Theory of Wing Sections: Including a Summary of Airfoil Data

mx

t t = - = -2- - ­

ax

cP = m In

2r

2r(x

+ y2)

vx + y2 2

(2.20)

SlJfPLE TWO-DIMENSIOlfAL FWWS

41

From Eqs. (2.19) and (2.20) it is seen that the streamlines for a source situated at the origin are radial lines and that the equipotential lines are circles about the origin. c. Doublets. A doublet is the limiting case of the flow about a single source and a sink of equal strength that approach each other in such a manner that the product of the source strength and the distance between the source and sink remains constant. The axis of the doublet is the line joining the source andsink. To obtain expressions for the flow about a doubtlet, consider the arrangement of "a source and a sink shown in Fig. 26. The source and sink are situated on y the x axis at - 8 and 8, respectively. The flow resulting from the combi­ nation is

1/1 = !!!.(tan-1 ~ 2r

x

+8-

tan-1 _ Y- ) X -

8

Since

a-b

tan:" a - tarr? b = tan-1 - ­ I +ab we have l/I = m tarr:' - 2ys 2r x 2 + y1. - 8 2

FlO. 26. Streamlines of flow for a source and sink.

For values of s small with respect to x and y,

1f = If the product of m and

8

;:

C-; ~y:~)

remains constant as

8

approaches zero, (2.21)

where p, = 2ms d. Circular Cylinder in a Uniform Stream. The flow about a circular cylinder in a uniform stream is obtained by superposing a uniform stream [Eq. (2.17)] on the flow about a doublet [Eq. (2.21)] with the uniform stream flowing from the source to the sink.

1f =

Y( IT -

21r(x:+

if»)

This equation may be written as follows in terms of polar coordinates: '" = Y r

where

a2

r2

(1 - ~) sin 8 = JJ/21r l'

= x 2 + y2

(2.22)

THEORY OF WING SECTIONS

42

It is obvious from Eq. (2.22) that part of the streamline'" == 0 is the circle r = a. At large distances from the origin the flow is uniform and parallel to the x axis. Equation (2.22) therefore represents the flow about a circular cylinder in a uniform stream. The streamlines for this flow pattern are shown in Fig. 27. The velocity distribution about the cylinder may be obtained by finding the tangential component of velocity on the circle T == a. 11

= - : = - V ( 1 + ~) sin (J = - 2V sin (J

(2.23)

i FIG.

27.

Streamlines for the flow about a circular cylinder in a uniform stream.

Because the circle r == a is a streamline, \ve may apply Bernoulli's equation (2.5) to obtain the pressure distribution about the cylinder. p

+ ~2P(4yt sin! 6) == H

If we define a pressure coefficient S as S=

H-p Mp VJ

(2.24)

the distribution of S over the surface of the cylinder is given by

S == 4 sin 2 8

(2.25)

Integration of the pressure distribution over the surface of the cylinder will show that the resultant force is zero.

e. Voriex. A vortex is a flow pattern in which the elements of fluid follow circular paths about a point, and the flow is irrotational at all points except the center. The equations defining the flow pattern may be derived directly from Laplace's equation for the stream function. Equation (2.16) may be written 88 follows in terms of polar coordinatesr" (2.26)

8IMPLE TWO-DIMEN8IONAL FWW8

43

Because the radial component of velocity is zero,

!ol/l = 0 ro8

and hence

lo~ =0

r afP Equation (2.26) therefore becomes

lPt/t+!d1/l=O dr

rdr

Integrating once with respect to r we obtain

d

ln 1/l + 1n r = c dr

or

d1/l

dr =

I' rc' = 2111 == -

,

v

(2.27)

where the constant of integration c' is assigned the value I' /21r. The veloc­ ity therefore varies inversely with the disy tance from the center of the vortex. The circulation I' about the vortex is equal to - 2rrv'. The circulation is considered to be positive in the clockwise direction, whereas v' is positive in the counterclockwise direction. The stream function may be obtained by in­ tegration of Eq. (2.27)

r

1/1 == 211" In r

+ e"

(2.28)

x

The pressure in the field of flow may be FIG. 28. Equilibrium conditions for a vortex. calculated from the condition that the pressure gradients must maintain the fluid elements in equilibrium with the acceler­ ations (d'Alembert's principle). The centrifugal force acting on each ele­ ment must be (Fig. 28) pv'" --rd8dr r The pressure force acting on the element is

Equating the pressure and acceleration forces, we have pV'2

r=

dp

dr

44

THEORY OF WIlvG SECTION8

Substituting the value of v' from Eq. (2.27) gives p~

47N8

p= -

=

dp dr

p~

srr+ H

where H = constant of integration This equation may be written in the form P+~pV'2==H

which shows that Bernoulli's equation may be applied throughout the field of flow,

FIG. 29. Streamlines for the flow about a circular cylinder in a uniform stream with eireu­ lation corresponding to a wing-eection lift coefficient of 0.6.

f. Circular Cylinder with Circulation. The flow pattern represented by a circular cylinder with circulation is the basic flow pattern from which the flow about "ring sections of arbitrary shape at various angles of attack is calculated. Such & flow pattern is obtained by superposing the flow pro­ duced by a point vortex upon the flow about a circular cylinder. Adding the flows corresponding to Eqs. (2.22) and (2.28), we obtain '" -=

vr(l - a~ sin 8 + 2r~ In !a ~J

(2.29)

where the constant e" in Eq. (2.28) has been taken equal to

-r 1n a 21r

A typical flow pattern for a moderate value of the circulation r is given in Fig. 29. The velocity distribution about the cylinder is found by differentiating the expression for the stream function [Eq. (2.29)] as follows:

(ar. 2

&/I r - = V 1+-) sin8+­

ar

2rr

SIMPLE TTflO-DIMENSIONAL FWWS

45

The tangential component of velocity v' (positive counterclockwise) at the surface of the cylinder is obtained from Eq. (2.9) and the substitution of r == a. Vi == -

2V sin 8 + ~

(2.30)

21ra

It is seen that the addition of the circulation I' moves the points of zero velocity (stagnation points) from the positions 8 = 0 and 11" to the positions 8 = sin-1 -

r­

4raV

The pressure distribution about the cylinder may be found by applying Bernoulli's equation (2.5) along the streamline 1/1 = o. p

+ 21 P (4 V2 SIll .

2

8

-

2Vr sin 8 +

ra

r

2 )

4ra2 ==

H

(2.31)

Setting

r

21Ta V = K

(2.32)

the pressure coefficient S [Eq. (2.24)] becomes S == 4 sin 2 8 - 4K sin 8 + K2

(2.33)

The symmetry of Eq. (2.33) about the line 8 = 1f/2 shows that there can be no drag force. The lift on the cylinder can be obtained by integration, over the surface, of the components of pressure normal to the stream.

fo2rSa sin 0 dO = 72PV fo2r (4a sin" 0 -

l = 72P V2

2

4aK sin" 9 + aJ(2 sin 8) dO

l == 72PV2ak[28 - sin 28]~r l == 2pV 2akr == PlTr

It can be shown that the relation l == pvr

is valid regardless of the shape of the body. 61

(2.34)

CHAPTER 3

THEORY OF WING SECTIONS OF FINITE TIDCKNESS

3.1. Symbols. A., B. H 8 V a

coefficients of the transformation from x' to z total pressure pressure coefficient (H ­ p)/J-ipVt velocity of free stream radius of circular cylinder Cr section lift coefficient e base of Naperian logarithms, 2.71828 i V=-i In logarithm to the base e m source strength per unit length

(~J

- (is

p

1-

p

local static pressure radius vector of z (modulus) component of velocity along x axis component of velocity along y axis local velocity at any point on the surface of the wing section complex variable Cartesian coordinate real part of the complex variable % or Cartesian coordinate magnitude of the imaginary part of the complex variable z complex variable complex variable in the near-circle plane complex variable for the flow about a circle whose center is shifted from the center of coordinates circulation, positive clockwise section angle of attack distance of the center of the circle from the center of coordinates complex variable mngnitude of the imaginary part of the complex variable r angular coordinate of the complex variable z (argument) angular coordinate of z' (see Fig. 32) aeA is the radius vector of z doublet strength per unit length real part of the complex variable r ratio of the circumference of a circle to its diameter mass density of air real part of the complex variable w potential function 46

r u

v v

w x x

y y z

z' z*

r ao E

r " 8 fJ

~ p.

~ 11'

p

q, q,

r

THEORY OF WING SECTIONS OF FINITE THICKNESS

" 1/1 1/1 1/1 1/10

angular coordinate of z magnitude of the imaginary part of the complex variable w stream function ae~ is the radius vector of z' (see Fig. 32) average value of 1/1

00

infinity

47

3.2. Introduction. It was shown in Chap. 2 that the field of flow about a circular cylinder with circulation in a uniform stream is known. It is possible to relate this field of flow to that about an arbitrary "ring section by means of conformal mapping. In relating these fields of flow, the circulation is selected to satisfy the Kutta condition that the velocity at the trailing edge of the section must be finite. Such characteristics as the lift and pressure distribution may then be determined from the known flow about the circular cylinder. The resulting theory permits the ap­ proximate calculation of the angle of zero lift, the moment coefficient, the pressure distribution, and the field of flow about the section under con­ ditions where the flow adheres closely to the surface. 3.3. Complex Variables. Conformal mapping, which makes the cal­ culation of'wing-section characteristics possible, depends on the use of complex variables. A complex number is a number composed of two parts, a real part and a so-called imaginary part. The real part is just an ordinary number that may have any value. The imaginary part contains the factor v=!, which is given the symbol i. A complex variable z may therefore be written in the form

z = x+ iy where the symbols in the expression x + iy obey all the usual rules of algebra. It should be remembered that~"2 is equal to - 1. Such a variable z can be represented conveniently by plotting the real part z as the abscissa" of a point and the magnitude y of the imaginary part as the ordinate. By De Moivre's theorem, z may also be written in the form

z

=

reiIJ

where rand 8 are the polar coordinates.

Let us consider the complex variable

w=+il/l and let w

=

f(z) , that is,

4> + il/l

=

f(x

+ iy)

Then

:2q,2 + i:~.. = /:"r(x + iy) == f'(z) uX u~t

and cP (j2 .a2;p ~ ( +.) f"( ) iJ 2 + ~ a y 2 = JJl1J X ~y = z y

THEORY OF WING SECTIONS

48 adding, we have

02
ax2

=

0

In any equation involving complex variables, the real and imaginary parts must be equal to each other independently. Therefore

iJ2
a*lq,

or + iJ y

2

=

0

and

iJ~+ay=o

ar

iJlI

Because these equations are the same as those of Eqs. (2.15) and (2.16), any differentiable function w = j(z) where and

i = x+ iy

may be interpreted as a possible case of irrotational fluid motion by giving the meaning of velocity potential and stream function, respectively. The derivative dw/dz has a simple meaning in terms of the velocities in the field of flow. dw = d+ i a1/l dz = dx + i dy Further a d
and'"

ax

ay a." dl/l = - dx + - dy ax ay 81/1

Therefore [(iJt/>/ax)dx

dw

dz

+ (o(j)loll)dy] + i [(iJ~/iJx)dx + (iJ1/I/iJy)dyJ d.x+ i dy

=

In order for dw/dz to have a definite meaning, it is necessary that the value of du: 'dz be independent of the manner with which dz approaches zero. If dy is assumed to be zero, the value of the differential quotient dw/dz is o
ax

a~l~

= dw dz

(3.])

according to Eqs. (.2.8) and (2.12)./ Similarly, if dx is assumed to be zero, the value of the differential Quotient duridz is

~ iJq, + 0'" = _ iv + 'It toy

ay

THEORY OF WING SECTIONS

OJl~

FINITE THICKNESS

49

The expressions for simple two-dimensional flows given in Sec. 2.6 may be expressed conveniently in terms of complex variables. Uniform stream parallel to x axis to

= Vz

(3.2)

Source at the origin w=m ln z 2r

(3.3)

Doublet at origin with axis along x axis w=~

(3.4)

2rz

Circular cylinder of radius a in a uniform stream

w=V(Z+:2)

(3.5)

Vortex at origin 10 =

ir -In z

2r

(3.6)

Circular cylinder with circulation w = V (z +

a2)ir -z + -21r In -az

(3.7)

3.4. Conformal Transformations. .1\ conformal transformation con­ sists in mapping a region of one plane on another plane in such a manner that the detailed shape of infinitesimal elements of area is not changed. This restriction does not mean that the shape of finite areas cannot be considerably altered. It has been ShO"·11 previously that the equipotential lines and streamlines intersect at right angles, thus dividing the field of flow into a large number of small rectangles. It has also been shown that the equation w = j(z)

represents a possible flow pattern. The equation w = g(r)

represents another flow pattern where ~

r is the complex variable

+ i11

The coordinates in the z plane are considered. to be x and y, and those in the r plane are ~ and 'I. If the equipotential lines and streamlines are plotted in either of the planes, they will divide the plane into a large num­ ber of small rectangles. These rectangles will be similar at .corresponding points in both planes. The corresponding points are found from the relation f(z) = g(t)

THEORY OF' WING SECTIONS

50

This equation accordingly represents the conformal transformation from the z plane to the r plane or the converse. In practical use, the Ho\v function in the z plane is known and the corresponding flow function in the r plane is desired. To plot the flow known on the z plane on the r plane, it is necessary to solve this relation for r and to obtain an equation in the form

r == h(z)

In the transformations we shall consider, this relation will be given, The velocities in the z plane, by Eq. (3.1), are dw _ dz ==u-~v The corresponding velocities in the dw

r plane are dwdz

(3.8)

dt == liz dr

As a simple example of a conformal transformation, consider the re­ lations

W=V(Z+~)=Vt These relations transform the flow about a circular cylinder on the z plane [Eq. (3.5)] to uniform flow parallel to the ~ axis on the plane [Eq. (3.2)]. Corresponding points of both planes are obtained from the relation

r

r == z+­atz " and are indicated in Fig. 30. 3.5. Transformation of a Circle into a Wing Section. A circle can be transformed into a shape resembling that of a wing section by substitution of the variable

a2

t==z+­z

(3.9)

into the expression for the flow about a circular cylinder having a radius slightly larger than a, and so placed that the circumference passes through the point x == a. If, in addition, the center of the larger cylinder is placed on the x axis, the transformed curve will be that of a symmetrical wing section (Fig. 31). In the present example, let the center of the larger cylinder be placed at the point x = - E where E is a real quantity. The radius of this cylinder will then be a + E. The equation of flow about the larger cylinder with circulation is then 10

==

V(z* +

E

+ (a + E)2) + Z*+E

ir In z* + E 211" a+E

THEORY OF WIl{G SECTIONS OF FINITE THICKNES8

- - - - - - - ----4~-- + - - ~

51

--------­

Z PLANE

I

{=Z+-j-

2

I

.I;

( PLANE

FIG.

30. Conformal transformation of the flow about a circular cylinder to uniform flow.

FIo. 31. Conformal transformation of a circle into a symmetrical wing section.

THEOR1' OF WING SECTIONS

52

The more general expression for the flow about the circular cylinder with the flow at infinity inclined at an angle aD to the oX axis is found by substitut­ ing the expression Z + E = (z* + E)eiaro 1»

= V

[(Z + E)e-iao + (a ~~):eiao] + ;~ In (z ~ 2eE- iao

(3.10)

Substitution of Eq. (3.9) into Eq. (3.10) would result in the equation for the flow about the wing section but would lead to a complicated ex­ pression. A simple way of obtaining the shape of the 'ling section is to select values of z corresponding to points on the larger cylinder and find the corresponding points in the r plane by the use of Eq. (3.9). The velocity of any point on the wing section can be found from Eq. (3.8). dw

dr

=

[v (e-

dw dz = dz dr

Wzo _

(a

e

+ E)2

iae )

(z + E)2

+

ir

] (~) z~ - a2

2r(z + E)

It can be seen from this equation that the velocity at the point z = a is infinite unless the first factor is zero. The point z = a corresponds to the trailing edge of the wing section. The Kutta-Joukowsky condition states that the value of the circulation is such as to make the first factor equal to zero, which is the condition that ensures smooth flow at the trailing edge. The value of the circulation satisfying this condition is found as follows: V + E)2eiaol + ir - 0 iao _ . (a + E)2 J 2"..(a + E) ­

[e-

(a

ir = 211'"(a + E) V(e iao - e- iao) Since

eiao - e- iao • • •• 2 == Sinh tao == t sin ao or

I'

== 4"..(a + E)V sin

Qo

The leading edge of the wing section corresponds to the point a2

r=-a-2E­ a + 2E neglecting powers of

E

greater than one

r=

- 2a

Because the trailing edge corresponds to the point

r == 2a

THEORY OF WING

8ECTIO~V8

the chord of the wing section is 4a. The lift on the wing section is and the lift coefficient is Ct = 211" ( 1

53

OF FINITE THICKNESS p

vr, .

+ :) sin «XcI

In the limiting case as E/a approaches zero, the slope of the lift curve dc,/da o is 2...per radian for small angles of attack. Detailed computations"

z-Pkne 1------....------.

z~Plme t - - - - - - f ! I i o - - - - - I

~=x+iy 32. Dlustration of transformations used to derive airfoils and calculate pressure dis­ tributions.

FIG.

will show that the thickness ratio of the wing section is nearly equal to (3V3/4) (E/a). For wing sections approximately 12 per cent thick, the

theoretical lift-curve slope is about nine per cent greater than its limiting value for thin sections. 3.6. Flow about Arbitrary Wing Sections. vious section that the transformation a2

s=z+­z

It was seen in the pre­

THEORY OF WING SECTIONS

54

transforms a circle in the z plane into a curve resembling a wing section in the plane. Most wing sections have a general resemblance to each other. If the aforementioned transformation is applied to a wing section, the resulting curve in the z plane will therefore be nearly circular in shape. Theodorsen recognized this fact and showed that the flow about the nearly circular curve, and hence about the wing section, can be derived from the flow about the true circle by a rapidly converging process. The basic method is presented in reference 122, and a detailed discussion of the method is given in reference 125. The derivation of Theodorsen's relation for the velocity distribution about the wing is divided into three parts (Fig. 32). 1. Derivation of relations between the flow in the plane of the wing section (r plane) and in the plane of the near circle (z' plane). 2. Derivation of relations between the flow in the z plane and in the plane of the true circle (z plane). 3. Combination of the foregoing relations to obtain the final expression for the velocity distribution in the r plane in terms of the ordinates of the wing section. The basic relation between the z' plane and the r plane is

r

r = r + --;~z The coordinates of

(3.11)

r are defined by the relation f=x+iy

and the coordinates of z' are defined by the relation

z' == tJt!I'+#} By Eq. (3.11),

Since e#}

and since

= cos () + i

sin

(J

r = a(~ + e~) cos 8 + ia(e1' - e-') sin () == 2a cosh '" cos (J + 2ia sinh '" sin 8 x y

t == x+iy = 2a cosh 1J' cos

= 2a

8}

sinh '" sin ()

(3.12)

Expressions for'" and fJ in terms of x and y are desired and may be obtained as follows: x cosh'" == 2a cos 8 sinh '"

= 2a SID ~

6

,

I

THEORY OF WING SECTIONS OF FINITE THICKNESS

55

Since l

cosh

l

sinh

'" -

'"

=1

(2a :OS S- (2a ~in 8Y= 1

!;

!

or

2 sin 8 = P + l

~r + (~y

(3.13)

where

Because sin 8, and hence cos 8, are known in terms of the ordinates x and y of the wing section, the values of sinh'" and cosh '" can be found from Eq. (3.12). The factor relating velocities in the z' plane to those in the r plane is dz'/dr. From Eq. (3.11), dr

dz'

=

1_

a2 = ,!(z, _ ~)

Z'2

= -= =

Z'

.!(aeY+ iB -

z'

~ a(eI' -

z

z~

Z'

ae-+- iB)

rI') cos 8 + ia(e" + e-I') sin 8

(2a sinh '" cos 8 + 2ia cosh '" sin 8)

(3.14)

The second step is to find the relation between the flows in the z' plane and those in the z plane. The coordinates of z are defined by the relation z = aeA+;'" The transformation relating the z' to the z plane is the general transforma­ tion

i; (A.+iB.)(l/p)

z'=ze 1 By definition Consequently

or

co

'"- }. + i«(J -

q» = "\'

4

(An + iBn)..!. (cos rIO

1

where z has been expressed in polar form z = r(cos

q>

+i

sin

q»

nq> -

i sin

nq»

THEORY OF WING SECTIONS

56

Equating the real and imaginary parts, we obtain the two Fourier ex­ pansions

~(A" 1/1 - ). = L, -;:;- cos

n" + B..) r" sm n"

1

and 00

(J-qJ=

1(Bn

(3.15)

A".)

-cosntp--Slnn'l' rn rn

1

These relations show that l/I - ~ and 8 - tp are conjugate functions. In order for the deviation of the near circle from the true circle to be a minimum, the value of ~ corresponding to the radius of the true circle is taken to be

1 2

1/10

). =

= -1

21r

.­1/1

0

df)

where the values of 1/1 correspond to points on the near circle. The required transformation from the near circle to the true circle is then (3.16)

If 1f/ is known as a function of tp, 8 - I{) can be found by a method de­ veloped by Naiman." The 1/1 function is assumed to be approximated by a finite trigonometric series of the form t/I(tp)

=

1/10 + Al cos

I{)

+ .. · + An-I cos (n - 1)1{) + A" cos

1

(n - l)cp

n-l

1/I(rp)

= 1/10 +

.--1

(A", cos m" + B", sin mrp) + A" cos

~

If '" is specified at 2n equally spaced intervals in the range 0 ~ cp ~21r: that is, 0, 1r/n, 2r/n, ...., [(2n - l)1r]/n then 2n-l

\11o

where 1/Ir = value of.p at

f/' =

= 2~

1

\IIr

r-=O

T7r/n 2n-l

.4", =

! \" \IIr cos m!:! nL, n rlCO

211-1

B", =

! " \IIr sin m nr nL, n r-O

A" =

~

L(-

2n-l

r-=O

l)r\llr

B.=O

THEORY OF WING SECTIONS OF FINITE THICKNESS

57

Now, by Eqs. (3.15),

2: 1I-l( 2: 2: "'r 11-1

f{J -

(J

= E(Ip) =

(A.. sin nkp - B m cos mlp) + A .. sin nip

m-l

2n-l

1.

= 17:

mf{J

SIll

m:=1

cos m

r=O

rT

n

2n-l

- cos mlp~ "'r sin m r; 11-1 2n-l

=

~

LL

2n-l

)

2~"'t sin nf{J ~ (- 1)r"'r

+

2n-l

"'r sin m(f{J -

r:) + 2~ sin nf{J 2:(- 1)r"'r

m=l r=O

r=-O

Upon interchanging the order of summation, there is obtained 2n-l

E(Ip) =

~

.-1

2: 2: "'r

r-O'

r:) + 2~ sin nip 2: (- 1)r"'r 2n-l

sin m(f{J -

m~l

r==O

If E is evaluated at the same points qJ at which 1f is given, that is, at the points qJ = r'r/n, the variable qJ - (r7T/n) becomes [(r' - r)r]/n = - (Kr/n), and the last term becomes zero.

"'n:

1fr = rr =,p

("r n + nK 1r) = 1/1 (~ + Knr) ==.pK 2n-1

E(Ip) = -

11-1

1~

~. n~ "'K ~ sm m n K-O

](r

fI&~l

The summation over K may also be taken from 0 to 2n - 1 because of the periodicity of 1/Ix. A simple expression can be obtained for the coefficients of 1/IK !(r [( odd . [(r cot -Slnm-= 2n { 11, 0 Keven and therefore

2:

2n-l

E(qJ) = - -1

n

.pK cot -K7r 2n

tc odd

E-l

or Kodd

(3.17)

THEORY OF WING SBCTIONS

58

In most cases a value of n equal to 40 gives sufficiently accurate results. Ordinarily 1/1 is known as a function of 6, and first approximations to the values of and (J - f' are obtained by substitution of 6 for fJ in the relations for IJ - fJ and The factor relating the velocities in the true-circle plane to those in the near-circle plane is dz' /dz. From Eq. (3.16),

"'0

"'0.

~' dz = ~

{Iz+ d [(1/1 -1J'0) + i(t) dz

fP)]}

z' ~ [If + i(B - tp) + In z]

=

But, on the true circle, z=~

from which

!s-== dz !i In z = !i (In a + 1/10 + if{J) == !i (kp) dz dz Therefore dz' = z'

dz

!! [1/1 + i(O dz

+ ifP]

,,)

dz' d - = z' - (l/I + i8) dz dz

This expression may be written dr dz

=

z'

!i (1/1 + is) dB dJJ dz

But

1

.df(J

z

dz

-=1­

or dz = i d((J = i d(f(J ­ 8) + i dB

z

and Hence

dz' dz =

z'd.

1

zdO (- 1,1/1 + 8) 1 + (dEjdIJ)

where or

z' 1 - i(tU/dD)

dz' dz

=

z 1 + (dE/dB)

(3.18)

THEORY OF WING SECTIONS OF FINITE THICKNESS

59

The flow in the plane of the true circle of radius a,eh is described by the following equation: W==

a2e2h) if Z +-Inz 2r aeh

V ( z+--

(3.19)

where the value of a in Eq. (3.7) is here taken to be ae/". The velocities in the true-circle plane are 2e2

dw ( 1 a- "") -=V dz Z2

ir +2rz

(3.20)

In order for the rear stagnation point of the circular cylinder to cor­ respond to the trailing edge of the wing section, the flow about the cylinder is rotated through an angle ao equal to the angle of attack of the wing sec­ tion. This process corresponds to the application of the Kutta-Joukowski condition. At zero angle of attack, the trailing edge corresponds to the point z = a,eh+~T It is therefore necessary for the circulation r to have a value corresponding to a rotation of the stagnation point by an amount ao + ET. This value may be seen from Eq. (2.30) to be I'

=

4r~V

sin (ao + ET)

Substituting the general value of z on the surface of the cylinder into Eq. (3.20) dw == V[I - e-2i (ae+.»

dz

+ 2i sin

(ao + ET)e-i
from which the absolute value of dw/dz may be obtained

I~;I = 2V[sin

(an + q» + sin (ao + ET)]

(3.21)

The third step is to obtain the velocities in the plane of the wing section from the expression for the velocities in the plane of the true circle. The expression for the velocity in the plane of the wing section is dw

dw dz dz'

dr = dz dz' dr

Multiplying Eqs. (3.14) and (3.18), we obtain drdz' dz' dz

=

1[I1-+ i(dl/l/dfJ)]. (dE/dO)

Z

.

·

(2a SInh ,pcos 8+ 2w cosh,p sm 8)

The absolute value of

Idrl == 2v'1 ++ (t#I/d8)2 v':sinh 1/1 + 8m 2

dz

eh[1

(de/dO)]

2

(J

THEORY OF WING SECTIONS

60

Dividing Eq. (3.21) by this expression,

Iddrwl= v = where

v

V [sin (ao + tp) + sin (ao + Er)][l + (dejdO)]e" -V(sinh2 '" + sinl 6)[1 + (dl/l/d8)I]

(3.22)

= local velocity at any point on surface of wing section

V = free-stream velocity The necessary calculations to obtain the pressure distribution for a given wing section are as follows: 1. The coordinates of the wing ~ction are found with respect to a line joining a point midway between the nose of the section and its center of curvature and its trailing edge for sections having sharp trailing edges. 122 The coordinates of these points are taken as (- 2a, 0) and (2a, 0), respec­ tively. It is usually convenient to let a have the value of unity. 2. sin 8 is found from Eq. (3.13). 3. sinh 11' is found from Eqs. (3.12). 4. '" is found from tables o( hyperbolic functions. 5. '" is plotted as a function of 6. 6. #0 is determined from the relation

"'0 =.1 (2" '" dB 2r Jo 7. A first approximation to

E

is obtained by conjugating the curve of

t/I plotted against 8 using Eq. (3.17). 8. From the curves of E and '" plotted against 8, dE/dO and th/I/dB are found. 9. Determine F by the relation

F ==

[1 + (dE/d6»)eJ't v(sinh2 '" + sin2 8)[1 + (dl/I/d8)2J

10. ; = F[sin (8 + ao + E)

+ sin (ao + ET)]

11. The pressure coefficient 8 == (V/V)2. For most purposes the first approximation to E is sufficiently accurate. If greater accuracy is desired, a second approximation may be found by plotting 1/1 against 8 + E and repeating steps 6 and 7 before proceeding. 3.7. Empirical Modification of the Theory. Although the foregoing theory for computing the pressure distribution about arbitary wing sections is exact for perfect fluid flow, the presence of viscous effects in actual flows leads to discrepancies. Even when the flow is not separated from the surface, the thickness of the boundary layer effectively distorts the shape of the section. One result of this distortion is that the theoretical slope of the lift curve is not realized. A comparison of the actual lift and moment.

THEORY OF WING SECTIONS OF FINITE THICKNE8S

61

characteristics with the theoretical ones is made in Fig. 33 for the NACA 4412 wing section.a This comparison is typical in that the experimental lift coefficient is lower than the theoretical value at a given angle of attack and the theoretical value of the pitching-moment coefficient is not realized. Several methods suggest themselves for bringing the theory into closer agreement with experiment. It is obvious that the lift coefficient can be 2.4

J

I

2.o~ lJsuoIlheofy

/

MbWned/~-------j

I---

Experiment

0

+

1/ J

/ V',l 1/ /1

/. e ~

It ~

.1! .8

§

~.4

J

......

i

l' v

~

l'

//

0+

+

)

I

-.4

J

0

-. 8

r IJI

Cl",..-o

J.6

0

If

-. -

-+- ....-+

emc.

r--

..... +

~

~

I

I -16

-8 0 8 /6 24 Effecfke mgIe ofoIlock,ao,degrees

FIo. 33. Comparison between theoretical and measured lift and pitching-moment coefficients, NACA 4412 wing section.

brought into agreement by reducing the theoretical angle of attack. This method seriously alters the flow about the leading edge, leading to dis­ crepancies in the pressure distribution over the forward portion of the wing section. Another method tried by Pinkerton'" is to reduce the circulation to the required value by disregarding the Kutta condition. This method leads to fair agreement over .the greater part of the wing section (Fig. 34) but, as would be expected, leads to infinite velocities at the trailing edge. Pinkerton- found that fairly satisfactory agreement with experiment could be obtained by effectively distorting the shape of the section (Fig. 34)~ This distortion is affected by finding an increment aET required to avoid infinite velocities at the trailing edge when the eirculation is 00.­

THEORY OF WING SECTIONS

62

justed to produce the experimentally observed lift coefficient. The altered function fa is arbitrarily assumed to be given by

E

fa

where

E

= E + aET 2 ( 1 - cos 8)

(3.23)

original value computed by methods of preceding section modified value used to compute flow about distorted section

=

Ea =

I I I I

-3

«=8"

x

---r--2

Experiment

lJ5uoIlhtf!OtY RedtK:ed Clrculolion Modified theory

~ r-; .. '=' ~

p

-I ~

~-~ ~ .... ~

-~ ~

~-.:::

~ ~

~

I

~~

.~ s;

o

If'

o

~

.-

-~

...:::~-

50

--

I'. J

,

~

,.\

100

Percenlchord 34. Compariaon between measured and various theoretical pressure distributions NACA 4412 wing section.

FlO.

