Steady vibrations of wing of circular plan form, and, theory of wing of circular plan form

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Material Information

Title:
Steady vibrations of wing of circular plan form, and, theory of wing of circular plan form
Series Title:
NACA TM
Portion of title:
Theory of wing of circular plan form
Physical Description:
93 p. : ill. ; 27 cm.
Language:
English
Creator:
Kochin, N. E ( Nikolaĭ Evgrafovich ), 1901-1944
United States -- National Advisory Committee for Aeronautics
Publisher:
NACA
Place of Publication:
Washington, D.C
Publication Date:

Subjects

Subjects / Keywords:
Aerodynamics   ( lcsh )
Airplanes -- Wings -- Testing   ( lcsh )
Genre:
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Abstract:
This paper treats the problem of determining the lift, moment, and induced drag of a thin wing of circular plan form in uniform incompressible flow on the basis of linearized theory. As contrasted to a similar paper by Kinner, in which the acceleration potential method was used, the present paper utilizes the concept of velocity potential. Calculations of the lift and moment are presented for several deformed shapes. It is shown that considerable deviations exist between the strip theory analysis and the more exact theory. The lift, moment, and induced drag are also determined for a harmonically oscillatory circular plan form wing. As constrasted sic. to a similar paper by Schade, in which the acceleration potential method was used, the present paper utilizes the concept of the velocity potential. Expressions for lift, moment, and induced drag are given and finally specialized to the case of a slowly oscillating circular wing.
Bibliography:
Includes bibliographic references.
Funding:
Sponsored by National Advisory Committee for Aeronautics
Statement of Responsibility:
by N.E. Kochin.
General Note:
"Report date January 1953."
General Note:
"Translation of "Ob ustanovivshikhsya kolebaniyakh kryla krugovoi formy v plane." Prikladnaya Matematika i Mekhanika, Vol. VI, 1942. and "Teoriya kryla konechnogo razmakha krugovoi formy v plane." Prikladnaya Matematika i Mekhanika, Vol. IV, 1940."

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Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003779524
oclc - 86223117
sobekcm - AA00006167_00001
System ID:
AA00006167:00001

Full Text
AT-132-1











NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS


TECHNICAL MEMORANDUM 1324


STEADY VIBRATIONS OF WING OF CIRCULAR PLAN FORM*

By N. E. Kochin

The nonvortical motion of an ideal incompressible fluid has been
?.solved. (reference 1) for the case of uniform rectilinear motion of a
Pi :.ng of circular plan form. The method developed in reference 1 may
1 8so be generalized to the case of the nonsteady motion of such wing.
iT'.he problem of the steady vibrations of a circular wing is solved
.herein. The results will be frequently referred to herein. The prob-
;. em of the steady vibrations of a circular wing was solved by another
; ethod by Th. Schade (reference 2).


Fundamental equations

S". The wing, the motion of which is under consideration, is assumed,
,... as in reference 1, to be thin and slightly curved; its projection on
Sthe .xy-plane has the shape of a circle ABCD of radius a with cen-
J. ter at the origin of coordinates. The principal motion of the wing is
iK, assumed to be a rectilinear translational motion with constant velocity
i.- ', parallel to the x-axis. The coordinate axes are assumed as displaced
*v i4'ilth the same velocity. On the principal motion of the wing is super-
::posed itsadditional harmonic vibration of frequency u, where the pos-
.bibility of deformation of the wing is not excluded. The equation of
1.:. the surface of the wing may then be represented in the form:

z(x,y,t) = ~o(x,y) + Sl(x,y) cos aut + 2(x,y) sin ot (1.1)
:..'where the ratios tk/a as well as the derivatives ok/ox and ak/y,
wheree k = 0,1,2, are assumed small magnitudes.

The fluid is assumed incompressible and the motion is assumed non-
vortical and occurring in the absence of external forces. The velocity



*"Ob ustanovivshikhsya kolebaniyakh kryla krugovoi formy v plane!'
Prikladnaya Matematika i Mekhanika, Vol. VI, 1942, pp. 287-316.








2 NACA TM 1324


potential will be denoted by (p(x,y,z,t) and steady vibrations of the
fluid will be assumed; that is, the velocity potential is represented
in the form:

((x,y,z,t) = o0(x,y,z) + pl(x,y,z) cos wt + p2(x,y,z) sin ut

It is evident that the functions (P0, P1, and 92 satisfy the
equations of Laplace

2 k 2 (k o2 k
+ +2 2 0 (k = 0,1,2)
ox2 oy2 ~z2

The velocity of the particles of the fluid near the leading edge
of the wing DAB is assumed to approach infinity as 5-1/2, where 8
is the distance of the particle from the leading edge, but the velocity
of the fluid particles near the trailing edge of the wing BCD is
assumed as finite. From this edge a surface of discontinuity passes
off on which the function p undergoes a discontinuity. As in refer-
ence 1, the problem will be linearized. Since the values of the functions
pk and their derivatives are assumed to be small quantities of the first
order, their squares and products are rejected. The functions
'@k(x,y,z) are further assumed to have discontinuities on the infinite
half-strip E situated in the xy-plane in the direction of the nega-
tive x-axis from the rear semicircumference BCD of the circle S to
infinity. The boundary conditions on the surface of the wing are
replaced by the conditions on the circle S located in the xy-plane.
Everywhere outside the half-strip S and the circle S the functions
Qk(x,y,z) are thus regular functions.

The boundary conditions which these functions satisfy are now set
up. On the surface of discontinuity E, the kinematic condition
expressing the continuity of the normal component of the velocity must
first of all be satisfied:


^2 =+0 -T z-0

from which is obtained the conditions


= _=- on S (1.2)
z=+O z=-O







IACA TM 1324


The dynamical conditions expressing the continuity of the pressure
in passing through the surface of discontinuity E are now stated.

If a stationary system of coordinates xlYlzl is employed, con-
nected with the coordinates xyz of the moving system of coordinates
by the relations

x = x1 ctl Y = Y1 z = z1 t = t1

then the pressure may be determined by the following formula:

p = p 2 \ + Y + \ J + F(tl) (1.3)

Since

=D (P 5x IV O a d (1.4)

the following equation will apply in the movable xyz system:

p p + PC [ ( 2 + + F(t) (1.5)

When small quantities of the second order are rejected and the
magnitude F(t) is not dependent on the coordinates,

3(p 5V
p = p + pc

or, on account of equation (1.2),

oK0o / 1 2 .09
p(x,y,z,t) = pc a- + pc p cos at + + p2ip sin 3t

(1.6)

For briefness, the following notation is introduced:

U/c = k (1.7)

The condition of continuity of the pressure on 3 then leads to the
three equations:




Ner


NACA TM 1324


z=+0 z=-O


= t 2z=-0

S 2 z
~23z-0


The condition on the circle S is now written.
the stationary system of coordinates has the form:


on S


(1.8)


Equation (1.1) in


I = ~O(xl-ctl,y) + S~(xl-ctlY1) cos wtl + t2(xl-ctlYl) sin 1tl
Hence, for the normal component of the velocity of the fluid parti-
cles adjacent to the surface of the wing,

dzl 1o 21 2
=t c x- c cos w s t ( sin sin ct + 4u2 cos ot
The notations
The notations


- 1
C l
(2d;x


3 o
- c = ^(x)


- k2) = Zl(x,y)


- +2


= z(x,y)


yield the boundary condition


zz=0


= ZO(x,y) + Zl(x,y) cos wt + Z2(x,y) sin wt


which must be satisfied on both the upper and the lower sides of the
circle S and which breaks down into the three conditions:


z z=o


= Zk(x,y) on S (k = 0,1,2)


(1.9)


The presence of conditions (1.2) and (1.9) permits consideration of
the functions ,pk(x,y,z) as odd functions of Z:


@k(xy,-z) = 'k(xyz)


Fx


\T-


- k) z=+0


+ kl )z=+0


(1.10)







NACA TM 1324


If it is assumed, in particular, that z = 0,

Pk(x,y,0) = 0 (1.11)
in the entire xy-plane with the exception of the circle S and the
half strip E on which 9k undergoes a discontinuity.

The conditions (1.8), because of equation (1.10), assume the form:

q0o a1q 692
= 0 k)2 =0 + kqi = 0 on S (1.12)

Finally, the absence of a disturbance of the fluid far ahead of the
wing leads to the evident conditions at infinity:

'Vk Ck9
lim T- = lim = lim = 0 (1.13)
x -+ x- +*-" X-++

The problem of determining the function 90(x,y,z) satisfying all
obtained conditions for this function was considered in reference 1.
*
The following equality is set up:

@(x,y,z) = Pl(x,y,z) + iP2(x,y,z) (1.14)
so that


cp(x,y,z,t) = 0o(x,y,z) + Re (x,y,z)e-it)


(1.15)


Also,


1(x,y) + i'2(x,y) = S(x,y)

Z1(xy) + iZ2(x,y) = Z(x,y) = c + ikt)

The shape of the wing will be determined by the equation

z(x,y,t) = o(xy) + Re (x,y)e-iwt
The functions 4(x,y,z) will then be a harmonic function,
in the entire half-space z>0 and satisfying the conditions:


(1.16)



(1.17)

regular








NACA TM 1324


= Z(x,y) on S


(1.18)


(1.19)


+ ikc = 0 on S
dX


following from equations (1.9) and (1.12). In the entire remaining
part of the plane xy the following condition must be satisfied:


P(x,y,0) = 0


(1.20)


Moreover, the following conditions must be satisfied at infinity:


lim N- =
x-* t


lim =y
X-* +-


lim N o
X+ = 0
x -+m


which are the boundary conditions of the first derivatives of the func-
tion P(x,y,z) near the rear semicircumference BCD of the circle S
and the condition that near the forward semicircumference DAB these
derivatives may become infinite to the order of 6-/2.


2. Fundamental formulas

In reference 1 an expression was constructed, which depended on an
arbitrary function fo(x,y), which determined a harmonic function
PO(x,y,z) satisfying all the conditions imposed in the preceding section


0P(x,y,z) = -


J fo(t,') K(x,y,z,,Ti) +
S -


(2.1)


G(xEyz, a2 -2 2 cos y dr dx
(2 2 + a2 2a& cos r 2aq sin f)j


The functions K(x,y,z,,rq) and G(x,y,z,y) for z>0


2 2 2 _- Aa2 x2 y2 -2 R
(x,y,z,,0) = ---arc tan--
nr _V7 ar


G(x,y,z,y) =


a2 x2 y2 z2 + R
x + y + z' + a" 2ax cos y 2ay sin y


(1.21)


1 1

22 n
4F


are given by


(2.2)


Sz=0o







NACA TM 1324


which are harmonic functions of x,y,z where


r = Ad(x )2 )2 + z2

R = \(a2 x2 y2 z2)2 + 4a2z2


In order to satisfy boundary condition (1.9)


=0P
b- = ZO(x,y)


on S


it is necessary to take


fo(x,y) = ZO(Xy) + g0(y)


where g0(y) is determined from a Fredholm integral equation of the
second kind.

The solution of the more general problem of steady vibrations may
be presented in a similar form.

Thus, fl(x,y) and f2(x,y) denote two arbitrary real functions,
continuous, together with their partial derivatives of the first and
second order, in the entire circle S;


fl(x,y) + if2(x,y) = f(x,y)


(2.6)


It will now be shown that the function


Q(x,y,z) =


IJ f( ,g) [K(x,y,z,,n) +


11
2


G(x,y,z,r) eik x 2 2 Y 2 cosy d d( dfx
(2 + T2 + a2 2ae cos y 2ay sin y) j
(2.7)


satisfies all the conditions of the preceding section except
condition (1.18).


(2.3)


(2.4)


(2.5)


1 -ikx
e22
n24 [


'








8 NACA TM 1324

The function G(x,y,z,T), as shown by equation (2.2), is harmonic;
hence the function


L(x,y,z) = e-ikx
%J+m


eikx G(x,y,z) dx


will be a harmonic function. In fact,


62L 62L 62L TG
A 2 + 2 +z2 x


G ikG -
TX -


- ikG + e-
x+


e-ikx


[y2


eikx (2G +
(x2


a2G
ai?


- k2G] dx


k2G dx


When this expression is integrated by parts, it is easily shown
that AL = 0, since both G and 6G/6x approach zero for x -+ .


It then follows that
Laplace equation


the function @(x,y,z) likewise satisfies the


A4 = 0


(2.8)


where from the form of equation (2.7)
regular everywhere outside the circle
exactly the same way it is shown that
and condition (1.20) are satisfied.


it is seen that s(x,y,z) is
S and half strip E. In
the conditions at infinity (1.21)


Furthermore,


6+ ik 1
T- + ik4 =2n
dx I


fJ f((,)TO 6+ ikK +
if I TX-s a


G(x,y,z,y) Aa2 g2 2 cos r dr d d
(t2 + n2 + a2 2a; cos r 2an sin y) d


1
2


(2.9)








NACA TM 1324


It is clear that if x2 + y2>a2 then


S+ ik4 = 0
Tx


for z = 0


(2.10)


so that condition (1.19) is likewise satisfied.

It thus remains to check the finiteness of the derivatives of the
function 4(x,y,z) at the points of the semicircumference BCD of the
circle S and to establish the behavior of these derivatives near the
forward semicircumference DAB. But near the forward semicircumference,
the inside integral in formula (2.7) evidently remains bounded, as do
its partial derivatives; since the first derivatives of the integral



Sf f(E,n) K(x,y,z,C,n) dC dl

as established in reference 1, and as will again be proven, have near
the contour of the circle S the order 5-12 (where 5 is the dis-
tance of a point to the contour ABCD of the circle S), it is clear
that the first derivatives of the function @(x,y,z) also have the
order &-1/2 near the forward semicircumference DAB of the circle S.

For determining the behavior of the function c(x,y,z) near the
rear semicircumference BCD, the right side of equation (2.9) is trans-
formed. Denoting it by M(x,y,z) and making use of formula (2.11) of
reference 1 and the formula of integration by parts (2.14) of
reference 1,


M(x,y,z) =
2n


i f(,) likK -
S


G(xy,z,r) Va2- 22 cos y dr
(2+Tn2+a2-2ac cos r-2aTj sin r)


S


3
2


1
2


I- K dC dq

(2.11)








10 NACA TM 1324


It is evident that this function remains finite near the rear
semicircumference BCD. But when the following equation is integrated,

+ ikt = M(x,y,z) (2.12)

there is obtained


4 (x,y,z) = e-ikx
0


eikx M(x,y,z) dx + q(0,y,z) e-ikx


whence it is clear that both the function 4 and its derivative with
respect to x remain finite near the rear semicircumference BCD. The
derivatives of M with respect to y and z will be of the
order 6-1/2 near BCD, as follows from a consideration analogous to
that which was adduced previously for determining the behavior of the
functioned (x,y,z) near the forward edge of the wing DAB. Since


eikx dM + (oyz) e-ikx
dy ay


it is clear that the derivative 8d'/y, and similarly MP/)z, remain
finite near the rear edge of the wing BCD.

