Tiven for the numerator; it is equal to h and
the denominator is equal to _-(L!_A:), as may be easily vcri-
lied. _Ye thus obtain
"_ h 0 A_
q=V"
which is just the formula given above for q.
Translation by Dwight H. I_iner,
National Advisory Committee
for Aeronautics.
87
N.A.C.A. Technical Memorandum No.336
Figol
88
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N.A.C.A° Technical _emorandum No.336
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Fig.3: f/z =1 s/_' = :]-
I0' _0
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N AoC.A. Technical Memorandum No.3S8
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Fig.4: 18/Z =--
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Graphic Construction of Joukowski Wings

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