The agreement obtained by this method is indicated by Fig. 34 and the moment coefficients of Fig. 33. 3.8. Design of WiDg Sections. Section 3.6 presents a method for obtaining the pressure distribution for arbitrary wing sections. A knowl­ edge of the pressure distribution is desirable for structural design and for the estimation of the critical Mach number and the moment coefficient if tests are not available. The pressure distribution also exerts a strong or predominant influence on the boundary-layer flow and, hence, on the sec­ tioJl. characteristics. It is therefore usually advisable to relate the aero­ dynamic characteristics to the pressure distribution rather than directly to the geometry of the wing section. In the experimental development of wing sections, it is accordingly desirable to have a method of determining changes of shape of the section corresponding to desired changes in the

7'BEORY OF WING SECTIONS OF FINITE THICKNESS

63

pressure distribution. A method that has been used successfully for the development of the newer NACA wing sections is presented in-reference 3. Equation (3.22) gives the relation between the velocity distribution over the wing section and the shape parameters 1/1 and 8. It is shown by Theodorsen and Garrick12l that an alternate expression is v

sin (ao + Ip)

+ sin

(ao + E'I')e4'o

V = v(sinh2 ~ + sin" 8) t[1 - (dE/d'P)]2 + [(dlY/d'P)]2}

(3.24)

Basic symmetrical shapes are derived by assuming suitable values of dE/dtp as a function of tp. These values are chosen on the basis of previous experience and are subject to the conditions that

(1r dE = ()

10 dqJ

and dE/dtp at qJ is equal to dE/dtp at - qJ. These conditions are necessary for obtaining closed symmetrical shapes. Values of E{qJ) are obtained by in­ tegrating (dE/d'P) dtp. Values of 1/I(tp) are found by obtaining the conjugate curve of E{tp) by means of Eq. (3.17) and adding an arbitrary value 1/10 sufficient to make the value of 1/1 equal to or slightly greater than zero at 'P = ".. This condition assures a sharp trailing-edge shape. If 1/1 equals zero at qJ = ..., the wing section will have a cusplike trailing edge. Slightly positive values of 1/1 result in more conventional trailing-edge shapes. Theodorsen'P has shown that small changes in the velocity distribution at any point on the surface are approximately proportional to 1 + (dE/dqJ). The initially assumed values of dE/dip are accordingly altered by a process of successive approximations until the desired type of velocity distribution is obtained. After the final values of y, and E are obtained, the ordinates of the symmetrical section are computed by Eq. (3.12). Although a similar procedure may be used to design cambered sections, it has been found convenient to design symmetrical sections which may then be cambered by a method to be described in Sec. 4.5. This method is satisfactory for thin or moderately thick sections. Very thick sections must be designed directly with the desired camber. Goldstein17C1 has also developed useful approximate methods for the design of cambered wing sections.

CHA~TER

4

THEORY OF THIN WING SECTIONS 4.1. Symbols. A A

B L PR S U V

a C

c, Clo

e'i C.L E c.e/4

e

k

1 In 1nLB

p

v v" l1v J1v"

x %1

'II

r ~ ai

a 1~ -0

fJo ../ 8

'h 1£,) r

p ~.FJ

coefficients of Fourier series multiplier for obtaining the angle of zero lift multiplier for obtaining the pitching-moment coefficient lower surface ordinate in fraction of the chord local load coefficient pressure coefficient (11 - p) fJAp 'Jl1 upper surface ordinate in fraction of the chord velocity of the free stream mean-line designation; fraction of the chord from leading edge over which loading is uniform at the ideal angle of attack chord section lift coefficient value of c, corresponding to calculated value of ~t·4/V ideal lift coefficient

section pitching-moment coefficient about the lending edge

section pitching-moment coefficient about the quarter-chord point base of Naperian logarithm, 2.71828 multiplier for obtaining the angle of zero lift section lift logarithm to the base e section pitching moment about the leading edge

local static pressure

local velocity over the surface of a symmetrical wing section at zero lift

component of velocity normal to the chon! increment of local velocityover thesurface of a wing section associated with camber increment of local velocity over the surface of a wing section associated with angle of attack distance along chord fixed point on the %axis

ordinate of the mean line

circulation

section angle of attack ideal angle of attack angle of zero lift quantity defined by Eq. (4.13)

circulation per unit length along chord

angle whose eosine is 1 - (2x/ c)

angle whose cosine is 1 - (2x1 / c)

quantity defined by Eq. (4.14)

ratio of the circumference of a circle to its diameter

mass density of air

infinity

THEORY OF THIN WING SECTIONS

65

4.2. Basic Concepts. Many of. the properties of wing sections are primarily functions of the shape of the mean line, The mean line is con­ sidered to be the locus of points situated halfway between the upper and lower surfaces of the section, these distances being measured normal to the mean line. Although the mean line of an arbitrary lying section is rather difficult to obtain, the construction of cambered lying sections from given mean lines and symmetrical thickness distributions is relatively easy. Among the properties mainly associated with the shape of the mean line are 1. The chordwise load distribution. 2. The angle of zero lift. 3. The pitching-moment coefficient. Other important results obtained from the theory of thin wing sections y

~~-6---------"""'-X 11' FlO.

35. Configuration for analysis of mean linea.

are a value of the slope of the lift curve and the approximate position of the aerodynamic center. The theory of thin lying sections was developed by Munk." Later contributions to the theory were made by Birnbaum, Glauert," Theodor­ sen,t21 and Allen." The present treatment follows that given by Glauert." For the purposes of the theory, the lying section is considered to be re­ placed by its mean line. The chordwise distribution of load is assumed to be connected with a chordwise distribution of vortices. Let 'Y be the difference in velocity between the upper and lower surfaces. The total circulation around the section is then given by the relation (4.1)

where c is the chord and the camber is assumed to be sufficieutly small for distances along the chord line to be substantially equal to those along the mean line. The configuration for the analysis is given in Fig. 35. The vertical components of velocity caused by an element of the vortex distribution along the mean line have the magnitudes [see Eq. (2.27)] 'Ydx

THEORY OF WING SECTIONS

66

where x is the position of the element and Xl is the point where velocity is being calculated. The total vertical component of velocity at. any point Xl is then

(4.2) where v" =- component of velocity normal to chord line The thin wing section must of course be a streamline. There is no flow through it. The equation expressing this condition is (4.3)

for small angles. Equations (4.2) and (4.3) are the fundamental relations between· the shape of the mean line and its aerodynamic characteristics. It is possible to obtain the characteristics of arbitary thin wing sections from these relations. The pressure distribution or the chordwise load dis­ tribution may, however, be obtained more accurately and just as con­ veniently by the method presented in Sec. 3.6. 4.3. Angle of Zero Lift and Pitching Moment. Simple relations for the angle of zero lift and the pitching moment of arbitrary thin wing sec­ tions may be obtained by the application of a Fourier method of analysis to Eqs. (4.2) and (4.3). Let x

c

= - (1 ­

2

cos 6)

(4.4)

then

Assume that 'Y may be represented by a trignometric series as follows: GO

'Y

=

2V(Ao cot ~ + LA. sin nO)

(4.5)

n-l

When "Y is expressed in this manner, it will be shown later that the coeffi­ cient Ao depends only on the angle of attack, and the coefficients Aftdepend only on the shape of the mean line. Since cot 2-

~ _ 1 + cos 8

"Y dz

sin s

= cV[Ao(l + cos 8) +

GO

LA. sin n8 sin 8]d8 1

THEORY OF THIN WING SECTIONS

67

The lift is

l

= J:pv-y dx = 10r~pV2[Ao(l + cos 6) + ~ .. sin nO sin 6]dB 1 rcpV2(A o + MAl)

=

(4.6)

and the lift coefficient C,

=

~:v2c = 2r(Ao+ ~ AI)

(4.7)

The expression for the pitching moment about the leading edge is given by the relation

Since sin 8(1 - cos 8) = sin 8 - sin 8 cos 8 = sin 8 - ~~ sin 28 mLB

1"'~ pV2 [ Ao(l -

=-

L 00

cos2 6) +

A .. sin nO(sin 6 -

~ sin 2tJ)] dB

1

Upon integration

(4.8) Equations (4.7) and (4.8) provide simple relations between the lift and moment and the first few coefficients of the Fourier series. The next step is to determine the relations between the coefficients of the Fourier series and the shape of the mean line. Substituting Eqs. (4.4) and (4.5) into Eq. (4.2), the expression for the vertical component of velocity becomes

V 111"{

Vn () 81 = -

11"

0

A o(1 + cos 8) COS 81 -

COS 8

HrA..

cos (n - 1)6 -'cos (n+ 1)61}d8 + - -1 - [--------­ cos 61

-

cos 8

since ~[CQS

(n - 1)8 - cos (n + 1)8]

==

sin nO sin 8

It is shown by Glauert" that

r

cos nO dB = Jo cos 8 - cos q,

'lI"

~ nq, SIn

q,

(4.9)

THEORY OF WIJ.VG SECTIONS

68 eonsequently

and, dropping the subscript,

v

V

Ao +

= -

L Aft cos 118 00

(4.10)

1

The required relation between the Fourier coefficients and the shape of the mean line is obtained by substitution of Eq. (4.10) in Eq. (4.3) dy dx

=

~ ao - A o + k A n cos n8

(4."11)

1

The standard expressions for the Fourier series coefficients are then

11'IT-dy dB

ao - Ao = r

0

dx

(4.12)

21'IT

An = -

r

0

dY d cos nO dB X

Equations (4.12) for A o, AI, and 411 2 can be transformed in such a manner that the angle of zero lift and the pitching moment are expressed in terms of the ordinates of the mean line rather than in terms of its slope. The quantities {jo and Po defined as follows are convenient in effecting the change. {jo -

and 11.0 =

~1" ll. dB ... 0 c 1+cos8

(4:.13)

1" o

ll. cos 9 dB

(4.14:)

C

The value of fJo in terms of the coefficients of the original Fourier series may be found by the following process. Integrating Eq. (4.13) by parts we obtain {J - 2[Y

0-; c

1cos 8]" 2 [" Idydx II - cos 9 d9 1+cos8 -;}o cdxd8V'l+cos8 0

(reference SO, formulas 296 and 578). The first term of this expression equals zero if y approaches zero at 8 = 1(' faster than v'C=X. This is true for nearly all wing sections. From Eq. (4.4), dx = !:. sin 9 = !:. V(1 d8 2 2

+ cos 9)(1 -

cos 9)

THEORY OF THIN lVING SECTIONS

Hence

69

li ftd

J(1 - cos 8)d8 dx

{jo = - -

.". 0

and, from Eq. (4.12), Po

= Ao -

ao

+ ~Al

(4.15)

The value of JJ.o in terms of the Fourier coefficients A" is found as follows: Integrating Eq. (4.14) by parts, JJo

. 6J1r = [·c~y mn 0

== -

lr 0

1 dy dx . - sm 8 dO cdxd8

- -

t'.!.

dy (l - cos 28)dO Jo 4dx

1)

(4.16)

+ Po)

(4.17)

1r( 4" Qo - Ao - 2 A 2

Comparing Eq. (4.7) with Eq. (4.15) 2".(ao

C, =

The angle of zero lift QL-O is therefore - {3o, and the slope of the lift. curve according to Eq. (4.17) is 211" per radian. The equation for the pitching-moment coefficient about the leading edge is found by substituting Eqs. (4.15) and (4.16) into Eq. (4.8) c",u

= (2#lO -

~ Po) -

i

Cl

(4.18)

The pitching-moment coefficient about a point one-quarter of the chord behind the leading edge Cm./ equals Cwa L B + (%)c,. From Eq. (4.18), 4

Cwa

1r

~/t

= 2#lO - --{3 2 0

(4.19)

It should-be noted that the pitching-moment. coefficient about the quarter­ chord point is independent of the lift coefficient. The quarter-chord point is therefore the aerodynamic center of thin wing sections. The aerodynamic center for most commonly used wing sections is found to be approximately the quarter-chord point at speeds where the velocity of sound is not reached in.the field of flow. By substituting the expression x

c

= -

2

(1 - cos 8)

into Eqs. (4.13) and (4.14) {30=

i

1y

o

-/1 (X)dX ­ C

C

C

(4.20)

THEORY OF "K'ING SECTIONS

70

and

i

IJo ==

1y (X)dx - f2 - c c

(4.21)

o c

where

fl(~) = r[l- (x/c)] ~(x/e)[l f2

1 - (2x/c) (Cx)= V(x/e)[l - (x/c)]

(x/e)i

In order to avoid infinite velocities at the leading edge, it is necessary that the coefficient A o of Eq. (4.5) be equal to zero. From Eq. (4.12), Ao equals zero when ao =

! ("dy d8

1f'Jo .d,x

Substituting the value of dx from Eq. (4.4) 1 {. Qo

= ;}o

d dB

(e/2)Ysin 8 d8

and integrating by parts ao == ![~y/cl·

sm S'Jo

1r

+.! rr2(y/c~ cos 8 dB 2 11"

Jo

sln 8

As previously discussed in connection with the integration of Eq. (4.13) the first term of this expression equals zero for most wing sections. Sub­

stitution of the expression x

= -2c (1 -

cos 8)

gives (4.22)

where

fa (

X)

c

1 - (2x/c) = 2-r{ (x/e)[l - (x/e)Jl ~

This angle of attack is called the "ideal angle of attack" by Theodorsen.P' The lift coefficient corresponding to this angle of attack [obtained from Eq. (4.1 i)] is called the "ideal" or ,., design" lift coefficient. In general the angle of sero lift, the pitching moment about the quarter­ chord point, and the ideal angle of attack may be calculated by graphical integration of Eqs. (4.20), (4.21.), and (4.22). For convenience in such calculations, values of the functions /1, !2, and fa corresponding to several values of x/c are presented in Table 5. Although the functions /1, j!, and fa become infinite at both the leading and trailing edges, the integrands of

THEORY OF THIN WING SECTIONS

71

Eqs. (4.20) and (4.21) approach zero at the leading edge for mean lines that approach zero faster than VXjC. Most mean lines satisfy this con­ dition. Similarly, the integrand of Eq. (4.21) approaches zero at the trailing edge in most cases. In order to avoid the difficulties at the trailing edge in Eq. (4.20) and at the leading and trailing edges in Eq. (4.22), parts of these integrals are evaluated analytically." The analytical determinaft,

TABLE .5.-VALUES OF FUNCTIONS

z/c

fl(x/c)

0 0.0125 0.0250 0.0500 0.0750

/2,

f2(x/c)

CO

AND

fa

f.{x/c) CO

00

2.901 2.091 1.537 1.306

1.179

8.774 6.085 4.131 3.226

113.15 39.73 13.84 7.403 4.716 2.447 1.492 0.980 0.662

0.1000 0.15 0.20 0.25 0.30

0.995 0.980 0.992

2.667 1.960 1.602 1.156 0.873

0.40 0.50 0.60 0.70 0.80

1.083 1.273 1.624 2.315 3.979

0.408 0 -0.408 -0.873 -1.502

0.271 0 - 0.271 - 0.662 - 1.492

-2.667 -4.131

- 4.716 - 13.84

1.049

0.90 0.95

10.61 29.21

1.00

00

00

-

00

tion of the increments at the critical points is accomplished by assuming the mean line near the ends to be of the form y x (X)2 -==A+B-+C c c c

Evaluation of the increment of the angle of zero lift gives

(xC = 0.95 to 1.0)

dy J1fJo == 0.964Yo.t, - 0.0954 dx 1

Where Yo." is the ordinate of the mean line at x/c = 0.95 and dy/dx1 is the slope of the mean line at x/c = 1.0. This increment is added to the value of 80 obtained by graphical integration from x/c = 0 to x/c = 0.95. The increments for the ideal angle of attack are dy

1:100i=

+ O.467Yo.05 + 0.0472 dx o dy { - 0.467Yo.9S + 0.0472 dx

1

(~= 0 to 0.05) (~ =

0.9,) to 1.0)

THEORY OF WING SECTIONS

72

These increments are added to the" values of Qi obtained by graphical integration from x/c == 0.05 to x/c == 0.95. By application of Gauss's roles for numerical integration to the expres­ sion for the angle of zero lift, Munk70 obtained a simple approximate solution. - ao' = klYl

+ kty2 + kaYa + k4'g4 + k,y,

where gl, gt, etc., are the ordinates of the mean line expressed as fractions of the chord at the points Xl, X2, etc., as tabulated together with corre­ sponding values of the constants kl , k i , ete., calculated to give the angle of zero lift in degrees. kl

0.99458 0.87426 0.50000 0.12574 0.00542

Xl

X2 X, %4

z,

1,252.24 109.048 32.5959 15.6838 5.97817

let k, ~

k,

This method is useful for quick calculation of the angle of zero lift when a high order of accuracy is not required. Approximate solutions for the angle of zero lift and the moment coefficient were also obtained by Pankhurst78 in the following form: (XL-O

= 2;A(U + L)

c..1. == '1:B(U + L) where U,L = upper and lower ordinates of wing section in fractions of chord A,B == constants given in following table x

0 0.025 0.05 0.1 0.2

0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.95 1.00

B

A 1.45 2.11 2.41 2.94

-

2.88 3.13 3.67 4.69 6.72

- 0.009 0.045 0.101 0.170 0.273

11.75 21.72 99.85 - 164.90

0.477 0.786 3.026 - 4.289

r.se

0.119 0.156 0.104 0.124 0.074

THEORY OF THIN WING SECTIONS

.78

Pankhurst's solution for the angle of zero lift is often more convenient than Munk's because the constants are given for stations at which the ordinates are usually specified. The solution is based on the assumptions that the mean-line ordinates are represented satisfactorily by (U + L)/2 and that the tangent of the mean line at the trailing edge coincides with the last t\VO points. The accuracy is accordingly expected to be limited for sections with large curvatures of the mean line near the trailing edge, or for thick, highly cambered sections. 4:.4. Design of Mean Lines. It is possible to design mean lines to have certain desired load distributions by means of the theory of thin wing sections. The fundamental relations are Eqs. (4.1), (4.2), and (4.3). A simple example of the manner in which a mean line is designed may be HI demonstrated by the design of a mean line to have uniform load. \ For a uniform chordwise load distribution, the value of -y in Eq. (4.2) is a constant over the chord. Equation (4.2) may therefore be written (4.23)

The lift is

and

Cl,

equals C,

,

21

2-y

pV c

V

= - 2- = ­

or

Ve"

'Y=T By substitution, Eq. (4.23) becomes

"ft(XI) Cl'1 (x/c)d(x/c) V C 411" - (xI/c) 1

=

(4.24)

0

Special means must be used to integrate Eq. (4.24) because the inte­ grand becomes infinite as x approaches Xl. The integration is performed as follows:

V"(Xl) V C

= =

cl'I· 4r ~

[l

wc

d(x/c)

>-,

0

(x/c) - (Xl/C)

+

r

d(x/c) ]

- (Xl/C)

}CZl/C)+e (x/c)

Cl, lim{[ln(~ - ~)~:Q-' IJo + [In(~C - ~)Jl

411" e-+O

C

C

= ~~[ln(l-~)-ln(:l)~

C

}

St+e

THEORY OF WING SECTIONS

74

The angle of attack at which the uniform load distribution is realized is zero because the assumed load distribution, and accordingly the mean line, is symmetrical about the mid-chord point. Equation (4.3) becomes

~ = :; [ In(1-~) -In(~)J

(4.25)

Integration of this expression gives y == -

~~[(l- ~)In(l- ~)+~ In ~J

(4.26)

where the constant of integration has been selected to make the ordinates zero at the leading and trailing edges. By similar but more complicated processes, the mean lines corre­ sponding to other types of load distribution may be calculated. In this con­ nection, it is interesting to note that the distribution of l' and the slope of the mean line are conjugate functions at the ideal angle of attack [see Eqs. (4.5) and (4.11)]. The slope of the mean line can therefore be obtained from the assumed load distribution by the proper application of Naiman's coefficientg8.73 as in Eq. (3.17). The mean-line ordinates can then be ob­ tained by graphical or analytical methods. Mean lines have been caleulatedv corresponding to a load distribution that is uniform from x/c = 0 to x/c = a and decreases linearly to zero at x/c = 1. The ordinates are given bythe expression

~ - ~r(:'+ 1) {I ~ a[~(a - ~rln \a - ~I- ~(1 - ~rln(l -~) +~(l-~r -~(a-~)1-~ln~+g-h~} where

(4.27)

J) +-4IJ

- 1 [ 0)(12

g:::z-- a· - lna--

I-a

h=

4

n

1~ a (1 - aYIn(1 - a) - ~ (1 - a)] + 0

The ideal angle of attack for these mean lines is Cl,}"

a;

= 21r(a + 1)

It \\;11 be noted that the ordinates of the mean lines are directly pro­ portional to the design lift coefficient. The fact that the load distributions and the ordinates of the corresponding mean lines are additive may be seen from examination of Eqs.. (4.5) and (4.11). This property is, of course, a direct consequence of the linearizing assumptions made in writing the fundamental expressions (4.1), (4.2), and (4.3).. This linearity is con­ venient because ordinates of a mean line having a load distribution made

THEORY OF THIN WING SECTIONS

75

up of the sums of load distributions for which the corresponding ordinates are known can be obtained by simply adding the ordinates of the com­ ponent mean lines. The linearizing assumptions lead to increasing errors as the slope of the mean line increases. For geometrically similar mean lines, the slope in­ creases with the design lift coefficient, and the errors are consequently greater for the larger design lift coefficients. In the case of the uniform­ load mean line [Eq. (4.26)], the slope is infinite at the leading and trailing edges, and it is not to be expected that the assumed load will be maintained 2.0 -"-­

/

I( °0

-~ ~

" .2

.6

.4

r--. r---.... .8

1\

LO

~ FIo. 36. Typical basic load distribution at ideal angle of attack.

near these points. The region in which the slope is large is extremely small, however, because of the manner in which the slope approaches infinity [Eq. (4.25)]. 4.6. Engineering Applications of Section Theory. The theory of wing sections presented in Sec. 3.6 permits the calculation of the pressure distri­ bution and certain other characteristics of arbitrary sections with con­ siderable accuracy. Although this method is not unduly laborious, the computations required are too long to permit quick and easy calculations for large numbers of wing sections. The need for a simple method of quickly obtaining pressure distributions with engineering accuracy has led to the development of a method! combining features of thin and thick wing-section theory. This simple method makes use of previously cal­ culated characteristics of a limited number of mean lines and thickness forms that may be combined to form large numbers of related wing sections. The theory of thin wing sections as presented in Sec. 4.3 and reference 121 shows that the load distribution of a thin section may be considered to consist of 1. A basic distribution at the ideal angle of attack (Fig. 36). 2. An additional distribution proportional to the angle of attack as measured from the ideal angle of attack (Fig. 37). The first load distribution is a function only of the shape of the thin wing section or (if.the thin wing section is considered to be a mean line) of the mean-line geometry. Integration of this load distribution along the

THEORY OF WING SECTIONS

76

chord results in a normal force coefficient which, at small angles of attack, is substantially equal to the design lift coefficient. If, moreover, the camber of the mean line is changed by multiplying the mean-line ordinates by a constant factor, the resulting load distribution, the ideal or design angle of attack as, and the design lift coefficient Ck may be obtained simply by multiplying the original values by the same factor. The second load distribution, which results from changing the angle of attack, is designated the U additional load distribution," and the cor­ responding lift coefficient is designated the Ct additional lift coefficient." 5.0 4.0 ~O

\

\ \ ~

1.0

00

"~ .2

~ .............

.4

~

.6

-

r--. r--.-

r-,

.8

LO

~ Flo. 37.

Additional load distribution 8880Ciated with anale of attack..

This additional load distribution contributes no moment about the quarter­ chord point and, according to the theory of thin wing sections, is independ­ ent of the geometry of the wing section except for angle of attack. The additional load distribution obtained from the theory of thin wing sections is of limited practical application, however, because this simple theory leads to infinite values of the velocity at the leading edge. This difficulty is obviated by the exact theory of wing sections presented in Sec. 3.6, which also shows that the additional load distribution is neither com­ pletely independent of the airfoil shape nor exactly a linear function of the angle of attack. Suitable additional load distributions can be calculated by the methods of Sec. 3.6 for each thickness form (symmetrical section) and a typical value of the additional lift coefficient.. The variation of the additional load distribution with the additional lift coefficient is arbitrarily assumed to be linear. In addition to the pressure distributions associated with these two load distributions, another pressure distribution exists which is associated with the thickness form, or basic symmetrical section, at zero angle of attack and which is calculated by the methods of Sec. 3.6. J

THEORY OF THIN WING SECTIONS

77

The velocity distribution about the wing section is thus considered to be composed of three separate and independent components as follows: 1. The distribution corresponding to the velocity distribution over the basic thickness form at zero angle of attack. 2. The distribution corresponding to the load distribution of the mean line at its ideal angle of attack. 3. The distribution corresponding to the additional load distribution associated with angle of attack. The local load at any chordwise position is caused by a difference of velocity bet,veen the upper and the lower surfaces. It is assumed? that the velocity increment on one surface is equal to the velocity decrement on the other surface. This assumption is in accord with the basic concept of the distribution of circulation used in the theory of thin wing sections. It can be shown that the relation between the local load coefficient P R 'and the velocity increment ratios !:1v/V or !:1vo/V is

!:1v V

~Va

or

1 PR

V =4v1V

where v/V is the velocity ratio at the corresponding point on the surface of the basic thickness form. In actual practice the value of !:"v IV used is that corresponding to the thin wing section or mean line where v/V equals

unity: The velocity increment ratios !J.v/V and I1vo / V corresponding to com­ ponents 2 and 3 are added to the velocity ratio corresponding to component 1 to obtain the total velocity at one point from which the pressure coefficient S is obtained, thus S = (::. ± .1v ± I1Va

V

V

)2

V,

,

(4.28)

This procedure is illustrated in Fig. 38. "Then this formula is used, values of the ratios corresponding to one value of x/c are added together and the resulting value of the pressure coefficient S is assigned to the surface of the wing section at the same value of x/c. The values of !:,.v/V' and I1vo / V are positive on the upper surface and negative on the lower surface for positive values of the corresponding local loads. When the ratio I1va/ V has the value of zero, integration of the resulting pressure-distribution diagram will give approximately the design lift coefficient Cli of the mean line. In general, however, the value of CI will be greater than ct, by an amount dependent on the thickness ratio of the basic thickness form. This discrepancy is caused by applying the values of !:1v/V obtained for the mean line to the sections of finite thickness where v/V is greater than unity over most of the surface. The pressure distribution will usually be desired at some specified lift coefficient not corresponding to cu: For this purpose, the ratio ava/V must

THEORY OF WIJtiG SECTIONS

78

1.2

E __---~ ~ C A LYJN;

SECT1CJ'.J ATZERO AtG...E OFATTACK

.8

v

V .4

.4 [ -

MEAN l.f£ AT IlEAL

ANiLE OFATTACK

E

CM'SEREO YiN; SECTOJAT lEAL ANiLE (F ATTACK

-

-

-

_

~

I~'

.4

0'------------­ ~

E·_u===:::::'-­ VELOCITY IfCREMENT RESlLTING

FRQ.1 ANGLE rF ATTACK FOR

SYtAETRCAL ¥lING ~

-

0[ 1.6

V

1.6 1.2

YJ..Ar.+- ~

V~

V -

V

.4

°0

.2

.4

.6

.8

LO

.Jt FIG.

38. Synthesis of pressure distribution.

be assigned some value by multiplying the originally calculated values oi this ratio by a factor J(a). For a first approximation, this factor may be assigned the value f{a) = C, - Cl, Clo

where CI is the lift coefficient for which the pressure distribution is desired and Clo is the lift coefficient for which the values of L1va / V were originally calculated. If greater accuracy is desired, the value of J(a) may be ad­

THEORY OF THIN WING SECTIONS

79

justed by trial and error to produce the actual desired life coefficient as determined by integration of the pressure-distribution diagram. Although this method of superposition of velocities has inadequate theoretical justification, experience has shown that the results obtained are

~ 1.

c/

~

~

-~

.:

r- Upper .UI"t'aoe

~

~Lowr.Ul"1"_

·1'"

.o,

\

- r-, 1\

~

~

-~OrJ'

"~ ~,

o Kx:perJ.-ent

o

.2

.,.

.&

.8

&.0

x/o

FIO.39. Comparison of theoretical and experimental pressure distributions for the NACA 66(215)-216 airfoil; Cl == 0.23

adequate for many engineering uses. In fact the results of even the first approximation agree well with experimental data and are adequate for at least preliminary consideration and selection of wing sections. A com­ parison of a first-approximation theoretical pressure distribution with an experimental distribution isshown in Fig. 39. Some discrepancy naturally occurs between the results of experiment and the results of any theoretical method based on potential flow because of the presence of the boundary layer. These effects are small, however, over the range of lift coefficients for which the boundary layer is thin and the drag coefficient is low.

€ H APTER

5

THE EFFECTS OF VISCOSITY 6.1. Symbols. A

constant velocity gradient skin friction total pressure outside boundary layer ratio of the displacement thickness to the momentum thickness constant L characteristic length £, length of mean free path of molecules B Reynolds number pVL/p Be Reynolds number pVc/p. R,. Reynolds number pur/ p. R z Reynolds number pVx/p. Ra* Reynolds number pV6*/p. or pU6*/p. R, Reynolds number pU6/p. U velocity outside boundary layer U« maximum velocity

UtI velocity outside boundary layer at the point Xo

U. velocity outside boundary layer at the laminar separation point U z value of U at the point :z; V velocity of the free stream c chord

Cd section drag coefficient

ep skin-friction coefficient

c/ local skin-friction coefficient

d diameter of pipe

" base of Naperian logarithms, 2.71828

l length

l wave length

In logarithm to the base e log logarithm to the base 10 n exponent p local static pressure q dynamic pressure go dynamic pressure of the free stream ~p yt r radius of pipe u . component of velocity in boundary la.yer along the x axis u average velocity of flow in pipe

v component of velocity in boundary layer along the y axis

v mean molecular velocity

11* friction velocity v:;jp v. velocity of slip

80

F F H H K

THE EFFECTS OF VISCOSIT1' z

81

Cartesian coordinate (distance along surface)

z. effective length of flat plate %0

11 a

position on surface at start of integration

Cartesian coordinate (distance normal to surface)

21r/l

fJi time rate of amplification of disturbance

a boundary-layer thickness

a* displacement thickness of the boundary layer

" nondimensional variable (y/x)Vifz

'1

8 80 ~

~ p.

~ 11"

p

.,. q, wit

1/11 GO

distance parameter plJ*y/ Il

momentum thickness of the boundary layer

momentum thickness at the point Zo

pressure-loss coefficient

laminar boundary layer shape parameter

viscosity of the fluid

coefficient of slip

ratio of the circumference of a circle to its diameter

mass density of the fluid

shearing stress at the wall (skin friction)

velocity parameter u/v·

stream function

nondimensional stream function

infinity

6.2. Concept of Reynolds Number and Boundary Layer. In order to be able to compare directly the forces acting on geometrically similar bodies of various sizes at various air speeds, it is customary to express the forces in terms of nondimensional coefficients, as explained in Chap. 1. For geometrically similar configurations, these coefficients tend to remain constant, and they would remain exactly constant if all factors influencing the flow about the body were properly accounted for. It is well known, however, that some characteristics such as the drag and maximum lift coefficients vary with the size of the wing for a given air speed, and with the air speed for a given size of wing. The two most important factors that are neglected in defining the, coefficients are effects associated with the compressibility and viscosity of air. An examination of the complete equations of motion- shows the significant parameters to be the ratio of the air speed to the speed of sound in air (the Mach number) and the Reynolds number pVL/p, where p is the mass density of the fluid, V is the velocity of the free stream, L is the characteristic length, and, "" is the viscosity of the fluid. It can be shown that similarity of the flows about different bodies is obtained only if the bodies are geometrically similar and if the Reynolds number and the Mach number are the same. At speeds where the pressure variations around the body are small compared with the absolute pressure (low Mach numbers) the effects of compressibility are negligible, and the viscous effects may be considered independently.