The function (2.7) thus satisfies all the imposed conditions. The
only condition not utilized was condition (1.18)


(z=0
dZ /)


= Z(x,y) on S


(2.14)


When the following formulas are employed:


(2.13)


=x
0 = e-ikx
5y0








NACA TM 1324


.lim K f(C,T) dg di = 2nf(x,y)
z T+0


lim r a2 x2
z-++0


_ y2 z2 + R =


0

aAE2

Tx2 +2 a2


for x2 + y2
for x2 + y2 > a2


it is found without difficulty that on S


z=0


= f(x,y) + g(y) e-ikx


22


e (x c 2 2- '-)-1 2 2 2 121/2 cos r f ) d, ax d dJ
(x2 2 a2 2x cos r 2sy sin y)L(C 92 2 a cUE r 2?1 aEl Y)


2. 16)


The following equation is thus obtained:


-f(x,y) + g(y) e-ikx = Z(x,y)


(2.17)


whence


f(x,y) = Z(x,y) g(y) e-ikx


(2.18)


Substitution of this value of the function f(x,y) in equa-
tion (2.16) yields, for the determination of the function g(y), an
integral equation of Fredholm


g(y) = (y) (y)
g(y) = N(y) +f- H(y,TI) g(Ti) dT
J-a


(2.19)


where


2 0
4 -


ei k _- 2- Gtx, yz,r) coE 1 Zt&,n) dy dx dE dTI
xr2 + y- .2(2 q2 a2 2aC coa r 2a sin r)


where


(2.15)


(2.20)


*
:id* :'*I: ~: ":'




-----------------------I --- .--



12 NACA TM 1324

with G(x,y,z,r) according to equations (2.2) and




_. xf ek (e )(x,2 2 2 a-1/ (2 t2 -2 2 12 cos r dr dx dC (2.)
2 (x2 y2 a2 ax cos 2ay sin y)(Y2 4 q2 -2 2ae co6 r 2aq sLn y)



3. Computation of forces

The pressure p may be determined from formula (1.6), which with
the notation (1.14) may be written in the following form:

p = + Re + i) e- (3.1)

For the computation of the forces acting on the wing, it is neces-
sary to know the pressure on the circle S.

Because of equation (1.10), the pressures above and below the wing
differ only in sign:

P = P+ (3.2)

For clarity, the signs of the functions on the wing will henceforth
be assumed to be the limiting values in approaching the wing from
above, that is, for z-.+ 0.

For the lift force P the following expression is obtained


P = (p- p+) dx dy = 2 dP d y =
S


2pc f(7-+ Re 5+ 1/ e-it] dx dy (3.3)
S O X | \X




I~


NACA TM 1324 13


But by formulas (2.9) and.(2.11), the following equation applies
on the utoer side of the circle S:

+ I k (xy,) K(x,y,O,,,l) d dq +



Irl i k K(x,y,0,Ce,T) (3.4)

-S


| (x2 + y2, a -2ax cos y 2ay sin T)(2 + I2 + a2 2aC cos r 2aq sin r)

2
This expression is integrated over the entire area of the cir-
cle S. The order of integration is interchanged and the two integrals
must be computed first of all by formula (4.13) of reference 1



a dx dy = 2na (3.5)
JI x2 + y2 + a2 2ax cos T 2ay sin y

It will be proven further that


fJ K(xy,O0,,l) = 4va2 2 -_ (3.6)
S

For this proof, the following function is considered:


F(x,y,z) = ff K(x,y,z,g,T) dg dTi (3.7)


Because of the definition of the function K, the function
F(x,y,z) is a harmonic function over the entire space outside the cir-
cle S. By formula (2.35) of reference 1, the following condition is
satisfied on the surface of this circle:

F= 2 on S (3.8)
Sz-


r r




w w -


14 IACA TM 1324


and therefore the function (3.7) is the potential of the nonvortical
motion of a fluid corresponding to the translational motion of a cir-
cular disk with velocity +2A along the negative z-axis normal to the
plane of the disk. This motion, however, belongs to those that have
been studied in classical hydrodynamics, from which can be taken the
corresponding expression of the function.


F(x,y,z) = K(x,y,z,k,l) dC dT


S2,/R + a2 2 -2

R + x2 + y2 + z2 -a2 ct + x2 +
2a ar


Passing to the limit z-.+ 0 yields the formula


4 K(x,y,0,C,j) da d = 4 ,a2 x2 y2


2
y2 +
2a2


z2 a2


(3.9)


on S


which is equivalent to equation (3.6), since K(x,y,0,Cq) is a symmet-
rical function with respect to the points M(x,y) and N((,q).

The-following formula is thus obtained:



pr k (xy,0) dxdy =2 r j2 + ddn -
S (rylr ), ik 4,,y,O. -d' d d2f


s)- ni
f


(3.10)


cos r dy rd dq
C2 i2 a- Pa k cos y 2ai sin r







NACA TM 1324


If this expression and similar expressions are substituted for the
function (P, obtained from equation (3.10) for k = 0, the final
expression of the lift force acting on the wing is obtained:


P = .2 f Re e-t + ikf -
iFfiin 7


a o + Re e-tf]
2x-i If


3
2


2


cos r dy ld dl
2 + 2 + a 2aC cos 2aq sin y


By integration by parts and with the aid of the following formula


2 Y
Scos dy 2xn
S2 + n2 + a2 2aC cos y 2aq sin y a(a2 E2 q2)

(3.12)

equation (3.11) may be rewritten in the form:


ikfe-it) +


3
2
2 + Re(fe-Cit]

2


cos y dy ldt d&
t2 + -2 + a2 2at cos y 2aq sin y

(3.13)


In a similar manner, the formulas for the moments of
about the x- and y-axes are obtained.


the forces


For the moment of the pressure forces about the x-axis


Mx = y(p- p+) dx dy = 2 fS yp+ dx dy
S S


(3.14)


(3.11)








16 NACA TM 1324


there is obtained


= 2pc y (yJ + Re[T + i k i) e-i ax dy (3.15)


The order of integration is interchanged by use of equation (3.4).
It is here necessary to compute two integrals. By formula (4.44) of
reference 1,


fI y a2 x2 -_y2 dx dy 4 2
dFd=dy = nta2 siny (3.16)
x2 + y2 + a2 2ax cos y 2ay sin y
S

It will now be shown that


y K(x,y,0,C,n) dx dy = a2 2 (3.17)


For this derivation, the following function is considered:


Fl(x,y,z) = -S K(x,y,z,,n) dt dT
S

By formula (2.35) of reference 1, the following equation applies
on the circle S:

= 2;ry (3.18)


and therefore Fl(x,y,z) is the potential of the motion of a fluid cor-
responding to the rotation of a disk about the x-axis with angular
velocity -2n, a case studied in classical hydrodynamics:








NACA TM 1324


Fl(x,y,z) = J IJ K(x,y,z,C,q) dt dn
S


= 2&2 yN


R +x


R + a2 x2 y2 z2 1 -


2 2 2
+ y +z -a
2a2


are ctn A4-J


3(R + x2 + y2 + 2 + a2)


2 2
x + y +
2a2


2 2
z a


Passing to the limit z- + 0 yields the formula



n K(x,y,0,C,n) dC dj = ya x
0


equivalent to equation (3.17).

As a result, the following formula is obtained



JJ + k 4] dx dy = 2 + d d -


1
2a2

-S -~2
2


(3.20)


sin r cos r dr dC d
E2 + T2 + a2 2aC cos r 2al sin y


Hence, ,for the moment of the pressure forces about the x-axis,
the following expression is obtained:


Mx jsJ


a2 2 _- 2


Re e- nt + kf -


f0 + Re(e-iCt


1
2



2


sin y cos y dy


(3.19)


on S


(3.21)


2







18 NACA TM 1324


or, on account of the formula

2 n
sin y cos y dy 2nr_
S 2 + 2 + a2 2a; cos r 2an sin a2a2a2 2 T2)

(3.22)

the equivalent expression


Mx = jc a 2 2 2 Re(ikfe-t) + [fo 0 + Re(fe-it) x



2 s
sin y cos y dr d d (3.23)
S2 + 12 + a2 2a0 cos r 2aT sin r


In the same way for the moment of the pressure forces about the
y-axis


My= -J x(p_ p+) dx dy = 2 xp+ dcx dy (3.24)
S S

there is obtained


M = 2pc x J + Re + i k ) e-i dx dy (3.25)


It is here necessary to employ the formulas
s xA, a2 x2 2 4 2
xa x2- y2 dx dy = na2 cos
x2 + y2 + a2 2ax cos y 2ay sin y 3
(3.26)







NACA TM 1324


C x K(x,y,0,C,n) dx dy = | a2 -2 -


As before, there is obtained


+ Re e-iont -+ ik -
~+kfJ i


1
2 + Re e-i

2
i [fo + =(e-Lt ( f)


cos2 y dy -r d
2 712 + a2 2ag cos y 2aTq sin r
(3.28)


Integration by parts and use of the formula yields


C2 + q2 + a2


cos2 y dy
- 2af cos r


- 2a7 sin y


a2 2
= +
a2(a2 2 2)


-iwt (- f ikC f] +


+ Re(e-i)t


3
2


2
tJ. i


cos2 y d d d
-. -- co r d-----> dr i
2 + 72 + a ,- 2aC cos y 2aT sin y
(3.30)


The value can now be computed for the frontal resistance W, which
is composed of two parts. First, the normal force (p_ p+) dx dy
acting on an element of the wing dx dy will have a component in the
direction of the x-axis:


8pc
MY 3-A


(3.27)


2 n

0


8pc
My 3


(3.29)


3
3f Re




w .


20 NACA TM 1324


(p_ P+) dx dy = (p_ p+) 0 + Re e-wt)] x dy

if

z(x,y,t) = ~o(x,y) + Re [(x,y) e-i
is the equation of the surface of the wing. Integration of this
expression gives the first part of the frontal resistance in the form:


w1 = f +) 0 + Re( e-iWt) dx dy



-2pc f+ Re + i k e- +Re e -t) dx dy
SI (3.31)

In fact, the frontal resistance W will be less than W1, since
a suction force W2 appears because of the presence of the sharp
leading edge of the wing DAB; therefore,

W = W1 W (3.32)

The suction force W2 is connected with the presence of a strong
rarefaction near the edge of the wing. This rarefaction is taken into
account principally by the square terms of the fundamental formulas (1.3)
or (1.5) for the pressure and it is therefore unnecessary to employ
these formulas here.

The suction force W2 is computed from the law of conservation of
momentum applied to a thin filament-like close region containing
the forward semicircumference DAB of the circle S; region T is
bounded outside by surface a and inside by part S' of the upper side
of circle S adjacent to the semicircumference DAB and the part S'
of the lower side of the circle S. Figure 1 shows a section of these
surfaces obtained by a passing plane through the z-axis.

The equation expressing the momentum law is projected on the
x-axis:

- W, p cos(o,x) dS =J d + p u dS+ p vx dS (3.33)
a s+s*







NACA TM 1324


The left-hand side is the sum of the projections on the x-axis of
all the forces acting on the volume of fluid considered, and on the
right-hand side is the total derivative with respect to time of the
component on the x-axis of the momentum of this volume; this derivative
consists of two parts, a volume integral connected with the local change
of velocity and a surface integral expressing the transfer of the momen-
tum of the particles of the fluid through the bounding surfaces of the
volume T.
Equation (3.33) may be written both for the stationary system of
coordinates OlxlYlzl and for the moving system of coordinates Oxyz.
For the stationary system of coordinates, expression (1.3) is used
for the quantity p; moreover,


X= =


3=
v n =T


(3.34)


By the theorem of Gauss


I O v


-dT =


d2
_ < dt r


(3.35)


=/ p cos(n,x) dS + p cos(n,x) dS
Cy S'+S"


From equation (1.3) and the equation just derived, the following
expression is obtained from equation (3.33) after a number of simple
transformations:


W2 -[= (2


+ P2


+ )


cos(n,x) dS p


I lq, dS -
Ja f 7 7


p J J OI cos(n,x) dS
S'+S"


S dS
S'+S"


Since cp/oti and cp/3x near the leading edge of the wing are of
the order 8-1/2 and 39/6n and cos(n,x) are finite on the surface of
the wing, the last integrals drop out when region T is extended to the
line DAB. The following limiting equation is therefore applicable:


(3.36)





22 NACA TM 1324



= f + + cos(n,x) dS p d

(3.37)

For computation of the suction force W2, the expressions must be
found for the components of the velocity near the leading edge of the
wing DAB. The velocity of the fluid particles near the leading edge
of the wing are shown to be of the order of 5-1/2 if 5 is the dis-
tance of the particle to the contour C of the circle S. From equa-
tions (1.15) and (2.7) it is evident that


p(x,y,z,t) = -0 fJ o(,) + Re f(CIe- ] K(x,y,z,,nt) de dt +


X(x,y,z,t) (3.38)

where the function X(x,y,z,t) and its derivatives remain finite near
the leading edge.

The behavior of the function is now examined more closely


U(x,y,z) = s f(c,T) K(x,y,z,,Tl) dt dT (3.39)


near the contour C of the circle S. Therefore,


f= (f(,q') dt dq


Since on C the function K becomes zero, the following equation
results



f(Jr,j) dC d = K dE d
jS


showing the finiteness of this integral. Therefore


* a








NACA TM 1324


where 0(1)
approaches


= 6K + C)f(,) dC d + 0(1)



denotes a magnitude which remains finite when 5
0. But


21F2 a va2
Sa2r2 + (a2 2 _

x Va2 2 n2
R


- x2 2 2 + R
q2)(a2 x2 y2 z2



a2 2 r2'


hence


Sa -x-2+ (a2 2q2) (a2,+x2-2-z2R)
22 (a2-C2-2)(a2-x2-y2-z2iH)


x a2- 2 + dC dq + 0(1)


The coordinates 5, G, and a are introduced

x = (a + 5 cos a) cos 0 y = (a + 5 cos a) sin 0


(3.40)


z = 5 sin a


(3.41)


Then


a2 X2 2 z2 = 2a5 cos a 52

R = 564a2 + 4ab cos a + 52 = 2ab + .

a2 x2 y2 2 + R = 2 sin ~Ea +


R a2 + x2 y2 +2 = 2 cos 5 M + .


(3.42)


ii;-i; ;
;;


K
2c -








24 NACA TM 1324

The point with coordinates (x,y,z) is brought into correspondence
with the point of the circumference C with the coordinates

x0 = a cos e yO = a sin e z0 = 0

and

r02 = (0 )2 + (YO T)2 = C2 + 12 + a2 2a cos 8 2an sin 0

(3.43)

Near the contour C, the principal part of the integral


J(xyz) 2a2 a2 2 2 f(Jg,) dC dd
1 2a2r2 + (a2 2 92)(a2 x2 y2 z2 + R)

(3.44)

is



r02
N(0) =fJf(,) va2 2 T2 dO d



ffi( ) a2 t2 -2 d dd d(3.45)
( 2 + 2 + a 2a cos e 2aq sin 0

For this purpose, the following difference is estimated:

A = Jl(x,y,z) N(e)

The circle S is divided into two parts: the circle Sl
of radius a &; and the ring S2 lying between the circumferences of
radii a e and a.



