Prandtl's method of sources and sinks for air foils is used to investigate the aerodynamics of circular wings in constant flow. Lift distribution, total lift, and pitching moment are investigated as well as the influence of changes in the angle of attack

Kinner, W.

English

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NASA T E C H N I C A L  MEMORANDUM NASA T M - 7 5 5 0 5  
%WINGS O F  CIRCULAR D E S I G N  
v - .-- - - 
W i l h e l m  K i n n e r  
, (BAS&-TI-75505) Otd IlIlGS OF CIBCDLAB D E U O l l  180-20229 
(MaLional Aeronautics and Space 
Adainistrat ion)  I /  p BC A O l / l P  1101 CSCL 01A UncZas 
T r a n s l a t i a n  of ' 5 8 b e r  T r a g f l l i g e l  m i t  k r e i s f d r m i g e m  
G r u n d r i s s  ,-',' Z e i t s c h M f t  ' f u r  a n g e w a n d t e  M a t h e m a t i k  und 
M e c h a n i k ,  to1 .- Z G - m ~ e c .  1936, pp. 349-352 
- - 
-I_ - - 
-- 
, I  
+ .- * 
NATIONAL AERONAUTICS AND S P A C E  A D M I N I S T R A T I O N  ; 
WASHINGTON, D . C .  20546 OCTOBER 1979 
RPRWED BY 
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STANDARD T l T L  E PAGE 
1. R e ~ o r t  No. 2. Government Access~on  No. 3. Rec~pient 's  Cofolog No. 
NASA TM-75505 
4. Title and Subtitle 5. Report Dote 
ON WINGS OF CIRCULAR DESIGN F- ' October 1979 6. Performing Orgonirot ion Code : -- 
7. A u t h o r ( ~ )  
Wilhelm Kinner 
8. Performing Orgonirot ion Report No. 
10. Work Un i t  No. 
C 
9. Performing Organization Name o ~ d  Address 
Leo Kanner Associates 
Redwood City, California 94063 
*-I. 
15. Supplementory Notes 
Translation of "Uber Tragfldgel mit kreisfermigem 
Grundriss," Zeitschrift fdr angewandte Mathematik und 
Mechanik, Vol. 16, No. 6, Dec. 1936, pp. 349-352. 
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NA~W-3x99 
13. Type o f  Report and Period Covered 
Translation 
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Administration 
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12. S onsozing Agency Name and Addrpss 
lfat~onal Aeronautics and Space I 
14. Sponsoring Agency Code 
Prandtlws method of sources and sinks for aerofoils 
is used to investigate the aerodynamics of circular 
wings in constant flow. Lift distribution, total 
lift, and pitching moment are investigated, as well 
as the influence of changes in the angle of attack. 
Unclassified-Unlimited 
, 
22. Pr ice  19. Security Clossif .  (o f  th is  report) 
Unclassified 
20. Security Classif .  (of th is  page) 
Unclassified 
21. No. of  Pages 
1 8 
In this article, it will be shown how Prandtlss method of 
2 sources and sinks for aerofoils can be used to investigate wings 
of circular design in constant flow. The equations are made linear 
in the usual fashion; in particular, we move the reference point 
- -.- 
of the acceleration potential C@* from the wing to the surface of 
& '  
the circular base. The center of the circle is made to lie at the 
origin, its surface in the xy-plane; the radius is c. Let the 
dimensionless Cartesian coordinates be 
The x-axis is directed to the rear, the y-axis to the right, and 
the z-axis is directed vertically. The flow velocity in the x 
direction is v. The liquid is assumed to be weightless, friction- 
less, and of constant volume. 
Since the calculation of the potential for the given lift 
distribution leads to difficulties in integration, we work back- 
wards, i.e. we use the "potential function of the first kind" (2) 
from potential theory, determine the appropriate aerodynamics, and 
attempt to satisfy the kinematic condition by linear combinations. 
For this we use the elliptical coordinates which are associated 
<* *- 
with the Cartesian coordinates by the fr+:;F':ing orthogonal transfor- 
mations s 
i. Excerpt from the author's dissertation. The investigation of 
circular wings is important for the theory of the autogiro. 
2. L. Prandtl, this volume, p.  360.  The author's oral presentation 
followed immediately after that of Prandtl. 
* Numbers in the margin indicate pagination in the foreign text. 
The reverse transformation becomes single-valued when one allows the 
, . 
hyperbolic coordinate:[& to change from positive to negative values 
on going from the upper half-space to the lower. In elliptical 
3 
m 
coordinates, the well-known solutions of ~ 9 2 0  have the following 
real, nomali zed forms 
s a constant, and pjlRrJ and 
". -. 
are the associated spherical 
ions of the first and second kinds, respectively. We are only 
interested in the functions (I.,'" wifh odd n + m, since those with 
n + m even are constant on the base surface. The lift density at 
. a point on the base surface is proportional to the potential 
differene f?,,"' 
The total, lift A = j j { ~ , - - & ~ ~ ) d F  is o 
---- --- 
4 f 4 4 q l . = T x e ~ a ~ , o ,  A , ~  = 0 for all other (n,m) 
. . -, 
(4)  
--  - 