THEORY OF WING SECTIONS

82

Some concept of the physical significance of the Reynolds number may be obtained by expressing this number in terms of the mean velocity and the mean free path of the molecules, as suggested by von Karman. 137 The kinetic theory of gases indicates that the coefficient of viscosity p, of a homogeneous gas may be expressed by a formula of the type66 p, =

KtWL,

where K = constant == mean molecular velocity LI == mean free path of the molecules Substituting this value of p. into the formula for the Reynolds number; we obtain 1 V L

v

R=--K V t;

where R = Reynolds number The Reynolds number is thus proportional to the product of the ratio of the speed of the body to the mean speed of the molecules and of the ratio of the size of the body to the mean free path of the molecules. For bodies of ordinary size moving at low Mach numbers in air of ordinary density, the ratio V (V is small and the ratio L/L/ is very large. Under these conditions, the flow around the body corresponds to the conditions under which this formula was derived, namely, that the velocities are not large compared with the mean velocity and that the spatial dimensions of the phenomenon are large compared with the mean free path. The mean velocity v is of the same order of magnitude as the velocity of sound, and consequently the ratio V IV is of the same order as the Mach number. It accordingly be inferred that the Reynolds and Mach numbers are not independent parameters and that the scale effects ex­ perienced at high Mach numbers would be different from those at low Mach numbers even though the Reynolds numbers were the same in each case. Moreover, in a rarefied gas, as at extremely high altitudes, the mean free path may be of the same order as, or even much larger than, the characteristic length of the body. The concept of Reynolds number as presented here presupposes a mean free path that is small compared with the thickness of the boundary layer. Consequently the significance of this concept as applied to flows in very rarefied gases or at high Mach numbers is uncertain. Under such conditions, it is preferable to compare flows on the basis of the independent parameters V IV and L/L,. The effects of viscosity are of primary importance in a thin region near the surface of the wing called the U boundary layer." Boundary layers, in general, are of two types, namely, laminar and turbulent. The llow in the laminar layer is smooth and free from any eddying motion. The flow in the turbulent layer is characterized by the presence of a large number of relatively small eddies. The eddies in the turbulent layer produce a trans­

may

THE EFFECTS OF VISCOSITY

83

fer of momentum from the relatively fast moving outer parts of the boundary layer to the portions closer to the surface. Consequently the distribution of average velocity is characterized by relatively higher velocities near the surface and a greater total boundary-layer thickness in a turbulent than in a laminar boundary layer developed under otherwise identical conditions. Skin friction is therefore higher for turbulent boundary-layer flow than for laminar flow. Aerodynamically, the concept of the boundary layer is based on the premise of a continuous homogeneous viscous fluid. According to this concept the velocity within the boundary layer varies from zero at the surface to the full local stream value at the outer edge of the layer. Such a concept is valid only if the density of the gas is sufficiently great to limit the mean free path of the molecules to a length very small compared with the thickness of the boundary layer. The kinetic theory of gases" indi­ cates that the velocity at the surface is not exactly zero but that there is a velocity of slip proportional to the velocity gradient. du v. = t dy

The coefficient of slip ~ has the dimension of length and may be con­ sidered as a backward displacement of the wall with the velocity gradient extending effectively right up to the displaced wall where the velocity is zero. It has been shown by Maxwell, Millikan, and others that, for most surfaces, the coefficient of slip is very nearly equal to the mean free path of the molecules. At ordinary altitudes, this distance is so small that it may properly be neglected. At very high altitudes, this slip velocity may have large effects even though the mean free path is still only a fraction of the boundary-layer thickness. At still higher altitudes where the mean free path is long, the entire concept of the boundary layer and viscosity as presented here becomes invalid. 6.3. Flow around Wing Sections. When the pressures along the wing surfaces are increasing in the direction of flow, a general deceleration takes place. At the outer limits of the boundary layer, this deceleration takes place in accordance with Bernoulli's law. Closer to the surface, no such simple law can be given because of the action of the viscous forces within the boundary layer. In general, however, the relative loss of speed is somewhat greater for the particles of fluid within the boundary layer than for those at the outer limits of the layer, because the reduced kinetic .energy of the boundary-layer air limits its ability to flow against the adverse pres­ sure gradient. If the rise in pressure is sufficiently great, portions of the fluid within the boundary layer may actually have their direction of motion reversed and may start moving upstream. When this reversal occurs, the boundary layer is said to be "separated." Because of the increased inter­

84

THEORY OF WING SECTIONS

change of momentum from different parts of the layer, turbulent boundary layers are much more resistant to separation than are laminar layers. Except under very special circumstances laminar boundary layers can exist for only a relatively short distance in a region in which the pressure increases in the direction of flow. After laminar separation occurs, the flow may either leave the surface permanently or reattach itself in the form of a turbulent layer. Not much is known concerning the factors controllmg this phenomenon. Laminar separation on wings is usually not permanent at flight values of the Rey­ nolds number except when it occurs on some wing sections neai the leading edge under conditions corresponding to maximum lift. The size of the locally separated region that is formed when the laminar boundary layer separates and the flow returns to the surface decreases with increasing Reynolds number at a given angle of attack. The flow over aerodynamically smooth wings at low and moderate lift coefficients is characterized by laminar boundary layers from the leading edge back to approximately the location of the first minimum-pressure point on both upper and Iower surfaces. If the region of laminar flow is extensive, separation oecurs immediately downstream from the location of minimum pressure and the flow returns to the surface almost immedi­ ately at flight Reynolds numbers as a turbulent layer. This turbulent boundary layer extends to the trailing edge. If the surfaces are not suf­ ficiently smooth and fair, if the air stream is turbulent, or perhaps if the Reynolds number is sufficiently large, transition from laminar to turbulent flow may occur anywhere upstream of the calculated laminar separation point. ­ For low and moderate lift coefficients where inappreciable separation occurs, the wing profile drag is largely caused by skin friction, and the value of the drag coefficient depends mostly on the relative amounts of laminar and turbulent flow, If the location of transition is known or as­ sumed, the drag coefficient may be calculated with reasonable accuracy from boundary-layer theory. As the lift coefficient of the wing is increased by changing the angle of attack, the resulting application of the additional type of lift distribution moves the minimum-pressure point upstream on the upper surface, and the possible extent of laminar flow is thus reduced. The resulting greater proportion of turbulent flow, together with the larger average velocity of flow over the surfaces, causes the drag to increase with lift coefficient. At high lift coefficients, a large part of the drag is contributed by pres­ sure or form drag resulting from separation of the flow from the surface. The flow over the upper surface is characterized by a negative pressure peak near the leading edge, which causes laminar separation. The onset of turbulence causes the flow to return to the S1ma~e LCJ a turbulent boundary

THE EFFECTS OF VISCOSITY

85

layer. High Reynolds numbers are favorable to the development of tur­ bulence and aid in this process. If the lift coefficient is sufficiently high or if the reestablishment of the flow following laminar separation is unduly delayed by low Reynolds numbers, the turbulent layer will separate from the surface near the trailing edge with resulting large drag increases. The eventual loss of lift with increasing angle of attack may result either from relatively sudden failure of the boundary layer to reattach itself to the surface following separation of the laminar boundary layer near the leading edge or from progressive forward movement of turbulent separation. Under the latter condition, the flow over a relatively large portion of the surface may be separated prior to maximum lift. 5.4. Characteristics of the Laminar Layer. The characteristics of the laminar boundary layer may be deduced from detailed consideration of the general equations of motion. For two-dimensional steady motion neglect­ ing the effects of compressibility, these equations are28 u au +. v au dX

u

ay

av + v iJv

ay

oX

= _

~ ap + ~ (a u2 + 02U)} p

iJx

= _

! dp + J!'

2 (iJ V

2

pax

p iJy

d y2

dX2

p

+ iJ2v)

(5.1)

iJ y 2

These equations are known as the "K"avier-Stokes equations." Their general solution is difficult. A simple integration can be performed only in special cases. These equations, however, may be simplified by the use of dimensionless variables and consideration of the order of magnitude of each term. Prandtl" has shown that, if the viscous effects are assumed to be confined to a thin layer over the surface, the following approximate relation applies in this region u au

ax

+ v au = ay

_

! iJp + ~ a2u p

ax

p

iJ y 2

(5.2)

This equation is a simplified form of the first relation of Eq. (5.1). Con­ sideration of the order of magnitude of the various terms of the second relation of Eq. (5.1) indicates that op;iJy is small and hence that pressures are transmitted unchanged through the boundary layer. An interesting conclusion is reached by substitution of the following nondimensional variables in Eq. (5.2). X Xl =-

c U

UI

=-y

PI = 1­2 pV

Yl

=

'U.VR, C

vl=VVR

THEORY OF WING SECTIONS

86

Equation (5.2) becomes V 8(Ul V) a(XIC)

11,1

+ VI V

a(UIV)

= _

VR a(Ylc/VR)

! a(PtPV 2 ) + a2 (Ul V) p a (XIC)

a(Ylc/VR)"

and, upon simplification, UI

au! + VI OUI = _ apl + a2ul aXI aYl ax! aYl2

(5.3)

It will be noted that the viscosity does not appear directly in Eq. (5 3). Solution of this equation will give ttl and VI in terms of Xl and 'Ut. Because YI equals (y/c)vo" the effect of variations of the Reynolds number will be merely to change y in a manner inversely proportional to the square root

u

y FIG.

40.

Effect of pressure gradient on velocity distribution in laminar bounda.ry IS1'er.

of the Reynolds number. It follows directly that the thickness of the laminar boundary layer varies inversely with the square root of the Rey­ nolds number. It may also be concluded that the shape of the velocity distribution through the boundary layer is independent of the Reynolds number, unless, of course, the pressure distribution is a function of the­ Reynolds number. It also follows that the position of the laminar separa­ tion point, characterized by the condition that au/ay is zero at the surface, is independent of the Reynolds number. A considerable amount of information concerning the velocity distribu­ tion in the laminar boundary layer can be derived from Eq. (5.3) without actually attempting a solution. _~t the surface Ul and VI must both equal zero. At the outer limit of the boundary layer Ul approaches a constant value. Consequently near the outer edge of the layer a2u/all must be negative and approach zero with increasing distance from the surface, as shown in Fig. 40. At the surface, Eq. (5.3) simplifies to the form 82u1

apt

a yl2 = aXI

When apI/aXI is positive (increasing pressure in the direction of flow), atujayt must be positive at the surface, and the shape of the velocity dis­ tribution will resemble that shown in Fig. 40 for this condition. Similarly the velocity distribution must be linear near the surface for zero pressure

THE EFFECTS OF VISCOSITY

87

gradient, and the velocity profile will be continuously convex upward as shown in Fig. 40 for decreasing pressures in the direction of HOlV. 5.6. Laminar Skin Friction. Blasius'vobtained a solution for Eq. (5.2) for the case where op/ox is zero, that is, for a flat plate with uniform velocity outside the boundary layer. Blasius found that Eq. (5.2) could be greatly simplified by assuming a variable oi the form

== v.. v'Rz x

11

where B; = pVx/p.

and by the introduction of the nondimensional stream function

"'1 =

fo~ Ul

dT/

= j(T/)

where Ul = u/V The true stream function is

,p= fo"UdY '" =

U

=

Vx

f"Ul

~.Io

dT/

at/! == at/! °11

oy

o"loy

=

=

Vx j(T/) =

v'R z

~J! xVj(,,) P

[~J!p XVj/(T/)] (VRz) = x

Vj'(l1)

The velocity v perpendicular to the flat plate is given by &p 01/1 0 71 v == - - == - - -.

ox

f)

071 ax

= ~ ~~["j/(T/)

- j(1I)J

Making similar calculations for au/dx, 8u/dy, and iJ2u/dy2 and substituting in Eq. (5.2) with op/iJx zero, we obtain 2

2xLr

- V 11f ' (11)! ',/ (1I) + V2 1I}'(1I) - j(1I)Jl"(71) == 2x

_

or

j(TJ)I"<',,)

+ 2/'''(1/) == 0

~pxp. V e/"'("1) 2

(5.4)

The fact that neither x nor y appears explicitly in Eq. (5.4) indicates that the initial assumption that 1fl is a function of TJ alone is correct, that is, Ul is a function of "I alone. The boundary conditions for the solution of Eq. (5.4) are that J(fJ} == 0 !'(71) == 1 I" (11) --+0

for TJ == 0 for 71--+ 00 for 1/ ~ ex>

THEORY OF WING SECTIONS

88

Blasius obtained the solution of Eq. (5.4) by an approximate integration in a Taylor series," The solution has not been given in explicit form but is presented in the following table.

" 0 0.2 0.4 0.6

0.8 1.0 1.2 1.4

1.6 1.8

!'(.,,)

."

-

f'{.,,)

."

0 0.0664 0.1328 0.1990 0.2647

2.0 2.2 2.4 2.6 2.8

0.6298 0.6813 0.7290 0.7725 0.8115

4.0 4.2

0.3298 0.3938 0.4563 0.5168 0.5778

3.0 3.2 3.4 3.6 3.8

0.8461 0.8761 0.9018 0.9233 0.9411

5.0 5.2 5.4 5.6 5.8 6.0

I

I I

4.4

4.6 4.8

I

I

!'(.,,)

0.9555 0.9670 0.9759 0.9827 0.9878 0.9916 O.9M3 0.9962 0.9975 0.9984 0.9990

A plot of these values is given in Fig. 41. These values define the velocity distribution in a laminar boundary layer in the absence of a pres­

1.0

/

.8

/

~

~

~

/~

V

-6

:/

%

J

.4

JV

/

.2

)

V

-00

FlO.

41.

2

3

4

5

6

7

~AR; Velocity distribution in the boundary layer over a flat plate according to Blasius.

sure gradient. The form of the variables in which this velocity distribution is specified shows that the boundary-layer thickness increases as the square root of the distance from the leading edgeand inversely as the square root of the Reynolds number and that the velocity distribution is similar at all points along a flat plate. In addition to the velocity distribution through the boundary layer,

THE EFFECTS OF VISCOSITY

89

the solution of Eq. (5.4) gives a value of fl/('1) at the surface equal to 0.332, that is, 0.332 = /,,(.,,) = d(ujV) = d(ujV) d'1 d[(y/x)v'Rzl For the skin friction, we have

.,. = Cf

=

p,(~~) = 0.332 p,V ~ dy *=-0 X T I L . rnlL V 2 = 0.664 -IT v R% 72P P x

} 0.654

= v- /R

(5.5)

%

The total drag coefficient for one side of a. fiat plate is then given by

11%

cr = -

x

CJ

0

r~4~dx = -l~xO~ . l!:- dx z 0 VX pV

= !(1.323 x

fia.) = 1.328 ~

"\}pV

(5.6)

The obvious definition of the boundary-layer thickness is the distance from the surface to where the local velocity defect due to viscosity is zero, that is, u/V == 1. Practically, this distance is not 'Yell defined because u/V approaches unity asymptotically. A definition that has frequently been used is that the boundary-layer thickness is the distance from the surface to where u/V = 0.995. The preceding table indicates that, ac­ cording to this definition, o/x 5.3/v'R%. A thickness parameter that is convenient for experimental 'York is the distance from the surface to the point where (U/V)2 = ~2 or o/x'" 2.3/v'llz. For theoretical work it is fre­ quently desirable to obtain a measure of the effect of the boundary layer on the outside flow. The reduced velocities within the boundary layer cause a displacement of the streamlines around the body. This displacement is called the U displacement thickness" of the boundary layer 0* and is calculated from the relation "'J

0* = -1 ~a (V - u)dy

1T

0

or in the general case ~. =

Y11 10t' (U -

u)dy

(5.7)

where U = velocity just outside boundary layer For the laminar boundary layer over a flat plate with zero pressure gradient ~*

1.73

x= VR%

(5.8)

THEORY OF WING 8ECTION8

90

6.6. Momentum Relation. The KarmlLn integral relation27• 111 pro­ vides a method for relating the boundary-layer thickness to the pressure distribution if the skin friction and the velocity distribution through the boundary layer are known or assumed. The relation is useful because good approximations to the skin friction and the velocity distribution can often be obtained from consideration of local conditions. This relation is applicable to both laminar and turbulent boundary layers. The Karman integral relation is based on the principle of mechanics that, if any closed surface is described within a steady stream of fluid, the time rate of in­ crease, within the surface, of momentum in any direction is equal to the sum of the components in that direction of all the forces acting on the fluid. H body forces such as weight are absent, the only forces to be eon­ fI4 sidered are those acting on the surface, dx fix that is, normal forces caused by presu sures and tangential forces caused by _p6# ~'!dx viscous shearing. Consider the equilibrium of the element shown in Fig. 42. The mo­ mentum from left to right entering /':;dx the left face per unit time is FlO. 42.

1"011/_. dy)

Forces on an element of the

Jo

boundary layer.

\flu,

while the momentum leaving the right face per unit time is

[ u(pu dy)

+

[:xl

u(pu dY)] d:c

The mass flow per unit time leaving the right face must be the sum of the mass flow entering the left face plus the mass flow entering the upper boundary, The mass flow per unit time through the upper boundary is

tx([

PUdy)d:c

This mass enters with the velocity U. The momentum entering the upper boundary per unit time is then

u(:Xl'pu

dY)d:c

The total rate of change of momentum from left to right in the element is then

mB EFFECTS OF VISCOSITY

91

The forces acting on the element are

pa

on the left face

- [PIS + ~(PIS)dx]

on the right face

P ::: da:

on the upper boundary

dx

on the lower boundary

-

T

The resultant force is

d

- - (p8)dx dx

do + p -dx dx -

T

dx

which reduces to - {) dp dx dx

T

dx

Equating the resultant force and the rate of change of momentum

(:Xl'PU dY)d:x 2

Dividing through by

p

u(~l'pu dY)d:x =

-

IS:

dx -

T

dx

dx, we obtain

r

d d f' 2 dx}o u dy - U dx}o u dy

0 dp

T

= - Pdx - -;;

This equation is not very convenient because the values of the integrands do not approach zero at the upper limit of integration. -To avoid this difficulty the integrands are expressed in terms of the variable (U - u). by making the substitution u=U-(U-u) -d

l'[

dx o

L'[

J2

U-(U-u) dy-Ud dx o

]

U-(U-u) dY=--..1!._! 0d pdx

p

Expanding and collecting terms dU

d

(0

d

t'

d

t'

UIS dx - dx U}o (U - u)dy + U dx}o (U - u)dy - tI.x}o (U.;... u)u dy odp

T

==ptb;-p At the outer limit of the boundary layer, Bernoulli's equation is valid; hence + !2 pU2) = dH = 0 dx dx and dp dU dx = - pU dz

.!!(p

THEORY OF WING SECTIONS

92

Substitution of this relation in the previous equation gives

d -dUl' - (U-u)dy-dx o

l'

dx o

(U-u)udy=--T

p

(5.9)

This equation is known as the "!{arman integral relation." In this form the integrals approach zero as the upper limit is approached. This equation is suitable for the experimental determination of the skin friction, although the accuracy of the determination may be affected considerably by the form in which this equation is expressed. Another form of Eq. (5.9) can be obtained by the use of the displace­ ment and momentum thicknesses. The displacement thickness is defined by Eq. (5.7). The momentum thickness is related to the loss of momentum of the air in the boundary layer and is defined as

8 = }o{'(I - ~) 1.£ dy = ~ r U U U 2 }0

(U -

1.£)1.£

dy

(5.10)

Substitution of tJ· and 8 in Eq. (5.9) gives UlJ* dU +.!!:... U2fJ = i dx

dx

UtJ. dU dx

p

+ 2U8 dU + tr d8 = ! dx

dxp

dB +.!!...dU(tJ. +2)=~2 U dx (J pU

dx

Introducing the shape parameter H = = %plJ!

a*If}

and the dynamic pressure

q

dB +(/1 + 2)~dq =..:!.­ 2

dx

q d»

(5.11)

2q

Equation (5.11) is in a convenient form for the calculation of the thick­ ness of the laminar boundary layer when the pressures are not constant, as was assumed in the Blasius solution. 132 In many cases of interest, the pres­ sure gradients are sufficiently small so that the type of velocity distribution through the boundary layer will not differ greatly from that in the bound­ ary layer of a flat plate.' Application of Eq. (5.10) to the Blasius distribu­ tion gives a value of 8/x equal to O.664/~. Consequently the value of H for the Blasius distribution is 1.73/0.664 = 2.605. From Eq. (5.5), T

=

p,U ~ ~ 0.332- vRs x

Substituting the value of 8 T

=

0.664 X 0.332

p,U

p,U

8 == 0.220 8

THE EFFECTS OF VISCOSITY

Substituting the values for Hand dO +

dx

T

93

in Eq. (5.11)

(4.605)(_8_) pU dU = O.220pU 2 dx pU ~pU2

28

By use of the integrating factor 28CJ9·210, this expression becomes iP = 0.4401L i" U 8 •210 dx pUz 9.210j o

(5.12)

where U z = velocity at limit x If U is known as a function of x, that is, if the pressure distribution is known, Eq. (5.12) permits the boundary-layer thickness to be calculated approximately if the pressure gradients are sufficiently small for the actual distribution of velocity through the boundar)' layer to be approximated by the Blasius distribution. For most wing sections at low and moderate lift coefficients, the pressure gradients satisfy this condition back to the position of minimum pressure. The skin friction may be assumed to be that corresponding to the Blasius velocity distribution for the calculated thickness of the boundary layer. A convenient way of performing these calculations is to find an equivalent length of flat plate'l' 2 which is given by

i

X. =

.,.

..

.

~~.-.

..

(:.

{%

U",8.~lOJo

lj8.210

dx

(5.13)

where x, is the length of fiat plate in a stream of velocity Us that will generate the same laminar boundary-layer thickness as actually exists on the body in question at the point z. 5.7. Laminar Separation. It was pointed out in Sec. 5.3 that the laminar boundary layer cannot ordinarily exist for long distances in a region where the pressures are increasing ill the direction of flow, and the extent of the laminar boundary layer under such conditions is limited by separation. The Blasius solution does not, of course, give any information about laminar separation because this solution is premised on the absence of pressure gradients. Calculations of the location of laminar separation require a solution of the boundary-layer equations in more general form. One such attempt was made by Pohlhausen." He assumed that the velocity distribution in the boundary layer could be expressed by a poly­ nomial of the form 0­

Y

(Y)2 + a3(Y)3 (Y)4 ~ + a. ~

U = al ~ + at ~

'U

0

Using plausible assumptions regarding the physical characteristics of the layer, Pohlhausen developed a method for calculating the velocity dis­ tribution in the boundary layer and the position of the separation point. This method has only limited applicability to the calculation of the lam­ inar separation point. 105 Karman and Millikan-" obtained more accurate

94

THEORY OF WING SECTIONS

solutions of Eq. (5.2). The solutions are involved and require much cal­ culation, but the results appear to be satisfactory. For most actual 'York on wing sections, the extent of the laminar boundary layer in regions of adverse pressure gradients is too short to require much detailed attention. It is frequently desirable, however, to know the possible extent of the laminar layer as determined by laminar separation. It has been possible to develop a method for applying the I{arman-Millikan solution to the rapid estimation of the laminar separa­ tion point132 which is sufficiently accurate for most applications. More detailed studies may, however, be required for the condition associated with maximum lift. The method for rapidly estimating the position of laminar separation is based on the assumption that it is possible to replace the actual velocity distribution effectively by a simplified velocity distribution consisting of a region of uniform velocity followed by a region of linearly decreasing velocity. The actual pressure distribution up to the point of minimum pressure is used to calculate the equivalent length of fiat plate from Eq. (5.13). Because dp/dx is zero at the point of minimum pressure, the velocity distribution in the boundary layer at this point must approximate the Blasius distribution very closely. The replacement of the actual pres­ sure distribution downstream of the point of minimum "pressure by a linear velocity gradient is usually justified by the relatively small extent of the laminar boundary layer in this region. Consequently, the laminar separa­ tion point may be estimated from positions calculated for a family of such simplified pressure distributions. The position of the laminar separation point was calculated by the Kinnan-Millikan method for a series of pressure distributions of the fol­ lowing type. u - = 1 for 0 < !. < 1 Us - z.­ and u == 1 + F x - X e for 1 < ~

Uo

x.

- x.

where x == distance along surface from leading edge x, == length of flat plate equivalent to length from leading edge to position of minimum pressure U = velocity outside boundary layer at any point x U« == maximum velocity F = velocity gradient (xc/Uo)(dU/dx) (constant for any given case) Irrhe values of U8/ Ue, where U8 is the velocity at the laminar separation POint, were calculated by the Kinnan-Millikan method for a series of values of F, and the results are shown in Fig. 43. In applying this method, the value of F is obtained from calculated values of x,,/Uo and the value of ulj/dx correspondiug to the estimated

THE EFFECTS OF VISCOSITY

95

equivalent adverse velocity gradient. The corresponding value of UB/Uo is found from Fig. 43 and is applied to the actual velocity distribution to locate the position of laminar separation. 5.8. Turbulent Flow in Pipes. The earliest information on the nature of turbulent flow was obtained from studies of the flow in pipes. Reynolds found that, if the diameter of the pipe and the velocity were sufficiently 1.00 ~

/

/

.96

/

~

~-

./

'/

V

I(

I .92 .90 /

.880

1/ /

-.04

-.OS

-.16

-.12

-.20

-24

-.28

F

I""JG. 43. Velocity ratio for laminar separation as a function of the nondimensional velocity gradient F.

small, the flow \'.&8 rectilinear or laminar, as ShO\VD by the behavior mf colored filaments of fluid which did not mix with the surrounding fluid. With larger diameters and greater velocities, the flow became turbulent, as shown by diffusion of the colored filaments. Reynolds found that, if the quantity pl'dlfJ. was less than about 2,000, the flow was always laminar. The upper limit of this Reynolds number for laminar flow was indefinite and depended mainly on the care taken to provide steady conditions and smooth surfaces. Values of the upper limit as much as ten to twenty-five times the lower limit have been found by various observers." With laminar flow, the pressure gradient. along the pipe increases linearly with the velocity; but, with turbulent flow, the gradient increases nearly as the square of the velocity. The velocity distributions across the diameter of the pipe for laminar and turbulent flows are shown in Fig. 44. For turbulent flow, the velocity distribution is more nearly uniform than for laminar flow, and the shearing stresses near the walls are considerably higher. These conditions result from the increased transport of momentum associated with the turbulent motion. It has been observed that the nondimensional velocity distribution

THEORY OF WING SECTIONS

96

across the pipe is affected only slightly by the Reynolds number. On the basis of dimensional reasoning similar to that used for the formation of dimensionless coefficients for wings, Blasius" expressed the pressure loss in coefficient form as follows: r_ ~ = Pi - 'P2_ l ~p:;z2 where PI - 112 = pressure drop in a pipe of length l and radius r with a fluid

of density p flowing at average velocity u.

For smooth pipes or pipes having geometrically similar roughness, the

coefficient ~ is a function only of the

:JIi!:~:~~~:~= =~~~:;:~:~~::E;:::::

in smooth pipes is

LAMINAR

:11

0.133

~= ~r

: ~£:~::iE:~1::~:~~:~~~:~

the pipe and the rate at which the pressure-loss coefficient changes with Reynolds number. The most im­ portant region in which to study the nature of the phenomena appeared to be in the vicinity of the wall of the pipe where the velocity gradient is high rather than in the middle of the pipe where the velocity is nearly uniform. It appeared reasonable to assume that the velocity distribution near the wall depended on the local quantities p and II- and the skin friction T and was independent of the radius of the pipe. It was assumed, in ap­ proximate agreement with experimental evidence, that the velocity dis­ tribution across the entire pipe could be expressed as a function of y/r alone where y is the distance from the wall of the pipe; that is, the velocity distribution remains similar for various mean velocities and pipe sizes. Equating the pressure loss to the skin friction T\.J8JLENT

44. Velocity distribution across pipe for laminar and turbulent flow. FIG.

(PI - Pt)rr! = 2n-lT

2lT

PI-P2=-r

From Blasius' formula

(5.14)

THE EFFECTS OF VISCOSITY

97

If it is further assumed that the velocity distribution may be expressed as a power function of y/r,

where V = velocity at center of pipe or

3/

1/

= K p7 -&p.7 4 r-

]/

1

7/

7 4: ' 7 4

(r)(~)R

11

where K = a constant factor dependent on ratio of maximum velocity to mean velocity Applying the assumption that the velocity distribution near the wall and hence the skin friction is independent of the radius of the pipe, the ex­ ponent of r in the foregoing expression must be zero,

or

~~+ ~n

=0

n= ~1 (5.15)

The assumption made in the derivation of this equation that the velocity distribution is a function only of local parameters near the wall would indicate that this equation would also apply to the velocity dis­ tribution in a turbulent boundary layer on a flat plate. In this case, r is interpreted as the thickness of the boundary layer and V as the velocity just outside the layer. The validity of the analogy between the flow in pipes and that over a flat plate is shown by the data of reference 19, some of which are presented in Fig. 45. 6.9. Turbulent Skin Friction. It is possible to calculate the skin friction on ~fiat plate with turbulent flow by application of the one-seventh power law [Eq. (5.15)] and the Blasius skin-friction formula [Eq. (5.14)] to the local boundary-layer conditions. In the case of the flow of an incom­ pressible fluid in a pipe, the Reynolds number along the pipe remains constant and the skin friction results in a loss of pressure. In the case of the flat plate, the pressure is constant along the plate and the skin friction causes a deceleration of the fluid particles, resulting in an increase of the thickness of the boundary layer and a corresponding increase of the boundary-layer Reynolds number, At any station along the plate, the skin-friction drag over that portion of the plate upstream of the station can be obtained from the rate of loss of momentum in the boundary layer. It is assumed that, for any station, the velocity distribution in the boundary layer corresponds to that in the pipe and that the local skin friction is the same as that in a pipe at the same Reynolds number. The particles in the boundary layer having an original velocity V up­

KI7$CMJ

CMftec eooo

2000

-----

........

-

L.l .....

-

~-

L~

~

1000

~~

1000

~-

~~,.....

,....

-~

~

500

40,,-(:)

~

~

~

<:S

ti

CS

II

400

<::i

lI\Yv'M

-~-

-

,...,....

----

.fOOt"' N

'"

......

--

~

...... .,.-

1000

_1000'

tS

I

~

I

I

"

a~

I

~

....

...--,

IIII

I

I

,

,

.....

~ ~

II

500

,

,

, I I!'"

I

,

I

'~')CM

8 '''';

It)

~

a

~

...:

~

~ ....:

a yl'M

t\i

~

---...... l'-'

~

.... ~

~

400S?

ro:.

'"

~

~

~

~

r..:. ,..:.,..:.c::s "....,....,

S;

C

~

.......

~ ~

~

-~,.

--

~

~ ~ ~ ~

-

..-...-'l .....

_

IIII

II1II1

~ () ()

~ ti

....

)I

-

~680CM

II

ij

-~

~

I

, ,

~

-~

...

~~

~

..-

x-I25CM

-

-'--

500

~

....

-~

2000

-~ -"I'"

,..--,

-

IioIIIOIl

CM/sec

-

2000

--~

~ ..--:

()

~

K'81$CM

CM,&tc

1000

500

8...:

...

~ 1""'""""1

~oIX·lOCM and X Il 60CM

~

'to

--

-

~

'-

~

K'IOOCM

CM/sec

g

~

8 y"CM

...:...:t\i

FIG. 46. Logarithmic sua]o dj~"l\mA of tho vptocity in the boundary )nyer as a function 011/ at the sections z = 75, 87.5, 100, and 125 ern from tho leading edgo of the flt,t plate. The solid Imes represent thu one-seventh power law.

THE BFFECT8 OF VISCOSITY

99

stream of the plate are decelerated to a velocity u at station z, The rate of loss of momentum at station z is given by the following expression and is equal to the skin-friction drag of that portion of the plate upstream of station x. ' F

= p loa u(V -

u)dy

From the assumption that

V=(~)M F

i2PV2c5

=

but dF dx =

If"

hence

7

2

dO

-r = 72 pV dx

A corresponding value of T may be obtained from Eq. (5.14) if the rels­ tion between ii and V is known. A commonly used experimental value of ii/IT is approximately 0.8. Using this value Eq. (5.14) rna)' be written in the form r' = O.0225p~~p.~~o-3-~ TT~

Equating the

t\VO

expressions for

T

(5.16)

and integrating

(;Vy~ z c5 = 0.37 (-tV)%x%

~ c5~ = 7; 0.0225 Solving for

a

(5.17)

Substituting this value of 8 into the expression for the drag

F = O.036pV%xG;x)% In coefficient form cr

=

lL F 7";.P

V2X

=

O.072R:r;-~

(5.18)

Equation (5.18) is a formula for the drag coefficient for one side of & smooth flat plate of length x with turbulent flow over the whole surface. Figure 46 shows the skin-friction coefficient for turbulent flow from Eq. (5.18) and for laminar flow from Eq. (5.6). . Nikuradse performed pipe experiments over a range of Reynolds num­ bers much larger than for previous investigations.76 • 7iG These experiments

THEORY OF WING SECTIONS

100

showed that the pressure losses departed from values predicted by the Blasius formula above Reynolds numbers of about 50,000, the experi­ mental values decreasing less rapidly than the predicted values as the Reynolds number was increased. As expected from the simplified analysis of the relation between the pressure-loss coefficients and the velocity dis­ tributions, the velocity distributions at the higher Reynolds numbers are approximated better by one-eighth, one-ninth, or smaller powers than .0080 ~ .0070

~

.0060

.0050

f)()40

<,

~

~

=---

~

~

.-.