C = 22 /2 42 n2 --2 -
T, 2 ar + a2r2 +2 C2 T"2) (a x2 x2 2 + R) ro)2


S f() 2 T2a2ro2 2a2r2 (a' 2 2)(,2 2 y 2 2 + R)
JJ r2 2 a~2r2 (a2 2 )(a2 x2 y2 z2 + B)


' F'i '







NACA TM 1324 25


ro2 r2 = +2 + T2 + a2 2a cos 6 2an sin e (x-,)2 (y-)2 z2

= 28 cos a(C cos e + n sin 8 a) 62
2a2(rO2 r2) (a2 g2 2)(a2 x2 y2 2 + R)
= 2abrp2 cos a (a2 + S2 + i2) 52 (a2 -2 2) R
Since
r02 < 4a2 R B 2ab + 52
therefore

2a2(ro2 r2) (a2 .2 2)(a2 x2 y2 2 + R)I< 2abro2 + 2a282 +
(a2 22 I2) R
Hence if If(,n) < M in the circle S then

-J (2 -2 2 -
6 2aM a2 2 dt dT
2a2r2 + (a2 k2 T2)(a2 x2 y2 z2 + R)
S1


2a2b2Mo m a2 2 2 d k da
S a JS ro022a2r2 + (a2 t2 2)(a2 x2 y2 z2 + R)] +


RM J I(aa2 2 12 2)3 dC dl
r02 [2a2r2 + (a2 2 -2)(a2 x2 y2 z2 + R)

But by equation (2.24) of reference 1

4raa2 2 y2 dg di
S2a2r2 + (a2 72 22)(a2 x2 y2 z2 + R) a

Since
2a2r2 + (a2 2 2)(a2 x2 y2 z2 + R)
= 2a2r02 + 2a8r02 cos a + (a2 + +2 + T2) 2 2 (a2 2 -_ 2) R









26 NACA TM 1324


hence for 5
2a2r2 + (a22_ 2 2)(a2 _2 22 2 2 + R) > a2r02 (3.46)


Sa2,a2-a2-,22 d d 222
< qa- d- did
ro2[2a2r2+(a2-_2-T 2)(a2-x2-y2-z2+R JJ rO4
S1 S1

The last integral evidently does not depend on 6; hence it may be
assumed that 0 = 0 and therefore


2n a-r
- a2 1 2 dt d =[
04r

p=a- C
4I2na2 2n
S(a2 p2)3 p=0


ap2 do do d
(p2 2ap cos b + a2)2


4 xa2
4nac
(2a& e)


Similarly


S M_(a2 22 -_ 2)3 d3 d
J ro2 a22 2 + (2 2 2)(a2-2 2 2 +2 R2


1P (a-t2 d2 2 d3
C


and


21r a-, a-a
S(a2 2 23 d d (a2 d2ap + p)2 2t(a2 + p2) p dp
J^J 04 Vl 7a 4 J, (a2 2ap cos + p2)2 F(.2 p2)3


= 2 p=a- a -
+2np=0 2


6a < 4
IF3


As a result, the following inequality is obtained


A,1j c 2nM+ 8M852 -a 4nM(2a + 62)
310 4


a-&

I-0


21(a2 + p2)p dp
-(a2 2)5



3a -


2n
-Ae a


L o








NACA TM 1324 27


The difference is estimated


Sa2r2 + a2 2 I( ) (a2 X2 2 dz
I J 2E2r2 + (,2 &2 712)(a2 x2 y2 z2
S2


On account of equation (3.46)


IA21' 3M I 7a2 2 t2 d dR

S2


But


2x a
dp 12p dp d
SJ oj 2 2ap cosO a2


f1-02 v2 2 d&
ro2


a
S2 p dp
Ja- r- p2


= 2n-a e2


and therefore


I2| < 6iM 2a


Thus for


A = A1 + A2


the estimate is obtained

Al <2nM f +

Assuming


465


252
-rare


+3 2a s


6=


yields


AI < 24nMA4a


Thus


2a 2 -2 -2 f(g,Tr ) dt d?
S2a2r2 + (a2 g2 T2)(a2 x2 y2 z2 + R)
S


= N(e) + o )

(3.47)


-+ R) Jf( ) a2 2 2 d, drl
S2


' a +
17







28 NACA TM 1324

where 0(a) denotes a magnitude, whose ratio to a remains finite
when 5 approaches zero.
An estimate of the second integral entering equation (3.40) is
given:.


J2(x,y,z) = jr-2- -(, -1) (a2 x -2-Z
j 1 a242T2 [2a2r2+(a2-2i2)(a2-x2y2z(2+R3
(3.48)


Again assuming 8< c/2 yields


IJ 21 M a dM dn
I a J 2 2 ,2 (2 + 2)

a
ao a- _


a-&

fo




a

Ja


a 2n
M" /
= / -
Y


p dp d#
a2 p2 2 2ap cos # + p2 + g2)


pdp
2,(a2 + p2 + 52)2 4a2p2


p1 a-1

-a2 P2 p= 2a 2


p dp
Va2 p72-(a2 + p2 + 52)2 4a2p2
v JU


a-B
0


p do
/(a2 p2)3


1 1
a Aae


a

(a


p dp
2ab-a2 p2


S2ae 2
2ab


but


i.
ii







NACA TM 1324


hence


and for 5 = 5


J21< 4.M
a Vao


From equation (3.40) and equation (3.42), the
on account of the estimates (3.47) and (3.49):


VIN(e) xa2 x2 y2 z2 +


following is obtained


R + 0(1)


(3.50)


In exactly the same way, there is obtained


= -
b y


V2 "N() yVa2 x2 y2 z2 + R + O)
7aaR


(3.51)


Finally,


Mu = 6K f(, T) d. dri
"Fz JJ Tz-


But


K -2z arc tan A+
z Itr3


2A _
n(l + A2) r3


z(a2 +
rR(a2 -


y2 + 2 R)
y2 z2 + R)


2 2 2 2 2 2 2
a a x y z + R
ar V2

Assuming z> 0,


I i d dj dl 2.n
JJ r3


IJ21 2 ( +


52-)


(3.49)


where


_ .- V ^ L t -







30 NACA TM 1324

hence


2zare tan A+ 2- +A2 f(J) (,) dt dl 2 2(n + 1) M


and therefore

6U 2= 2-az(a2 t x2 + 2 + 2 z- R) f f )a2 _n2 d d( + 0(1)
nRa2- 2 -2 z2+R J i a2r2 +(a2 2 22)(2 2 2 -_ 2 z+R)


Again use is made of equations (3.47) and (3.42) and the fact that
for z >0

z R a2 + x2 + y + z2
\a2 y2 2 + R 2a

without difficulty:

6u a2 + x2 +2 + z2 R 2 2 2 2
Sy2 + z2 N(e) R a2 + x2 + y2 + z + O(1)
= Ra2 ff
(3.52)

From what has been said previously about equation (3.38) it is
evident that if

F(C,n,t) = fo( ,) + fl(E,T) cos at + f2(E,q) sin cut (3.53)


N(8,t) = F(,,t)Va2 (3.54)
h2 t+ 2 + a2 2ar cos e 2al sin e

the following results











a xJ -2 x2 y2 z2 + N(,t) o()
=x r-2a

a( = a2 x2 y2 ,2 + R N(e,t) + 0(1)
- N(e,t) + i)aR
A iv aR


aP 1 (a2 + 2
S2"VT x2a2R

SR a2 + x2 y2 +


+ y2 + z2 R) x


z2 N(e,t) + 0(1)


or, in the coordinates 5y e, a


N(e,t) cos e


sin( a.)


N(e,t) sin 0 sin( a)


+ 0(1)



+ o(1)


4, N(6,t) cosQ a)
-=+ o(i)
2a

The computation of the suction force W2 by equation (3.37) is
considered. An arc D'AB' of the circumference C is taken symmetrical
with respect to the x-axis with subtending angle 2e0 face a, the part 0O is taken of the surface determined by equa-
tions (3.41) for constant 50, where 0 changes from -80 to + 80 and
a from -n to +x and two bases, one of which, al, corresponds to
8 = 80 and the other, 02, corresponds to 0 = -00, where on these
bases 6 varies from 0. to 60 and a from -n to +n.

On the toroidal surface:

cos(n,x) = cos a cos e cos(n,y) = cos a sin e cos(n,z) = sin a


= cos(n,x) + cos(nN(,t) sin a)+
0= cos(nx) + 64 cos(n,y)+ 2* cos(n,z)= t s- + 0(1)


-w
t~- ...I -


NACA TM 1324


(3.55)


(3.56)








NACA TM 1324


Hence simple computation shows that


+ cos(n,x) dS f as
2 o


= 00 1 2(e't) cos a cos e d8 do + + j N2(e,t) cos a sin2 a da +
9o f-" 04-e

0(o) = / N2(e,t) cos a de + o(/)
2 0



In the same manner, the integrals taken over the bases ao and 02
have the order 0(50). Hence if 80 approaches zero, for the suction
force developed along the arc D'AB', the following expression is
obtained

80
p N2(8,t) cos 0 de
T -30


Now when
suction force


80 approaches r/2, the required expression for the
W2 is obtained in the following form:


1
2


W2 2 /

2


N2(e,t) cos 8 de


The mean value of the frontal resistance is found.
shows that for the mean value of W1


1 =- 2pc 4 12 Tx "2
S


(3.57)


Equation (3.31)



k -J dx dy

(3.58)




~w w w- w


NACA TM 1324


In the same way, for the mean value of the suction force

1
2
142 3 NO2() + N12(8) + 1 N22(e) cos e de

2


where


fk(C,) Ia -2 4 2 dC d
Nk(e) = + +a a cos -2a sin
Sf t2 + q2 + a2 2ag cos e 2al sin e
s


(k = 0,1,2)

(3.60)


For the mean value of the frontal resistance

W = w2 (3.6]


4. Example

If a plane wing varies its angle of attack periodically according
to the harmonic law so that the equation of its surface is


z = (00 + 01 cos ct) x


(4.1)


in the notation of section 1, the following is obtained


CO(x,y) = POx


and therefore


Zo(x,y) = CPO


1(x,y) = PJx



Zl(x,y) = cl.


2(x,y) = 0


Z2(xy) = ckPlx


(4.2)
Z(x,y) = Z1 + iZ2 = cp0(l + i x)
The function f(x,y) corresponding to this value of the function
Z(x,y) is determined by equation (2.18) where g(y) is the solution
of integral equation (2.19).

Consideration is restricted to the solution of the inverse problem
by assuming that


fo(x,y) = A0 f(x,y) = A + Bx


(3.59)









34 NACA TM 1324


where A and B are constant complex numbers and AO is a constant
real number and the shape of the wing is determined corresponding to
this function. By such a method it is possible to obtain also an
approximate solution of the direct problem of the nonsteady motion of
a wing according to the law (4.1) for the case of small frequencies of
vibration.


The forces acting on the wing are determined.
the lift force P, use is made of equation (3.13).
tions are used


a2 2 2 d d = a3
S


For determination of
The following rela-


JJ Va2 -_ 2 -_ 2 dS d = 0
S (4.


as are equations (3.5) and (3.26), yielding without difficulty

p 4= cRe ( inka3 Ae-t) +
n 3


3
2
a2 [AO + Re(Ae-i]Jt

2i


3
2
cos dy + 2 a3 Re(Be-iwt)

2


cos2 dr


8pca2 AO + a2 ReAe-i-t


- 4ca3 Re(Be-iLt)


The moment of the pressure forces about the x-axis equals zero on
account of symmetry:


M = 0


(4.5)


If the moment of the pressure forces about the y-axis is determined
by equation (3.28) and, in addition to the previously mentioned formulas,
use is made also of the formula


2 V2 2 2 d d = na5
S


(4.4)







NACA TM 1324 35


8pc 2 t)
M = na5 Re(ikBe t) -

1 1
2 2
a3 [A + Re(Ae-itiJ cos2 Y dy a4 Re(Be-ict) cos3 r dyl
a + Re(Ae- 3 n

2 2

or

4pca3 4pca3 Re(Ae t) 64pca4 r '
M AO 4 Re(Ae-rot) 642ca4 Re [-hot i2
MY-3 A- L -2
(4.6)
The frontal resistance is computed. First the suction force is
computed:

If

A = A1 + iA2 B = B1 + iB2
according to equation (3.53)

F(c,l,t) = AO + (A1 + Bl() cos ot + (A2 + B2() sin wt
If equation (3.54) is applied and use is made of equations (3.5)
and (3.26),
4
N(6,t) = 2nra(A+Ai cos ,t+ A2 sin wt) + 3 na2 cos e (B1 cos ut+ B sin ot)

Equation (3.57) yields without difficulty the expression for the
suction force:

W2 = 2 8[2 a2(AO + A1 cos wt + A2 sin ut)2 +

_8 i3 a3(AO + A1 cos ct + A2 sin ct)(B1 cos ct + B2 sin ot) +

642 4(B cos t + B sin t)2
Y7







NACA TM 1324


W2 = A2 + Al2 + A'2 + aAiBI + | aA2B2 + a2B2 + a2B22 +

2AAl + aAoBl) cos at + (AnA2 + aAoB2) sin mt +

,,4Aa2B12- a2 cos t +
A12 A2 2 + aA1BI aA2B2 + a2B- 2 2B2 Cos 2t +
2 6 6 27 27

1A2 + aA1B2 + I aA2B1 + a2BlB2 sin at4

The total frontal resistance is obtained by the equation
W = W1 W2
where W1 is determined by equation (3.31)


(4.7)


Wl= -2p c + Re + ik e-t L-0 + Re e-i dxt) 4dy8
L Wo't -+ / o 1) (4.8)
For the mean value of the frontal resistance the following is
obtained:
W = W1 W2 (4.9)
where
4 4 (A2 + 1A2 + A22 + aAB1 + 6 aA2B2 a2B12 + 2

(4.10)


1 = 2p 0 + Re + ik a dy (4.11-)
S 1\ /
For determination of the functions So(x,y) and g(x,y) character-
izing the shape of the wing, equation (1.16) is used.

x = ZO(x,y) c( + ik) = Z(x,y) (4.12)








NACA TM 1324


where by equation (2.17) in this case


Zo(x,y) = AO + g0(y)


Z(x,y) = A Bx + g(y)e-ikx


(4.13)


and the functions go(y) and g(y) in this case according to equa-
tion (2.16) have the form:



aAo
0(y) 2= x


-y 3 n2


isr+- It2


1 1
(x2 + y2 a2) 2 (a2 5g2 2)2 cos y dr dx dd dr
(x2 + y2 + a2 2ax cos y 2ay sin r)(C2 + 12 + a2 2_8 cos y 2aTj sin r)


g(y) = a
2n


1 1
eik x(x2 y2 a2) 2 (a2 _- 2 92)2 (A t BE) cos y dy dx d2 dy
(x2 +y2 t a2 2ax cos y 2ay sin r)(C2 + T12 + a2 2aC cos r 2aq sin y)


2-y2 3

is~f7+f~l I


Equations (3.5) and (3.26) yield


Sa2A0 f f cos T dy dx
12 J J1 4x2 + y2 a2 (x2 + 2 + a2 2ax cos r 2ay sin r)


A/a2-y2 x
g(y) = 2 e'ik(A + 2/3 aB cos r) cos ry dx
X2 J x2 + y2 a2 (x2 t y2 t a2 2ax cos r 2ay sin y)
so 2-


Integration of equations (4.12) yields


(4.14)








NACA TM 1324


0(x,y) = 1
,g0(y) x+ h0(y)
(4.1

(x,y)= ( 1 k e-ikx Bx 1 g(y) xe-ikx + h(y) e-ikx

where h0(y) and h(y) are arbitrary functions of y.

