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

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a. a l

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5 o^>*3 E ".^.Ss s

50~"~0

0. LC -

o, u. 0U1

a g c a

0 s S .

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3 U ~ ~ -0 M u T N

0 ~ ~ Mr 4 ru

0OaOO~ 1 W

^ Iyill2^ Ipi" D

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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

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