I!.......

-

~.- Turbulent flI=I~-Eq.5.20

£q.5.18./' f'~

"'~

"'i~

~ ~lt'"

.........

............. .:::::::

~

~

.0007 .0006

"" """

,....

"" ""lIlli, ~~

~~

fJOO5

"'~

.0004

os

.ooaJ

~ ~ ""- .... ,.....

~LomintT(Eq.5.6)

:"r-...

0010

~~

1..5 2 Z53

4 5 6 78906

1.5 2 25.3

4

5 6 78 9d

Rx FIG.

46.

Laminar and turbulent skin-friction coefficients for one side of a fiat plate.

by the one-seventh power, On the basis of an elaborate and extended dimensional analysis, a logarithmic expression was obtained28 • 128 for the velocity distributions corresponding to Nikuradse's values for the pressure losses.

q, == 2.5 In 1J + 5.5

(5.19)

where

u

<1>=v* pt,*y

11=­ p.

v*={;

and is called the U friction velocity." The skin friction on a flat plate may be calculated using this expression by a process using the same general

THE EFFECTS OF VISCOSITY

101

principles as that for the derivation of Eq. (5.18).28 The tabular solution to this problem may be approximated by the expression 0.472

cp=---­

(5.20)

(log.1l:,;)2.ss

Skin-friction coefficients calculated from this formula are shown in Fig. 46. Formulas (5.18) and (5.20) are in convenient form for the calculation of skin friction on flat plates, For application to bodies having arbitrary pressure distributions, it is more convenient to relate the local skin-friction coefficient to a Reynolds number based on the momentum thickness of the boundary layer. For the one-seventh power velocity distribution, such an expression may be obtained from Eq. (5.16). For this case, according to Eq. (5.10),

_T _ C/1 L U "2 72P

- ()

0 · 02~1 (-.!!-)~ ;J ["'0 - . 02r-·lR-~ a, p :

(5.21)

For the case of the logarithmic velocity distribution, approximate formulas have been ohtuined relating t he local skin-friction coefficient to the boundary-layer Reynolds number Ro• Thut given by Squire and ) . . oung!" is

er~ = [5.8HO log (-l.O~5Jlo)f

(5.22)

T

It is known that Eq. (5.21) is valid only for moderately small Reynolds numbers, whereas Eq. (5.22) is a much better approximation- at large Reynolds numbers. Equation (5.22) is, hO\VCVCf, inconvenient for use in connection with the momentum equation (5.11). Interpolation formulas have therefore been developed in the form

c, = .ilR,-n

(5.23)

where the coefficients and exponents nrc selected to correspond with the desired range of Reynolds numbers. For any particular application, the two constants of Eq. (5.23) may be found by making Eq. (5.23) agree with the assumed skin-friction law at t\VO values of R, including the range under consideration. If the subscript 1 corresponds to conditions at the low value of R s and the subscript 2 to conditions at the high value of R" then (T / Pl]2)l

log (T jp(]2)2 n-­ - log (R 8,./RIJJ

A

=

c/lRS1"

THEORY OF WING SECTIONS

102

The assumed skin-friction law may be Eq. (5.22) or the following formula · 120 suggested by T etervm T 1 (5.24) 2 pU = [ R 2.5 In 2.5(1 _ 5~) + 5.5

J2

The values of cJ corresponding to Eqs. (5.22) and (5.24) are plotted in Fig. 47. 6.10. Calt;ulation of Thickness of the Turbulent Layer. A knowledge of the thickness of the turbulent boundary layer is required for several .007 .006 f){)5

I -~~ ~!"""o

.004

------

~.OO3

.......

......... ~I""-~

....

Eq.5.22 ~

Ef~·524 ~"-

..

~

.002

2

s

4

5 6 78 9 f()4

--.

~~

2

....

3

~ 1-1-.

4

.-

r-

5 6 78 gl()5

R(J FIG. 47. Local turbulent skin-Iriction coefficient as a function of the boundary-layer Reynolds number.

purposes. The profile drag of the wing is, of course, intimately associated with the thickness of the boundary layer at the trailing edge. It is also desirable to know the thickness of the boundary layer at various points on the wing surface in connection with the study of control surfaces, high-lift devices, air intakes, and protuberances. The thickness of the turbulent boundary layer can be calculated from the momentum equation (5.11) if a relation between the local skin-friction coefficient and the dependent variables 8, q, and H is knO\VD. Values of the skin-friction coefficient have been found by assuming that the relation between the boundary-layer Reynolds number R, and the skin-friction coefficient is the .same as the corresponding relation found from pipe experiments. This relation has been plotted in Fig. 47. Although this relation may not be very accurate, .particularly for values of the shape parameter H differing considerably from those found in pipes, such marked differences usually occur only when dqld» is large negatively. In this case Eq. (5.11) shows that the contribution of the skin-friction term to the rate of increase of the boundary-layer thickness dO/dx is relatively small. It may accordingl~ybe assumed for practical calculations that CJ is independent of the value of H. Further simplification of the calculations is obtained by assuming a

c,

THE EFFECTS OF VISCOSITY

103

skin-friction law of the type indicated by Eq. (5.23). (5.11) may be written in the form de + (H + dx U

2) dU dx

(J =

A~n

(pU)r&

e»

In this case, Eq.

. ' ,;,-."

The value of the factor (H + 2) is not greatly affected by the value of H, which usually has values of about 1.4 or 1.5 with extreme variations from about 1.2 to 2.5. Detailed calculations have S110\Vn that excellent agreement can be obtained between experimental and calculated values of 6 if H is assumed to be constant and to have a value"? between 1.4 and 1.6. "~ith H assumed constant, this equation is seen to he of the Bernoulli type, and it can be integrated to give the value of 6 us follows:

(~)

C ",/c

=

1 (UjV)H+2

[(1

+

-t:

2Rc "

(U)(l/+l)

Xo!c

V.

(u+lH-l

~

d c

+ (~)"+l( ¥~)rH+2)("+I] 1/<1+..>

A

(5.25)

6.11. Turbulent Separation. To determine the turbulent separation point, detailed information is needed concerning the velocity distribution in the boundary layer and the effects on the -velocity distribution of such factors as the pressure distribution and the skin friction. Although the value of the shape parameter H can be found for any given velocity dis­ tribution, the value of H in itself does not define the velocity distribution. Thedata collected by Grusehwitz'" and by Tetervin-" show experimentally that the velocity distributions in turbulent boundary layers actually form a one-parameter family of curves and that the velocity distributions can be specified by the value of H. Velocity profiles for turbulent boundary layersl 35 are presented for various values of H in Fig. 48. The value of H increases as the separation point is approached. It is not possible to give an exact value of H corresponding to separation, because the turbulent separation point is not very well defined. The value of H varies so rapidly near the separation 'point, however, that it is not necessary to fix accurately the value of H corresponding to separation. Separation has not been observed for a value of H less than 1.8 and appears definitely to have been observed for a value of H of 2.6. An empirical expression was found 135 for the rate of change of H along the surface in terms of the ratio of the pressure gradient to the skin friction and the shape parameter itself.' The following equation was derived to fit the experimental data: 8 -dH = e4.680 (H dx

2.975)

Q2q - 2.03::>(H [8d ,.. - 1.286) ] - - - -

qdx

(5.26)

T

The skin friction corresponding to a given value of R6 may be obtained from Fig. 47 or from Eqs. (5.22) or (5.24). Equations (5.11) and (5.26) are

THEORY OF WING SECTIONS

104

simultaneous first-order differential equations that can be solved by step­ by-step calculation. It is usually necessary to use such a method, although, 8 7

J/.~9

1.0

! ~. d

.Q

r-

.

~

..

=­

'"U"'o

~

Q~

l7U

r'ii:Pc Cilt't

~

117'

-4

~

~~

~~

~

,

~~ ~ -.

.4

fP'~

..,

+

~ ~1(

~

o-Q

r4-...~

.

~ ~

-­

~ r--...

~~

.2

/2

/.6

1.4

.5 ~

~

s-;

N

~

,r--..

..

~

r--­ B"'-­ 'I

-~

~

~~

-

2.2

~

~AJ

K oht r--

)(~2

~~

-

0' ~ ~

2.0 H

1.8'

r-­

IOoz

~~

---.... r---..

~

--.

0

o

..,~

... 1>

---v-

Q

~~ A

It:.

~

~

04:

v

0

6

........

..

h.

n

r ..

I\IIl

~A ~

~~ ~

.6

bh

''''V

ft_

.. ~

.....

"U

61.

2 ........... ~

+ I

02

D3 ~

N

~~

~

rw

v4 F--

.

2.4

~5

46 I--

2.6

q

7

08 ~9

2.8

(0) • varia/ion 01 'rU wi/h H lor various values 01 ~

H ~r--"""--;

/.286 1---+--'--;---;r---1 A /.4

I:"

.,.....-..--....--.. 0 1--,~,r-,..,..e.,~'7"-:::~~~~--t--t--t--.-,t--t

1.5

a 1.6 0

tr

~~..,4.~~~~~,I'-+---+--t--t-~t-., + l8 x 1.9 1fU.~~~'#7~~-+--t---t---.,t---t---;r---t <J ~.O ~ 2./ ~~~~~t----t--+--"""""'--t----t--'t---t b.

2.2

~

2.3

~~o

'---ir---+--t---t--i---t---t---t---t 17

2.4

v 2.5

2.6 -0-2.7 6

~-+--t---+--t----+--t---+--t--t--1t-"""1-o-

1

2

4

~8

7

9

(b) - velocity profiles for turbulent botndory layers correspcnding 10 various KJlues of H FIG.

48.

Velocity diatributions in turbulent boundary layers.

for SOme particular cases, the equations may be integrated directly. The method of calculation is as follows. The values of the variables entering into the computation at the initial station are substituted in the momentum equation (5.11) and the equation for dH/dx (5.26). Values for d8/dx and dH/dx are thus obtained at the initial station. An increment of the length

THE EFFECTS

or

VISCOSITY

105

along the surface of the body x is then chosen and multiplied by d8/dx and dH /dx to give aO and 1::Jl, respectively. These increments of 8 and H are added to the initial values and result in values of 8 and H for the ne-w value of x, The process is repeated until the separation point is reached, as shown by the value of H. For the purposes of the computations the value of H corresponding to separation may be taken as 2.6. The choice of the increments of x is a matter for the judgment of the investigator. AB a general rule, the increment of x should be made small when d8/dx or dH/dx changes rapidly from one value of x to the next. 5.12. Transition from Laminar to Turbulent Flow. Experience has shown that transition from laminar to turbulent flow takes place at a Rey­ nolds number that depends upon the magnitude of the disturbances. For viscous flows at very low Reynolds numbers, as in oil, all disturbances are damped out by the viscosity, and the flow is laminar regardless of the magnitude of any disturbance. .As the Reynolds -number is increased, a condition is reached at which some particular types of disturbances are amplified and eventually cause transition. This value of the Reynolds number is called the U lower critical." Further increase of the Reynolds number causes amplification to occur for a greater variety of disturbances and increases the rate of amplification. Under these circumstances the Reynolds number at which transit.ion occurs depends 011 the magnitude of the disturbances. Transition can he delayed to high values of the Reynolds number only by reducing all disturbances such as stream turbu­ lence, unsteadiness, and surface roughness to a minimum. For ftO\VS in a pipe, the lower value of the critical Reynolds number based on the diameter and tho mean velocity is about 2,300.98 The upper value appears to depend only on the care taken in conducting the experiment, and values twenty times as great as the lower value have been obtained. I t has been found possible 99 • 100. 127,128 to compute the rate of amplification or damping of disturbances of various frequencies and wave lengths. Such computations's" permit. curves of constant amplification coefficient {:ji~* IV to be plotted on coordinates of all· against the Reynolds number R,. = p 1l8*/p. where a = 21rII and l is the wave length. Such a plot is given in Fig. 49. Experimental points obtained by Schubauer and Skramstad'P are also plotted in Fig. 49. The data of Fig. 49 were obtained for a Blasius profile corresponding to flow over a flat plate. It will be noted that there is a Reynolds number below which no disturbance will be amplified. This value of the Reynolds number is the lower critical. The agreement between the experimental and theoretical results is good considering the difficulties of the experimental work and the simplifying assumptions made in the development of the theory. Similar calculations have been made by Schlichting and Ulrich'?' for Pohlhausen's velocity profiles corresponding to various pressure gradients.

THEORY OF WING SECTIONS

106

The results are presented in Fig. 50, which is a plot of the lower critical Reynolds number against the parameter X that defines the velocity distri­ bution through the laminar layer. According to the Pohlhausen method, .44r--.....---.....--........----.---.....,.---,----.---,.---,--,-----...

I

.40

__9110-4

\

.36

\

\

-35

"

I

-/9

.32 . 28 .24­ a6* .20

./6 ./2 .08 .04

°0

400 800 /200

16fX)

2000 2400 2800

~ 3600 4000 4400

R Contours of equal amplification according to Schlichting. V &lUC8 of fji~· i U v (aU valuee to be multiplied by 10-4 ) opposite points are amplifications determined by experiment.. Faired experimental contour of zero amplification shown by broken curves. FIG.

49.

the velocity distribution through the layer is given by the following formula and A = p02dU

p,dx

Figure 5Q shows that decreasing pressures in the direction of flow are favorable to the stability of the laminar boundary layer and that increasing pressures are unfavorable.. The theory has not been developed to such a point that it can be used to predict transition, although it is useful in pointing out the fundamental reasons for the instability of the laminar layer at high Reynolds numbers. The theory does not relate the boundary-layer disturbances to either the

THE EFFECTS OF VISCOSITY

107

.surface irregularities or the turbulence of the air stream. The theory also does not relate the transition to the magnitude of the fluctuations of veloc­ ity in the boundary layer. Under these circumstances, it is necessary to

t--+--+--+--+--+--J~,--I 8

I

p(

I

V

t--~~--+---+I--"'I03jl+--+--+-~~-"""~~ I

I I

I

I

-6 -4

-2

I

I

'f

1/

0

2

4

6

8

50. Critical values of lU- as 11 function of the shape parameter '" aceording to Schlichting and Ulrich. I OI

FIG.

rely on empirical results to predict the location of transition. The empirical data are discussed in Chap. 7. 6.13. Calculation of Profile Drag. In cases where it is known that the. flow is not separated from the surface, the boundary-layer thickness at the trailing edge of a wing section and the profile drag can be calculated from a knowledge of the potential-flow pressure distribution and the location of transition from laminar to turbulent flow, The momentum thickness of the laminar boundary layer at the transition point can be calculated from Eq. (5.12). The initial momentum thickness of the turbulent boundary layer is taken to be the same as that of the laminar layer at the transition point. The thickness of the turbulent boundary layer at the trailing edge can be calculated from Eq. (5.25) . If the pressure at the trailing edge were free-stream static pressure, as in the case of a flat plate, the profile drag would be

THEORY OF WING SECTIONS

108 .0/6

x

Experimenta! meoSU:;menIs ~ R-2.9xlO T R=6.0xlOs

.0/4

~

-

lC/

Theoreticol colcu'r/J8"5

.0/2 -

--R=2.9x ----R-6'Ox106

I

/

.010

c>

/

,

~

~

ioo""~

"r;;'

/~/

iJl

x

\

+

\\

.006

J J.

'X

'"

~'

~-- --.-iIk

1-

,~~

+

o

.2

-t

.-:l~ ~t

.002

-.2

C,

.4

.8

.6

LO

FIG. 51. Comparison of calculated and experimentally measured polars for N.ACA 67,1-215 airfoil.

.014

.010

I

t

.012 '-

+~

~.008

~..........

~

.........

,

J

/ /'

¥

!~ "'--

...~

...... ~+~- ... .-:. ~

---

-~

~

+ Experimedol

.004

measurement -

.oO?:s

Theoretical

Or--'"

I---

f - - I---

colcu/olion

-.6

-.4

-.2

o

.2

Cz.

.4

.6

.8

1.0

/2

1.4

Comparison of calculated and experimentally measured polars for NACA ~015 airfoil at R = 5.9 X 10'.

FIG. 52.

where 8 = sum of momentum thicknesses on upper and lower surfaces In the usual case, the pressure at the trailing edge is not the same as free-stream static pressure, and some means must be found for finding the effective momentum thickness at a point far downstream in the wake

THE EFFECTS OF VISCOSITY

109

where the pressure has returned to the free-stream static value. Squire and Y oung'P derived the required relation by setting the value of T in Eq. (5.11) equal to zero in the wake and finding an empirical relation between Hand q/qo- The resulting expression for the profile drag is

(8)

ctl=2- (U~ - (Jl+fl)/2 c t 11 t

(5.27)

where the subscript t designates conditions at the trailing edge. The value of H is the value used in calculating the boundary-layer thickness. The agreement to be expected between experimental and calculated profile drag coefficients is indicated by Figs. 51 and 52. The calculated 14 --...r-­ r-­ drag coefficients presented in these .... 8 """'­ --. .............. ~ figures were obtained by a method" r-----. ........... A r---... r--­ fundamentally the same as that pre- ~ 12 r--­ senteel here. ¢ 1.0 5.14. Effect of Mach Number on Skin Friction. Solutions for the velocity distribution through the 080 1 4 2 3 5 laminar boundary layer in corn­ Mach number pressihle flow and for the correspond- FIG. 5~1. Skin-Iriction eoeffir-ienrs, ( ...4) No ing skin friction have been obtained heat transferred to wall. (Ill wall tern perature one-quarter of Iree-streurn temperature, by von Karman and Tsien1ft for the case of the flat plate. The results of the skin-friction calculations are pre­ sented in Fig. 53 for the case of no heat transfer to the plate and for the case of a plate whose absolute temperature is one-quarter of that of the free stream. These results show a moderate decrease in laminar skin friction with increasing Mach number. Although these results indicate that the laminar skin friction increases with beat transfer from the fluid to the plate, this increase is small even for the extremely 10"" plate temper­ atures for which these results were obtained. For the turbulent boundar)' layer, Theodorsen and Regier!" found ex­ perimentally that the skin friction is independent of the Mach number (at least up to a value of 1.69). The experimental data are presented in Fig. 54. These data were obtained by measuring the torques (moments) required to rotate smooth disks in atmospheres of air and Freon 12 (CCI2F2) at various pressures. Data obtained by Keenan and X eumannf for the skin friction in pipes appear to confirm Theodorsen's conclusion for the fully developed turbulent flow. Analytical studies by Leeg6% on the effect of Mach number and heat transfer on the lower critical Reynolds number for the case of the flat plate indicate that increasing th~ Mach number has a destabilizing in­ fluence on laminar flow when there is no heat transfer to the plate. Heat

---

THEORY OF WING SECTIONS

110

transfer to the plate has a stabilizing effect, while the opposite is the case for heat transfer from the plate to the fluid, This effect becomes stronger I

-1.2

I I I Moch P1essute (in.HgobsJ tvmber Q24toQ62 r-­ 1.8 fo.JO Air .4010 .96 Freon /2 30 t-­ 14 .5J 10/.42 FI1!OII 12 to.77101.69 Fr«Jfl 12 5

I

I

Symbol 0

-1.4

04­

x D

~ IIlloo....

~ ~ ~ ........ 100.... """"­

4.8

5.0

52

~

KDrrnDns -Ys

r--. ........

"""­

~~ 4 ~ KcrnD1s /ominar-f/oW"'/ i" ~ -2.2 fomvlo, C",=.J.871r~ ~,

-2·~.6

-

futtJu/enI-f/oN formub, /"" Lm CM=O.I46R

-~

"'"",,­

-2.0

[,I

I

60s

5.4

5.6

a -"'1fI­

M=O.53 ~=l69 ~

~ ~ ~.

~

5.8

6.0

6.2

6.4

....:..; ~ ....... + it-+

6.6

6.8

-ZO

LDgI()R FIG. M. Moment coefficient for disks as function of Reynolds number for several values of Mach number with air and Freon 12 as mediums, Maximum Mach number, 1.69.

as the Mach number is increased, and, at moderate supersonic speeds, comparatively small amounts of heat transfer to the plate stabilize the laminar layer to very high values of the Reynolds number.

CHAPTER 6 FAMILIES OF WING SECTIONS 6.1. Symbols. Ps resultant pressure coefficient R radius of curvature of surface of modified NAC.A four-digit series symmetrical sections at the point of maximum thickness IT velocity of the free stream a coefficient a mean-line designation; fraction of the chord from leading edge over which loading is uniform at the ideal angle of attack

e chord

ci. section design lift coefficient

c.e/. section moment coefficient about the quarter-chord point

d coefficient k1 constant m maximum ordinate of the mean line in fraction of the chord p ehordwise position of 111 r leading-edge radius in fraction of the chord r, leading-edge radius corresponding to thickness ratio t maximum thickness of section in fraction of chord v local velocity over the surface of a symmetrical section at zero lift !:Av increment01local velocity over the surface of a wing section associated 'lith camber Av.. increment of local velocity over the surface of a wing section associated with angle of attack % abscissa. of point on the surfaee of a symmetrical section or a chord line XL abscissa of point on the lower surface of tl. wing section Xu abseisaa of point on the upper surface of a wing section XC abscissa. of point on the mean line YL ordinate of point on the lower surface of a wing section Yu ordinate of point on the upper surface of a wing section Yc ordinate of point on the mean line y, ordinate of point on the surface of a symmetrical section Qi design angle of attack 8 tan-I (dyc/dzc ) T trailing-cdge angle

6.2. Int:"oduction. Until recently the development of wing sections has been almost entirely empirical. Very early tests indicated the desirability of a rounded leading edge and of a sharp trailing edge. The demand for improved wings for early airplanes and the lack of any generally accepted wing theory led to tests of large numbers of wings with shapes gradually improving as the result of experience. The Effie! and early RAF series were outstanding examples of this approach to the problem. 111

THEORY OF WING SECTIONS

112

The gradual development of wing theory tended to isolate the wing­ section problem from the effects of plan form and led to a more systematic experimental approach. The tests made at Gottingen during the First World War contributed much to the development of modem types of wing sections. Up to about the Second World War, most wing sections in common use were derived from more or less direct extensions of the 'york at Gottingen. During this period, many families of wing sections were tested in the laboratories of various countries, but the 'York of the NACA was outstanding. The NACA investigations were further systematized by separation of the effects of camber and thickness distribution, and the experi­ y ./0

°u(xy,Yy)

e -./0

'I p!!_.LJ_-_~-=---

~(XLIYLJ 'Rodius fhrough end of chord (mean line slopeOf a5 % chord)

Xy=X-Jf sin9 XL=X"'t

sinS

Yu=~

+11 cos9

JL 7C -JI cos e

Sample calculationA lor derivation 01 the NACA 86.3-818 airfoil (0 -= 1.0)

-

tan'

- ­ ---1----1

sin"

FIG. 56. Method of combining mean lines and basic-thickness forms.

mental work was performed at higher Reynolds numbers than were generally obtained elsewhere. The wing sections now in common use are either N .~CA sections or have been strongly influenced by the NACA investigations. For this reason, and because the NACA sections form consistent families. detailed attention will be given only to modem NACA wing sections. 8.3. Method of Combining Mean Lines and Thickness Distributions. The cambered wing sections of all NACA families of wing sections con­ sidered here are obtained by combining a mean line and a thickness distribution. The process for combining a mean line and a thickness distribution to obtain the desired cambered wing section is illustrated in Fig. 55. The leading and trailing edges are defined as the forward and

FAMILIES OF WING SECTIONS

113

rearward extremities, respectively, of the mean line. The chord line is defined as the straight line connecting the leading and trailing edges. Ordinates of the cambered lying sections are obtained by laying off the thickness distributions perpendicular to the mean lines. The abscissas, ordinates, and slopes of the mean line are designated as Xc, Ye, and tan 8, respectively. If Xu and Yu represent, respectively, the abscissa and ordi­ nate of a typical point of the upper surface of the wing section and y, is the ordinate of the symmetrical thickness distribution at ehordwise position %, the upper-surface coordinates are given by the following relations: Xv

= :r -

y, sin 9 }

yv = Yc + y, cos "

(6.1)

The corresponding expressions for the lower-surface coordinates are XL = X + Yt sin 8 } YL = Ye - Yt cos 8

(6.1)

The center for the leading-edge radius is found by drawing a line through the end of the chord at the leading edge with a slope equal to the slope of

the mean line at that point and laying off a distance from the leading edge along this line equal to t.he leading-edge radius. This method of con­ struction causes the cambered wing sections to project slightly forward of the leading-edge point. Beeause the slope at the leading edge is theoreti­ cally infinite for the mean lines having a theoretically finite load at the leading edge, the slope of the radius through the end of the chord for such mean lines is usually taken as t.he slope of the mean line at x/c equals 0.005. This procedure is ~ustified by the manner in which the slope

increases to the theoretically infinite value as x/c approaches o. The slope increases slowly until very small values of x/c are reached. Large values of the slope are thus limited to values x/c very close to 0 and may be neglected in practical wing-section design. ­

or

The data required to construct some cambered wing sections are pre­ sented in Appendixes I and II, and ordinates for a number of cambered sections are presented in Appendix lll. 6.4:. NACA Four-digit Vmg Sections. a. Thickness Distributions. "'hen the N ACA four-digit wing sections were derived.P it was found that the thickness distributions of efficient wing sections such as the Gottingen 398 and the Clark Y were nearly the same when their camber was removed (mean line straightened) and they were reduced to the same maximum thickness. The thickness distribution for the NAC~4\. four-digit sections was selected to correspond closely to that for these wing sections and is \' given by the following equation: ±'Yt,=_t- (O.29G90·~-O.12600x-O.35160x2+0.28430r-O.l0150zC)(6.2) 0.20

THEORY OF WING SECTIONS

114

where t == maximum thickness expressed as a fraction of the chord The leading-edge radius is r, = 1.1019t2

(6.3)

It will be noted from Eqs. (6.2) and (6.3) that the ordinate at any point is directly proportional to the thickness ratio and that the leading-edge radius varies 88 the square of the thickness ratio. Ordinates for thickness ratios of 6, 9, 12, 15, 18, 21, and 24 per cent are given in Appendix I. b. Mean Line«. In order to study systematically the effect of variation of the amount of camber and the shape of the mean line, the shape of the mean lines was expressed analytically as two parabolic arcs tangent at the position of maximum mean-line ordinate. The equations" defining the mean lines were taken to be m

Yt: :;: p;. (2px - x 2 )

forward of maximum ordinate

and

(6.4) Yc

=

m

(1 _ p)t [(1 - 2p)

+ 2px -

r]

aft of maximum ordinate

where m = maximum ordinate of mean line expressed as fraction of chord p == chordwise position of maximum ordinate It will be noted that the ordinates at all points on the mean line vary directly with the maximum ordinate. Data defining the geometry of mean lines with the maximum ordinate equal to 6 per cent of the chord are presented in Appendix II for chord,vise positions of the maximum ordinate of 20, 30, 40, 50, 60, and 70 per cent of the chord.. c.. Numbering System. The numbering system for NACA wing sections of the four-digit series is based on the section geometry. The first integer indicates the maximum value' of the mean-line ordinate Yc in per cent of the chord. The second integer indicates the distance from the leading edge to the location of the maximum camber in tenths of the chord. The last two integers indicate the section thickness in per cent of the chord.. Thus the NACA 2415 wing section has 2 per cent camber at 0.4 of the chord from the leading edge and is 15 per cent thick. The first two integers taken together define the mean line, for example, the NACA 24 mean line. Symmetrical sections are designated by zeros for the first two integers, as in the case of the NACA 0015 wing section, and are the thickness distributions for the family. d. Approximate Theoretical Characteristic8. Values of (V/V)2, which is equivalent to the low-speed pressure distribution, and values of v/V are presented in Appendix I for the NACA 0006, 0009, 0012, 0015, 0018, 0021, and 0024 wing sections at zero angle of attack. These values were cal­ culated by the method of Sec. 3.6. Values of the velocity increments AIl.IV induced by changing angle of sttaok are also presented for an

FAMILIES OF WING SECTIONS

115

additional Iift coefficient of approximately unity. Values of the velocity ratio v/V for intermediate thickness ratios may be obtained approximately by_linear scaling of the velocity Increments obtained from the tabulated values of v/V for the nearest thickness ratio; thus

(!!.V.)" = [(~) 1

II

1J ~ + 1 t1

(6.5)

Values of the velocity increment ratio ~vc/V may be obtained for inter­ mediate thicknesses by interpolation. The design lift coefficient. Cl i , and the corresponding design angle of attack oa, the moment coefficient Cm.lf) the resultant pressure coefficient P R , and the velocity ratio ~/V for the NAC.L~ &2, 63, 64, 65, 66, and 67 mean lines are presented in Appendix II. These values were calculated by the method of Sec. 3.6. The tabulated values for each mean line may be" assumed to vary linearly with the maximum ordinate Ye; and data for similar mean lines with different amounts of camber, within the usual range, may be obtained simply by scaling the tabulated values. Data for the NACA 22 mean line may thus be obtained simply by multiplying the data for the NACA 62 mean line by the ratio 2:6, and for the X.~CA 44 mean line by multiplying the data for the N.t\CA 64 mean line by the ratio 4:6. Approximate theoretical pressure distributions may be obtained for cambered lying sections from the tabulated data for the thickness forms and mean lines by the method presented in Sec. 4.5. 6.6. NACA Five-digit Wlng Sections. 4. Thickne&8 Distributions. The thickness distributions for the NAC.A~ five-digit wing sections are the same as for the NACA four-digit sections (see Sec. 6.4a). b. Mean. Lines. The results of tests of the N.&-\.CA four-digit series wing sections indicated that the maximum lift coefficient increased as the posi­ tion of maximum camber was shifted either forward or a~t of approximately the mid-chord position. The rearward positions of maximum camber were not of much interest because of large pitching-moment coefficients. Be­ cause the type of mean line used for the NACA four-digit sections was not suitable for extreme forward positions of the maximum camber, a new series of mean lines was developed, and the resulting sections are the NACA five-digit series. . The mean lines are defined 46 by two equations derived so as to produce shapes having progressively decreasing curvatures from the leading edge aft. The curvature decreases to zero at a point slightly aft of the position of maximum camber and remains zero from this point to the trailing edge. The equations for the mean line are Yc = ~kl[x3 - 3mx2 + m2(3 - m)x] from x = 0 to z = ml y.; = %k1m3(l - x) from x = m to oX = c = l~

(6 6) .

THEORY OF WING SECTIONS

116

The values of m were determined to give five positions p of maximum camber, namely, 0.000, O.IOc, O.ISe. O.2Oc, and O.25c. Values of k1 were initially calculated to give a design lift coefficient of 0.3. The resulting values of p, m, and k1 are given in the following table. Mean-line designation

Position of camber p

""

kl

210

0.05 0.10 0.15 0.20 0.25

0.0580 0.1260 0.2025 0.2900 0.3910

361.4 51.64 15.957 6.643

220

230 240

250

3.230

This series of mean lines was later extended" to other design lift coefficients by scaling the ordinates of the mean lines. Data for the mean lines tabulated in the foregoing table are presented in Appendix II. c. Numbering System. The numbering system for wing sections of the N..\ CA five-digit series is based on a combination of theoretical aerodynamic characteristics and geometric characteristics. The first integer indicates the amount of camber in terms of the relative magnitude of the design lift coefficient; the design lift coefficient in tenths is thus three-halves of the first integer. The second and third integers together indicate the distance from the leading edge to the location of the maximum camber; this distance in per cent of the chord is one-half the number represented by these .-integers. The last two integers indicate the section thickness in per cent of the chord. The NACA 23012 wing section thus has a design lift coefficient of O.3 J has its maximum camber at 15 per cent of the chord: and has a thickness ratio of 12 per cent. d. Approximate Theoretical Characteristics. The theoretical aerody­ namic characteristics of the N ACA five-digit series wing sections may be obtained by the same method as that previously described (Sec. 6.4<1) for the NACA four-digit series sections using the data presented in Appen­ dixes I and II. 6.6. Modified NACA Four- and Five-digit Series Wing Sections. Some early modifications of the NACA four-digit series wing section· included thinner nosed and blunter nosed sections which were denoted by the suffixes T and B, respectively. Some sections of this family49 with refiexed mean lines were designated by numbers of the type 2R112 and 2R212, where the first integer indicates the maximum camber in per cent of the chord and the subscripts 1 and 2 indicate small positive and negative moments, respectively. Another series of N ACA five-digit wing sections" is the same as those previously described except that the mean lines are reflexed to produce theoretically zero pitching moment, These sections are distinguished by

FAMILIES OF JV[.;V'G SECTIONS

117

.he third integer, which is always 1 instead of o. These modified sections lave been little used and will not be discussed further. More important modifications common to both the NACA four-digit md five-digit series wing sections consisted of systematic variations of the hickness distributions. 1M These modifications are indicated by a suffix -onsisting of a dash and two digits as for the NACA 0012-64 or the 23012-64 sections, These modifications consist essentially of changes of the leading­ edge radius from the normal value [Eq. (6.3)] and changes of the position of maximum thickness from the normal position at 0.3Oc [Eq. (6.2)]. The first integer following the dash indicates the relative magnitude of the leading-edge radius. The normal leading-edge radius is designated by 6 and j, sharp leading edge by o. The leading-edge radius varies as the square of this integer except for values larger than 8, when the variation becomes ]xbitrary.U6 The second integer following the dash indicates the position of maximum thickness in tenths of the chord. The suffix -63 indicates sec­ tions very nearly but not exactly the same as the sections without the suffix. The modified thickness forms are defined by the following two equations:

± y,

:=

aov'X + alX + ll2X2 + a~

ahead of maximum thickness

} () 7'

:l:Yt==do+dl(1-x)+d2(1-x)~d3(1-x)3 aft of maximum thickness (.)