The function go(y) was obtained in reference 1, where, however,
errors slipped into the computations. Setting


y = a cos 6


2
Ho(e) = sin e go(- a cos e)
A0


gives in place of equation (4.22) of reference 1


2
Ho(e) = sin e
4


1
+ sin 0
8


1
i 1 sin 0
cos e in +
1 + sin +
2

Hence setting ho(y) = 0 and
of reference 1 yields


1 1 cos g
+ sin 8 In 2
1 + cos
( -2


S 1 cos 0
sin In 1 (4.17)
1 + cos e

A0 = ac in place of equation (4.23)


1
8n2


2n2 4 42a Pr+ ^a- -
2n2 _\Fa +y \Fa + \F +Y


"V za +
V2a-


2n2 V 'rTT
2 ir 2 y r/2a + Afa -Ty


In particular for y = 0 and y = a/2 the following values are
obtained in place of those given in reference 1:


(0o< r< )


,4.18)








NACA TM 1324


(x,0) = ax ln2(C +1) + .2) 1n(- + 1-



S 2c ) 1 h23 2 n(2 +1
ax In2 (2 -FV-) +- ln23 + n(2+^A+ In
[ 2n2 812 nV 2

In the same way, the expansion given in reference 1
HO%() in a trigonometric series in the interval 0-oe
replaced by the following:

(i = 2 ) sin(2k + 1) e 1
HO(e) = sin e + k(k + +
2 k(k + 1=l
k=1


0.9263 ax





S" 0.9146 ax

of the function
should be



(4k+ )

(4.19)


that is,


HO(e) = 2k+l


sin(2k + 1) 0


01 = 0.9348
03 = 0.2667


05 = 0.1312
07 = 0.0796


09 = 0.0504


In connection with this, corrections should also be applied to
the numerical values,which are given in reference 1, of the coef-
ficients Bn of the trigonometric series for the circulation obtained
by the usual theory


B1 = 2.2125 mcca

B3 = -0.0934 aca


B5 = -0.0296 aca

B7 = -0.0133 aca


B9 = -0.0067 aca


Hence for the lift force in place of equation (4.29) of refer-
ence 1, the following is obtained:

PO = npca B1 = 3.4755 pc2a2c
2g


where








40 NACA TM 1324


which exceeds the accurate value by 36 percent. For the induced drag,
in place of equation (4.30) of reference 1, the following is obtained

WO r 1.9350 pc2c2a2

which exceeds the accurate value by 87 percent.

Corrections are made in the third example given in reference 1.
The value of the definite integral is:


D arc tan y _2 1 2

V41 -y2 8 2
Hence in equation (4.52) of reference 1 the coefficient of
sin 0 cos 9 is simplified and assumes the value -3n2/8. In equa-
tion (4.53) the coefficient of sin 20 was incorrectly computed;
its correct value is
352 32
2 + = -0.14555

In this connection, the value of the coefficient B2 should also
be corrected:

B2 = -0.7436 aca2

For the induced drag and the moment of the forces about the x-axis,
in place of the values of equation (4.55) of reference 1, the following
is obtained:

W = 0.4343 p2c2a4 M = 0.5840 pc2c2a4

the first gives an error of 140 percent; the second of 55 percent.

The shape of the wing obtained

z(x,y,t) = x gO(y) x +


Re e-t (A + ) e-ikx x 1 g(y) xe-ikx (4.20)

depends on the frequency of the vibrations and is deformed during the
vibrations. The rigid wing is of greater interest.

It is possible with the aid of the results obtained to obtain an
approximate solution of the problem of the vibrations of a plane cirT
cular wing for small frequencies of vibration.







NACA TM 1324


The case is now considered of a wing varying its angle of attack
periodically according to the harmonic law (4.1), so that equation (4.2)
holds-


f0(x,y) = AO


f(x,y) = A + Bx


equation (4.2) yields


Zo(x,y) = AO + g(y)


Z(x,y) = A Bx + g(y) e-ikx


Go(y) a2
a2


ik 3 n


a 2
+f + 2


cos y dr dx
x2+y2- a2 (x2+ y2 + a2 2ax cos y 2ay sin r)


ta2


G2(y) a2


a2
G2(Y) -


eikx cos y dy
x2 y2- a2 (x2 +y2 + a2 2ax




eikx cos2 y dy
x2+y2 -a2 (x2 + y2 + a2 2ax


dx
cos 2ay sin y)




dx
cosy 2ay sin y)

(4.22)


Then


go(y) = AOGO(y)


In place
the wing:


g(y) = AGl(y) + BG2(y)


(4.23)


of Gk(y), their mean values are taken over the area of


SGk(y) 2 y2 dy: a -Y2 y2 dy = 2 Gk () 2 dy
( ~-a = ,-a12
(k = 0,1,2)


(4.21)


1


(4.24)








42 NACA TM 1324


If the frequency of the vibrations is assumed small, or more
accurately, the magnitude ka is assumed small, the expansion

e-ikx = ikx 1 k2x2 .
2

may be limited to the first two terms.

From equation (4.21), the following approximate expressions were
obtained


ZO(x,y) A A0 + AOGO

Z(x,y) A Bx + (1 ikx)(AGi + BG2)

Comparison with equation (4.2) results in:

c -C A= + AOGO

cP1 = A + AG1 + BG2

cPIik = B ik(AGj + B2)


(4.25)


whence


AO= 00
1- G0
= G


c i(l + 2ikG2)
A = --k2
1- G, + ik62


cPlik(l 21)
1 G, + i
i- G1 + ikO2


The following is computed


a
2G
GO -
na2
U-a


GO(y) /a2 y2 dy =
At


0 GO(- a cos e)sin2 e de


But by equation (4.16)


sin e GO(- a cos e) = sin e


go,(- a cos e) 1
AO x2 B0


hence, expansion (4.19) is used, yielding


(4.26)


r







NACA TM 1324


=C
2t


H0(9) sin e de = 3 4 = 42


and therefore


= 1
=00 -! -0.4053 = 0.0947
2


A0 = 1.105cp0


Equations (4.26) show that in computing 01 it is sufficient to
use the terms of first-order smallness relative to ka, while in com-
puting G2 it is sufficient to use the principal term not depending
on k. For small ka the following results


GI 0 + ikll + 0 k2a2 In-


G2 G20 + O(ka2)


where G a1 and 020 are the mean values over the area of the
of the functions


x cosE y dy dx
x2 + y_2 a2 (2 y2 + a2 2ax cos y 2ay sin r)


G20 ) = 2 a cos2 y dr dx
2oy 3 ) 2 1 f A/x2 + y2 a2 (x2 2 2 2ax cos r 2ay sin g)
S+ 2- 2ax


(4.28)

circle S


(4.29)


(4.30)


GI Go ikGl


a



fa


Y2


G*(x,y,r) dy dx dy

(4.31)


G*(x,y,y) = (eikx 1 ikx) cos r/a2 ),2
x + y2 a2 (x2 2 + a2 2ax cos y 2ay sin T)


(4.27)


In fact,


where







44 NACA TM 1324

The interval of integration with respect to x is divided into two
parts: from a2 y2 to 2a and from 2a to -. Since for a>0
Ieix 1 iJ

in the interval 'Va2 y x< Z, eikx I ikxl (2ka)2
therefore


and


13 /21
a x


JOa J2a I

On the other hand, for
ity holds

x2 + y2 a235 x2

As


-


a2 y2 dy


G*(x,y,y) dy dx dy < (2ka)2 G 0O.38k2a2

(4.32)

x>2a, Iyl

(x a cos y)2 + (y a sin y)2? x2


3
2


cos y dy = -2


= 2 eikx 1 ikx = cos kx 1 + i(sin kx -
2


kx)


the following inequalities are obtained when, for clarity, ka is
assumed C1,
a 2a 3
2a i G*(x,y,r) dy dx dy < 1 Co in k dx
j-a j- n Z 2a x 2a

4a2k2 1 osdu + u sin


< k t 2
4a2k2 /
4- 2L


< 0.25a2k2 + O.12a2k2 In -L
ak


1du 2 du + 4a 2 In 1 +
2u U 3 45 f3- (2 "")' )


(4.33)







NACA TM 1324 45

Combining inequalities (4.32) and (4.33) yields, on account of
equation (4.31),

I1 CO ikll< 0. 63a2k2 + 0.12a2k2 In -

from which the first of the estimates (4.30) follows.

In an entirely analogous manner, since, for a>0

ei e 1 j
from the inequality

a a2-2 3
a-.,- 2= 4a (eik 1) cos2 y a2-y2 dy dx dy
S- 3X x2 + y2 a2 (x2 + y2 + a2 ax cos y 2ay sin r)
J-a J. x


the inequality is obtained

a a -
G a2 2ka dy dx 2ka dy dt 2ka2
3nj2 -a xx2 + y2 a2 t2 tTi 3

which proves the correctness of the second estimate (4.28).

The integral (4.30) was considered in reference 1. The function
H1(0) of'reference 1 is obtained if

7j-2 sin 0 G20 -a cos 0) = H(e) (Oe(< .)
.a 2

For this function the expression was obtained (equation (4.36)
of reference 1 with the correction of the error appearing therein)

1 +(1C1
1+ cos 0 l+ sin 0
3nn ) ( sin 0 -
Hl(e) = sin 1 sin cos + n n +
-- cos -1 sin -
2 2

(l + cos 1 ) (- sin 0
sin 8 cos a In tan + 1 LIn c n(4.34)
S4 -cos + sin 0
2 2









NACA TM 1324


The expansion of this function in the interval Os nometric series has the form


Hi(e) = Y
k=O


in a trigo-


2k+l sin(2k + 1) 6


17
i= =n 3


1

in tan- dx = -0.69314
2
0


320 20(-a coB e)



The mean value
to r yields


sin2 6 dO = 4 sin


is computed.


8 H(9e) de = 2_a Y1 = -0.0468a
3n2


Integrating (4.29) with respect


x x Y In x (y a)2
12 + y2 a2 2(x Y2) 2(x2 y2) x2 + (y + a)2

x(a2 + x2 2) arc tn x y2 a2
(x2 +y2)(x2 + y2 a2) 2ax
(x2 + y2)(x2 + y2 a2) 2&x I


y = -a cos 0


2
-- sin e Gll(-a cos 8) = H(0)
a


where


Hence


(4.35)


cG(y) = a
1 n2


x = at


then


(4.36)








NACA TM 1324


sin e sin 0 rt2 O + 4 cos 1
H(t) = i- t2 t cos n 2
t2 sin2 2(t2 + cos2 e) 2(t2 + cos2 e) t2 + 4 sin4 0

t2(t2 + 1 + co 2 ) 2 e_ sn2
(t + e)( i are tan -dt t
(t2 + cos2 e)(t2 sin2 ) 2t


Computation of this integral results in


aH() = sin e (1


- sin Cos + 4 sin a

n2 2

4 sin 2 Ln tan 2 In
4- 1


Further,


CGl(-a cos e) sin2 a de


H(e) sin e de


2a
0S


The computation of the last integral leads to the result


=-a J2
-j2 T


1
2
38
+ 2
9
0


u a
u du = 1.536 a- = 0.1556a
sin u J 2


Thus for small ka


81 = 0.0947 + 0.1556ika


G2 = -0.0468a


(4.38)



(4.39)


Substituting these values in (4.26) gives


ik(0.8106 0.311ika)
B = c01 0.9053 0.202ika


(4.37)


Gl


1 0.0936ika
A = 0.9053 0.202ika


(4.40)








48 NACA TM 1324


Thus for small frequencies of vibration, to a first approximation:

A0 = 1.105c0 A = (1.105 + 0.144ika) c31

B = 0.895ikcp1 (4.41)

For the periodic vibrations with small frequency, in accordance
with the law (4.1) of a plane circular wing, the previously derived for-
mulas may be used for the forces where the values A0, A, and B have
the values just given. For the lift force, the approximate expression
is obtained from equation (4.4)

P = pc2a2 2.81300 + i0(2.813 cos Wt -1.766ka sin wt (4.42)

The fluctuation in the lift force due to the vibrations of the
wing thus leads the latter in phase, the maximum value of the lift
force being greater than the value which was obtained in the computa-
tion for the steady motion.

In the same way, equation (4.6) leads to the following expression
for the moment of the pressure forces about the y-axis:

My = pc2a3 1.473po + 01(1.473 cos wt + 0.867 ka sin wt (4.43)

The component of the frontal resistance W1 is determined in the
given case by the evident formula

W1 = P(PC + -P cos Wt)

that is,

1 = pc2a2 2.813002 + 1.406p12 + P0P1(5.626 cos at -1.766ka sin at) +

1.406012 cos 2ut -0.883012 ka sin acot (4.44)

The suction force is obtained from equation (4.7), restricted to
the first powers of ka,

W2 = pc2a2 1.554802 + 0.777012 + P001(3.107 cos at + 1.888ka sin at) +

0.777Pi2 cos 2wt + 0.944ka 012 sin 2a (4.45)

The following expression is obtained for the total frontal
resistance:







NACA TM 1324 49


W = W1 W2 = pc2a2 1.259302 + 0.630112 + P013(2.519 cos at -

3.653ka sin wt) + 0.630p12 cos aut -1.827p12 ka sin 2Ut) (4.46)

For the mean value of the frontal resistance

W = pc2a2 (1.259002 + 0.63012) (4.47)

The flapping wing is considered such that

z = P0x + pj cos (at (4.49)

In this case

Zo(x,y) = cP0 Z(x,y) = ikcp (4.49)

Comparison of these expressions with equation (4.25) shows that
in the case considered it is necessary to take

c0o ikcPl(l + ik52) k2c.l 1
A= ---- A= B= (4.50)
SI G + ikG2 1 I + ikG2

that is,

1-ika 0.0468 k2c11(0.0947 +0.156ika)
AO = 1.105co A = ikc 0.9053- 0.202ika 0.9053- 0.202ika

(4.51)

or, by restriction to small terms of the second order with respect to k,

A0 = 1.105c0o A = ikcpl(l.105 + 0.195ika) B = 0.105k2c13

(4.52)

For the lift force

P = pc2a2 (2.8153 + 2.813kB1 sin ct + 0.301k2a1p cos ua (4.53)

and for the moment of the pressure forces about the y-axis

My = pc2a3 (1.4730 + 1.473kpl sin ot -0.181k2apl cos co (4.54)


I~








50 NACA TM 1324


The component of the frontal resistance

l = P0 = pc a2 i 2.813o02 + 2.813kooo3 sin t + 0.301k2.o00 cots

(4.55)

The suction force will be, with an accuracy up to terms of the
second order with respect to ka:

W2 = pa2c2 (1.554P02 + 0.777k2P32 -0.376op30k2a cos cut +

3.107kp0o1 sin wt -0.777k2012 cos 20t) (4.56)

For the total frdntal resistance

W = pa2c2 (1.25902a -0.777k2 12 -0.294kPo01 sin at +

0.677k2ap031 cos ut + 0.777k21312 cos 2) (4.57)

Its mean value will be

S= p2c2 (1.25900o2 -0.777k2 012 (4.58)

so that a decrease is obtained in the frontal resistance as compared
with the wing which does not execute a flapping motion.