The moment about the y-axis ' ,= (pU-,e d 5 a ~  is 
- 
4 
.If,' = a R Q eL C:, MAm, = 0 for all other (n,m) (5) 
I 
The downdraught on the wing follows from w{z ,~ )= :  \ g d g  by 
- O D  
- w> 
integrating in the plane of the wing parallel to the direction of 
flow. One obtainss 
- 
In general, ,,n_mb:,) may be represented by tesseral spherical func- 
An - 1 c;y&-- 
tions, to which for odd n the factor 
- is attached. 
- --  
P 3 .  Ms J. 0. Xtrutt, Lame, Mathieu, and Related Functions in Physics 
and Technology, Ergebnisse der Mathematik und ihrer Grenzgebiete, 
Vole 1, 1932, Po 590 
In this manner, the downdraught for the functions concerned is made 
to change logarithmically. 
Z 
1 The shape of the wing is given by . s ~ s , g l ~ - \ w ~ s . ,  
2, 
The drag w,"' , according to the separation law of Munk, corres- 
. ,< <.- 
ponds to its one dimsnsionak malog. For odd m, the 2if-t density 
-I. rn 
+ 4 
'I & ~ l ~ E l & - p , , ) d s  
- p h  
(with =,=v- ) vanishes identically; for even m 
."<<< 
one obtains: 
For the function enm,, there is no suction pressure at the 
leading edge since the induced speeds (except at the wing ends) are 
finite. It follows that these functions alone are insufficient to 
permit investigation of the influence of changes in the ahgle of 
attack. 
The "potential function of the second kind," @, necessary for 
- -2- - 
this is obtained from the %+2 (n = 0, 1, 2,. . . . ) by the limiting 
process 
from which 
- . I -  . 
The '4; become infinite at the edges of the circle; they have no 1351 
physical meaning in and of themselves since the outflow condition at 
the rear edge (that the lift density approach zero) is not fulfilled. 
The lift is 
* .  . 
A , = 4 s e c Y C , ,  A ,  = 0 for n = 1, 2, 3,.... (11) 
The moment about the y-axis is 
- 4 
' q = ~ ~ e ~ ,  Hn = 0 for n = 0, 2 ,  3 , . .  .. (12) 
To calculate the downdraught on the wing, one integrates to 
a line parallel to the x-axis outside the plane of the wing base, 
and, after the integration has been carried out, shifts the terminus 
to the base surface. The downdraught is independent of inside 
the base circles 
Thus the wing surfaces are ruled surfaces. 
The drag vanishes for odd n and becomes infinite for even n; 
for a curved section of the front edge the suction pressure becomes 
infinite for even n if the integration is extended to the ends 
of the wing. 
For the corresponding one dimension&l equation, one has 
With the mathematical apparatus we have developed the second 
basic problem can be solved in the following manner, From the 
(3 8 ('gjj) given wing form s[gs$j one calculates the downdraught ~ @ , ~ = ~ T z -  
i > 
and separates from the result the downdraught function of the first 
kind that is contained in it. The remaining function' ~ ( g ~ ~  is a 
function of 3: alone; one manipulates this as a downdraught function 
of the second kind so that the outflow condition at the rear edge 
is fulfilled. The potential coefficients are determined by an 
infinite system of linear equations; from these follow the + by 
means of the outflow condition. 
Fig. 1. Polar Curve of the 
Theoretical Drago 
1. Planar circular disce 
2. Surface with fixed 
aerodynamic center 
t 4 + with cppr c ,~ ,   
&g,~ =-ILj8 C,a ;5",d '"'" z z  ~b*i& 
''$ * - ,  . 
3 .  As in 2. with ' : : z R ~ * . ~ .  
. - 
solid curves: line drag 
stippled curves: surface drag 
The distribution of lift, the total lift, and the pitching 
moment are thus known. The influences of changes in the angle of 
attack can be investigated by the linear superposition of a planar 
1 " 
circular disc . The "surf ace drag" !I;. =-- \(p,, t t ~ ( r ,  t ~ )  , I P ~  is greater 
219 
than the actual drag by the amount of the suction pressure. The 
actual drag is identical with the "line drag" ,tLwm- j ql.fd) rp,s $!I LSr 
- e 
since the outflowing vortex sheet, and thus also its kinetic energy, /352 
the equivalent of the drag work, is not changed by displacement of 
the lift elements. According to their analytical form, the lift and 
its moment appear as linear expressions, the two drags and the 
suction pressure as quadratic expressions in the two parameters 
Pk (camber parameter) and tg a. 
According to the procedure outlined, the planar circular disc, 
the calotte, a wing with fixed aerodynamic center, and S-flap ( ? )  
were investigated (see illustration); also, the rolling moment of a 
warped wing was calculated, using the appropriate antisymmetric 
potential function (trigonometric factor sin+). 
Experiments to check the theory are in preparation at the Insti- 
tute for Aerodynamic Research in GBttingen. A theoretical investi- 
P gation of elliptically shaped wings in a constant stream, using Lame 
functions, is underway. 


On wings of circular design

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