The four coefficients d-o, dt , d~, and ds arc determined from the following conditions: 1. Maximum thickness, t. 2. Position of maximum thickness. In. 3. Ordinate at the trailing edge x = 1, y, = O.Olt. 4. Trailing-edge angle, defined by the following table: m

dl/ --:., (x

dx

0.2 0.3 0.4 0.5 0.6

= 1)

1.000t

1.170l 1.575t 2.325t

3.500t

The four coefficients 00, at, a2, and U3 are determined from the following conditions: 1. Maximum thickness, t. 2. Position of maximum thickness. m, 3. Leading-edge radius, r, == ~2/2. T, = 1.1019(tI/6)2, where 1 is the first integer following the dash in the designation and the value of I does not exceed 8. 4. Radius of curvature R at-the point of maximum thickness

R=

1

2d: + 6d3(1 -

11t)

118

THEORY OF WING SECTIONS

Data" far some of the modified thickness forms are presented in Appendix I. The data presented may be used together with data for the desired type of mean line to obtain approximate theoretical characteristics by the method presented in Sec. 4.5. The family of N ACA wing sections defined by these equations has been studied extensively by German aerodynamicists, who have applied designa­ tions of the following type to these sections: NACA

1.8

25

14-1.1

30/0.50

where 1.8 = maximum camber in per cent of chord 25 = location of maximum camber in per cent of the chord from the leading edge 14 = maximum thickness in per cent of the chord 1.1 = leading-edge radius parameter, r/t,2 30 = location of maximum thickness in per cent of the chord from the leading edge 0.50 == trailing-edge angle parameter, (l/t)(tan T/2) and t = thickness ratio T = leading-edge radius, fraction of chord T = trailing-edge angle (included angle between the tangents to the upper and lower surfaces at the trailing edge) For wing sections having this designation, the selected values of the leading-edge radius and of the trailing-edge angle are used directly with the other conditions in obtaining the coefficients of Eq. (6.7)~ The selected values for the maximum camber and location of maximum camber are applied to Eq. (6.4) to obtain the corresponding mean line. 6.7. NACA l-series W-mg Sections. The NACA l-eeries wing sec­ tionsG- 113 represent the first attempt to develop sections having desired types of pressure distributions and are the first family of NACA low-drag high-critical-speed wing sections. In order to meet one of the require­ ments for extensive laminar boundary layers, and to minimize the induced velocities, it was desired to locate th~ minimum-pressure point unusually far back on both surfaces and to have a small continuously favorable pressure gradient from the leading edge to the position of minimum pres­ sure. The development of these wing sections (prior to 1939) was ham­ pered by the lack of adequate theory, and difficulties ,,:ere experienced in obtaining the desired pressure distribution over more than a very limited range of lift coefficients. As compared with later sections, the N .~C.A. I-series wing sections are characterized by sOla).lleading-edge radii, com­ paratively large trailing-edge angles, and slightly higher critical speeds for a given thickness ratio. These sections have proved useful for pro­ pellers. The only commonly used sections of this series have the minimum pressure located at 60 per cent of the chord from the leading edge, and data "ill be presented only for these sections.

FAMILIES OF WING SECTIONS

119

o. Thickne88 Distributions. Ordinates for thickness distributions with the minimum pressure located at 0.6c and thickness ratios of 6, 9, 12, 15, 18, and 21 per cent are presented in Appendix I. These data are similar in form to those presented for the KACA four-digit series. These sections were not developed from any analytical expression. The ordinate at any station is directly proportional to the thickness ratio, and sections of intermediate thickness may be correctly obtained by scaling the ordinates. b. Mean Lines. The NACA l-series wing sections, as commonly used, are cambered with a mean line of the uniform-load type [Eq. (4.26)]. Data for the mean line are tabulated in Appendix II. Thi, type of mean line was selected because, at the design lift coefficient, it does not change the shape of the pressure distribution of the symmetrical section at zero lift. This mean line also imposes minimum induced velocities for a give~ design lift coefficient. c. Numbering System. The N.A.CA l-series wing sections are designated by a five-digit number as, for example, the N ACA 16-112 section. The first integer represents the series designation. The second integer represents the distance in tenths of the chord from the leading edge to the position of minimum pressure for the symmetrical section at zero lift. The first Dum­ ber following the dash indicates the amount of camber expressed in terms of the design lift coefficient in tenths, and the last t\VO numbers together indicate the thickness in per cent of the chord. The commonly used sections of this family have minimum pressure at 0.6 of the chord from the leading edge and are usually referred to as the Nl\.CA 16-series sections. d. Approximate Theoretical CharacleristiC8. The theoretical aerody­ namic characteristics of the NACA l-series wing sections may be obtained by application of the method of Sec. 4.5 to the data of Appendixes I and II by the same method as that described for the NACA four-digit wing sections (Sec. 6.4d). 6.8. KACA &-series Wmg SectioDS. Successive attempts to design wing sections by approximate theoretical methods led to families of wing sections designated NACA 2- to 5-series sections." Experience with these sections showed that none of the approximate methods tried was sufficiently accurate to show correctly the effect of changes in profile near the leading edge. Wind-tunnel and flight tests of these sections showed that extensive laminar boundary. layers could be maintained at comparatively large valdes of the Reynolds number if the wing surfaces were sufficiently fair and smooth. These tests also provided qualitative information on the effects of the magnitude of the favorable pressure gradient, leading-edge radius, and other shape variables. The data a~ showed that separation of the turbulent boundary layer over the rear of the section, especially with rough surfaces, limited the extent of laminar layer for which the lying sections should be designed. The wing sections of these early families generally showed relatively low maximum lift coefficients and, in many

120

THEORY OF WING SECTIONS

cases were designed for a greater extent of laminar flow than is practical. It w~ learned that, although sections designed for an excessive extent of laminar flow gave extremely low drag coefficients near the design lift coefficient when smooth, the drag of such sections became unduly large when rough, particularly at lift coefficients higher than the design value. These families of lying sections are accordingly considered obsolete. The NACA 6-series basic thickness forms were derived by new and improved methods described in Sec. 3.8, in accordance with design criteria established with the objective of obtaining desirable drag, critical Mach number, and maximum-lift characteristics. a. Thick"neB8 Distributions. Data for NACA 6-series thickness dis­ tributions covering a wide range of thickness ratios and positions of minimum pressure are presented in Appendix I. These data are com­ parable with the similar data for wing sections of the NACA four-digit series (Sec. 6.4a) except that ordinates for intermediate thickness ratios may not be correctly obtained by scaling the tabulated ordinates proportional to the thickness ratio. This method of changing the ordinates by a factor w ill, however, produce shapes satisfactorily approximating members of the family if the change of thickness ratio is small. b. Mean Lines. The mean lines commonly used with the NAC_.\. 6-series wing sections produce a uniform chordwise loading from the leading edge to the point x/c = a, and a linearly decreasing load from this point to the trailing edge. Data for NACA mean lines with values of a equal to 0, 0.1, 0.2, 0.3, 0.4, 0.5,0.6, 0.7, 0.8, 0.9, and 1.0 are presented in Appendix II. The ordinates were computed from Eqs. (4.26) and (4.27). The data are presented for a design lift coefficient e" equal to unity. All tabulated values vary directly with the design lift coefficient. Corresponding data for similar mean lines with other design lift coefficients may accordingly be obtained simply by multiplying the tabulated values by the desired design lift coefficient. In order to camber NACA 6-series wing sections, mean lines are usually used having values of a equal to or greater than the distance from the leading edge to the location of minimum pressure for the selected thickness distributions at zero lift. For special purposes, load distributions other than those corresponding to the simple mean lines may be obtained by combining two or more types of mean line having positive or negative values of the design lift coefficient. c. NtUmbering System. The NACA 6-series wing sections are usually designated by a six-digit number together with a statement showing the type of mean line used. For example, in the designation NACA 65,3-218, a == 0.5, the 6 is the series designation. The 5 denotes the chordwise posi­ tion of minimum pressure in tenths of the chord behind the leading edge for the basic symmetrical section at zero lift. The 3 following the comma

FAJIILIES OF n'IlVG SECTIONS

121

ives the range of lift coefficient in tenths above and below the design lift oefficient in which favorable pressure gradients exist on both surfaces. 'he 2 following the dash gives the design lift coefficient in tenths. The .LSt two digits indicate the thickness of the lying section in per cent of the horde The designation a = 0.5 shows the type of mean line used. When he mean-line designation is not given, it is understood that the uniform­ oad mean line (a = 1.0) has been used. When the mean line used is obtained by combining more than one mean ine, the design lift coefficient used in the designation is the algebraic sum )f the design lift coefficients of the mean lines used, and the mean lines are Jescribed in the statement following the number as in the following case:

{a

cs,

NAC.A. 65 3-218 = 0.5 = 0.3 } , a = 1.0 ci, = - 0.1 Wing sections having a thickness distribution obtained by linearly increasing or decreasing the ordinates of one of the originally derived thick­ ness distributions are designated as in the following example: N.~CA 65(318)-217 a = 0.5 The significance of all the numbers except those in the parentheses is the same as before. The first number and the lust two numbers enclosed in the parentheses denote, respectively, the low-drag range and the thickness in per cent of the chord of the originally derived thickness distribution. The more recent XACA 6-series wing sections are derived as members of thickness families having a simple relationship between the conformal transformations for 'ling sections of different thickness ratios but having minimum pressure at the same chordwise position. These wing sections are distinguished from the earlier individually derived wing sections by writing the number indicating the Jew-drag range as a subscript, for example, N4~C.A. oor 218 a == 0.5 Ordinates for the basic thickness distributions designated by a subscript are slightly different from those for the corresponding individually derived thickness distributions. As before, if the ordinates of the basic thickness distributions arc changed by a factor, the low-drag range and thickness ratio of the original thickness distribution are enclosed in parentheses as Iollows:

X.:\C..\ 65(311)-217

a = 0.5

For wing sections having a thickness ratio less than 12 per cent, the low-drag range is less than 0.1 and the subscript denoting this range is omitted from the designation, thus N .t\C..-\ 65-210 or

122

THEORY OF WING SECTIONS

The latter designation indicates that the 11 per cent thick section was obtained by linearly scaling the ordinates of the 10 per cent thick sym­ metrical section. If the design lift coefficient in tenths or the thickness of the wing section in per cent of the chord are not whole integers, the numbers giving these quantities are usually enclosed in parentheses as in the following example: NACA 65(318)-(1.5)(16.5)

a == 0.5

Some early experimental wing sections are designated by the insertion of the letter x immediately preceding the dash as in the designation 66,22:-015. Some modifications of the NACA 6-series sections are designated by replacing the dash by a capital letter, thus NACA 641A212 -In this case, the letter indicates both the modified thickness distribution and the type of mean line used to camber the section. Sections designated by the letter A are substantially straight on both surfaces from about 0.8c to the trailing edge. d. Approximate Theoretical Characteri8ticB. Approximate theoretical characteristics may be obtained by applying the methods of Sec. 4.5 to the data of Appendixes I and II as described for the NACA four-digit series sections in Sec. 6.4d. If t\VO or more of the simple mean lines are com­ bined to camber the desired section, data for the resulting mean line may be obtained by algebraic addition of the scaled values for the component mean lines. 8.9. NACA 7-series W-mg Sections. The NACA 7-series wing sections are characterized by a greater extent of possible laminar flow on the lower than on the upper surface. These sections permit low pitching-moment coefficients with moderately high design lift coefficients at the expense of some reduction of maximum lift and critical Mach number. The N ACA 7-series 'ling sections are designated by a number of the following type: NACA 747...\315 The first number 7 indicates the series number. The second number 4 indicates the extent over the upper surface, in tenths of the chord from the le&ding edge, of the region of favorable pressure gradient at the design lift coefficient. The third number 7 indicates the enent over the lower surface, in tenths of the chord from the leading edge, of the region of favorable pressure gradient at the design lift coefficient. The significance of the last group of three numbers is the same as for the NAC..~ 6-series wing sections. The letter A which follows the first three numbers is 8 seria1letter to distinguish different sections having parameters that would

FAMILIES OF WING SEC7

1/ONS

123

correspond to the same numerical designation. For example, a second' section having the same extent of favorable pressure gradient over the upper and lower surfaces, the same design lift coefficient, and the same thickness ratio 88 the original lying section but having a different mean­ line combination or thickness distribution would have the serial letter B. Mean lines used for the NACA 7-series sections are obtained by combining t\VO or more of the previously described mean lines. The basic thickness distribution is given a designation similar to those of the final cambered wing sections. For example, the basic thickness distribution for the NACA 747A315 and 747A415 sections is given the designation NACA .747A015 even though minimum pressure occurs at O.4c on both the upper and lower surfaces at zero lift. Data for this thickness distribution are presented in Appendix I. The NACA 747A315 lying section is cambered with the following com­ bination of mean lines: a == 0.4 { a == 0.7

e'i == 0.763 } eli == - 0.463

The NACA 747..\415 wing section is cambered with the following com­ bination of mean lines: a == 0 4 . a == O~7 { a == 1.0

C" == 0.763 } ci, = - 0.463 c.; = 0.100

8.10. Special Combinations of Thickness and Camber. The methods presented for combining thickness distributions and mean lines are suffi­ ciently flexible to permit combining any thickness distribution, regardless of family, with any mean line or combination of mean lines. Approximate theoretical characteristics of such combinations may be obtained by ap­ plication of the method of Sec. 4.5 to the tabulated data. In this manner, it is possible to approximate a"'considerable variety of pressure distributions without deriving new lying sections.

CHAPTER 7 EXPERIMENTAL CHARACTERISTICS OF WING SECTIONS 7.1. Symbols. R U V

Reynolds number, pVc/p. velocity just outside the boundary layer velocity of the free stream a mean-line designation; fraction of the chord from leading edge over which loading is uniform at the ideal angle of attack ac aerodynamic center c chord

Cd section drag coefficient

minimum section drag coefficient c/ local skin-friction coefficient c, section lift coefficient ci, design lift coefficient ClanR'I: maximum section lift coefficient section pitching-moment coefficient about the aerodynamic center c../4 . section pitching-moment coefficient about the quarter-chord point k height roughness t time ~ abscissa measured from the leading edge 11 ordinate measured from the chord line Go section angle of attack

" angle of flap deflection

p. viscosity of air

p mass density of air

c.r.m.

e_.

of

7.2. Introduction. The theories presented in Chaps. 3 to 5 permit reasonably accurate calculations to be made of certain characteristics of wing sections, but the simplifying assumptions made in the development of these theories limit their over-all applicability. For instance, the perfect­ fluid theories of Chaps. 3 and 4 permit the pressure distribution to be calcu­ lated," provided that the effects of the boundary-layer flo,v on the pressure distribution are small. Similarly the viscous theories of Chap. 5 permit the calculation of some of the boundary-layer conditions, provided that the pressure distribution is known, Obvious interactions between the viscous and potential flows are the relatively small effective changes of body shape caused by. the displacement thickness of the unseparated boundary layer and the seriously large changes caused by separation. N one of the wing characteristics can be calculated with confidence if the flow is separated 124

EXPERIMENTAL CHARACTERISTICS OF WING SECTIONS

125

over an appreciable part of the surface. Under these circumstances, wing­ section characteristics used for design are obtained experimentally. Wind-tunnel investigations of wing characteristics were made before airplanes were successfully flown and still constitute an important phase of aerodynamic testing. Until recently wing characteristics were usually obtained from tests of models of finite aspect ratio, The development of lying theory led to the concept of wing-section characteristics that were derived from data obtained from tests of finite-aspect-ratio wings, These derived data were then used to predict the characteristics of wings of different plan forms. The systematic investigations in the N'AC..~ variable­ density wind tunnel46, 48, 49 were examples of this type of investigation. This method of testing is hampered by the difficulties of obtaining full-scale values of the Reynolds number and sufficiently low air-stream turbulence to duplicate flight conditions properly without excessive cost for equip­ ment and models. Other difficulties were experienced in properly correcting the data for the support tares and interference effects and in deriving the section characteristics from tests of models necessarily having varied span­ load distributions and tip effects.Go In order to avoid some of these difficulties and to permit testing of models that are large relative to the size of the wind tunnel, two-dimensional testing equipment was built by the X.t\C.i\. The X.:\(~..\ two-dimensional low-turbulence pressure tunneP33 provides facilities for testing wing sections in two-dimensional flowat large Reynolds numbers in an air stream of very )O\V turbulence, approaching that of the atmosphere. The wing-section data presented here were obtained from tests in this tunnel. This tunne}1D has a test section 3 feet wide and 7~~ feet high and is capable of operation at pressures up to 10 atmospheres. The usual models are of 2-foot chord and completely span the 3-foot width of the test section. The lift is measured by integration of pressures representing the reaction on the floor and ceiling of the tunnel. The drag is obtained from wake­ survey measurements, and the pitching moments are measured directly by a balance. "~ing-section characteristics can be obtained from such measurements with a high degree of accuracy. The usual test.s were made over a range of Reynolds numbers from 3 t.o 9 million and at Mach numbers less than about 0.17.· This range of Reynolds numbers covers the range where large-scale effects are usually experienced between the usual low-seale test data and the large-scale flight range. Individual tests have been made to provide some data at lower Reynolds numbers applicable to small personal airplanes and at much larger Reynolds numbers to indicate trends for very large airplanes­ The tunnel turbulence level is very low, of the order of a few hundredths of 1 per cent; and, although it is not definitely known that the remaining turbulence is negligible, the tunnel results appear to correspond closely to

126

THEORY OF WING SECTIONS

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56. Measured section angles of aero lift for a number of N..t\CA

127

EXPERIMENTAL CHARACTERISTICS OF WING SECTIONS

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EXPERIMENTAL CHARACTERISTICS OF WING SECTIONS

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127

THEORY OF WING SECTIONS

128

those obtained in flight. Application of these wing-section data to the prediction of the characteristics of wings of finite span depends on the adequacy of three-dimensional \V~g theory. These data are not applicable at high speeds where compressibility effects become important. 7.3. Standard Aerodynamic Characteristics. The resultant force on a "ring section can be specified by t"90 components of force perpendicular and parallel to the air stream (the lift and drag, respectively) and by a moment in the plane of these t\VO forces (the pitching moment). These forces are functions of the angle of attack of the section. The standard method of presenting the characteristics of 'ling Sections is by means of plots of the lift, drag, and moment coefficients against angle of attack or, alternately, plots of angle of attack, drag, and moment coefficients against lift coefficient. Plots of wing-section characteristics are presented in Appendix IV for a wide range of shape parameters. On the left-hand side of each plot, the lift coefficient and the moment coefficient about the quarter-chord point are plotted against the angle of attack. On the right-hand side of each plot the drag coefficient and moment coefficient about the aerodynamic center are plotted against the lift coefficient. In most cases, the data indicated in the following table are presented. Surface condition

Charneteristie

Split flap deflection degrees

Reynolds number, millions

Left-hand side Lift tift Lift

. . .

I4Iift..•......... Moment

.

~Io~ent

.

Smooth Rough­ Smooth Rough­ Smooth Smooth

0 0 60 60 0 60

3,6,9 6 6 6 3,6,9 6

0 0 0

3,6,9 6 3,6,9

Right-hand side

Drag Drag Moment

. . .

Smooth Rough" Smooth

*O.011-ineh amin carborundum lipread tbinJy to cover 5 to 10 per cent of the area from the leadinc ed«e to 0.0& alung boUa surfaces of a IleCtion wit.h a chOld of 24 inchee.

7.4:. Lift Characteristics. 4. Angle of Zero lift. As indicated in Chap. 4, the angle of zero lift of a wing section is largely determined by the camber. The theory of wing sections provides a means for computing the angle of zero lift from the mean-line data presented in Appendix II. The agreement between the calculated and the experimental angles of zero lift

EXPERIMENTAL CHARACTERISTICS OF WING SECTIONS

129

depends on the type of mean line used. Comparison of the theoretical data given in Appendix II 'lith the experimental data of Appendix IV shows that the agreement is good except for the uniform-load type (a = 1) of mean line. The angles of zero lift for this type of mean line are generally closer to 0 degrees than predicted. The experimental values of the angles of zero lift for a number of NACA four.. and five-digit and NACA C)-series wing sections are presented in Fig. 56. The thickness ratio of the wing section appears to have little effect on the angle of zero lift regardless of the type of thickness distribu­ tID

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; -4 II

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~ 8 12 16 Alrton thicknes.. percent

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(e) MACA 66- .eries. FIG.

56. (Cond1ldal)

tion or camber. For the NACA four-digit series wing sections, the angles of zero lift are approximately 0.93 of the value given by the theory of thin wing sections. For the NACA 23o-serics wing sections, this factor is approximately 1.08; and for the X.\Cl\ f.)-~(~r ies sections with the uniform­ load type of mean line, this factor is approximately 0.74. b. L~Jt-curve Slope. Lift-curve slopes fOI" a number of NACA four- and five-digit series and KACA fJ-8Cri~ wing ~~·t.jons are plotted against thick­ ness ratio in Fig. 57. These values of the lift-curve slope were measured for a Reynolds number of 6 million at vulues of the lift coefficient approxi­ mately equal to the design lift coefficient of the wing sections. This lift coefficient is approximately in the center of the low-drag range for the NACA f>-Series wing sections. In the range of thickness-ratios from (i to 10 per cent, the NACA four­ and five-digit series and the XAC.-\ 64-series wing sections have values of the lift-curve slope very close to the value given by the theory of thin wing sections (2... per radian, or 0.110 per degree). Variation of Reynolds Dum­

THEORY OF WING SECTIONS

130

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131

EXPERIMENTAL CHARACTERISTICS OF WING SECTIONS

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22

THEORY OF WING SECTIONS

132

ber between 3 and 9 million and variation of camber up to O.04c appear to have no systematic effect on the lift-curve slope. The thickness distribu­ tion appears to be the primary variable. For the NACA four- and five­ digit series wing sections, the lift-curve slope decreases with increase of thickness ratio. For the NACA 6-series wing sections, however, the

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rl~~

" 44-5eries

24-Series (4-Digif) -

I

~

~

~j /

PkJin >airfoil

/ ~ ~~ ---:

-

~)

--fbJgh

--5rnodh I J I Symbols with fJogs correspond 10 sinllbfed splil fbp deflected 60-

.4

I

00

4

8

I

I /2

I

I

I

16

Airfoillhick~pen;:enI

20

24

of chotrI

(a) N ACA four- and five-digit series. 58. Variation of muimum section lilt coefficient with airfoil thickness ratio and camber for .veral NACA airfoil sections with and without simulated split ftaps and standard rough­ ness. B, 6 X loa. FIG.

lift-curve slope increases with increase of thickness ratio and with forward movement of the position of minimum pressure on the basic thickness form. The effect of thickness ratio is comparatively small for the NAC.\ 66-series sections. The thick-wing-section theory (Chap. 3) shows that the slope of the lift curve should increase with increasing thickness ratio in .he absence of viscous effects. For wing sections with arbitrary modifica­ tions of shape near the trailing edge, the lift-curve slope appears to decrease with increasing trailing-edge angle. Some KACA6-series wing sections show jogs in the lift curve at the end

.

EXPERl}.{ENTAL CHARACTERISTICS OF WING SECTIONS

133

of the low-drag range, especially at lo,v Reynolds numbers. This jog be­ comes more pronounced with increase of camber or thickness ratio and with rearward movement of the position of minimum pressure on the basic thickness form. This jog decreases rapidly in severity with increasing Reynolds number, becomes merely a change of lift-curve slope, and is practically nonexistent at a Reynolds number of 9 million for most lying I

-

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~

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.... L:..-

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ft

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......,s:: o

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:

C&l

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L.. iIl.............

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..... ~

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~-

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Plain

a1rt'o11

-6

~

\ \\ ~ '"' \\ ~'( -vo.6-4 C~l

8IDOotb

\

\.- -6

~

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I

ltouab

I--SJlDbola with 1"l-p correepood ~o 8!'-w..ated- ap11t 1'1ap :Serlect~d 600

o o

A1rt'oll W1tb ep11t. ~J.ap

~ 8 12 16 A1rtoll th1ckDe... perceD t or (6) NAC.A 63-series. }"'IG. 58. (Continued)

20

Cbor4

sections that would he considered for practical application. This jog may he a consideration in the selection of wing sections for small low-speed airplanes. An analysis of the flow conditions leading to this jog is pre­ sen ted ill reference 134. The values of the Iift-curve slopes presented are for steady conditions and do not necessarily correspond to the slopes obtained in transient con­ ditions when the boundary layer has insufficient time to develop fully at each lift coefficient. Some experimental resultglOG indicate that variations of the steady value of the lift-curve slope do not result in similar variations of the gust loading,

THEORY OF WING SECTIONS

134

c. Mazimum Lift. The variation of maximum lift coefficient with thickness ratio at a Reynolds number of 6 million is shown in Fig, 58 for a considerable number of NACA wing sections. The sections for which data are presented in this figure have a range of thickness ratios from 6 to 24 per cent and cambers up to 4 per cent of the chord. From the data for the NACA four- and five-digit wing sections (Fig. 58a), it appears that the

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Airtoll wltn split flap

'°1 1

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with tlaS8 correspond to ted .pllt flap detlected 600

"

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.1

-00

8 12 k 16 20 A1rto11 thiCknes., percent or chord (c) NACA 64-eeriea. FIG.

58.

(Continued)

maximum lift coefficients are the greatest for a thickness ratio of 12 per cent. In general, the rate of change of maximum lift coefficient with

thickness ratio appears to be greater for sections having a thickness ratio less than 12 per cent than for the thicker sections. The data for the KACA 6-seri~ sections (Figs. 58b to d) show a rapid increase of maximum lift coefficient with increasing thickness ratio for thickness ratios less than 12 per cent. The optimum thickness ratio for maximum lift coefficient increases with rearward movement of the position of minimum pressure and decreases with increase of camber. For wing sections having thickness ratios of 6 per cent and for wing sections having thickness ratios of 18 or 21 per cent, the maximum lift

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Airfoll with spllt.flap 2.0 ..---+--f--I---4-~-+-~~ ........~~~~~

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.... 1.6 ~

rot

g :: o

••

J

Plain airfoil

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i

t--4---t-~--+-~-+------t-~-.-_Smooth - - + - - + - - - + - - t __ _ flougl1

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SJlDbola with tlasa correspond to 0 .1aIula~e4 spllt rlap d.flected ·60

I&. 8 12 J.6 20 Airtoil thlCkne.-;, percent or chord (tl) NACA 65-series. FIG. 58. (Continued)

2.8

Cl

l

no.q- -'"

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.

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8 12 16 20 A1rf'oll th1ckD.... perc.ati or chord (e) NACA 66-eeriea. FIG. 58. (C01&eluded)

J&.

135

~

THEORY OF WING SECTIONS

136

coefficients do not appear to be very much affected by the position of minimum pressure on the basic thickness form. The maximum lift coeffi­ cients of sections of intermediate thickness, however, decrease with rear­ ward movement of the position of minimum pressure. The maximum lift coefficients for the NACA 64-series wing sections cambered for "a design lift coefficient of 0.4 are slightly higher than those for the NACA 44-series 2.0

..I 1.' ...•.. ...'"' 0

a

C)

I0

1.2

... ~

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.8

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., 2Jpe of o-.ber.

.8 a

59. Variation or maximum lift coefficient with type of camber lor some NACA 65a-418 airfoil sections from testa in the Langley two-dilnensional low-turbulcnce pressure tunnel.

FIG.

sections. The NACA 230-series sections, however, have somewhat higher maximum lift coefficients. The maximum lift coefficients of moderately cambered sections increase with increasing camber (Fig. 58). For wing sections of about 18 per cent thickness, the rate of increase of maximum lift coefficient with camber is largest for small cambers. The effect of camber in increasing the section maximum lift coefficient becomes progressively less as the thickness ratio increases above 12 per cent. The variation of maximum lift with type of camber is shown in Fig. 59 for one condition. No systematic data are available for mean lines with values of a less than 0.5. It should be noted, however, that lying sections such as the NACA 230-series with the maximum camber far forward show large increments of maximum lift as compared with symmetrical sections. Wipg sections with the maximum camber far

137

EXPERIAIENTAL CHARACTERISTICS OF lVING SECTIONS

forward and with normal thickness ratios stall from the leading edge with large sudden losses of lift. A more desirable gradual stall is obtained when 2.0

..

'----. -.... -a... <, -......

1.6

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.8

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.

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e

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MACA 24-serlea (4 digit)

c ~ 0

w4 .... ....

8 0

1.6

~ ~'

.p

R

G 9.0 x 106

ir

~ ....

1.2

s::

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.8

i

1.6

V

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J

JlACA

e 6.0 Q

V

'.0

at. Stendard rougbne,s

14-•• rl•• (4

6 )( lOb dl&1t)

R

G 9.0 x

s JV

1.2

I.{/

.r

1&.

--A

a 6.0 e

'.0

A StaDdard

l'OUSbDeaa

MACA CO-aeries

8 12 16 Alrto1l thickness. percent (a)

106

(4-

d1a1t)

6 x

106

20

or chord

MACA tour-dlg1 t aeries.

60. Variation of maximum section lift coefficient wid. airfoil thickness ratio at several Reynolds numbers for a number of NACA airfoil sections of different cambers.

FlO.

the location of maximum camber is farther back, as with the NACA 24-, 44-, and 6-series sections with normal types of camber. The variations of maximum lift coefficient with Reynolds number (scale effect) of a number of XACA wing sections are presented in Fig. 60 for a range of Reynolds numbers from 3 to 9 million. The scale effect for the NACA 24-, 44-, and 23D-series wing sections (Figs. 60a and b) having

THEORY OF WING SECTIONS

138

2.0

- ---" ... ~ P) ~ ~

l~ ~

1.6

~

100­

Jl

I-

e Z_O B

I

.. 1.2

~

e

I- 4.

0

..•­ 1!

~.

.0

V

»c 1.06-/

/

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r~.a

, x

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V

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ataDdard

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~ ...............

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.8

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... 0

IfACA 2,O-a.rl•• (5 digit,

.4

... ~

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2.0

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••

I

R

1.6

.9.0 x 10' B 6.0 ------. G

'.0

~___JI~-+-M..-~:aiII'~-+-I~~

. . .--I

AStan4ard

ro\IIbDell8 -t-~t---+--~~--I-~--I----I 6 )( 106

1.2

XACA

4

~-.erl.8·(4

dlglt)

8

12 16 20 Airtoil thickness. percent of chord

(b) RACA rour- and rl".-dlg1~ eerie•• FIG.