REFERENCES

1. Kochin, N. E.: Theory of a Wing of Circular Plan Form. Prikladnaya
Matematika i Mekhanika, vol. IV, no. 1, 1940, pp. 3-32.

2. Schade, Th.: Theorie der schwingenden kreisf6rmigen Tragflache auf
potentialtheoretischer Grundlage. I Analytischer Tell.
Luftfahrtforschung, bd. 17, Ifg. 11/12, 1940, pp. 387-400.








NACA TM 1324


THEORY OF WING OF CIRCULAR PLAN FORM*

By N. E. Kochin

A theory is developed for a wing of circular plan form. The dis-
tribution of the bound vortices along the surface of the wing is con-
sidered in this theory, which has already been applied in a number of
papers. In particular, the case of the circular wing has been examined
by Kinner in reference 1.

A second method is considered herein which Dermits obtaining an
expression in closed form for the general solution of this problem.
The wing is assumed infinitely thin and slightly cambered and the problem
is lipearized in the usual manner.

Comparison of the results of the proposed theory with the results of
the usual theory of a wing of finite span shows large divergences,
which indicate the inadequacy of the usual theory of the case under
consideration. For the wings generally employed in practice, which
have a considerably greater aspect ratio, a more favorable relation
should be obtained between the results of the usual and the more accurate
theory.


1. Statement of the Problem

The forward rectilinear motion of a circular wing with constant
velocity c is considered. A right-hand system of rectangular
coordinates Oxyz is used and the x-axis is taken in the direction
of motion of the wing. The wing is assumed thin with a slight camber
and has as its projection on the xy-plane a circle ABCD of radius a
with center at the origin of the coordinates (fig. 2, in which a section
of the wing in the xz-plane is also shown).

Let

z = t(x,y) (1.1)

represent the equation of the surface of the wing, where the ratio Q/a
as well as the derivatives 3/ox and 6t/5y are assumed to be small
magnitudes.


*"Teoriya kryla konechnogo razmakha krugovoi formy v plane."
Prikladnaya Matematika i Mekhanika, Vol. IV. No. 1, 1940, pp. 3-32.








52 NACA TM 1324


The coordinate axes are assumed to be immovably attached to the wing.
The fluid is considered incompressible and the motion nonvortical, steady,
and with no acting external forces. The velocity potential of the
absolute motion of the fluid will be denoted by 9(x,y,z) so that the
projection of the absolute velocity of a particle of the fluid is deter-
mined by the formulas

v v vZ (1.2)
x ox y y' a oz

The equation of continuity

bVy y by
+ -- + = 0
Bx by bz

shows that the function cp must satisfy the Laplace equation


+ + -- o (1.3)


At the leading edge of the wing the velocity of the fluid particles
is assumed to become infinite to the order of 1/-/F where B is the
distance of the particle to the leading edge; at the trailing edge the
velocity is assumed finite. From the trailing edge of the wing a
surface of discontinuity is passed off on which the function p suffers
a discontinuity. The function cp(x,y,z) and all its derivatives over
the entire space bounded by the said surface of discontinuity and the
surface of the wing are continuous.
The problem is linearized in the following manner. The function i
is assumed to suffer a discontinuity on an infinite half-strip Z located
in the xy-plane in the direction of the negative x-axis from the rear
semicircumference BCD of the circle S to infinity. In the same
manner, the condition on the surface of the wing is replaced by the
condition on the surface of the circle S located in the xy-plane
and in this way the function p(x,y,z) is assumed to be regular in the
region obtained by cutting the infinite half-strip Z and the circle S
from the entire infinite space.


The boundary condition must be satisfied on the surface of the wing.


= c cos(n,x) (1.4)
=c







NACA TM 1324 53


where n is the direction of the normal to the surface of the wing.
The direction of this normal, because of the assumption of small curva-
ture of the wing, differs little from the direction of the z-axis. If
small terms of the second order are rejected according to the formula


cos(n,x) = (1.5)

+ (a)2 (+ )2

in place of equation (1.4),


= c-

This condition must be satisfied on the surface of the wing, but
it is assumed satisfied on the surface of the circle S, that is, for
z = 0; this again reduces to the rejection of small terms of the second
order by comparison with those of the first order.

The boundary condition is obtained:


= c a(xy) for x2 + y a2 (1.6)
6z z=0 ox

which must be satisfied on both the upper and lower sides of the
circle S.
The boundary conditions are set up which must be satisfied on the
surface of discontinuity Z. On the surface of discontinuity at the
trailing edge of the wing, the kinematic condition expresses the con-
tinuity of the normal component of the velocity, that is, the magnitude
o9/on must remain continuous in passing through the surface of dis-
continuity. Since on the surface of discontinuity the direction of the
normal differs little from the direction of the z-axis, transfer of
the condition on the surface of discontinuity to the half-strip Z,
gives the equation


(~z) (z)z for y < a; x2 + y2> a2; x< 0 (1.7)
6z 7=+O 6z_-O l '

which expresses the continuity of p/6z in passing through the surface
of discontinuity Z .








NACA TM 1324


The dynamical condition expressing the continuity of the pressure
in passing through the surface of discontinuity e is considered.
In order to determine the pressure p, the formula of Bernoulli is
applied to the steady flow about a wing obtained by superposing on the
flow considered, a uniform flow with velocity c in the direction of
the negative x-axis. In this steady flow the velocity projections are
determined by the equations


v =-c+ v- v-
x x y y' Z

and therefore the formula of Bernoulli reduces to the form


p -= c + + 2 + + constant (1.8)

Rejection of small terms of the second order results in


P = PO + pc (1.9)
3x

where pO is the value of the pressure at infinity.

Since the pressure must remain continuous in passing through the
surface of discontinuity at the trailing edge of the wing, the equation
obtained shows that 8p/6x does not suffer a discontinuity on the
surface of discontinuity. Transfer of this condition to the surface E
yields the condition


f ( for ly < a; x2 + 2 > a2; < 0 (1.10)
z=+0 z=-0

which expresses the continuity of Op/ix in passing through E.

The function (p suffers a discontinuity on the surfaces S
and z, which means that along the surfaces S and 2, surface
vortices are located as shown in figure 2. The direction of such a
surface vortex is perpendicular to the direction of the relative velocity
vector of two particles of the fluid adjacent to the surface of dis-
continuity on its two sides. In particular, on the surface Z, on
account of equation (1.10), only <9/ty suffers a discontinuity and
therefore the vortex lines on Z are directed parallel to the x-axis
as shown in figure 2.







NACA TM 1324


Since all the vortices lie in the xy-plane, at two points symmetri-
cal with respect to the xy-plane, the values of 89/8z will be the same,
whereas the values of OP/ox and NcP/ty will differ only in sign.

It may therefore be assumed that

P(x,y,-z) = (p(x,y,z) (1.ll)

Assuming in particular z = 0 yields


cp(x,y,o) = 0

in the entire xy-plane with the exception of the circle S and the
strip Z (on which P suffers a discontinuity).

Since on the strip Z both condition (1.10) and the condition
derived from equation (1.11) must be satisfied



z = ())z-0

and

C(O)_ =( =- = 0 for y < a; x2 +y2 a2; x < 0 (1.12)
z=+0 z=-O

Finally, since the fluid far ahead of the wing is assumed to be
undisturbed, the condition at infinity is


lim lim lim 0 (1.13)
x-=- 6x X-#- 6y X-H-+ Wz

In the hydrodynamic problem under consideration, account is taken
of the distribution of the vortices along the surface of the wing. It
is this circumstance which makes the treatment more accurate than
the usual wing theory.

The hydrodynamic problem is thus reduced to the following mathe-
matical problem: To find a harmonic function P(x,y,z) regular over
the entire half-space z > O, which on the circle S satisfies the
condition


( == -c (1.14)








56 NACA TM 1324


on the strip 2 the condition


()z O =0 (1.15)


on the entire remaining part of the xy-plane, the condition


q,(x,y,O) = 0 (1.16)

and the derivatives of which remain bounded in the neighborhood of the
rear semicircumference BCD, while in the neighborhood of the forward
semicircumference BAD they may approach infinity as 1/-/f where 8
is the distance of a point to the semicircumference BAD. Finally
the condition at infinity (1.13) must be satisfied.

An expression for the harmonic function p(x,y,z) is given in
closed form depending on an arbitrary function f(x,y) satisfying all
the imposed requirements besides equation (1.14). The function ((x,y)
can be determined from this condition, that is, the shape of the wing
corresponding to the function f(x,y). An integral equation is also
given, the solution of which is reduced to the determination of the
function f(x,y) for the given shape of the wing, that is, for a
given function L(x,y).

2. Derivation of the Fundamental Equation

Inside the circle ABCD, the point Q with coordinates E, i is
taken and the function K(x,y,z,C,T) constructed, where x,y,z are
the coordinates of the point P, according to the following conditions:

(1) The function K, considered as a function of the point P, is
a harmonic function outside the circle ABCD.

(2) The function K becomes zero at the points of the plane xy
lying outside the circle ABCD.

(3) The derivative 3K/3z becomes zero at all points of the circle
ABCD, except the point Q.

(4) When the point P approaches the point Q, remaining in the
upper half-space z > 0, the function K increases to infinity but
the difference K (l/r), where


r= (x ) (y )2 + z2
remains bounded.







NACA TM 1324 57


(5) The function K remains finite and continuous in the neigh-
borhood of the contour C of the circle ABCD.

Because of the second condition, the values of the function K at
two points situated symmetrically with respect to the plane xy differ
only in sign:

K(x,y,-z,,TT) = K(x,y,z,,Tq) (2.1)

as follows from the principle of analytic continuation. It is.then
evident that if the third condition is satisfied on the upper side of
the circle ABCD it will be satisfied also on the lower side, since
according to equation (2.1) the derivative 3K/6z has the same value
at two points situated symmetrically with respect to the xy-plane.
It is evident further that when the point P approaches the point Q
from below so that z < 0 then K(x,y,z,,qT) will behave as 1/r.

Because of the third condition, the function K can be continued
into the lower half-space through the upper side of the circle ABCD
as an even function of z. Thus a second branch of the function K is
assumed, again determined over all the space outside the circle ABCD
and differing only in sign from the initial branch of the function K.
It is then evident, however, that at the points of the upper side.of
the circle ABCD, the values of the second branch of the function K
and its derivatives coincide with the values of the first branch of
the function K and its derivatives at the points of the lower side
of the circle ABCD. That is, in the analytic continuation of the
second branch of the function K through the upper side of the circle
ABCD into the lower half-space, the initial branch of this function
is again obtained.

A two-sheet Riemann space is considered for which the branching
line is the circumference ABCD. In this space K(x,y,z,E,q) is a
single-valued harmonic function remaining finite everywhere with the
exception of the two points Q having the same coordinates (Cl,0),
but belonging to two different sheets of space; in one sheet the
function K behaves near the point Q as 1/r and in the other sheet
as 1/r. Such a function K(x,y,z,C,r) can readily be constructed
by the method of Sommerfeld (reference 2). In this way for the case of
a two-sheet Riemann space having as branch line the z-axis, a harmonic
function V(p,p,z) (p,p,z being the cylindrical coordinates of
the point) is determined which is single-valued and continuous in the
entire two-sheet space with the exception of the points Q and Q'
having the cylindrical coordinates (p',P',z') and (p',-q',z'),
where near the point Q the function V behaves as 1/r and near
the point Q' as l/r, where


r =p2 + p'2 2pp' cos( ') + (z z)2

r' = p2 + p,2 2pp' cos(p + 4') + (z z')2








NACA TM 1324


This function V has the form:


V = arc tan ----
it r a T


1
- arc
r'


where


p')2 + (z z')2


p ; T = COS j r '= Cos
2 2


Setting, in particular,


= ; r =02 + p'2 + 2pp' cos p + (z z')2


yields


2
V =-- ar c
Sirnall

or finally


l + T a 2 T_
tan arc ta= ar tanc
a -+T + 7 nr 4o2 r2



2 2-/p sin '
V = -arc tan
rr r


An inversion with respect to the point with coordinates p = a,
P = 0, z = 0 is carried out.


p cos 9 = a +


a2(x1 a)

(xl a)2 + yl2 + z12


2a2 (1 a)
; p' = a +
(Rl a)2 + 12


2a2yl

(xl a)2 + y12 + z12


2a2z1

(x1 a)2 + y12 + z12


2a2 t
Za =
(1 a)2 + 12


The function


v =
1


2a2V

(x a)2 + y12 z12 (c1 a)2 + 12


1 0
0 = --- -- + +


p sin 9 =


- T'








NACA TM 1324


expressed in the variables xl, yl, El is then, as is known, a harmonic
function. Computing it and replacing xl', y, Zl by y, z, x and (1, 5
by T1, E yield the required expression of the function K(x,y,z,r,:):

K(yz,2 a2 2 2 a2 2 + R (2.2)
K(xyzavC) = are tan (2.2)
nr -V ar

valid for z > 0, where


r =V(x )2 + (y )2 + z2
(2.3)
S=V(a2 2 y2 z2)2 + 4a2z2 a2 + x2 + y2 + z2)2 4a2(x2 + y2)

That this function satisfies all the above set requirements is
easily verified; the arc tangents must be taken between 0 and t/2;
for z < 0 the value of the function K is obtained by equation (2.1).