60. (Continued)

24

EXPERIMENTAL CHARACTERISTICS OF WING SECTIONS

S,mbols with tlase correspond to

2.0 0'1

=0.4

aad

0.6

I'~ ~

r

::::t~

R ~ r-;

e-, ~ ~~

r"""

.........

= 0.6

~

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-

c~l

139

~

-.......

....

~

~

............

106

v

9.0 )(

A

Standard rouatmeaa

e 0.0 e

~

~

'.0 6

)C

10 6

~ .8

.." ..• o

.. 2.0.---.--......~.-.......-..-......- -.--..-....- ......- ...-..... c

.p

~ ::

•

9.0 )( 106 e 6.0

0

8 .p ::s

e

roushn•••

6 )( 106

c

4J

~

•g

'.0

4 Stand.ard

~ 1.2

...o

R

1.61--+~-+-~t---4--4.,...g~~:&-.d---4--+--~-I

.8 ......

......__a.--....._

......-....I_....a-

....... _~....

!

a 1.6 ...-....-........~......- -..-..-.............,....-............-..--......- -.

:c

e

R

9.0 x 106

6.0 1.2.I-----r-t--;--n~-r_;_-=-t_;~'__f::;~'a==:t() 0 ;.0 ~+---t-~

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.8 ..-........_ ......--.

o

--------I-""""'--..... 12 8 10

--a.-....... -~-01

20

Alrtol1 thickneas, percent or chord

(e) NACA 63-aerles. Flo. 60. (Continued)

Standard

rougbnea 8

6

x 106

THEORY OF WING SECTIONS

140

SJlDbola nth flag e~reapond to et.i

2.0

0Il

=0.4 an4 0.6

=0.6 R

~-..............-....o 9.0

1-'--'86.0

x 106

1. 6 t---t""--t--t--+--t---+--~~"""'~rIl'-l.'---I-~ 0 3.0 A t----t---t--+--+--+--+---+-~~.a.----a.--'.....I-..'--I

~

...e

Standard

r~e'8

6

x

100

1.6 It

~ 9.0 ,. 1.2 t---t"--t--'t"-~~-+--~.....+-a. l?J 6.0 .....-...a-~;::a.-+:a~(:) 3.0

0

x 106

~

c

. -......-

~ t..e

~

4D

0

...A Standard

t--t--t-~~~r:::;;....-+--+--+--4---I-~--+---I

S

rougbne, a 6 x 100

.8

0

~

~

.... c

1.6

...

R

0

1'----+--......-.---..-.... <2> 9.0

.p

0

••

-,--~-+--4---+El

1.2

)C

10 6

03.0 Standard

I

.-a-~t--.......- - I - - - - L 6

rougbne,• 6 x 100

.-t

i

6.0

.8

1.6 R

e 9.0 x 106

fl 6.0 1.2 t---T--r--t---+-,.f"--+-~-4-~o-~~-I---I 1-.....;;;.--.--.... 0 3.0

A t---t--t--'t--i'#--f--r,,,,+--+--01--4"':::'-+...--1

.8

Standard roughne,s

6 x 111' 0

4

8 12 J.6 hlrrol1 thickness. percent (dl

NACA FIG.

20

or chord

64-serles.

60. (Continued)

24

EXPERIMENTAL CHARACTERISTICS OF WI}/G SECTIONS

8J11lbola with flap correspond to

01

1

= 0.6

4

.p

!...o t

•oo

.p

....""... C

...o .p

o

••

J 1.6 r---,...-r--~r---"Ir---~.----...---~~"-' U

=­

.8 "-......_~-.liil- ......_..a..---a._...a.---IIo.--...a._..&........._ - I o 4 8 u ~ ~ 6 Airtoil thlckn•• s. percent at Chord

(e)

MACA

6S-8erl•• ~

FIG.60 (Continued)

141

THEORY OF WING 8BO'l'ION8

142

...I

o

1.2 ""-.........

--a_-a..._"--....... _.-.....~_....... _..a.___._..

,

J 1.6

a 9.0 x 106

'.0 . rougme,_

8 &.0 1.2 .--r---,r-I---,r--o"'L..-;---t-;---,~::t::::::l0 t---+--+--t--­

.8.0

6. Standard =EJ-+-+---t---+-~~c:;;p-~~-~

6 x 100

4

8 12 16 Air£oU thickne.s. percent (r)

IIACA

o~

66- •• rl•••

FIG. 60. (Concluded)

20 chord

24

EXPERIMENTAL CHARACTERISTICS OF WING SECTIONS

143

thickness ratios from 12 to 24 per cent is uniformly favorable and nearly independent of the thickness ratio. Increasing the Reynolds number from 3 to 9 million results in an increase of the maximum lift coefficient of approximately 0.15 to 0.20. The scale effect on the NACA 00- and 14­ series sections having thickness ratios of 0.12c and less is very small. The scale-effect data for the NACA 6-series wing sections (Figs. 60c to e) do not show an entirely systematic variation. In general, the scale effect is favorable for these sections. For the NACA 64-series wing sections, the increase of maximum lift coefficient with increase of Reynolds number is generally small for thickness ratios less than 12 per cent, but it is some­ what larger for the thicker sections. The character of the scale effect for the NACA 65- and 66-series wing sections is similar to that for the NACA 64-series, but the trends are not so well defined. The scale effect for the NACA 6-series wing sections cambered for a design lift coefficient of 0.4 or 0.6 is greater than that for these sections with less camber. The data of Fig. 61 show that the maximum lift coefficient for the NACA 63(420)-422 section continues to increase with Reynolds number up to values of at least 26 million. The values of the maximum lift coefficient presented are for steady con­ ditions. It is known that the maximum lift coefficient increases with the rate of change of angle of attack. tOto The significant par~eteris (daoldt) (cjV). Even such low rates of change of angle of attack as those encountered in landing flares produce increases of the maximum lift coefficient. d. Effect of Surface Condition on Lift Characteri8tics. It has long been known that surface roughness, especially near the leading edge, has large effects on the characteristics of lying sections. The maximum lift coeffi­ cient, in particular, is sensitive to leading-edge roughness. The effect on maximum lift coefficient of various degrees of roughness applied to the leading edge of the NACA 63(420)-422 wing section is shown in Fig. 62. The maximum lift coefficient decreases progressively with increasing roughness. For a given surface condition at the leading edge, the max­ imum lift coefficient increases slowly with increasing Reynolds number (Fig. 63). Figure 64 shows that roughness strips located more than about O.2Oc from the leading edge have little effect on the maximum lift coefficient or lift-curve slope. It is desirable to determine the relative effects of leading-edge rough­ ness on various wing sections. In order to make a systematic investigation of this sort, it is necessary to select a standard form of roughness. The standard leading-edge roughness selected by the NACA for 24-inch chord models' consisted of 0.01I-inch carborundum grains applied to the surface of the model at the leading edge over a surface length of 0.0& measured from the leading edge on both surfaces. The grains were thinly spread to cover 5 to 10 per cent of the area. This standard roughness is considerably

THEORY OF WING SECTIONS

144

2.8

2.0

1.6

... ...

A

.,.

1.~

.:s::: ~

...

.8

C)

;

"­ to. ~

0

()

1'4

~

0

II J'

I

ri

C

~..r.,..

r

..."'"

.

..,.­

.JP

0

...

-- -

~~ ~ ~"E) -­ PJ '-' -JJ

0

~

0

• co

j

- .4

v

j

- .8 j

J:t

-1.2

~~ -~

:A

~

Il G

6.0

lt0

x

106

a • 0 A 20.0 • 26.0

If

r-fF •

-1.6 ~

-1'

0 , J6 24­ or attack, Clo ' des FIG. 61. Lift and drag e~acteristiC8 of the NACA 63(420)-422 airfoil at high Reynolds number; TDT tests 228 and 255. Section

-8

anel.

EXPERIMENTAL CHARACTERISTIC8 OF WING SECTION8

·032 .02 a

-

-:

-"""""

0

.2

- r--.

---...

<, ...........

-­ --­ .,

- ...-­ -4 zJ"

............... ............

~

.8

1.0

c

.02."

"d

o

R

,

I

6.0 x 106 0 B 10.0

.01

14.0

V

26.0

6

~

:~ ~

,

e

II J

'~-.6

20.0

l~j

~ r-, ~

~~

f!A V 4'1

~

~ .~

......

r:

~~

,

v

o

-1.2

-.8

-.1,.

o

&eo'lon lUt ooettlclent. FIG. 61. (Concluded)

.8 cl

145

THEORY OF WING SECTIONS

146 I

I

1

1

I

B O.0Q4-1ftOh-sra1n

- X roUIbD... ..1.2 - roqJme..

Oft

L.S.

~~

O.ooe-1aOh~

• i•

.:= : ::.

... ~

...•=.. tI

-

OIl

('-­

L. L

~~

• O.Oll-hoh-ara b ro~••• OIl

.8 ~ + lb.l1ao

~

~

~

-..­

CL.. ~~

~ID

r

L.L

OIl 1,•••

~r

elllootil

,

,1

.~

j

II

0

/

­.... -) -.8

r

Airfoil. DCA

~, c

2' Ie 10' " In. T.ata. m!' 2'1) an4 262

III

Chord I

I I I

-1.'

~

"(WO)~a

-al.

0 8 16 01 4e. FIG. 82. Lift characteriatica of a N ACA 63(420)-422 airfoll with varioua at the leadiq edae.

-16

-8

I I

I

""loa

.'taok. .0..

I

•

dtcreee of

roughnea

2.0

...-01 ~

~

~

"

,.. I

...

.

~ \.

"

----- ­ ..

. '0 \\ ...... ~ .~ 1-..

i-a. !~ ~•• M'!It'=••

.8

'-­ -6

.p

-.om

o

o

o

8

12

16

20

~14a""r ••

FIG. 63. Effects of Reynolds number on maximum section lift coefIicient 63(420)-422 airfoil with roughened and smooth leading edge.

28 CI

of the NACA

EXPERIMENTAL CHARACTERISTIC8 OF WING SECTIONS

-

..

1.2

o

~

.

1; .8

:: : 44 44

I

-

-

I

I

I

o.~

G IaooUl

. a

...

~

rw~

9Ill

I'

,

1

J(

~

...o

......-:.;;

~,

...

... I

~

rc

IV

LeE. + BoqbM.. atrip OIl

....

o

r

at 0.200 • BoqbM.. atnp at

?

I

I

1(1Iou. . . . .vip

147

0

I

/

-.'­

r ~o111 a. 2' x1,p

DCA ,,(!a20 J-Ja22

j

•• 8

-1.2

~~

CboN. ~ 111•


255

~.tl

) 'if

~

~ 0 8 ""loa aq1e of .tWk.

~

.0.

•

del

M. Lift characteriatice of a NACA 63(420)-422 airfoil with O.Oll-inch-grain rouahnesa at various ehordwiae locatioDL

FIG.

I .,j

.•­

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148

THEORY OF WING SECTIONS

more severe than that caused by usual manufacturing irregularities or deterioration in service, but it is considerably less severe than that likely to be encountered in service as a result of accumulation of ice, mud, or damage in military combat. Maximum-lift-coefficient data at a Reynolds number of 6 million for a large number of NACA wing sections are presented in Figs. 58 and 59. The variation of maximum lift coefficient with thickness for the NACA four- and five-digit series wing sections with standard roughness shows the same trends as those for the smooth sections except that the values are considerably reduced for all these sections other than the NACA OO-series of 6 per cent thickness. The values of the maximum lift coefficient for these rough sections with thickness ratios greater than 12 per-cent are sub­ stantially the same for a given thickness. Much less variation of maximum lift coefficient with thickness ratio is shown by the NACA 6-series wing sections in the rough condition than in the smooth condition. The varia­ tion of maximum lift coefficient with camber, however, is about the same for the wing sections with standard roughness as for the smooth sections. The maximum lift coefficients of rough wing sections decrease with rear­ ward movement of the position of minimum pressure; and, as in the smooth condition, the optimum thickness ratio for maximum lift increases with rearward movement of the position of minimum pressure. The NACA 64-series wing sections cambered for a design lift coefficient of 0.4 have maximum lift coefficients consistently higher than the NACA 24-, 44-, and 23o-series sections of comparable thickness when rough, with the exception of the NACA 4412. For normal wing sections, the angle of zero lift is practically unaffected by the standard leading-edge roughness. The results presented in Ap­ pendix IV show that the effect of roughness is to decrease the lift-curve slope for wing sections having thickness ratios of 18 per cent or more. The effect increases with increase of thickness ratio. For lying sections less than 18 per cent thick, tile effect of roughness on the lift-curve slope is relatively small. '1.6. Drag Characteristics. a. Minimum Drag of Smooth: Wing Sections. The value of the minimum drag coefficient for smooth lying sections is mainly a function of the Reynolds number and the relative extent of the laminar boundary layer, and it is moderately affected by thickness ratio and camber. If the extent of the laminar boundary layer is known, the minimum drag coefficient may be calculated with reasonable accuracy by the method presented in Sec. 5.13. The effect on minimum drag of the position of minimum pressure that determines the possible extent of laminar flow is shown in Fig. 65 for some N J.~CA 6-series wing sections. The data show a regular decrease of drag coefficient with rearward movement of minimum pressure. The variation

EXPERIMENTAL CHARACTERISTICS OF WING SECTIONS

149

of minimum drag coefficient with Reynolds number for several wing sec­ tions is shown in Fig. 66. The drag coefficient generally decreases with increasing Reynolds number up to Reynolds numbers of the order of 20 million. Above this Reynolds number, the drag coefficient of the NACA 65(Gl)-420 section remains substantially constant up to a Reynolds number of nearly 40 million. The earlier increase of drag coefficient shown by the NACA 66(2%15)-116 section may be caused by surface irregularities because the specimen tested was a practical construction model. It may be noted that the drag coefficient for the NACA 65,-418 lying section at low Reynolds numbers is substantially higher than that of the NACA 0012 section, whereas at high Reynolds numbers the opposite is the case. The higher drag of the NACA 65,-418 wing section at low Reynolds numbers is caused by a relatively extensive region of laminar separation downstream from the point of minimum pressure. This region decreases in size with increasing Reynolds number. These data illustrate the inadequacy of low Reynolds number test data either to estimate the full-scale characteristics or to determine the relative merits of lying sections at flight values of the Reynolds number. The variation of the minimum drag coefficient with camber is shown in Fig. 67 for a number of smooth 18 per cent thick N .A.CA 6-series wing sections. These data show little change of minimum drag coefficient with increase of camber. A considerable amount of systematic data is included in Fig. 68 showing the variation of minimum drag coefficient with thickness ratio for some NACA wing sections ranging in thickness ratio from 6 to 24 per cent of the chord. The minimum drag coefficient is seen to increase with increase of thickness ratio for each series of wing sections. This in­ crease, however, is greater for the XACA four- and five-digit series lying sections (Fig. GSa) than for the NACA 6-series sections (Figs. 686, c, and d). b. Variation oj Profile Drag with Lift Coefficient. Most of the variation of drag with lift for "rings of finite span results from the induced-drag coefficient, which varies approximately as the square of the lift coefficient for a given wing configuration. It is important to keep the induced drag in mind when considering the variation of profile drag with lift, because the variation of the wing drag coefficient with lift coefficient will be largely determined by the induced drag, which is a function of the aspect ratio. At low and moderate lift coefficients where there is no appreciable separation of the flow, the drag is caused almost completely by skin fric­ tion. Under these circumstances, the value of the drag coefficient depends upon the relative extent of the laminar boundary layer and the induced velocities over the surfaces of the section and may be calculated by the method of Sec. 5.13. As the lift coefficient increases, the average square of the velocities over the surfaces increases, resulting in a. small drag increase even though the relative extent of laminar flow was not affected. The

THBORY OF WING SECTIONS

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EXPERIMENTAL CHARACTERISTICS OF WING SECTIONS

151

more important effect of lift coefficient is to change the possible extent of laminar flow by moving the minimum-pressure points. In many of the older types of wing section, the forward movement of transition is gradual and the resulting variation of drag with lift coefficient occurs smoothly" The pressure distributions for the XACA 6-series wing sections are such as to cause transition to move forward suddenly at the end of the low-drag range of lift coefficients. A sharp increase of drag coefficient to the value corresponding to a forward location of transition on one of the surfaces results. Such sudden shifts of transition give the typical drag curve for these wing sections with a II sag" or ((bucket" in the low-drag range. The same characteristic is shown to a smaller degree by some of the earlier wing sections such as the NACA 23015 when tested in a low-turbulence air stream. The data presented in Appendix I for the NACA 6-series thickness forms show that the range of lift coefficients for low drag varies markedly with thickness ratio. It has been possible to design wing sections of 12 per cent thickness with a total theoretical low-drag range of lift coefficients of 0.2. This theoretical range increases b)" approximately 0.2 for each 3 per cent increase of thickness ratio. Figure 69 shows that the theoretical extent of the low-drag range is approximately realized at a Reynolds num­ ber of 9 million. Figure 69 also shows a characteristic tendency for the drag to increase to some extent toward the upper end of the low-drag range for moderately cambered wing sections, particularly for the thicker ones. All data for the KACA 6-serics wing sections show a decrease of the extent of the low-drag range "pith increasing Reynolds number. Extrapolation of the rate of decrease observed at Reynolds numbers below 9 million would indicate a vanishingly small low-drag range at flight values of the Reynolds number. Tests of a carefully constructed model of the NAC"'A 65(421)-420 section showed, however, that the rate of reduction of the low-drag range with increasing Reynolds number decreased markedly at Re)·001<18 num­ bers above 9 million (Fig, 70). These data indicate that the extent of the low-drag range for this wing section is reduced to about. one-half the theoretical value at a Reynolds numberof 35 million. The values of the lift coefficient for which low drag is obtained are de­ termined largely by the amount of cumber, The lift coefficient at the center of the low-drag range corresponds approximately to the design lift coefficient of the mean line. The effects on the drag characteristics of various amounts of camber are shown in Fig. 71" Section data indicate that the location of the low-drag range may' be shifted by even such crude camber changes as those caused by small deflections of 8 plain flap," The location of the low-drag range shows SODle variation from that pre­ dicted from the simple theory of thin wing sections. This departure ap­ pears to be a function of the type of the mean line used and the thickness

THEORY OF WING SECTIONS

152

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EXPERIAfENTAL CHARACTERISTICS OF' n'ING SECTIONS

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THEORY OF WING SECTIONS

154

ratio. The effect of thickness ratio is shown in Fig. 69 from which the center of the low-drag range is seen to shift to higher lift coefficients with increasing thickness ratio. This shift is partly explained by the increase of lift coefficient above the design lift coefficient for the mean line obtained when the velocity increments caused by the mean line are combined with

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FIG.

the velocity distribution for the thickness form according to the first­ approximation method of Sec. 4.5. At the end of the low-drag range, the drag increases rapidly with increase of the lift coefficient. For symmetrical and low-cambered wing sections for which the lift coefficient at the upper end of the low-drag range is moderate, this high rate of increase does not continue (Fig. 71). For highly cambered sections for which the lift at the upper end of the low-drag range is already high, the drag coefficient shows a continued rapid in­

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EXPERIMENTAL CHARACTERISTICS OF WING SECTIONS

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THEORY OF WING SECTIONS

156

mean line with data for sections cambered to carry the load farther forward shows that the uniform-load mean line is favorable for obtaining low-drag coefficients at high lift coefficients (Fig. 72). Data for many of the wing sections given in Appendix IV show large reductions of drag with increasing Reynolds number at high lift coefficients.

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with various amounts of camber. R, G X 10'.

This scale effect is too large to be accounted for by the normal variation of skin friction and appears to be associated with the effect of Reynolds number on the onset of turbulent flow following laminar separation near the leading edge.1M A comparison of the drag characteristics of the XACA 23012 and of three NACA 6-series wing sections is presented in Fig. 73. The drag for the N.~CA 6-series sections is substantially lower than for the NACA 23012 section in the range of lift coefficients corresponding to high-speed flight, and this margin may usually be maintained through the range of lift

EXPERIl.{ENTAL CHARACTERISTICS OF rVING SECTIONS

157

coefficients useful for cruising by suitable choice of camber. The NACA 6-series sections show the higher maximum values of the lift-drag ratio. At high values of the lift coefficient, however, the earlier NACA sections .generally have lower drag coefficients than the N ACA 6-series sections. c. Effect oj Surface Irregularities Oil Drag Characteristics. Numerous measurements of the effects of surface irregularities on the characteristics of wings have shown that the condition of the surface is one of the most important variables affecting the drag. Although a large part of the drag increment associated with surface roughness results from a forward move­ ment of transition, substantial drag increments result from surface rough­ ness in the region of turbulent flow." It is accordingly important to maintain smooth surfaces even when extensive laminar flow cannot be expected. The possible gains resulting from smooth surfaces are greater, however, for wing sections such as the XA(~A G-series than for sections where the extent of laminar flow is limited by a forward position of mini­ mum pressure, No accurate method of specifying the surface condition necessary for extensive laminar flow at high Reynolds numbers has been developed, although some general conclusions have been reached, It may be presumed that, for n given Reynolds number and chordwise location, the size of the permissible roughness will vary direct ly with the chord of the lying section. It is known, at one extreme, that the surfaces do not have to be polished or optically smooth. Such polishing or waxing hus shown no improvement in tesu,.3 in the Kf\CA two-dimensional low-turbulence tunnels when ap­ plied to satisfactorily sanded surfaces. Polishing or waxing a surface that is not aerodynamically smooth ,,;U, of course, result in improvement, and such finishes may be of considerable practical value be cause deterioration of the finish may be easily seen and possibly postponed. Large models having chord lengths of 5 to 8 feet tested in the NACA two-dimensional low-turbulence tunnels are usually finished by sanding in the ehordwise direction with Xo, 320 carborundum paper when an aerodynamically smooth surface is desired," Experience has shown the resulting finish to be satisfactory' at flight values of the Reynolds number. Any rougher surfuee texture should be considered as a possible source of transition, although slightly rougher surfaces have appeared to produce satisfactory results in some cases.. l.;oftin6S 8ho\\'00 that small protuberances extending above the general surface level of an otherwise satisfactory surface are more likely to cause transition Ulan are small depressions. Dust particles, for example, are more effective than small scratches in producing transition if the material at the edges of the scratches is not forced above the general surface level. Dust particles adhering to the oil left on wing surfaces by fingerprints may be expected to cause transition at high Reynolds numbers.

THEORY OF WING SECTIONS

158

Transition spreads from an individual disturbance with an included angle of about 15 degrees. H • 42 A few scattered specks, especially near the 2.8

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leading edge, will cause the flow to be largely turbulent. This fact makes necessary an extremely thorough inspection if low drags are to be realized. Specks sufficiently large to cause premature transition can be felt by hand. The inspection procedure used in the N ACA two-dimensional low-turbulence

EXPERIMENTAL CHARACTERISTICS OF WING SECTIONS

159

tunnels is to .feel the entire surface by hand, after which the surface is thoroughly wiped with a dry cloth. 0J2

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THEORY OF WING SECTIONS

160

transition when located at 5 per cent of the chord with its axis normal to the surface" is shown in Fig. 74. These data were obtained at rather low 2.8

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values of the Reynolds number and show a large decrease of the allowable height with increase of Reynolds number. Analysis'S of these data showed that the height of the protuberance that caused transition depended on the shape of the protuberance and on the Reynolds number based on the height of the protuberance and the local

EXPERIMENTAL CHARACTERISTICS OF It''lNG SECTIONS

161

velocity at the top of the protuberance. This Reynolds number is plotted against the fineness ratio of the protuberance in Fig, 75 for protuberances located at various chordwise positions on two wing sections. •0,2

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The effect of Reynolds number on permissible surface roughness' is also indicated in Fig. 76, in which a sharp increase of drag at a Reynolds number of approximately 20 million occurs for the model painted with camouflage lacquer. Experiments with models finished with camouflage paint" in­

162

THEORY OF WING SECTIONS

dicate that it is possible to obtain a matte surface without causing pre­ mature transition but that it is extremely difficult to obtain such a surface sufficiently free from specks without sanding. ~

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-JDs BaJD014a m.ber. .. FIo. 74. Variation with wing Reynolds number of the miDimum height of a cylindrical protuberance neceusry to cause premature transition. Protuberance has O.035-incb di­ ameter with axis nonnal to the wing surface and is located at 5 per eent chord of a 9O-ineh­ chord symmetrical6-series airfoilaection of 15 per cent thickness and with minimum preseure at 70 per cent chord. Low-drog airfoil numIJer 2

~

0.035 in. .20 .035 iii. 015 in. .20 .50 OJ5 in. Low-dtog oirfrJllllllnber I

+ JC

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............... r---~ 1

~~

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sa 40 Projeclion fineness ratio, ~

50

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75. Variation of boundary-layer Reynolds number factor with projection fineness ratio for two low--drag airfoils.

FlO.

The magnitude of the favorable pressure gradient appears to have a small effect on the permissible surface roughness for laminar flow. Figure 77 shows that the roughness becomes more important at the extremities of

163

EXPERIMENTAL CHARACTERISTICS OF WING SECTIONS

the low-drag range where the favorable pressure gradient is reduced on one surface.' The effect of increasing the Reynolds number for a surface of

...


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Reynolds nmoe; R (b') LACQUER CAMO..FLAGE UNIMPROVED AFTER ~INTING 76. Variation of drag coefficient with Reynolds number for a 6O-inch-chord model of the NAC.4 e.S(m)-420 airfoil for two surface conditions. FIG.

Smooth cond1 tion

enamel ca:nourlase wIth all specl:a ~~lt orr with bl&.de

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cI section lUt coefficient. c

1

FIG. 77. Drag characteristics of NACA 65 l 4n )-420 airfoil for two surface conditions.

marginal smoothness, which has an effect similar to increasing the surface roughness for a given Reynolds number, is to reduce rapidly the extent of the low-drag range and then ~o increase the minimum drag coefficient

THEORY OF WING SECTIONS

164

(Fig. 77). The data of Fig. 77 were especially chosen to show this effect. In most cases, the effect of increasing the Reynolds number when the sur­ faces are rough is to increase the drag over the whole low-drag range. The effect of pressure gradient as shown in Fig. 77 is apparently not very powerful and tends to be evident only in cases of surfaces of marginal smoothness. 17 .;,5 I---~

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78. Drag characteristics of a NACA 63(420)-422 airfoil with various degrees of rougb­ at the leading edge.

More difficulty is generally encountered in reducing the waviness to permissible values for the maintenance of laminar flow than in obtaining the required surface smoothness. In addition, the specification of the required fairness is more difficult than that of the required smoothness. The problem is not limited to finding the minimum wave size that will cause transition under given conditions because the number of waves and the shape of the waves require consideration. H the wave is sufficiently large to affect the pressure distribution in such a manner that laminar separation is encountered, there is little doubt that such a wave will cause premature transition at all useful Reynolds num­ bers. Relations between the dimensions of a wave and the pressure dis­ tribution may be found by the method suggested by Allen." If the pressure

EXPERIMENTAL CHARACTERISTICS OF WING SECTIONS

165

distribution over the wave is known, the criteria for laminar separation discussed in Sec. 5.7 may be applied. The size of the wave required to reverse the pressure gradient increases with the pressure gradient. Large negative pressure gradients would therefore appear to be favorable for +

.o"

A1rtol1 I

DCA

6,lLzO)-422

III 2& x 106 ChoN, ~ 111. Teat. '1' 255

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FIG. 79. Drag characteristics of a at various chordwise Iocatione,

~

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I(

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+ Bouatme••

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63(420)-422 airfoil with O.Oll-inch-grain roughness

Experimental results have shown tills conclusion to be qualitatively correct. For the t}1JCS of waves usually found on practical-construction wings, the test of rocking a straightedge oyer the surface in a ehordwise direction is a fairly satisfactory criterion." The straightedge should rock smoothly without jarring or clicking. The straightedge test will not show the exis­ tence of waves that leave the surface convex. Tests of a large number of practical-construction models, however, have shown that those models

wavy surfaces.

THEORY OF WING SECTIONS

166

which passed the straightedge test were sufficiently free from small waves to permit low drags to be obtained at flight values of the Reynolds number. It does not appear feasible to specify construction tolerances on or­ dinates of wings with sufficient accuracy to ensure adequate freedom from dt ~

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·.'"......

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FIG. SO. Drag characteristics of two NACA 6-...~rics airfoils with O.Oll-ineh-grain roughnese at O.3Oc.

waviness. If care is taken to obtain fair surfaces, normal tolerances may be used without causing serious alteration of the drag characteristics. If the wing surface is sufficiently rough to cause transition near the leading edge, large drag increases are to be expected even if the roughness is confined to the region of the leading edge. Figure 78 shows that, although the degree of roughness has some effect, the increment of minimum drag coefficient caused by the smallest roughness capable of producing transi­ tion is nearly as great as that caused by much larger grain roughness when the roughness is confined to the leading edge.! The degree of roughness has a much larger effect on the drag at high lift coefficients. If the roughness is sufficiently large to cause transition at all Reynolds numbers considered,

EXPERIMENTAL CHARACTERISTICS OF WING SECTIONS

167

the drag of the wing with roughness only at the leading edge decreases with increasing Reynolds number.' The effect" of fixing transition by means of a roughness strip of car­ borundum of O.011-inch grains is shown in Fig. 79. The minimum drag

32 x 10'

28

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0:

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'I

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16 12

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coetrl~1ent.

01

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o 8ec~1oa

lift

coet~lcleDt.

01

FIG.81. Comparison of section drag coefficients obtained in flight on various airfoils. Tests

of NACA 27-212 and 35-215 sections made on gloves.

increases progressively with forward movement of the roughness strip. The effect on the drag at high lift coefficients is not progressive; the drag increases rapidly when the roughness is at the leading edge. Figure 80 shows that the drag coefficients for the X.A.C.-\. 65(223)-422 and 63(420)-422 lying sections' are nearly the same throughout most of the lift range when the extent of the laminar flow is limited to O.30c. These data indicate that,

THEORY OF WING SECTIONS

168

for wing sections of the same thickness ratio and camber, the drag coeffi­ cients depend almost entirely on the extent of the laminar boundary layer if no appreciable separation occurs. The variation of minimum drag coefficient with thickness ratio for a number of NACA four-digit, five-digit, and 6-series wing sections with

.0/2

I

o Surfaces 05 received excepllighllysmded I- 0

Both surfocespointedandfinished10 rear spar

-..

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/6

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24

28

32

36x/0 6

Rf!I'1OIds number, R FIG. 82. Drag scale effect on l00-inch-ehord praetieal-eonstruction model 65(216)-3(16.5)(approx.) airfoil section. C1 = 0.2 (approx.).

or

the NACA

leading edges rough is shown in Fig. 68. The minimum drag coefficients increase with thickness ratio but are substantially the same for all the sections of equal thickness ratio. The increments of drag coefficient caused by leading-edge roughness are correspondingly greater for the sections having the lower drag coefficients when smooth. The section drag coefficients of several airplane wings have been measured in flight l 60 by the wake-survey method, and a number of practical­ construction wing sections have been tested in the NACA two-dimensional low-turbulence pressure tunnel! at flight values of the Reynolds number. Some of the flight data obtained by Zalovcik l 60 are summarized in Fig. 81.

EXPERIMENTAL CHARACTERISTICS OF WING SECTIONS

169

All wings for which data are presented in Fig. 81 were carefully finished to produce smooth surfaces. Care was taken to reduce surface waviness to a minimum for all the sections except the NACA 2415.5, N-22, Republic

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FIG. 83. Variation of the drag coefficient with 1I"Y1101
8-3, 13, and the XACA 2i-212. Curvature gauge measurements of surface waviness for some of these sections are presented by Zaloveik.P" These data show that the KACA 2-,3-, and 6-sel;es sections that permitted ex­ tensive laminar flow had substantially lower drag coefficients when smooth than did the other sections. Data obtained in the XACA two-dimensional low-turbulence pressure tunnel' for typical practical-construction sections are presented in Figs, 82

THEORY OF WING SECTIONS

170

to 86. Figure 87 presents a comparison of the drag coefficients obtained in this wind tunnel for a model of the NACA 0012 section and in fiight for the same model mounted on an airplane.' For this case, the wind-tunnel and flight data agree to within the experimental error. The wind-tunnel tests of practical-construction wing sections as de­ livered by the manufacturer showed minimum drag coefficients of the order of 0.0070 to 0.0080 in nearly all cases regardless of the type of section

.016

' 1

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Flo. 84. Drag characteristics of the NACA 65.2-417 (approx.) and NACA 23015 (apllrox.) airfoil eectiODB built by praetical-conatruction methods b)~ the same manufacturer. R, 10.23 X lOS.

(Figs. 82 to 86). Such values may be regarded as typical for good American construction practice during the Second World War. Finishing the 800­ \ tions to produce smooth surfaces always resulted in substantial drag re­ ductions, although considerable waviness usually remained. None of the sections tested had fair surfaces at the front spar. Unless special care is taken to produce fair surfaces at the front spar, the resulting wave may be expected to cause transition either at the spar location or a short distance behind it. One practical-construction specimen tested with smooth surfaces maintained relatively low drags up to Reynolds numbers of approximately 30 million [NACA 66(2%15)-116 wing section of Fig. 66]. This specimen had no spar fonvard of about 35 per cent chord from the leading edge and no spanwise stiffeners forward of the spar. This type of construction resulted in unusually fair surfaces. Few data are available on the effect of propeller slipstream on transition or wing drag; the data that are available do not show consistent results.

0 11&0&

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Scale effect on drag of the NAC..-\. 66(215)-116 and NACA 23016 airfoil sections 88 received.

built by practical-eonstruction methods by the same manufacturer and tested •016

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86. Drag scale effect for a model of the N AC.A... 65-serics airfoil section 18.27 per cent. thick and t.he Davis airfoilllOOtion 18.27 per cent thick, built by practical-construction nletbods by the same manufacturer. c, == 0.46 (approx.) . FIG.

0

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FIG. 87. Comparison of drag coefficients measured in flight and wind tunnel for the NACA 0012 airfoil tteCtiOll at zero ilit.

171

THEORY OF WING SECTIONS

172

This inconsistency may result from variation of lift coefficient, surface con­ dition, air-stream turbulence, propeller advance-diameter ratio, and num­

t A1rtoU ...'1. . Boot. KACA "(2X15)-en.8

!!P. ileA 67.1-(1.,)15 .....\ ..~, 5.98

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88. The effect of propeller operation on section drag coefficient of a fighter-type airplane. from tests of a model on the Langley 19-foot pressure tunnel. CL = 0.10; R, 3.7 X 10'.

FIG.

ber of blades. Some early British investigation Sl51 , 158. 159 showed that transition occurred at 5 to 10 per cent of the chord from the leading edge in the slipstream. Similar results were indicated by tests in the N..~CA 8-foot high-speed tunnel. 4I Drag measurements in the N.~CA 19-foot pressure tunnels (Fig. 88) indicated that only moderate drag increments

EXPERIJIENTAL CHARACTERISTICS OF WING SECTIONS

173

resulted from a windmilling propeller. Although these data are only quali­ tative' because of the difficulty of making wake surveys in the slipstream, they seem to preclude very large drag increments such as would result from a movement of transition to a position close to the leading edge. A

I

20 x 106

........

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10

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Fro. 89. Flight measurements of transition on a NACA

............

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c~

f)()"~rics wing

within and outside

the slipstream.

similar conclusion is indicated by X ..\.C..\ flightdata/ (Fig. 89), which show transition as far back as 20 per cent of the chord in the slipstream. Even fewer data are available on the effects of vibration on transition. Tests in the X . .\ CA 8-foot high-speed tunnel 43 showed negligible effects, but the range of frequencies tested may not have been sufficient.ly wide to represent conditions encountered on airplanes. Some K .~(~.~ Hight data" showed small but consistent rearward movements of transition outside the slipstream when the propellers were feathered. This effect was noticed even when the propeller on the opposite side of the two-engined airplane was feathered, and it was accordingly attributed to vibration or noise. In

THEORY OF WING SECTIONS

174

some cases, increases of drag have been noted in wind tunnels when the model or its supports vibrated. A tentative conclusion may be drawn from the meager data that vibration may cause small forward movements of transition on airplane wings, The skin friction associated with a turbulent boundary layer on a smooth surface decreases with increase of Reynolds number, as shown in Chap. 5. This favorable scale effect is not obtained at high Reynolds

,

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90. Coefficient of resistance of rough and smooth tubes dependent on Reynolds number.

numbers on rough surfaces. An explanation of this effect is that the pres­ sure drag of the individual protuberances constituting the roughness con­ tribute to the skin friction when they project through the laminar sublayer, The shapes of the individual protuberances are usually such that the scale effect on the pressure drag should be small. The result at large Reynolds numbers for generally roughened surfaces is that the skin-friction coeffi­ cient is essentially constant. The general nature of the scale effects for rough surfaces has been indicated b)' experiments in pipes." These data (Fig. 90) show that at lo"r Reynolds numbers the pipe-loss coefficient decreases with increasing Rey­ nolds number at the rate expected for laminar flow, When transition occurs, the pipe-loss coefficient increases to the value expected for tur­ bulent flow over smooth surfaces. For very small roughness, the pipe-loss coefficient decreases with increasing Reynolds number along the turbulent skin-friction curve for smooth surfaces until large values of the Reynolds number are reached. For any given size of roughness, however, there ap­

EXPERIMENTAL CHARACTERISPICS OF liTING SECTIONS

175

pears to be a Reynolds number beyond which the coefficient increases slightly to a constant value. The value of the Reynolds number decreases and the constant value of the coefficient increases as the grain size of the roughness increases. At large values of the Reynolds number, the rough­ ness causes large increments of skin friction over the values corresponding to smooth surfaces. On the basis of the pipe experiments and reasoning similar to the fore­ going, von KarmAn139 obtained a formula for the grain size just sufficiently large to affect turbulent skin friction. This formula may be written pl:k=3.
f!

V4

Reynolds number based on grain size of roughness and local 'vhere -pUk - ::: velocity outside boundary layer JI.

c,

= local skin-friction coefficient For a 'ling of approximately 9-foot chord at a speed of 300 feet per second, the limiting grain size is approximately 0.0004 inch and varies only slightly over the surface, increasing toward the trailing edge. The effect of roughness on the drag of wing sections with largely turbu­ lent flow is analogous to the effect in pipes as ShO\\·113 by Fig. 83. At the lower Reynolds numbers, the drag coefficient decreases with increasing Reynolds number at about the rate expected for turbulent flow over smooth surfaces, In this case, the drag remains essentially constant at Reynolds numbers above about 15 million. ...~t a Reynolds number of 70 million the increment of drag caused by the moderate roughness was of the order of one-quarter of the drag that would have been obtained had the favorable scale effect continued. The sensitivity of the turbulent boundary layer to roughness is so great that airplanes should not be expected to show favorable scale effects at large Reynolds numbers unless considerable care is taken to obtain smooth surfaces. It should be noted that, in contrast to the situation with respect to the laminar layer, it is relatively easy to obtain reductions of drag by attention to the surface conditions for turbulent flow. Even though the allowable size of the roughness is very small, each speck of roughness presumably contributes onI)" the drag of itself and does not have any appreciable effect on the skin friction over the surface downstream. Consequently favorable effects may be expected from careful finishing of the general surface of modem high-performance airplanes even though imperfections such as rivets and seams may be present. d. Unconsenxuioe li'ing Sections. The need for low drags in order to obtain long range for large airplanes flying at speeds below the critical Mach number leads to designs having high wing loadings to reduce the wing area and profile drag together with relatively )O\V span loadings to

THEORY OF WING SECTIONS

176

avoid high induced drags. These tendencies result in "rings of high aspect ratio that require large spar depths for structural efficiency. The large spar depths require the use of thick root sections. The comparatively high lift coefficients corresponding to the cruise condition for such designs leads to the use of large cambers to obtain low profile drags. Such designs are encouraged by the drag characteristics of modem smooth wing sections

~

..........

.yr

r=­

/

4 ~

o

•

OOO} (l~ cUBit) ",,\~

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644

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FIG. 91. Variation of the lift coefficient eorresponding to a drag coefficient of 0.02 with thick­ ness and camber for a number of NACA airfoil sections with roughened leading edges. R. 6 X lOS.

which show relatively small increases of drag coefficient with increasing thickness ratio and camber (Sec. 7.00). Unfortunately airplane 'lings are not usually constructed with smooth surfaces, and, in any case, the surfaces cannot be relied upon to stay smooth under all service conditions. The effect of roughening the leading edges of thick wing sections is to cause large increases of the drag coefficient at high lift coefficients. The resulting drag coefficient may be excessive at cruising lift coefficients for heavily loaded high-altitude airplanes. Wing sections that have suitable characteristics when smooth but have exces­ sive drag coefficients when rough at lift coefficients corresponding to cruis­ ing or climbing conditions are called "uneonservative." -The decision as to whether a given lying section is conservative will depend upon the power and wing loadings of the airplane. The decision may be affected by expected service and operating conditions. For ex­

EXPERIMENTAL CHARACTERISTICS OF U·ING SECTIONS

177

ample, the ability of a multiengine airplane to fly with one or more engines inoperative in icing conditions or, in the case of military airplanes, after 1.1,

<,

1.2

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1.0

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5

a

e

)

1

.8

~/

)

,..

~)

.J"

\

'I\, el

t

00 00.1 00.2

.Oe2

c/ 4

~

<,

V

/

t

'" -, -, "-----

".-..:.r

/0

! 0

//

/

~ ........6

A

VO•

8

16

12

A:'rCol1 thlckne:.s, percent (C ) FIG.

i;ACA

64-

or

ZO

2~

c.hord

se rlea.

91. (Continued)

suffering damage in combat may be a consid~ration. As an aid in judging whether the sections are conservative, the lift coefficient corresponding to a drag coefficient of 0.02 was determined from the figures of Appendix IV for a number of NACA wing sections with roughened leading edges. The variation of this lift coefficient with thickness ratio ar.d camber' is shown

THEORY OF WING SECTIONS

178

in Fig. 91 for a Reynolds number of 6 million. These data show that, in general, the lift coefficient at which the drag coefficient is 0.02 decreases with rearward movement of the position of minimum pressure and with .1.2 ~~

~ o

~

1.0

~~

•

...Ii

ef

..-=

.8

~7

t Z o

,

~

.6

I

/

..V /

~~ ~

V

~l

r-, ~

~

<,

o,!. GO

~

~O.2

AO.~

jO.

II .c

...o

.:•

.~

~

(4)

i

MACA

65- .erie.

CJ,l

-

-00

e

0.2

/

40.4

~

/

&

:: .8 o

•

O'J

(

/

/

7

----

/

~ i': -e1.

~~

r-, 2\

~~

\

~

0

•

n

12

1)

20

2"

Alrtol1 thickness. percent or chord (e) FIG.

NACA 66- aeries. 91. (Concluded)

increased thickness above thickness ratios of about 15 per cent. For wing sections thinner than approximately 18 per cent, the effect of camber is to increase this lift coefficient. For the thicker sections, however, increasing the camber becomes relatively ineffective and may even be harmful in extreme cases. The highest values of this lift coefficient for wing sections having thickness ratios greater than 15 per cent are obtained with the NACA 64-series sections having a design. lift coefficient of 0.4.

EXPERIltfENTAL CHARACTERISTICS OF WING SECTIONS

179

7.6. Pitching-moment Characteristics. The variation of the quarter­ chord pitching-moment coefficient at zero angle of attack with thickness SlDa1e •1

n.aAId

•pIbola

o

,

aN t or (,tf' dada eel .p,1It

......

aerlea

......­

....

~

ioS :I ...""I

-.1

,.o

-

-

-

r---Ii

OOr 41&1t)

e :.~

A J,.Ii • 250 (5 41&1t)

..

it

-.2

t

~ .l..

i -.,

iI

~

4 (a)

tbl..... 8

AutoU MACA

12

to---.. r---,

k

1& 20 peroent or c:bor4

tour- and tlYe-dl81t aerlea •

•1

--..

"...

i

...

.....o t

:

o

-v --&

:r­

-.1

V­

°ls.

~~

00 .0.2

~ ..........

-.2

r- ~ h

""c

I

..,

h 'h_

-·5

··!JO

~ Io.......~

-

~

~

~

""V­ -.::Ill

4

8

12

Alr1"o11 thiekn•••.•

16 pe~ent

\..

-, I...-.­

f'---.

Ao.h .0.'

~

--

20

24

or chord

(b) HAC. 63- eerlell. 92. Variation of section quarter-chord pitching-moment coefficient (measured at an angle of attack of zero degrees) with airfoil thickness ratio for several NACA airfoil sections of dift'erent camber. R. 6 X I ()I.

FIG.

ratio and camber is presented in Fig. 92 for a large number of NACA wing sections," The pitching-moment coefficients of the NACA four- and five­ digit series sections become more negative with decreasing thickness ratio. Comparison of the experimental data in Fig. 92 with the theoretical values obtained from the theory of thin wing sections shows that the absolute

THEORY OF WING SECTIONS

180

magnitudes of the pitching-moment coefficients for the NACA four- and five-digit series sections are somewhat less than those indicated by the theory. The pitching-moment coefficients for the NACA 6-series wing .1

-

.

~

......• ~

,,:..

~

.A.

.A.

..,;,.

..;.

......

- - --- -

-.1

o

I

o

...

~.~

-.2

~o~

~ '"--

B 0.1

----

.~:~-L I-&-

i -.,

.....

64-

oJ--

-

0.2

o.L.

" 0.6

H1

~ ~~

~ 8 12 16 Alr1'o11 tblClmeaa. percent lIACA

-

~

.A

'V

"V"'" ~

4J-- ~

(c)

c&l

eo

'-

20

or cborcl

aeries.

$lasle t"lasP4 sJIIIbols are £or 6fP siMulated split 1'lap

.1

1 .....j t ..

...

0

• • o. 5

~.aa

-. ....

~

'"

---

-

.....

~~

C

~

~

0

«-: oJ-.....

a

I -.,

,,:..

r«: ~

o~ .~

~h

--*

A..

-.1

I0 -.2

.l1De,

~ ~

~

......, f'--- -<J

.....

L. 8 12 16 20

Airfoil thlckness. percent ol chord

(d)

65- aeries. 92. (Continued)

MACA FIG.

sections show practically no variation with thickness ratio or position of minimum pressure. As indicated by the theory of thin wing sections, in­ creasing the amount of camber causes a nearly uniform negative increase of the pitching-moment coefficient. In general, the absolute magnitude of the quarter-chord pitching-moment coefficients for the K AC.~ 6-series wing sections having mean lines of the type a = 1.0 are approximately three­

EXPERIltlENTAL CHARACTERISTICS 0// WING SECTIONS

181

.1 ~

0

I

..

~

c;

.:

7"

"':"

"":"

~

A.

~

h­

~~

-.1

0

:: '-c

•a 0

-.2

~R Nt... ~ ~

.pi

c:

I :.I

-.,

~

~h

r«.. r---., .h­

-­rc;r--...

-..,

"'"

--1&.0

~

~~

4 8 12 16 20

AlrtoU thlcJ.:nes.•, percent or chorcl (e) HACA FIG.

66- aeries.

92. (Concluded)

.­ •.

c:

.:

V

~

"s... o o

-­

V

..•

v

65,-618/

I»

­

~

• ~ o

I

.

V

c

,i

lL

-'1--66(21 «; )-216 • •

2'012

-.02

-.oL

I

j -.06

= 0.6

-.08

....10

-.12

-.l~

-.16

1beoretlcal IDoment coefficient tor ~t.e airfoIl

. . .n liae about quarter-chord point

FIG. 93. Comparison of theoretical and measured pitching-moment coefficients for eome

NACA airfoils. R, 6 X l()S.

182

THEORY OF WING SECTIONS

quarters of the theoretical values. The pitching-moment coefficients for sections having mean lines of the type a < 1.0 are equal to or slightly more negative than the theoretical values as shO\VD in Fig. 93. Consequently, changing the type of mean line from a = 1.0 to a < 1.0 to reduce the magnitude of the pitching-moment coefficient is relatively ineffective un­ less the value of a is reduced to a small value. The variation of chordwise position of the aerodynamic center at a Reynolds number of 6 million for a large number of NACA wing sections is presented in Fig. 94. From the data presented in Appendix IV, there appears to be no systematic variation of ehordwise position of the aero­ dynamic center with Reynolds number between 3 and 9 million. For the N ACA 24-, 44-, and 23o-series wing sections with thickness ratios ranging from 12 to 24 per cent, the chordwise position of the aerodynamic center is ahead of the quarter-chord point and moves forward with increases of thickness ratio. The data for the NACA 00- and 14-series sections show that the aerodynamic center is at the quarter-chord point. The aero­ dynamic center is aft of the quarter-chord point for the NACA 6-series wing sections and moves rearward with increase of thickness ratio. There ap­ pears to be little systematic variation of the chordwise position of the aero­ dynamic center with camber or position of minimum pressure for these sections. For the thick cambered sections, the chordwise position of the aerodynamic center appears to move forward as the design position of minimum pressure moves back. For wing sections with arbitrary modifica­ tions of shape near the trailing edge, the trailing-edge angle appears to be an important parameter affecting the chord,vise position of the aero­ dynamic center. For such sections, the aerodynamic center moves forward as the trailing-edge angle is increased."

EXPERIMENTAL CHARACTERISTICS OF WING SECTIONS

183

.26

.24

~

HACA

r---

r---.­

~~

2~- •• rl.a

"'"