The following function is set up:


(1(x,y,z) = J1 K(x,y,z,C,Ti)f(c,T)ddTj (2.4)
S

where f(x,y) is an arbitrary function, which is continuous together
with its partial derivatives of the first and second order in the entire
circle S, and the integration extends over the entire area of the
circle S. Evidently, P1(x,y,z) is a harmonic function in the entire
space outside the circle S. Because of the first property of the
function K, the function pl(x,y,z) becomes zero at all points of
the plane xy which are outside the circle S. Hence equations (1.15)
and (1.16), which must be satisfied by the solution p(x,y,z) of the
problem posed in section 1, will be satisfied for the function 91(x,y,z).
The function ip(x,y,z) does not in general satisfy the condition of
the finiteness of the derivatives of this function on the rear half of
the contour of the circle S. For this reason, a function such that
the obtained function P(x,y,z) also satisfies this condition is added
to i~(x,y,z).

The following equation is evident:


S- f(fr)dFd
)x 2n V6x
s








60 NACA TM 1324


The character of the approach of the function )K/?x to infinity is
considered as a point approaches the contour C of the circle S. As
may be easily computed

6K 2(x t) ar a 2 2 -_ 2 2 y2 _z2 + R
-arc tan- -
3x r3" -V ar

2-J2a a2 2 -1 2 Va2 x2 y2 z2 + R x -
x 2a2r2 + (a2 2 2)(a2 x2 y2 z2 + R) Hr2
(2.5)
If a point with coordinates x,y,z is near the contour C of the
circle S the distance of this point to the contour C is denoted
by 8; then

6 =Va2 + x2 + y2 2 2aVx2 + y2 (2.6)

Hence near the contour C, the approximate equation holds:

R 2ab (2.7)

When the fixed point C,q lies inside the circle S while the
point with coordinates x,y,z lies near the contour C of the circle,
then, as follows from equation (2.5),

K V2 2 V2 2 x2 y2 z2 + R + o(1) (2.8)
=x = tarRP.

where the symbol 0(1) denotes a magnitude which remains finite when
8 approaches zero. Thus /K/6x has the order 1/-/%. The principal
part of 6K/ox is not a harmonic function. It is not difficult, however,
to find a harmonic function having the same infinite part near the
contour C as oK/ox. For this, it is sufficient to form, after the
analogy of equation (2.5), the derivative 3K/ C; this derivative
remains finite near the contour C of the wing; moreover it is easy
to see that
aK )K 2_/2 a/ a2 x2 y2 z2 + R
ix --. 2a2r2 + (a2 -2 -_ 2)(a2 x2 y2 z2 + R)]



x Ra2 (2.9)




=: ". '




NACA TM 1324 61


This function is harmonic and differs from 6K/ax by a quantity
which remains finite near the contour C.

By computation, it is further shown that the function just described
is represented in the form of the integral


aK 6K 1 x a2 2 2 -1 ( z2 J Y2 +22 R coa y dr
n (x2+ +22 z 2 a 2ax cos r 2ay sin r)(C2 f 2 + a2- 2at cos r- 2aT din r)


where the function

a2 x2 y2 2 + R
x2 + y2 + z2 + a2 2ax cos r 2ay sin y

is a solution of the equation of Laplace having the circumference C as
the branching line and the point with coordinates (a cos y, a sin y, 0)
as a singular point. From this it follows that the function

3n
SK 4 ,a2 r -2 -4.2 X2 2 2 R cos r dy
i J (x2 y2 + z2 a2 2ax cos y 2ay sin y)(2 2 a2 2aC cos r 2an sin y)



2r (x2+2 + 2 + a 2- 2ax cooy 2ay sin r)(&2 + 2 ,+,2 a c.)S r 2&q sin r)



remains finite near the points of the rear semicircumference of the
circle S.

Therefore it is assumed


-a" i f/ ,r(, Ja
3w S

S + a2 4-7 2y 2 y-- 2 2 coa y dy d dl (2.12)
+x a 22ax COB y 2ay sin y)(C2 + a2 a 2a s cos 2ax sin r'l
9 i2. Ba? 2_ z 0








NACA TM 1324


Integrating with respect to x and considering the condition at infinity
(1.13) yield the final equation



*(x,y,z) = f( R,q l K(x,y,z,C,n) +
3. S
21 Y2 2 2 2 Y2 a- 2 + R cos y dy x dC (2.13)x
T ,r2 (x2 + y2 22 a2 2ax cos y 2ay sin r)( + 2 + a2 2a cos r 2ai sin r)
2


This equation may be written in somewhat different form. Because of
equation (2.11)


a = n r f(,r)dSC d 1
3x 2, 3 2 3
0
r2 ff 2- 2 22 2 2 + OE dy- fio )dr d
(x2 + / z2 + a2 2ax cos r 2ay sin r)(J 2 a 2a cos r 2ai1 sin y)
S

Since the function K becomes zero on the contour C

Since the function K becomes zero on the contour C


-S r ( c ) d -c = i i
at
S S

Introduction of further notations


K dL d d
aE


1 I l a2 t2 2 f(C,j)dg da = C(y);
2n35/2 S t 2 + T12 +2 2aC cos y 2ai sin y


results in


ax 2 JIS
S


K(x,y,z,C,n)F(E,n)dE dr +


ja2 x2 y2 z2 + R G(r) cos r
x2 + y2 + z2 + 2 2ax cos y 2ay sin y


dr (2.16)


(2.14)


= F(C,)
(2.15)


n




2







NACA TM 1324


and after integration with respect to x
x
S(x,y,z) = j1 K(xy,z,,tl)dx F(,Tl)dt dT +
S +0


Tx a2 x2 y2 z2 + R G(r) cos ry
-- dy (2.17)
x2 + 2 + a2 2ax cos y 2ay sin y

2

The given functions F(c,n) in the circle S and the function
G(y) in the interval (-n/2, n/2) completely determine f(,7l), so
that the equations (2.13) and (2.17) are equivalent.

The equation cp(x,y,z) obtained satisfies the conditions imposed
in section 1.
This function is evidently a harmonic function in the entire space
exterior to the circle S and satisfies the conditions at infinity,
equation (1.13).. From equation (2.12) it follows, that in the plane xy
for x2 + y2 > a2 the condition is satisfied:


fx z =0


and from equation (2.13) it follows that


P(x,y,O) = 0

in that part of the plane xy which lies outside the circle S and

the strip E.

It remains to prove the finiteness of the first derivatives of
the function q(x,y,z) at the points of the rear semicircumference C
and to determine the behavior of these derivatives on approaching the
points of the forward semicircumference C.

In considering the neighborhood of the rear side of the circum-
ference C, equation (2.16) may be used. The latter shows that oP/ox
remains continuous at the points of the rear half of the circumference C
and becomes zero at these points.

The behavior of the derivatives with respect to y and z of the
following function is considered:







NACA TM 1324


-(x,y,z) = f K(xy,z,CTl)F(Q,1)dCdi j
S


near the contour C.


f i F(C,r)dt dl9
S


Similarly to equation (2.9),


K K
-y + -
oy Ti


[2 2af+ (aV/a2
n112a2r2 + (a2 t2


Sx2 y2 z2 + R
- 2)(a2 x2 y2


- 2 +R)]


yVa2 2 2
R


and similarly to equation (2.14),


F(K ,j)d di JfK dg dj
S S

where this part of the integral remains finite everywhere
contour C becomes zero.


(2.21)


and on the


In order to evaluate the remaining part of the integral equa-
tion (2.19), the following two integral are considered:


Jl(x,y,z) =


2/ E2 -2 2 d& d_
s 2a2r2 + (a2 C2 2)(a2 x2 y2 z2 + R)


J2(x,y,z) = dC d
J a2 2 T2 2a2,2 (2 ( 2 12)(a2 x2 2 Z2 + R)


Both, on account of the symmetry, depend only on x2 + y2 and
hence without restricting the generality, it may be assumed that y =
x > 0. The distance 5 of a point with coordinates (x,0,z) is
introduced to the contour C:


(2.18)


(2.19)


(2.20)


> tl


C-'-,


r


r "'T',


V 7j






NACA TM 1324


8 = f(a x)2 + z2


R > Ix2 + z2 a2|


the following relation will hold:

J1(x,0,z)4 ff
S


a2 2 12 d dP
2a2[x- C)2 + 2 + z2]


Polar coordinates are introduced

C = p cos# ; T = p sin O


whence


2a2[p2 2px cos


dp d6
S+ x2 + 2]


d,
2 2 +
p 2px cos O + x + z


( +2 2
(p


2n
+ z )2 4p2 x


J1(x,o,z)< 4
afeqV


p a2 p2 dp
+ x2 + z2)2 4p2 x2


For x > a


J1(x,0,z)< -
a2


a


f


p a2 p2 dp
V(p2 + x2)2 4p2 x2


p a2 p2 dp <
x2 p2 a2


J


a
pdp _
0 4Ja2 p2 a
/ P2=^
f ^ ?'


Since


Jl(x,0,z)


Since


2A

0


hence


2


2oro
O fJ






NACA TM 1524


While for x 4 a, use is made of the inequality

R ; a2 x2 z2
to obtain
1 Va2 _2 1 d ddT
Jl(x,0,z) < c
2 S [(x )2 + + T2 ] + z2+ (a2 2 2)(a2 x2 2)


1 p a2 p2 d dp
(2.23)
2 a4 2a2 xp cos 5 + p2(x2 + z2) (2
/. _B. ______ a

-=p a2- p2 dp pVa2 p2 dop pdp
f [ a4g[p + f i~J '
a4+p2( 2 + z2)] 44x22 4 a -x22 a2 2_ -2 p2

The following inequality results:

Jl(x,y,z) < (2.24)

The second integral is considered. As before,

a p dp
J2(x,O,z) 4 -2 dp
a V2 p2 (2 +x2 + z2)2 4p2x2

For x > a

J2(x,O,z)4 2 p dp p dp
o f V(a2- P 2) + 2] a3 f/ 2 V(a x)2 + z2 a

For x4 a an inequality of the type in equation (2.23) is used:

p dp
J2(x,O,z) p dp--
Jo (a p2)[a4 + p2(x2 + z2) + 2a2xp][a2 xp)2 + p2z2]

< p dp
a2 V(a2 p2)[2 xp)2 + 2 2]


It
a







NACA TM 1324 67

If z >a x and therefore 84 z-i, then


Si dp it2 ,2
J2(x,0,z) = -
a2Q p a2 2 2a2 & Z 2
fo 4a aP


but if 0 4 z 4 a x, and therefore 4 (a x)-/2, then

pp pd p p dp 2
J2(X (a2 -xp) 2 -p2 Vaa (a-x)a2-p2 a2( -x) a2B


The following approximation is obtained:

2
J2(x,y,z) 4 (2.25)
a2-(V2


where


B = (a x2 + y2)2 + z2


(2.26)


Near the contour C


R = 2a&


If this relation, the evident inequality

|a2 x2 y2 z21 R


and the obtained approximations are used, the following
is obtained from equation (2.20):


approximation


3+ F(C,T)dC d4 =( 0o
S

It is evident from equations (2.19) and (2.21) that near the
contour C


(2.28)


=0
6 (


(2.27)







NACA TM 1324


The following derivative is formed:


4= ffSBut F(g)dg d
S

But


2z
= -- arc
nrr3


2 A
tan A +
v 1 + A2


I-L
z
r3


+ z(a2 + x2 +
rR(a2 x2


y2 4
- y2


,2 R)
- z2 + R)J


Ta2 2 2 2 -2 y 2 + R


Hence if


IF(,Tn) I < M


then, on account of the inequality

A 1<
1 + A2 2

for z > O the approximation results:


L 2M
kZ


rz
s dy +
S


2-.2 azM(a2+ x2 +y2 + 2 R)

nR Va2- x2-_ y2 z2+ R


/ 4 a2_ 2_ dgT

S 2a2r2 T2+(a2_ -2)(a2 -x2 y2- z + R)


Noting that


f -L3 dC drj 21
S


and making use of approximation (2.24) yield


where






NACA TM 1324


Ia 4 7+ z(a2 + X2 + 2 + Z2 R)
in 4 + 2 or M
zI ]a. a2 x2- y2 z2 + R

Since for z > 0


a2 x2_22+R
N/2TX2- 2 +R


z\R- (a2 2_ y2_ z2)
VR2_- (a2- X2- 2-z2)2


AIR- a2+x2+y2+z2


S4M + 2- (a2 + x2 + y2 +
aR


z2 R)/VR a 2 + y2 + z2


Now when the point P(x,y,z) is near the contour C, then because of


R 2a6 ; x2 + y2 + z2 a2 < R


there is obtained


S= 0


(I)


(2.29)


Equation (2.16) is again considered. Since the derivatives


= 2 7 2 x2 Y2 g R
- y2 2 + = a x2 z + R
R

S2 z2 + R = z(a2 + x2 2 + z2 R)
R a2 x2 y2 z2 + R


(2.30)


= -- (a2 + x2 + y 2 + R)/R a2 + x2 + y2 + z2
2aR


have near the contour C the order 1/-5, it is clear from equa-
tion (2.16) and the obtained equations (2.28) and (2.29) that at the
points of the rear semicircumference of C there is the estimate


o29 1 )c
)xby 0 ^V&/


o 0 (-\
zxbz


hence


I z I


(2.31)


Va2 x2
-z







70 NACA TM 1324


But is is then evident that the derivatives )p/6y and 9p/6z are
finite at the points of the rear semicircumference C.

The behavior of the derivatives of the function P near the for-
ward semicircumference C can readily be determined, starting from
equations (2.12) and (2.13).


The first


qx 2n
ox 2n


of these equations may be written in the form:
3
f( ,)d dn a x2 y2 z2 +R G(y) cos r
n x2 +y2+z2 +a2- 2ax cos y- 2ay sin y
2


(2.32)


But on the one hand, the estimate


K f(,T)dE d = o0
xU:


sf
S


holds for the neighborhood of the entire
on the forward semicircumference C, the
evidently remains finite. Hence for the
first of the estimates is obtained


- = 0
C) (-\
dx b^S


T = 0
)qy b:1


contour C; on the other hand,
second integral of equation (2.32)
forward semicircumference C the


z = 0
oz Q50


(2.33)


while the latter two of these estimates are obtained in a similar manner
from equation (2.13).

In this manner all the conditions which must be satisfied by the
function p(x,y,z) are satisfied.

The shape of wing to which the obtained solution corresponds is
explained. By equation (1.14)


c -- ( z
dx .z=0


(2.34)


Hence it is necessary to find the value c(p/z in the plane of the
circle S. Both sides of equation (2.13) are differentiated with respect
to z and then z set = 0. On account of the very definition of the
function K,







NACA TM 1324


lim /
Z-" fJ
S


Sf(L,C)dcdn = rim JL f( ,)dt dj = 2nf(x,y)
s (2
S (2


Moreover, on account of equation (2.30),


0
- 2 + R = a- V/2
1x2 + y2 a2


for x2 + y2 < a2

for x2 + y2 > a2


If this is taken into account,


= f(x,y) + g(y)


(2.36)


where



g(y) 2 rjf(E ar t e ol ol)n exp des dn d9is oun :
V2 + 2 .2 h a r Zac sin r)((2 T) h Z- ocoa r n l r)



For the function t(x,y) the following expression is found:


x

C(x,y) =
0


f(x,y)dx g(y x + g(y)
c


where gl(y) is an arbitrary function of y.

Thus, for the assumed degree of approximation, the bending of the
wing in the transverse direction produces no effect on the form of the
flow.

It is assumed that the shape of the wing is given, that is, the
function t(x,y) and therefore the following function are given:


c = M(x,y) (2.39)
ax


.35)


(2.38)


Va2 x2 y2


z=0








72 NACA TM 1324


From equations (2.34) and (2.36) it is clear that


f(x,y) = M(x,y) + g(y) (2.40)

Substituting this value in equation (2.37) and introducing the
notations


[(y) X= ,
371 2,3


TJJ V y2 -ia25 y2.a2 2ax c r 2- ay sin r)(C2 + t2 4 a2 2a cos y 2ad d d r)





2- | s, 2_Ia2 -2 2 COE r dr d d d
S 2z Vx2 2 i Z -2 a2 t, 2 2ax coE 2ay sin 2 + n + 2 2a cos 2at sin y)
2


give an integral Fredholm equation of the second kind for the determi-
nation of the function g(y):
a

g(y) = I(y) + H(y,r)g(r)dr (2.42)
-a



In consideration of examples, a function f(x,y) shall be given
and the shape of the wing then determined by equation (2.38). For the
obtained shapes of the wing it is not difficult to find a solution by
the usual theory, a fact which provides the possibility of evaluating
the degree of accuracy of the usual theory.


3. Computation of the Forces Acting on the Wing

The fundamental equation determining the motion of the type under
consideration is recalled:


= f J'J'{Ku,,,z .,,7.I
3 s3
(X2 _____ y2 z 4lL2 42 x2 y2 z2 + R Cos y dr dx tI
(? y2* z2, a Zax coe 2ay sin y)(E2 9* 2 + s2 2a& coB T 2a sin r) f
m )







NACA TM 1324


The value of the function p for the points of the half-strip Z
is computed. Since at the points of the half-strip Z



-= 0
Tx

this value is a function only of y. The notation is introduced


4(y) = lirm P(x,y,z) for lyl a, x2 + y2 a, x< 0 (3.2)
z-O+0

Then evidently


lim q (x,y,z) = c(y) for lyl < a, x2 + 2 > a2, x < 0 (3.3)
z--0

The circulation over the contour M'NM (fig. 2) connecting the
two points M and M' of which point M' lies on the lower and point

M the upper side of the half-strip Z, both points M and M' having

the same coordinates x,y,0, is denoted by r(y). It is then evident that


r (y) =4 (M) -(M') = 2((y) (3.4)

Since in the plane xy outside the circle S both the function K
and the function


a2 2 y2 z2 + R

become zero, it is clear that


c(y)
.^ 2 __(3.5)
1 Pa2 -. 2 2 aZ x,2 y2 fr(,jl) cos y dy dx d& d
2 3 (x2 y2 + a2 2ax c06 2ay sin r)(t2 O + 72 + a2 2accos r 2a? sin y)








NACA TM 1324


Computation shows that


a -x2-2 d a(1 sin ) 1 (3.6)
x2 y2 + 2 2ax cos r 2ay sin r IVa sn yj

where the plus sign is taken for y < a sin y and the minus sign for
y> a sin y.

The following expression is written for the distribution of the
circulation in the vortex layer formed behind the wing:


0 ff Va2 2 12 f(tj) a( B sin ) ) cosy d d dd (3.7)
r(y) = (( f J2 ,82 2 os c sin a sin r y (3.
S


The forces acting on the wing are
pressure at a point of the wing S on
by p_ the pressure at the same point
basis of equation (1.8)


computed. Denoting by p+
the upper side of the wing
on the lower side gives on


p = 2pc
ox

where the value of (p/dx is taken on the upper side of the wing.

For the lift force P, the following expression is obtained:


P= f(p- P+) dx dy =
S


2C= f [( -V2 y2,yO)
U-a


- 200c a dx dy = 2c dx dy
f ax ax



- p(- y2,y,o)]dy = 2pc (y) dy
J-a


The following formula is obtained:
a

P = pc rp(y)dy
-a


the
and
the



(3.8)


(3.9)







NACA TM 1324


having the same form as in the usual theory of a wing of finite span.
But the distribution of the circulation F(y) by the present theory is
somewhat different from that obtained by the usual theory. The derviva-
tion given is not connected with the shape of the wing.

With the aid of equation (3.6) P may be directly expressed through
f(C, ):
3ff

p Zpac 2 r2 q2 f(r,l) cosy dr dd dl (3.10)
2 JJ J E2 + + a2 2a(cos r 2aq sin r
s

The expression for the induced resistance W in terms of the
circulation F(y) likewise has the same form as in the usual theory:

a a
W 4- d(y') dy dy' (3.11)
a -a dy y y'
-a -a

because the origin of the induced resistance is due to the fact that
behind the wing a region of disturbed motion of the fluid is formed;
the kinetic energy of this disturbance is determined on the other hand
exclusively by the distribution of the circulation at distant points
from the wing.

The expression for the induced resistance is obtained from the
momentum law.

A surface enclosing the wing S is denoted by B; the momentum
law applied to the wing in a steady flow then leads to the expression


W =Y p cos(n,x)da + JJ pV xVd (3.12)
B B

where n is the direction of the outer normal to the surface B and
Vx, Vy, Vz are the components of the velocity in the relative motion
of the fluid about the wing. Thus


vx = c + ; vn = cos(n,x) +



p = 2+ \y/ + z/J







76 NACA TM 1324

Substituting these values in the preceding formula and noting that

f Jcos(nx)da = 0; 5' J a = 0
B B
results in

W j \2 2 + (2cos(nx)do + px a (3.13)
B B

The surface B consists of a hemisphere of large radius with
center at the point x = x0 < -a of the x-axis enclosing the wing, and
of the circle cut out by this hemisphere on the plane x = xO. With
increase in the radius of the hemisphere to infinity the corresponding
parts of the integrals entering the preceding formula approach zero. On
the surface x = x0

cos(n,x) = 1; -
an 3x
therefore
W = J f 2 2 2 y dz (3.14)
L= Lf ) () nQ.T)J d

where the integration extends over the entire plane x = xO. For
x0 the following equation is obtained:

W = f )2 )2 dz (3.15)

where 4(y,z) denotes the velocity potential of the plane-parallel
flow which is established in the transverse planes far behind the wing.

The usual transformations by Green's formula yield
a
W= p 4(y) L dy (3.16)
-a

where the integral is taken over the upper side of the segment (-a,a)
in the plane yz.
Since a (y)
r(y) = 24(y); F= a ad(y ) (3.17)
-equata Y( y
equation (3.11) is obtained.







NACA TM 1324 7

In order to find the center of pressure, the principal moments of
the pressure forces about the Ox and Oy axes are determined.

For the moment about the Ox axis,


MX = ff (p_ P+)y dx dy
S

from which


Expressing


4
M -
3


S- 2pc j y dx dy = 2pc J (y)y dy
S -a

a
Mx = pc Y yr(y)dy (3.1
-a


8)


Mx in terms of f(x,y) yields


3y
pea2
S j
s 2


Va2 C2 -_ 2 f(C,rq)sin r cos r dy dC dn (3.19)
t2 + r2 + a2 2ak cos y 2a1 sin y


For the moment about the Oy axis,


M,= p fP -
S


p+)x dx dy = 2pc x dx dy
S


Substituting the value QP/x and integrating yield


3n
.3 2Z + 2 y 2co s y adr s
3 2 + 2 + a2 2a& cos aTj s
2


a2. -2 c (S,n)dg dq (3.21)


The following values are obtained for the coordinates of the center
of pressure:


x = ; YC = --
= '^


(3.22)


4. Examples

NACA comment: Errors in these examples are referred to and cor-
rected in the paper "Steady Vibrations of Wing of Circular Plan Form".


MY -c f
S-
%-=-,, sb


(3.20)








78 NACA TM 1324


The equations just obtained are presented again:

The velocity potential for z > 0 is determined by the equation



3(xyzj = -1- s CK(XJy,2A) + 1 x
*' (4.1)
2x % ,2 2 y2 z2 + R a2 2 2 cos dy dx
(x2 + y2 + z2 + a2 2ax cos y 2ay sin y)(C2 + n2 + a2 2at COE r e2a sin )J



where
2 Va-2 _2 24 a2 x2 2 2 + R
K(x,y,z,C,n) = -arc tanZ T2--
Vr -/2 ar (4.2)


R = (a2 x2 y2 z2)2 + 4a2z2 r =V/(x E)2 + (y 1)2 + z2

For the circulation distribution in the vortex strip formed behind
the wing,
1


ra2 2 ,2 2 y fr(,l)cos y dr dx d(d d.3)
(x2 + y? a" 2ax oE y 2ay sin y)(2 +q + a2 2aCg os r 2a' sin 4)
5 2
3v
__g7 2-- a i2 (_ 2 C(,)cos r Ba( L : sin rjY) d1 d d
fJJ J ( +2 2 2ag cos y 2an sin r) a sini 1 y J
S

where the plus sign is taken for y < a sin y and the minus sign for
y > a sin r

The following expression gives the lift force:
3n
P2pca 2 a2 -_2 -_ f(i,T)cos r dr dg dT
S= OC r(y)dy -2 2 + i2 + a2 a- 2a cos y 2an sin y
-a S

(4.4)

The usual expression for the induced resistance is
a a
W=- r(y) dr(y') 1 dy dy' (4.5)
4j dy' y y'
-a -a







NACA TM 1324


The coordinates of the center of pressure are determined by the
equations


My Mx
xc = -; =y -
P P


(4.6)


S2 = yry) 2 f(crd)1in r Cas y dr a& d'
J- 5 J JJ, e" *- + n 2 2aC C B r 2arT sin y
s y

my-a2 cos2 y dy a2 O
J J 2 ,2 a~ a e r 2zs sin r
s (


If y is set equal to -a cos e and F(y) is r(
form of a trigonometric series,


r(y) = Al sin 0 + A2 sin 20 + ... (0 <

P, W and Mx are directly expressed in terms of the c
this series by the formulas


:f((,rl)d d (4.8)




presented in the


S< tn)


(4.9)


!oefficients of


npca 1i
P = -- A ; W= ~ tp
n=l


nAn2


My = npca2A2


Finally, the shape of the wing is determined by the equation


x
l(x,y) = 1
0


f(x,y)dx g x + gl(y)
c


where


g(y) = -
'2 2 20 C dd
Vff "Va2 c2 2 fft(,j)cs r dy a dt d'I

J h x2 y2 + a 2 2 y2 + a2 Zax cos r 2ey sin y)(E2 + 92 2 2 a. o ro 2a' sin r)
S
2


where


(4.10)


(4.11)







NACA TM 1324


The examples are now considered.

1. First

f(x,y) = ca

where a is a small constant.
Polar coordinates are used and the following integral computed:

I 2 : a2 2 d T d-
fS J 2 + 12 + a2 2at cos y 2aT sin y
0


a2 2 p2 d dp J 2irp dp
a2 + p2 2ap cos(4 y) IO a2 p2


Substituting this value in equation (4.3) yields
3 T

P (y) = 2S3i cos r ( sin )
n -LVLa sin y yI
2

If the integral is tal.en,


r(y) = 4a + 2 2a(a y)


(a + y) log -I2- I (a -
A2a + ra y

Setting y = -a cos 0 and expanding
metric sine series in the interval 0 < 6
computations r

F(-a cos e) = 4 4+
it


(1 cos e) log


1 cos
2 (1
1+ cos e
2


= 2na (4.13)


- 11 dy


+ 2 2a(a + y) -


y) log yr (4.14)

r(-a cos 8) in a trigono-
< A give after simple

e 8
cos + 4 sin -
2 2

1 sin
+ cos e) log


= A sin 0 + A3 sin 30 + A5 sin 59 + ...


r 2n
o0


(0 4 4 it) (4.15)







NACA TM 1324


where

A. = .. A+1 ---- +- + ---
A1 l6aca A2k+1 4aca 1 1+ +
R2 A+ 2k(k + 1)(2k + 1) 3 5 4k+ 1

(k = 1, 2..) (4.16)

so that

A 16aca 496aca
A3 = A5 ...
45-2 4725n2

The distribution of the circulation obtained is very near that of
an elliptical distribution.

The lift force and the induced drag are obtained by application of
equations (4.10).


P = pcaA1 = 8 pa2c2a w 2.5465 pa2c2a
2 i
(4.17)
1 np(A12 + 3A32 + ...) 1.034 pa2c2a2


In order to determine the position of the center of pressure, My
must be computed by equation (4.8).
Equation (4.13) gives
4 M v
My = g pc23 Xc = -= a (4.18)
y = a XP 6

The distance from the center of pressure, which evi.de:tly lies
on the Ox axis, to the leading edge of the wing thus constitutes about
0.238 of the diameter of the wing.

In order to determine the shape of the wing corresponding to the
assumed function,it is necessary to form the function g(y) by equa-
tion (4.12). If equation (4.13) is considered,

35
a2ca -y cosy dr dx
g(y) = -
r 7 1 Vfx2+y2 a2(x2 +y2+a2- 2ax cos r- 2ay sin r)


(4.19)







82 NACA TM 1324


The computation shows that for x > a y2
3rY

cos y dy = x +
J x2 + y2 + a2 2ax cos y 2ay sin r 2a(x2 + y2)
2


y og + (y a)2
2a(x2 log + y2) 2
2a(x2 + y2) x2 + (y + a)2


x(a2 + x2 + y2)
a(x2 + y2)(x2 + y2 a2)


x2 + y2 a2
-arc tan
2ax


y = a cos ; a2 y2 = a sin e

2
o(e) = C- sin eg(-a cos 8)


for 0< e <
HOie) = n

Ho(0) =


sin ) At
Vt2-sin2e 2(t2+cos2e)


t(t2 + 1 + cos2e)
are
(t2 + cos2e)(t2 sin20)


Computation of this integral gives


cos 8
2(t2+cos2) log


t2 sin2 d
tan dt
2t


t2+4 cos4

t2+4 sin4 8
2


dy) sin 0 +


1 sin
- sin 8 log
8


1\2
1 + sin 0

1 sin 2/
2


+- sin (log
8


e
1 sin
cos log
2 1 + sin -
2


1 cos -
+ sin log 2
2 8
1 + cos-
2


(4.20)


(4.21)


HO(e) =


0\2
1 + cos -
--2 +
1 cos -
2


(4.22)








NACA TM 1324


The shape of the wing is thus determined by the equation


Y(x,y) =


81
S2r+


2 a2,

21c2.,a3 +y


2
,
n


log 2-


1
82


2a y
rrt2Ja -


are tan y
dy -
4[1- y2



g-/2a + /a y
r--



log V2- Va-y (4.23)
~/2~:;j4.~


This wing differs little from a plane wing inclined to the xy-plane
by a small angle a and may be obtained from such a plane wing by
twisting. 'The values of the function C(x,y) for the mean value
y = 0 and for the values y = a/2 are


t(x,o) = ax


arc tan y 1
Sdy -
11 y2 n-


- 0.8452 ax


(x1 a) 1


2
+ n


01


are tan 1 9
r_ tan ,dy-_ log2(2 + -3)
J1 y2 2n2


_ log23 +
872


2 log(2 + -3i) + -log 3 1 0.8335 ax
2 j2

It is of interest to consider what results for the obtained wing
are given by the usual theory. The circulation obtained by this theory
is denoted by ro(y); if the expansion of this circulation in a trigono-
metric series is


Fo(-a cos 0) = B1 sin e + B2 sin 20 + ...


(0 < 0 < n) (4.24)


then the usual theory gives an equation for determining the coefficients
Bn, which in the case considered reduces to the form


log2(q-f + 1) +


*I








84 NACA TM 1324


. Bn sin ne = 2nca sin e g(-a cos e) 1 Bn
n= c 4ca 1n,


sin ne
sin ef (4.25)


Equation (4.21) yields


SBn(l + ) sin ne = 2ncaa sin e 2aac HO() (4.26)
n=l

Expansion of the function H0(8) into a trigonometric series is
sufficient to determine the coefficients Bn. Despite the complicated
form of the function HO(8), it can be expanded and in the interval
0 < 0 t A


HO(O) = sin 8


sin(2k + 1)9 -
k(k + 1) 1


H.(e) =
k=0


01 = 0.1389 i

07 = 0.0460 ;


1 tan y
-2 f are tan y ) +
o 1 y 2 +


1 1 2(2k + 1)2 + 1 ,
+-1 4k + (k + )(4k + (4.27)
3 4k + 1 (4k + 1)(4k + 3)]


52k+l sin(2k + 1)8


03 = 0.5048 135 = 0.1213


39 = 0.0212, ...


Equation (4.26) shows that

4a'ca(n2 P1i)
B1 = -- B2k = 0
n(n + 2)


4Lcag2k+l
B2k+1 = (k =
2 + r(2k + 1)


1, 2, ...)

(4.28)


The numerical values of the first coefficients will be


B1 = 2.4784 aca ; B3 = 0.0562 aca ; B5 = 0.0087 aca


B7 = 0.0024 aca ; B9 = 0.0009 aca, ...


k=l


that is,


where








NACA TM 1324


The following value is obtained for the lift force:


PO = npcaBl = 3.8932 pc2a2a

exceeding the accurate value by 53 percent.

For the induced drag,

WO a 2.416 pa2c2a2

with an error of 134 percent.

2. If a is assumed to be small, f(x,y) = 2cax is taken.


The circulation r(y) is computed. First the
lowing integral is found.


SFva2 2 -2 dc di
S2r2 + 2+ a2 2ac cosy 2aq sin r

Equation (4.3) gives
3"
8ca2 2 a( sin r)
r(y) = cos2y
3In gVIn vi vl


value of the fol-


4
-- na2cos y
3


- 1 dy


The computation of this integral leads to the very simple expression


r(y) = 2ca(a2 y2)


(4.32)


Thus in the case considered, a parabolic distribution of the circu-
lation was obtained. For this reason the computation of the forces can
be easily carried out:


8
P = pc | r(y)dy = opc2a3 = 2.667 apc2a3
-a


(4.33)


W = pc2z2a4 = 1.2732 pc2a4a2
11


(4.29)


(4.30)


(4.31)







86 NACA TM 1524

Equation (4.31) is used in the computation of My by equation (4.8):

128 2 4 1.5O9pc2 4M, x 16
My 128 pc a4 1.509 pcaa i xc = My a (4.54)
27n P 9A

In order to determine the shape of the wing it is necessary to
compute the function g(y); equation (4.12) yields
3x
3 2nVa 2 3 2
g(y) = 4aca p cos r dy dx
3n2 + J x2 +y2 a2(x2+ y2+a2- 2ax cos r- 2ay sin r)
2

Setting

3x2
H1(e) = sin eg(-a cos e) (0 4 9 t) (4.35)
4aca
and carrying out the integration with respect to y yield
sin
HI(0) =s nsin (t2+ cos28) +
(t2 +cos20)2 ot2 sin2)

1 t2+ 4 cos4 8
S(cos2 t2)(t2+ 1+ cos20)- t cos 0(t2+ + cos2e)log 4
4 2 t2+ 4 sin4
2

2(t2 cos2e)2+ (t2- cos2a)[1+ (t2+ cos2e)2] t2- sin29e .
+---- ------ arc tan ------- dt
2(t2- sin2e) 2t J

Integration yields

I = 3 n i e 1 i + cos ) (1 + sin )
(n) i 0 sin cos ) + log +
2 2 2 12 (1 Cos sin 8\

1 + cos c 1- sin i
sin e cos e 1Dg tan- + log 2 (4.36)
2 cos ) + sin

In equation (4.11), the following is taken:

gl(y) = a(a2 y2)







NACA TM 1324


Then for the function ((x,y), which determines the shape of the wing,
the following expression is obtained:


t(x,y) = a(a2 x2 y2) +


Ara + y /a y
- -V +
2a V 2a


Y logal + y
a Va y


1 (l
-log (
12 (V2a


+ /a- -y)(-/E2 + -Va + y)
-%fa --y)(-Ma -yna + )


Y log ( + /--y)(-V2 -/a Ty)
4a (-v'2a -%/ 5(v2a + -JVs7y) j


(4.37)


This wing is thus obtained as a deformation of the wing:


(x,y) = a(a2 x2 y2)

which for small c differs little from a segment of a sphere.

In particular, for y = 0,


((x,0) =a(a2 x2) + 2a [1-F2 log(fl+ i) a(a2- x2-0.0767ax)


In order to apply the general theory to the
is-expanded into a trigonometric series:


Hi(C) =


obtained wing


log tan dx sin +
2


sin(2k + 1)8
4k(k + l)(2k 1)(2k + 3)


- 12nk(k + 1) +


2(16k2 + 16k 3) 1 4 + 1... + 1
3 54k + 1


= ZT2k+lsin(2k + 1)0
k=0


+ 6(2k + 1)


(4.38)


Tl = 0.6931 ; T3 = 0.1783 ; Y5 = 0.0812


77 = 0.0463 ; y9 = 0.0300, ...


H1(e)


k-l


Where







NACA TM 1324


For the case considered, the usual theory gives for the determination
of the circulation


ro(-a cos 8) = B1 sin e + B2 sin 20 + ...


the equation

Bn sin nB = Zca sin (aa sin 0 g(-a cos e)
n=1 c


(04 e it)


4ca n=l
4ca n=i


sin n9
nBn, (
sin e8

(4.39)


Equation (4.35) and


sin20 = -


8 sin(2k + 1)G
" k=0 (2k 1)(2k + l)(2k + 3)


(O < e < it)


(4.40)


give from equation (4.39) the equation


Bn(l + )


16aa2c
0 -sin(2k + 1)8
2k+ (2k )(2k + 1)(2k + 3) +
(4.41)


from which without difficulty Bn is obtained, in particular


B2k = 0 ; BI = 1.8457 aca2 ; B3 = 0.2132 aca2,


B5 = 0.0250 aca2 ; B7 = 0.0075 aca2 ; B9 = 0.0032 aca2,

The following value is obtained for the lift force:


P= I pcaB1 = 2.899 ac2a3p
2


(4.42)


exceeding the accurate value by 8.7 percent.

The induced drag


W = 1.3927 pa2c2a4


(4.43)


exceeds the accurate value by 9.4 percent.


n=1


[ SGLan2c
k=O




w


NACA TM 1324


3. In order to give an example of a nonsymmetrical wing,


f(x,y) = acy

In this case it is first necessary to compute the integral


In acc/unt a2+ 2 _
(J 2 + T2+ ,2 2aC cos

On account of equation (4.3),


sin y cos


31

4aca2
T(y) = -2
3n
2


2 -


{ d- 4= 2A 2sin r (4.44)
y 2ai sin y 3


r a(l1 sin y) d
a sin y y J


After computing the integral,


r (y)aylgJ
r(y) = -- a + y) F2a(a y) (a y) 2a(a + y) +


1 (a + y)(a 3y) log ,f
6 -%z + -'a y


1 (a y)(a + 3y) log -V a~
6 -.,2a + -%/a +y


is obtained.


Assuming y


r(-a cos e)


= -a cos 0 and expanding in a trigonometric series

aca2 F 0 0
=a- 2(1 cos )cos -2(1 + cos e)sin +
L 2 2
9


1 (1 cos 0)(1



S(1 + cos e)(1
6


1 cos -
+ 3 cos 0) log 2
8
1 + cos -
2
1 sin e
- 3 cos e) log 2
1 + sin -
2


=A2 sin 29 + A4 sin 4e + ... (0 4 0 4 A)


give


(4.45)


c


(4.46)







NACA TM 1324


where


4aca
A2k= 4a2_
t2 6k(k2


8k2 + 1
- 1)(4k2


128 aca2
A2 =
2732

+
- 1+-+ ...
- 1) 5

(k = 2,3, ...)


+ 4k
4k 1)


(4.47)
2k
(k2 1)(4k2 1)J


so that


A2 = 0.4803 aca2 ; A4 = 0.00549 aca2


A6 = 0.00234 aca2 ; A8 = 0.00123 aca2

Evidently there is no lift force, whereas for the induced drag
the following value is obtained:


W = 0.1813 pa2c2a4


(4.48)


The moment of the forces about the Ox axis is:,


MX = 1 pca2A2 = 0.3772 pac2a4
4x -


(4.49)


The moment
aid of equation
and it is found


of the forces about the Oy axis is computed with the
(4.8), where use must be made of the result (4.44),
that


M =0


(4.50)


The following function is now computed:


2a3ac
g(y) -
312


sin y cos y dy dx

xx2+ y2 a2(x2+y2 +a2 2ax cos y- 2ay sin r)


Setting


3t2
H2(0) = sin eg(-a cos 8)
2aac


(04 9 4 3)


(4.51)




w w


NACA TM 1324


and carrying out the integration with respect to y give


H2() Isin 9
H2(e) = f


sin 8 cos e(t2 + cos2e) +
(t2 + cos2e)2 4t2 sin2


St cos e(t2 + 1 + cos2e) +
2


1 (t2 + 1 + cos2)(cos2 t2)
4


t cos elb + (t2 + cos28)2]
t2 -arc
t2 s in26


= sin 8 cos 8




sin e(1 3 cos )
16


t2 + 4 cos4
2
log -
t2 + 4 sin4 2
2


tan t2 sin29 dt
2t


arc tan y
V y2


3dy
dy--4


2 log2(2 + 1) +3
I


(og


1+ 3 cos 9
sin
2 2


1 + cos -
2o
1 cos -
2


8e
1 + cos
2
log
81 -
1 cos -
2


Ssin 8(1+3 cos 8) (,lo
16



1 5 cos 9
cos log
2 2


(4.52)





1 + sin e
g 2


0
1 sin


1 + sin -
2
1 sin -
2


Expansion in a trigonometric series gives


E2(e) = sin 2e [ log2(-2 + 1)


(8k2 + 1) 1
(4k2 1)(4k2 4) 3


5
+-


arc tan y
i -2 dy


912
16


2159
630 -


1 12k2
+ ... + 1- ) sin 2k
4k 1 8k2 + 1


= 52k sin 2k8
l=1


(4.53)


F=2







NACA TM 1324


where

52 0.27412 ; B4 = 0.08127 ; b6 = 0.05198 j; 8 = 0.03641, ...

The usual theory for determining the circulation


rn(-a cos e) = / Bn sin ne
n=l

gives the equation


Bn(l + sin ne = 2 ca sin e -a cos 8 a cos
n=l1
4acca
= ica2 sin 28 3- H2(8) (4.54)


from which without difficulty


B2k+l = 0 (k = 0,1,2, ...)

B2=- 0.7304 aca2 ; B4=0.0047 aca2 ; B6=0.0021 aca2 ; B8 =0.0011 aca2,...

The lift force is found equal to zero and the induced drag and
moment of the forces about the Ox axis are


W = 0.4191 p,2c2a4 ; MI = 0.5737 pa2c2a4 (4.55)

The first gives an error of 131 percent, the second of 52 percent.

By a combination of the obtained solutions it would have been
possible to obtain further examples. From the examples given it is
clear that for the case of a circular wing considerable deviations are
obtained between the usual and the exact theories.

Translated by S. Reiss
National Advisory, Committee
for Aeronautics
REFERENCES
IIf
1. Kinner, W.: Die Kreisformige Tragflache auf potential theoretiscner
Grundlage. Ingenieur Archiv, Bd. VIII, 1937, pp. 47-80.

2. Sommerfeld, A.: Uber verzweigte Potentiale im Raum. 'Proc. Lond.
Math. Soc., 1897, pp. 395-429.


*.I .







NACA TM 1324


Figure 1.


D' D


Figure 2.


NACA-Langley 1-5-53 1000




:1







I-i .2 ci
1) C M u



a. a l

S a
?S ar5 OTJ


- i @4 0 6 U rH


-* 2 -
SQ ~ c M, M, '3
-3n | 0 3
5 o^>*3 E ".^.Ss s
50~"~0
0. LC -


o, u. 0U1
a g c a

0 s S .
El^ cis g '0WCC0
3 U ~ ~ -0 M u T N


0 ~ ~ Mr 4 ru
0OaOO~ 1 W
^ Iyill2^ Ipi" D



E r a i i | >p Li -: c
o z > Z d -0 v 0,


U>CI4S i sC QuC u
0 n 3 a !, 0


-S3 a o




4 1 eyy 'Ccw -cg o. a. .
c"-> EE 2U>
L CL rd Cd bpO
u C;6



,Z z 4 5 A. o"d55 0 ciE u


- C.4, 0

a- -s 0.
1 0 ) 0 "

-- ;%
mC

S. 5 a >d d z s

M n q* n _; i


j" S. ^-
c


So NOa,
33~
^SlE"6^0-"3
W. :.% f .
0" 9 E

0
5* a 5: ~" Ci c -


0 MO
8 a C -
ISP25 IS
""'
o C) a C





I- S5 I C -a.

Z z In a. 0
"Scs S^1 ^ "^
a5>ii1l 3
,o ^I I
i S i"^-
,ra iS:z ai


_- I




u j u ac 00
E L. -5 .c -5
- 5 a -5
C

MCs 9-aa i 5
^^^ *0







E .5 o ag
0 00







F. E sa 6 E g,
aMt 1< '



C mr

a.
-a 2 -- 0 i


"t % 5 a
B 4 1 .C
5sQ 5I


A a N1 a'a


O. a -a' a & a
a Ec i *c
Ogja S-"
E-B 10 Q BL


3.



a
z .
w |z p
5Si -

lg &


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