~~

.22

.2&

~~~

IlACA

~ ~.)

-..... re.-.

---­ ---­ _G

e

~ a,

~

.

JIACA 14-aerl••

.

MCA OO-aerle. .22 0

I

~

I

I

.

8 12 16 Airfoil thieme.a. percent ot (a)

r-, 4·)

Hl.CA 24... erlea

.22

~

44-aerl••

20 ch~rd

NACA four- and five-digit series.

94. Variation of section chordwise position of the aerodynamic center "pith airfoil thickness ratio for several NACA airfoil sections of different cambers. R, 6 X 10'.

FIG.

THEORY OF WING SEC7'IONS

184

SJIIlbo18 with f'lap corre8poDd to ~

c11 ~

~

.

=0.6 ~ ~

~~

G

Cl.,

=0.4 and 0.6

A

.

V~

V

.-...

C11

W ~

0 ~

-21,­

l~

~

=0.2

. 4~

....­

~~

.......

~

L..e

~

G

ell =0

o

8 12 16 20 Alrl'o11 thicknesa. percent ot chord

J,.

(b) FIG.

NACA 63-series. 94. (Continued)

2JI.

EXPERIMENTAL CHARACTERISTICS OF IVING SECTIONS

.~

=0.6

Bplbo18 with tlas8 correspond to 01 1 .--t.~

o

:::i--~ •

•24"-.......­ ......-----......~-------.a..-...a.---.l~~

~ .28 c.~

.-......~ ~

.~.

01

1

~.,

--­

= 0.2

..",J0) ~ ..",-

Cl

-­

c·,.........

---­ ----­ Cl

4

8

=0.1

l

l

o ~..,­ ,...".,.....

~.

-

=0

12

16

20

Alrroll th1ckness, percent or chord (c) N ACA 6·l-serics.

FIG.

94. (Continued)

185

THEORY OF WING SECTIONS

186

8Jmbo18 with

nasa

cCll'reapord to 0Il

.28

=0.6

~ .~

··L

~ .26

-

...

• •

01

~ .~

1

= 0.4

and

~.)

0.6

0

0

1 .28

f:.~ c

..•

.2&

...

.24

0

.28

0

.p

a.

e

e,....­

a

~

j.:L..-­

.,~

~Il

=0.2

.~

••

~

~

~

CI

~~

to­

e

.2&

~

0Il ~

w

4-)

4·)

8

=0

-

-­

.­

v

~

.

12

16

A1r.toU thlolme••• perOeDt (d) NACA 65-series. FIG.

•.)

....

94_ (Continued)

20 ~

chord

EXPERIlYfENTAL CHARACTERISTICS OF WING SECTIONS

.28

l...--J.) \;I

s:• .24

c, 1 = 0.4

.p

i o

o

1.28 ~

2 : .26

~

o

--.

4r

c~l

5 .....24 .p

.. •

...:..

v

t:\

=0.2

8. .28

....••

t• .26

a

G

:.-.--­ ~

~

r--­

~

...;..

,...­

ct.

l

=0

4 8 12 16 20 AirtoU thickness. percent or chord (e) NACA 56-series.

FIG.

94.

(Concluded)

187

CHAPTER 8 mGH-LIFT DEVICES 8.1. Symbols. CD C DOlin CLmax /i.C L DUlX

drag coefficient minimum lift coefficient maximum lift coefficient increment of maximum lift coefficient C N normal-force coefficient eN/flap normal-force coefficient based on the area. of the flap CQ volume-flow coefficient, Q/VS E flap-chord ratio, cllc G moment arm, in terms of the chord, of the basic normal force about the quarter­ chord point

u ~-Iach number P Q& incremental additional load distribution associated with flap deflection P b" incremental basic load distribution associated with flap deflection Ps incremental load distribution associated with flap deflection Q volume of flow through slot R Reynolds number, pVcJp. S wing area V velocity of the free stream c chord Cd section drag coefficient CI flap chord CA section flap hinge-moment coefficient, h/~pytcr Cl section lift coefficient

e'l section flap lift coefficient, ll/>~pVtcl

Cl mas maximum section lift coefficient

~Imax increment of maximum section lift coefficient

em section pitching-moment coefficient, m/~p lltc2

Cm~/, section pitching-moment coefficient about the quarter-chord point

c.1 section pitching-moment coefficient about the quarter-chord point with flap neutral ~ section pitching-moment coefficient about the quarter-chord point with flap deflected ~c". incremental section pitching-moment coefficient about the quarter-chord point Cal section normal-force coefficient with flap neutral Cnt section normal-force coefficient with flap deflected ~a incremental additional normal-force coefficient associated with flap deflection Ca" incremental basic normal-force coefficient associated with flap deflection C-n/a incremental flap normal-force coefficient At:. incremental section normal-force coefficient (cp)J center of pressure of the load on a flap measured from the leading edge of the flap k section flap hinge-moment coefficient (positive in direction of positive flap deflection) 188

HIGH-LIFT DElT/CES

l, m n/ a q r

v ~I'a

x y a

ao &to "Yael' "Yb&

8 8/ a

a/_

~cl p T

Tn. T..

189

lift force acting on the flap per unit span

section pitching moment

incremental flap normal force per unit span

dynamic pressure, ~p VI

leading-edge radius

thickness of wing section local velocity over the surface of a symmetrical section at zero lift increment of local velocity over the surface of a wing section associated with angle of attaek

distance parallel to chord

distance normal to chord

angle of attack

section angle of attack increment. of section angle of zero lift ratios of the flap normal force to the section normal force for the incremental additional and the incremental basic normal forces flap deflection

deflection of fore flap or vane

deflection of main flap of a double-slotted flap configuration

increment of flap deflection

mass density of air turbulence factor, effective Reynolds number /tcst Reynolds number

see Eq. (8.8)

8.2. Introduction. The auxiliary devices discussed in this chapter are essentially movable elements that per­ mit the pilot to change the geometry and aerodynamic characteristics of the wing sections to control the motion of the airplane or to improve the per­ formance in some desired -manner. The desire to imp ove performance by increasing the wing loading while maintaining acceptable landing and take-off speeds led to the developmen t of retrae able devices to improve the maximum lift coefficients of wings without changing the characteristics for the cruising and high-speed flight conditions. Some typical high-lift de­ vices are illustrated in Fig. 95. The aerodynamic characteristics of some typical high-lift devices are presented and discussed in the following sec­ tions. Primary emphasis is given to the capabilities and relative merits

SPlIT FLAP

C----~:::--=­

~

EXTERNAl AIRFOIL RAP

c:

~

DOUBLE· SLOTTED FlAP"

/

~-----:::-::--==LEADING EDGE SlAT FIG.

95. Typical high-liit devices.

THEORY OF WING SECTIONS

190

of the various devices. Numerous references to the literature are given to provide design information.

•8 I

I

I

I

I

.,

~

/~

1
./

4

..

CD

to

>

.,.f ~

0 CD Cf-t

ft-t

CD

D.

as

.-l fH

~

/

V

./ ~,

.2

CD

0

~

/

.~

s::CD

.L

~" ri.7

,-..

~

10-'"

r\ I

BxperlaenU.l

~ ~

~

C

~!J

j~

/rfJ !/ r 0

(a)

.8

range f'rom 0° to 10°.

I I I I I I

.,

4fI#'

~~

-:lheoretlcal

l ~",

-'~

AI ~ ~

,/

/~

.4

IAl

,

"L I

I

y

~

'Iii'

\ I

BxperSmental -

-

~

_l/~

II

.2

-

~~

itf

/

til

~ ~,

.JIll'

...,~

",~

~

0

0 CD

,.

~

....... ~

"'" V

/- ~

s::

......,

"'~ ." ",.

r-, ,-~

dOloca

I

'1heoret1cal

/~ ~~

,

J~

~

o 0V

.1

.2

.,

.0

•

flap-chord ratio, E (b)

6

range from 0 0 to 200

•

Flo.96. Variation of section flap effectiveness with flap chord ratio for true-airfoll-eontour flaps without exposed overhang balance on a number of airfoil sections; gaps sealed; Cz -= o.

8.3. Plain Flaps. Plain trailing-edge flaps are formed by hinging the rearmost part of-the wing section about a point within the contour. Down­ ward deflections of the trailing edge are called "positive-flap deflections." Deflection of a plain flap with no gap effectively changes the camber of the wing section, and some of the resulting changes of the aerodynamic ehsrae­

Sym­ hoi

-

0

+­ X 0 -

o -­ V

Air-flow characteristics

Type of flap

Basic airfoil

NACA 0009 NACA0015 NACA 23012 NACA 66(2x15)-009

NACA 66-009

NACA-

-­

63,4-4(17.8) approx,

~

66 (2xl5)-216, a = 0.6

-­ <1

E>

-­ ~

-­ b. -­ ~

17 $)

-­ JJ -­ -0 -­ b

S>

-­ ~

-­
M

R

Plain Plain Plain

1.93 1.93 1.60

0.08 0.10 0.11

1.4 X lOS 2.2 X lOS

Plain, straight contour

1.93

0.10

1.4 X 10'

Plain

1.93

0.11

1.4 X 10'

Internally balanced

Approach­ ing 1.00

0.17

2.5 X 10'

Internally balanced

Approach­ ing 1.00

0.18

5.3 X 10'

Intemally balanced

Approach­ ing 1.00

0.14

6.0 X 10'

NACA NACA

Piain

64,2-(1.4) (13.5) NACA 65,2-318 approx, NACA 63(420)-521 approx, NACA 66(215)-216 a ,. 0.6

NACA 66(215)-014 NACA 66(215)-216 a == 0.6

NACA

NACA

8.0 X IO'

Approach­ ing 1.00

0.20 to 0.48

2.8 X lOS

Plain

1.93

0.09

1.2 X lOS

Plain

Approach­ ing 1.00

.....

6.0 X 10'

0.13

6.0 X 10'

0.13

6.0 X 10'

0.13

6..0 X 1()6

0.14

8.0 X lOG

0.13

6.0 X 106

0.13

6.0 X 106

Internally balanced

Internally balanced

NAC.A

Internally balanced

Internally balanced

KAC4~

64,3-1 (15.5) approx.

Approach­ ing 1.00 Approach­ ing 1.00

....

Internally balanced

12)-213

745A317 approx. NACA ~3-013 approx.

.... 13.0 X loa

mg 1.00

6..0 X 10'

Plain

NACA

Approach­

....

0.14

Plain

65.-421 65 (

Internally balanced

Plain

65r415 NACA 65r418

-

.,.

NACA

66(2x15)-116, a == 0.6

-

Intemally balanced

Approach­ ing 1.00 Approach­ ing 1.00 Approach­ ing 1.00 Approach­ ing 1.00 Approach­ ing 1.00 Approach­ ing 1.00 A:pproach109

(c) Supplementary information.

Fla. 96.

(CQTJditded)

191

1.00

to 6.8 X 106

I 0.13

I

6.0 X lQ6 •

THEORY OF WING SECTIONS

192

teristics may be calculated from the theory of thin wing sections (Chap. 4) if the flow does not separate from the surface. For most commonly used wing sections, this condition is satisfied reasonably well for flap deflections of not over 10 to 15 degrees. The theory permits calculation of the angle of zero lift, the pitching-moment coefficient, and the ehordwise load dis­ tribution with reasonable accuracy. The flap loads and hinge moments may also be obtained, but the accuracy of these quantities is relatively poor because the effects of viscosity are particularly pronounced over the aft portion of the section. The effectiveness of the flap in increasing the to maximum lift coefficient cannot be (6 in rOdor/s) calculated. ~'Ch I Glauert" calculated the effect of .8 do plain flaps in changing the angle of zero lift, the pitching moment, and dcm e / ' ~ <, the flap hinge moments. The cal­ ~_~4 .6 If d4 r-, I culated effect of flap deflection on the angle of zero lift is shown in Fig. '~ "", dch _ 96, where the flap effectivenessJ l1aol alJ is plotted against the ratio I ~ -, dcz~ .2 of the flap chord to the section chord ~~ ~ -, L.-­ ~ cllc. Numerous experimental points ~ r-, are also shown for flap deflections of 6 .2 .4 c· .8 I.IJ 0 to 10 degrees and 0 to 20 degrees.

N -Z-

"

/ I

-:

'" I'< -,

-,

~,.....

In general, the flap effectiveness is less than that indicated by the the­ ory, and the discrepancy increases with increased flap deflection for small chord flaps. The calculated ef­ fectiveness of a plain flap in changing the section pitching-moment coeffi­ cient about the quarter-chord point dc./dO is shown in Fig. 97. The calculated values of the rate of change of hinge moment with lift coeffi­ cient dch/dc, and the rate of change of hinge moment with flap deflection dcA/tU are also shown in Fig. 97. For symmetrical wing sections, the theo­ retical pitching-moment and hinge-moment coefficients are

97. Theoretical hinge moment and pitching moment characteristics of plain trailing-edge flaps.

FIG.

em =

a(~)

_ (den) + ~ (dC\)

lU

en. - C, -

0­

dCI

The linearity of the theory of thin wing sections permits the applica­ tion of the values obtained from Figs. 96 and 97 to cambered as well as Symmetrical sections. In the case of cambered sections, these values are applied as increments to the corresponding values for the unflapped section.

HIGH-LIFT DEVICES

193

Pinkerton" calculated the flap loads that are presented in Fig. 98. The rate of change of the flap lift coefficient with the section lift coefficient dell/dc l and with flap deflection dc,l/dJ are plotted against the ratio of flap chord to section chord. 3.0 2.8

1

I

W/Ild

dir«:lion

.....

L.I

1

~''''''''''':

I

\oc

2.6

I

2.4 2.2

ik

CII:;;;'

r\..

~

2.0

"

1.8

~~ ~I~ a

"c, +::;"6 de:

I\.

1.6

1.4

'\.

1.2

"r\.-, !f:JL 1,,<6

1.0

~

.6

.4

.2

)

~

V oo ./

~

~

~

~

~

/

V

~ t<~

r\

.2 .3 .4 .5 .6 .7 .8 £, Ratio of flap chord 10 totatchord

l_

'\~

-,

.9

1.0

FlO. 98. Theoretical variation of flap lift coefficient with section lift coefficient and 1lap deflection.

The pressure distribution over a lying section with a deflected plain flap may be calculated by the "ring-section theory of Chap. 3, but such calcula­ tions for a number of flap deflections would be unduly laborious. Some knowledge of the ehordwise load distribution is given by the theory of thin lying sections. The change of load distribution caused by flap deflection can be resolved into t\VO components, one of which is simply the additional load distribution associated with angle of attack. The other component is the difference bet,veen the load distribution on the original and deflected mean lines at their respective ideal angles of attack, These load distribu­

THEORY OF WING SECTIONS

194

tions may be calculated by the theory of thin wing sections (Chap. 4) as explained by Allen,' but the results are not very accurate. For example, the simple theory predicts infinite load concentrations at the leading edge and hinge location. The infinite load concentration at the leading edge may be avoided by use of the additional types of load distribution associ­ ated with angle of attack as obtained from thick-wing-section theory and presented in Appendix I. Allen' obtained empirical load distributions analogous to those associated with changes of the camber that permit calculation of reasonable approximations to the load distributions of flapped 6

- - - R.N. .356 x 10 - - R.N.6.10 x/OS

-3

6=50­

8·:40· 8=30 0 +/

0

4=20 . . 4=/0 0

ConsIont angle of o~/ack 20%cflop. art"

FIG.

99.. Distribution of pressure over a wing section with a plain flap.

wing sections. Allen's method is presented in Sec. 8..8. Once the load distribution is obtained, the pressure distribution may be calculated by the method of Sec.. 4.5. Some typical pressure distributions for sections with plain flapsfl are presented in Fig. 99. The lift, drag, and moment characteristics of a typical NACA 6-series wing section with a O.2Oc plain ft ap3 are presented in Fig. 100 for several flap deflections up to 65 degrees. These and other data" show that the angle of maximum lift coefficient with the flap deflected is generally some­ what less than for the plain wing section. Curves of the maximum lift coefficient plotted against flap deflection for two sections" are presented in Fig. 101. Plain flaps of 0.2Oc appear to be capable of producing incre­ ments of the section maximum lift coefficient ranging up to about 1.0 and are more effective when applied to sections with small amounts of camber. Addit~onal data on plain flaps are presented byJacobs," Abbott," and Wenzmger. l 48, 1., 116

HIGH-LIFT DEVICES :~::

~

e ••

.... ... ..

i!::

I~:: :i:·

i~"

. . .•.

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THEORY OF WING SECTIONS

196 ....;g;

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HIGH-LIFT DE1'ICEB

197

When the turbulent boundary layer over the aft portion of the wing section is thin and resistant to separation (as when extensive laminar flow is obtained), small deflections of a plain flap do not cause separation but shift the range of lift coefficients for which low drag is obtained.' If the extent of laminar flow is not large or if the flap deflection is sufficiently

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Plap detlection. 6 . des FIG. 101. Aiaxirnuill lift coefficients for the NACA 65,3-618 and NACA foils fitted with O.20..airfoil-chord plain fiaps. R. 6 X 1(;'.

. ()6(215)-~16

air­

large to be of interest as a high-lift device, the flow over the flap separates and large drag increments result. .A. typical variation of drag with lift coefficient for the NAC..\ 23012 wing section with a O.20c plain flap de­ flected the optimum amount at each lift coefficient/ is shown in Fig. 102. 8.4. Split Flaps. The split flap is one of the simplest of the high-lift devices. The usual split flap is formed by deflecting the aft portion of the lower surface about a hinge point on the surface at the forward edge of the deflected portion (see Fig. 95). One variation of the simple split flap has the hinge point located forward of the deflected portion of the surface in

THEORY OF WING SECTIONS

198

such a manner as to leave a gap between the deflected flap and the wing surface. In another variation the leading edge of the flap is moved aft as the flap is deflected, either with or without a gap between the flap and the wing surface. Split flaps derive their effectiveness from the large increase of camber produced and, in the case of some of the variations, from the effective increase of wing area.

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~

The lift and moment characteristicsl for typical NACA 6-series wing sections with O.2Oc split flaps deflected 0, 40, 50, 60, and 70 degrees are pre­ sented in Figs. 103 and 104. Similar data for a flap deflection of 60 degrees are presented for most of the wing sections of Appendix IV. The lift-curve slope with split flaps is higher and the angle of maximum. lift is somewhat lower than for the plain section. There appears to be a tendency for the lift-curve slope for large flap deflections to be less than that for moderate deflections.v 154 Inspection of Figs. 103 and 104 shows that the split flap is more effective in increasing the maximum lift coefficient when applied to the thick than when applied to the thin "ring sections. Figure 58 shows that the maximum lift coefficients of NACA 6-series wing sections with O.20c split flaps de­ flected 60 degrees increase with thickness ratio up to ratios of at least 18 per cent. Figure 58 shows comparatively little variation of maximum lift coefficient for NACA four-digit wing sections with O.2Oc split flaps for thickness ratios greater than 12 per cent, but the maximum lift coefficient

HIGH-LIFT DEVICES

199

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eectiOD

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THEORY OF WING SECTIONS

200

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HIGH-LIFT DEli''lCE8

~1

decreases for smaller thickness ratios. These results are in essential agree­ ment with those1M of Fig. 105, which shows that, for the NACA 23o-series sections, the maximum lift coefficient generally increases with thickness ratio for large-chord split flaps but decreases for small-chord flaps. 3.2

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32

FIG. 105. Effect of airfoil thiCknC88on maximum lift coefficient of N AC~\ 2.1O-8eriea airfoils 1\ith and without split flaps.

The variation of tile increment of maximum lift coefficient resulting from deflection of split flaps of to, 20, 30, and 40 per cent chord 1i4 is shown in Fig. 106. For wing sections of normal thickness ratios, substantially full benefit is obtained for flap deflections of 60 or 70 degrees. The effect of the chord. of split flaps on the increment of maximum lift coefficientl M is shown in Fig. 107. For wings of normal thickness ratio, comparatively little benefit is obtained by increasing the flap chord to more than 20 or 25 per cent of the section chord, although large flap chords are more effec­ tive on thick than on thin sections. Deftection of a split flap produces a bluff body, and large drag incre-

THEORY OF WING SECTIONS

202

ments are to be expected. Figure 102 shows that the drag of the NACA 23012 section with a 0.2Oc split flap deflected the optimum amount at each lift coefficient is about the same or less than that for a plain flap.! The pressure distributions over wing sections with highly deflected split ftaps are similar to those for plain flaps (Fig. 99). The pressure distribu­ tions with deflected split flaps may be predicted by the semiempirical method of Allen.'

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(a)-NACA23012 ARFOIL FIG.

020406080 Flog delleclion,66 deg.

(b)-MACA 23021 AIRFOIL

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80

sx

(c)-NACA 23030AIRFOIL

106. Effect of split-flap deflection on increment of maximum lift coefficient for various

airfoils and flaps.

Data required for the structural design of split flaps are given by Wenzinger and Rogallo.l11 The normal force coefficients and centers of pressure for a 0.2Oc split flap at various deflections are shown in Fig. 108. The data of Wenzinger and Rogallo show that the chordwise position of the forward edge of the flap has only a small effect on the flap loads. The effect of a gap between the wing surface and a split flap with a nominal chord of 0.2Oc hinged at O.SOc from the leading edge of the sec­ tionllO is shown in Fig. 109. The data show that the lCBJ of maximum lift coefficient associated with the gap is considerably greater than that caused by removing the same area from the trailing edge of the flap. Figure 110 summarizes the effect 1," on maximum lift coefficient of moving a O.2Oc split flap toward the trailing edge from its normal position. These data show that the percentage increment of maximum lift coefficient obtained by moving the split flap to the rear is about the same as the percentage increase of wing area measured as projected onto the original

HIGH-LIFT DEVICE8

203

chord line. Similar results have been obtained!" for O.3Oc and O.4Oc split flaps. 8.6. Slotted Flaps. G. Description oj Blotted Flaps. Slotted flaps pro­ vide one or more slots between the main portion of the wing section and the 2.0

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Flap chotrJ,pen:enI C

Flo. 107. Effect of chord of split flap on inerement of maximum lilt coefficient for three airfoil thicknea8eL

deflected flap; and they derive their effectiveness from increasing the camber and, in some cases, from increasing the effective chord of the sec­ tion. The slots duct high-energy air from the lower surface to the upper surface and direct this air in such a manner as to delay separation of the flow over the flap by providing bdundary-layer control. The numerous types of slotted flap are classified by their -geometry. Several types are shown in Fig. 111. The primary classification is the number of slots. The single-slotted flap is the simplest and most generally

THEORY OF WING SECTIONS

204

used type. Double-slotted flaps have been used to some extent, and multiple-slotted or venetian-blind flaps have been investigated. An im­ portant consideration in the design of slotted flaps, especially of the single-

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FIo.108. Normal force coefficients and centers of pressure of a O.2Oc split flap at O.8Oc on a NACA 2212 wing.

slotted type, is the extent to which the flap moves aft as it is deflected. The movement of the flap may vary from a simple rotation about a fixed point to a combined rotation and translation that moves the leading edge of the flap to the vicinity of the normal trailing-edge position. Rearward movement of the flap requires an extension of the upper surface over some or all of the flap in the retracted position. This extension of the upper surface serves to direct the flow of air through the slot in the proper diree­

HIGH-LIFT DEVICES

205

tion and is called the "lip." The external-airfoil Hap (Fig. Ill) may be considered as a special case of the single-slotted flap with the distinguishing feature that it does not retract within the section. The flow about a wing section with a deflected slotted flap is very complicated, and no adequate theory has been developed to predict the aerodynamic characteristics. Consequently the information required for

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109. (4) Details of split flaps with gaps tested on Clark Y wing.

design is obtained entirely by empirical methods. Although many experi­ mental data have been accumulated, the large number of configurations possible and the sensitivity of the chacteristies to small changes in the slot configuration make the design problem a difficult one. b. Single-sloUed Flaps. One important parameter in the design of single­ slotted flaps is the chordwise position of the lip. Although completely comparable data are not available for configurations with varied positions of the lip, the maximum lift coefficients appear to increase as the lip position approaches the trailing edge for lying sections of moderate thick­ ness. This effect19• 153 is shown in Fig. 112, where increments of maximum lift coefficient are presented for the NAC£.\. 23012 section with single­ slotted flaps having lips located at O.82ic, O.9OOc, and 1.000c. The con­

THEORY OF WING SECTIONS

figuration with the lip located at 0.827c is not exactly comparable with the others because the chord of this flap is 0.2566c as compared with 0.3Ocfor the others. It is apparent, nevertheless, that the increment of the max­ imum lift coefficient is considerably higher for the configuration having the lip at the normal trailing-edge position than for those with a more forward lip location. It is uncertain whether any of the difference between the 2.4

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Reduclial ofchotrJ,percenl flcp chad FIo. 109. (Conduded) (b) Effect on CLmas and on CD at CLmaz of reducing the chord of a 0.2Oc aplit flap. a.. 60 dep'8e8.

other two configurations should be attributed to the difference in lip posi­ tion. Envelope polars for these flaps are presented- in Fig. 113. When the lip is located at or near the normal trailing-edge position, the thickness of the flap is necessarily less than that for a more forward location of the lip20 (Fig. 114a). In the case of thin wing sections, especially of the NACA 6-series type, the flap thickness may become too small with a rearward location of the lip to permit favorable slot configurations. Under such conditions, the favorable effect on the maximum lift coefficient of moving the lip toward the normal trailing-edge position may not be realized. CahillJO shows (Fig.. 114b) that the maximum lift coefficients for

HIGH-LIFT DEVICES

207

c

FIo. 110. Contours of

eL..... for various positions of trailing edge of 20 per cent flap.

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SlDTTED FLAPWITH LONG LIP

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HIGH-LIFT DEl"ICEB

209

the NACA 65-210 section are essentially the same for lip positions of O.84c, 0.9Oc, and 0.975c. The structural difficulties presented by a long thin lip extension and the mechanism necessary for the corresponding large rear­ ward movement of the flap are such as to discourage the use of rearward locations of the lip unless such configurations result in substantial improve­ ment of the maximum lift coefficient. :> Q30c sIoIfed llop with extended lip; -w---+--+-t--+---+----f gop=O.02c .20~

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113.

Envelope polar curves for three slotted flaps on a NACA 23012 wing section,

The effect of flap chord on the increment of maximum lift~· U3 is in­ dicated by Fig. 115. The data presented .for the O.. 25(j(ic and O.4Oc flaps are reasonably comparable in that the shapes of the slots are generally similar. Figure 115 shows that larger increments of maximum lift coeffi­ cient are obtained with the larger chord flap, but the increased effectiveness is small compared with the increase of flap chord, The slightly higher maximum lift coefficients obtainable with large chord flaps do not appear to justify the structural difficulties encountered with such flaps, and flap chords in excess of O.25c to 0.3Oc are seldom used. The maximum lift coefficients obtaineiP" 96. 1&3 with various arrange­ ments of slotted 8ap on NACA 23012, 23021, and 23030 wing sections are plotted in Fig. 116. The flap chord was 25.60 per cent of the section chord in all cases. These data indicate little variation of the maximum lift coeffi­ cient with thickness ratio from 12 to 30 per cent for this type of wing section. A few data3, . are also shown in Fig. 116 for comparable slotted flaps on NACA 6-series wing sections, In this case, the flap chords are 25 or 30 per cent of the section chord. These limited data indicate that, for the NAC.A. 6-8eries sections, the maximum lift coefficients obtainable with

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~

>---- l---'

J

.4

~ gj es

c1 (degJ

~

o Ph/no/rfoI1. ---~ e SbHfd fk¥J 1_ _ 45 A '1

~

SIdled/Iff' 2 _ _ 41.3 SIdled IlqJ 3._ _ 35

I

0

-

Sio/led flop 3

FlO.

~ ~

i

.8

(0) CONFIGURATICWS

"'<

.--

-

..

_.-:

-

Reynolds trr!1Iw, R (b)

MAXI~

LFT OlTA

114. Variation of maximum eect.ion lift cooflicjent. with .RM)Onoldi' number Cor several slotted flaps on the NACA

66-210 whig 8(wtion.

HIGH-LIFT DEVICBS

211

1.4

I'

c

/

D.2566cflap

V

f

/

I

,

........... .-'

-,'

-

-­ -­

~-

-­

~'

~o.4Oc flap

i. i

I I ,, I

c-------­

I

./

'I

l

"" O.2566c flop

I

I'J " l' If 00

(0) CONFIGURATIONS

D

20 .JO 40 Flop *,~,dtJg.

!JO

60

(b) INCREMENTS OFMAXIMUM UFT COEFFICIENT

FIG. 115. Effect of flap chord on increments of section maximum lift coefficient for the N ACA 23012 wing section.

3.2 I

+

--- ---..

§

1

.............,

4

E

.~

2.0

~

.~

~

1.6

-

Symbol Kfng section series 0

x

...

~

I 1.2

6

10

,

N.A&.A.230N.AC.A. 6 N.A.CA 6 -

14

Flop chord ratio O.2566c

--

o.ese

1 i

,

B..

22

O.JOc

1

--

I 26

30

section Ihickness rolio, Pc, percent FIG.

116. Maximum lift coefIiciente for various arrangements of slotted flaps.

212

THEORY OF WING SECTIONS

slotted flaps on 10 per cent thick sections are appreciably less than those obtainable with thicker sections. The effects on the maximum lift coefficient of some variations of shape of the slot are illustrated'" in Fig. 117. In configurations l-a and l-e, the slot is only slowly converging, if at all, at the end of the lip, and these COD­

[ Skilled flop 1-0

8,640~. 62.~6

Czm..

[ SIoIIedflop I-b

6,6SOdtlg. clmax.· 2.76

l

SJolltJd flop t-c

6,.:55 de9.

"1mcr.'X. ·2.75

[ SIrJIIedfkJp '-ee

6i 645 t!tJ9.

'f"...•2.il9

l

SIdled f/cfJ e-« 6,.:50~.

'ir..

·2.81

l

SbIt«J fltJp2- i 4,·60dtlg. Czm... • 2.674

[ SItJIIt1d fltJp 3-1 t1,.50~

'"'-w. .:2

I

SItJIItId fltJp 3-g

6,.55~

c'--.·

.60

FIG. 117. l\laximum lift coefficients attainable with various ammgements of slotted flaps on the NACA 23012 wing section.

figurations have the lowest maximum lift coefficients of the I-series configurations. A short extension of the lip as in configuration I-b, which makes the slot definitely convergent and directs the air downward toward the flap surface, is effective in increasing the maximum lift coefficient. It may be concluded from these and other data that the slot should be definitely convergent in the vicinity of the lip and shaped to direct the air downward toward the flap. The effects of changing the radius of curvature at the entry to the slot from the lower surface arc shown by configurations I-b and l-c of Fig. 117 for a flap having a comparatively small rearward displacement when deflected. Decreasing the radius of curvature from about 0.0& to O.04c did not produce a significant difference in the maximum

HIGH-LIFT DEV"ICES

213

lift coefficient. Other data indicate that this radius of curvature is of little importance when the flap is displaced rearward enough to produce 8. large area for the entry of air into the slot. It is difficult to draw general conclusions about the proper shape of a slotted flap. Figure 117 shows that the highest maximum lift coefficients were obtained with flap 2, which is shaped more like a good wing section than flaps 1 or 3. The difference between the maximum lift coefficients produced by flaps 1 and 2 is small and may be caused by the difference in slot shape and lip extension rather than by the difference in flap shape.

FlO. 118. Contours of flap location for 23012 wing eeetion.

Q lllar

Slotted flap 2-11., &, = 60 degree. on NACA

Flap 3, however, appears to be too blunt with a too small radius of curva­ ture on the upper surface aft of the lip in the deflected position. Typical cont.ourst61 of the maximum lift obtainable with various flap positions at one flap deflection are shown in Fig, 118. In' general, the optimum flap position for good flaps at large deflections appears to he that which produces a slot opening of the order of O.. Olc or slightly more and which locates the foremost point of the flap about O.Olc forward of the lip. The maximum lift coefficient, however, is frequently sensitive to the flap position, and the optimum position is best determined by test. A complete set of section eharacterisrics'P for a typical single-slotted flap configuration is shown in Fig. 119. This figure illustrates the charac­ teristic ability of slotted flaps to produce high lift coefficients with com­ paratively small profile drag coefficients. The increment of moment coefficient associated with the use of single­ slotted flaps" is illustrated by Fig. 120. This figure shows the ratio of the increment of the section pitching-moment coefficient" to the increment of the section lift coefficient at an angle of attack of 0 degree for three wing sections and several flaps. The moment coefficients used in this case are

~

o

~

t9 ~

-

~~

0, d6g.

o

.20

,

--b.

~

2O--v 30--0 50--1> 60--6

--

d I.

_nose

-

[ ~

I:lJ"

/

~

I'V

for deflecfed flop

II ~

J~v

~

.A~

rr

....

J~

~

lD'"~!o"""'"

10-~1IIr'

-

o

~~ ~

" - ID-

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~

-'

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l--"""

l-=== ~~ I

/6

~

~

~~

~-

~I-"""

L...;l~

~I""'" ~~

lJW-­

-~

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o

.4

..... ~

~v

l/~

I...oo"'~ L..--'~

10<'"

1.2

.8

"J" N''''

~ 'C

l...oo"'~ ....~ ~t,I' .~

~,;IIiT

;/..".

l/~

~Io"""

"

1/ ......

1.6

20

2.4

2.8

Section lifl coefficienll cl.

Path of flap nose for various fiap deflections. lip in per cent airfoil chord c. 81,

degrees

0 10

20

30

x

y

8.36 5.41 3.83 2.63

3.91 3.63 3.45 3.37

Distances measured from lower edge of

If, degrees

x

y

40

50

60

1.35 0.50 0.12

2.43 1.63 1.48

I

FIG. 119. Section aerodynamic characteristics of NACA 23012 airfoil with slotted flap 2-A.

214

HIGH-LIFT DElTICB8

215

based on the total chord with the flap deflected. The ratio plotted appears to be fairly constant at yaIues between about - 0.17 and - 0.21, instead of varying with flap-chord ratio as in the case of plain flaps at small deflections. The normal-force coefficient, pitching-moment coefficient, and center of pressure for the flap of the configuration of Fig. 119 (configuration 2-h of reference 153) are shownlU in Fig. 121. All the coefficients are based on the flap chord. The pitching moments are taken about the quarter-chord point of the flap, and the center of pressure is given in per cent of the flap

o -.04 1'---

-.08

"~I" ~ ~

Airfoi/ sections

~-

o a

1-0---

o

NAG4 23012

/

N4CA 2J02I N4G4 65-2/0

~

/'

<3-./2

/V

-./6

V •10

~o()

.20

~

~/

~T/le(relcol (plain flop)

V

0

~)

V-

V

i-'-

-

0

. .30

.40

.50

.60

F/q) -chotdratio, £ FIG. 120. Variation of ratio of increment of aection pitching-moment eoefIicienf, to increment of section lift coefficient with flap-ehord ratio for several sections with elotted flaps. GO = 0 degrees.

chord from the leading edge of the flap. These data are useful in determin­ ing the loads on the flap and the flap linkages, and they show that the normal force coefficients on the flap are less than those for the wing section. c. External-airjoil Flaps. The external-airfoil flap investigated by Wragg: 56GPlatt,838.f86 and Wenzinger161 may be considered as a special case of the single-slotted flap in which the ftap does not retract within the wing section. The maximum lift coefficients'4 obtained at an effective Reynolds number of 8 million for a N ACA 23012 wing section with & O.2Oc external­ airfoil flap of the same section are shown plotted against flap deflection in Fig. 122. These maximum lift coefficients are based on the combined areas of the wing and flap. The values presented were obtained from tests of a

... ~

0)

§I

L.E.

~~20 't5 '.-: et

i i~

~t

40

0

~

-./

~ ~

j{-.2 I)

~3

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2.0

~8

1.6

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.4

.8

1.2 1.6 2.0

.4

.8

1.2 1.6 2.0 2.4

0

.4

Norma/-force coefficient of-combination,cn

.8

1.2 1.6 2.0

0.f =40() (c) is =50 0 I Seolion chnrBcteristiclt of the flap alone or the NACA 23012 airfoil with a 0.25660 slotted flap.

(0) FlO. 121.

2.4 0

0.f

=30 D

(b)

2.4 28 '3

~

~ ~ ~ ~

S ~

HIGH-LIFT DEVICES

217

finite span model and were only partly corrected to section data. The corresponding section maximum lift coefficients are judged to be about 4 per cent higher than those presented. If these maximum lift coefficients were based on the chord of the wing section, as is customary for other types of flap, the resulting values would be about the same as those for single­ slotted flaps. Although the extemal-airfoil flap appearedu, 8& to offer some

2.4

..:­

a/f""

.i

/

V

./

'/

i-r

~~

~

\)

~ 1.6

1 ~

C",

~ 1.2

~

1§

~

.§ .8

~ P,iKJ/~

f 4

o

JO 20 XJ Flop defleclion l degrees

40

50

FIG. 122. Variation of maximum section lift coefficient. based on chord of wing section plus flap, with flap deflection for the NACA 23012 wing section with 0.20 c. external-airfoil flap.

advantages as a full-span flap, it has not been used extensively because of its failure to show definite advantages over retractable Haps and because its use would probably aggravate the icing hazard. d. Double-slotted /laps. Double-slotted flaps3 (Fig. 123) produce sub­ stantial increments of maximum lift coefficient over that obtainable with single-slotted flaps. The fore flap or vane of the double-slotted flap assists in turning the air downward over the main flap, thus delaying the stall of the flap to higher deftections. It has frequently been found possible to develop flap-vane combinations that can be retracted into the wing section without relative motion between the vane and the flap. Such an arrange­ ment is desirable because the linkage system is much less complicated than when relative motion is required between the flap and vane.

218

THEORY OF WING SECTIONS

Investigations by Harris,40 Purser," and Fischel' ! of approximately O.3Oc and 0.4Oc double-s1otted flaps on the NACA 23021 and 23012 wing sections indicated maximum lift coefficients of the order of 3.3 and 3.5 for the two sizes of flap. Typical results for the NACA 23012 section are shown in Figs. 124 and 125. The fore flap and the main flap did not deflect together as a unit for these configurations. Aerodynamic characteristics' are pre­ sented in Fig. 126 for the NACA 65,-118 section with the O.309c double­ slotted flap shown in Fig. 123. For this configuration, the vane and flap

.....-------lOCLf----------..I

...----.78/----------1...

moved together as a unit up to deflections of 45 degrees. At higher angles, the flap rotated about a pivot and the vane remained fixed. The variation of maximum lift obtained for thin NACA 64-series seetions" with approximately 0.3Oc double-slotted flaps is shown in Fig. 127. The type of fiap used for these tests is illustrated in Fig. 128. The :8ap and vane deflected as a unit. These data. (Fig. 127) show that the maximum lift coefficient decreases rapidly 88 the thickness ratio of the wing section is decreased to values below 10 or 12 per cent. The maximum lift coefficient obtained for the NACA 1410 wing section21 is also plotted to indicate the effect of type of section. The maximum lift coefficient ob­ tained for the NACA 65r118 wing section with a flap deflection of 45 de­ grees (Fig. 126) is also plotted to indicate the effect of larger thickness ratios. The NACA 65,-118 data are not exactly comparable, but the indicated gradual rise of the maximum lift coefficient as the thickness rano is increased to 18 per cent is believed to be representative. The effect of the design position of minimum pressure on the maximum lift coefficients obtained 'lith double-slotted flaps on 10 per cent thick NACA 6-series wing sectionsD is shewn !D Fig. 129. The type of flap used

HIGH-LIFT DEVICES

~ I-

-

~

/Chad -

.

219

~~'I

I

~~

~'''~~

-

v~~ +~

40'i;~

-

~

I-I--

I--

--~~,... .28

6~~

-~ --

~

X2

40 50 60 70 0.60 - 40 /.60 1.60

Y2

.05 205 2.05 1.05

'--

..........

/

~

·'1

11.

.24 ~

Of2'd!g.

.

D

~.20

70~~

~

.~

~ ./6

§

t·/2 :tl

6fV ~

I V

•

l~

1..os

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~

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V tJ ~

~,

A

.<.J

)

VI P

..-10~ VO ~

l

o

t

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i

8

~ 6'2,deg

".;;

~ \ .\ .\ .V' ~ ~ \J~ v ot'r~ 40 50 60 70

li

~

; I

0

~

~

~ -8 -.4

o

.4

.8

1.2

1.6

2.0

2.4

2.8

,

)~ J

3.2

3.6

5ec/lon lifl coefficient, 'i

FIG. 124. Aerodynamic eection characteristics of the NACA 23012 airfoil with a O.3Oc double-elotted flap. ala == 25 degrees; %1 :: 0.41; III == 1.72. (Values of %1, 111. %It and 1/1 are given in per cent of airfoil ehord.)

HIGH-LIFT DEVICES

221

was similar to that shown in Fig. 128. These data show that the highest maximum lift coefficient was obtained for the NACA 64-series sections and that the maximum lift coefficient decreases rapidly as the minimum-pressure

I~ ~

3.2

11) ~n r-, 45

6

(degJ

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0

8

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2~;

Section angle of oIl~ tXo,deg. FIG.

126. NACA 653-118 airfoil section with O.309c double-slotted flap.

position moves farther aft. It is thought that the rather large variations shown in Fig. 129 are associated with the fact that the thickness ratio of the sections \\"88 in a critical range, as indicated by Fig. 127. Load data for typical double-slotted flaps are given by Cahill.!! These data show that a disproportionately large part of the load is carried by the vane. Normal-force coefficients for the vane based on the vane chord and the dynamic pressure of the free stream reached a value in excess of 5.0.

THEORY OF WING SECTIONS

222

r­

I

.028

e

-/ V

6 .020

) ~

V 1,/

....

...... ro-..

'~

~

'/ I ~ -....

,~

1,

~

~ r-,

~ ~P-- ~ p.­ ~

'"

........tI.

)

~

V~

J

LJ

...-It!

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v

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10

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-

1;.35

J

J

(dtJg)

r--­

20

p

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r-­

~

~

r-­

r;JJ1

-

/ 'I

-

J

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~

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.004

o

,

.....

f--45

~

r-,

r........ >-­~

.....

~

o

IO "V'll

,..... ~

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.4

t'­ .-...,

~

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.....

-

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.8 /.6 Seclion lifl txJeffici~Ct FIG.

I

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fd8g.)

~ ~ ......

~ ~ .....

r--c:... ~ ~

-.4

-

.. ---­

~

lIo.... ....

65~ ~

-~8

..D..

A

r­

5~ rd:::: r---..

-

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126. (Condtule4)

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1 r-i l:.,,;R

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2.4

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.,

HIGH-LIFT DEVICES

223

I I ',,..A) N.A.CA. 65., -/18 A!A.aA./4/0~ )

(~

4

FIG.

I l.­ --N.A.C.A.64-series (f

_

8

20

/2

/6

SecIiaIlhickness lf1fio, ~ lpercenf 127. Effect of thickness ratio and type of wing section on maximum lift with double.

slotted 1laps.

Ftpdud

line

~---------

. T 1 ' 5 c - - - - - - - - -....

(o)AIRFOL WITHRAP

XI

c:: Airfo7chord line~

,,;

r

12·

8~~

FIG.

128. Typical airfoil and flap configuration.

THEORY OF WING SECTIONS

224 3.2

I

I

I

I

Double slottedflop wiIh fore tip .~ r---...... ......~

01t75c

2.8

J

.>---

2.4

~--

r----... ~

~_J.

---,---

~) ----4)

~

J...

~ 2.0

i

Plain wing section

fA

~ 1.6

~

.~

/1

~

..............

11

:~

~

...............

J

1.2 ~--

p-' ......

)-_1

-

."'4

.8

--Smooth ---- Leading edge rough .4

':'2

.3

.4

.5

.6

Position ofmimil1lU71 pteSSIIe, % maximum section lift coefficient with position of

.7

129. Variation of minimum pressure for 80rne NACA tHeries wing aectiODS of 10 per cent tbiclmees and a design lift coefficient of 0.2 .. R, 6 X 10'.. FIG.

FIXED AUXILIARY WING SECTION (AXED SLAT)

~
130.. Examples of fixed and retractable slats.

225

HIGH-LIFT DE"VICES

8.6. Leading-edge High-lift Devices. a. Slats. Leading-edge slats are airfoils mounted ahead of the leading edge of the wing in such an attitude as to assist in turning the air around the leading edge at high angles of attack and thus delay leading-edge stalling. They may be either fixed in I.lir--r-.---,r--...,..-...,...-..~-....-..-----.-------

~

2/.2t----r--t---+-~~-+-_I_______1.~~~-_J___J,.-__1

~

10

20

30

40

50

60

'Flop deflection, 6, ,deg. FIG. 131. efficient.

Eft"eet of flaps and leading-edge slat on increment of maximum section lift co­

position or retractable (Fig. 130). The fixed leading-edge slat consisting of an auxiliary airfoil mounted ahead of the wing leading edge has been investigated in detail by Weick and Sanders.s" This investigation showed that leading-edge slats of this type with chords varying from 7.5 to 25 per cent of the wing chord and with various sections all produced substantially the same maximum lift coefficient when located in the optimum position for the ratio CLmax2 ( Dmin

THEORY OJ! WING SECTIONS

226

The value of this maximum lift coefficient was about 1.64 for the rather low Reynolds numbers of the tests. It is doubtful whether such configura­ tions would experience much beneficial scale effect. The effectiveness of the retractable leading-edge slat107 shown in Fig. 130 in increasing the maximum lift coefficient and the angle of attack for 24

I I I I I I I I I - - ­ Q2566c sidled fk:p

---O.20c

- ­ O.2566c --Q2Oc

split fkp stxled f~arJB:xJing. sial splif flopmd 1eodiIg. sbf

6

2 .JI"'-

8~

~

~

~

--

_...­

~---

-~--.

........

­

,,/

4

n.

-

.......... ~~

'--r--­-... ------......!

4

o

10

20

30

..

---..IL_

~~ 40

~

50

r-, 60

Fkp deflechOnl 6 , ,deg. FIG.

132. Effect of 1lape and leading-edge slat on angle of attack for maximum lift.

maximum lift is shown in Figs. 131 and 132. These data, which were obtained on a K.~CA 23012 lying section, indicate an increment of about 0.5 for the maximum lift coefficient and about 8 degrees for the angle of attack for maximum lift. The increment of maximum lift decreased to about one-half of its value for the unflapped section when either a split or slotted flap was deflected to optimum deflections despite readjustment of the slat to optimum positions with flap- deflected. Weick and Platt!· obtained considerably larger increments of maximum lift coefficient with a special retractable slat (Fig. 133) having a shape providing a rounded entrance into the slot. The increments of maximum lift coefficient with

HIGH-LIFT DEVICES

227

this configuration were 0.81 for the unflapped section and 0.45 with a deflected slotted flap. b. Slots. Slots to permit the passage of high-energy air from the lower surface to control the boundary layer on the upper surface are common features of many high-lift devices. The most common application is the slotted flap. When the slot is located near the leading edge, the con­ figuration differs only in detail from the leading-edge slat. Additional slots may be introduced at various chordwise stations. Weick and Bhortal'" made a systematic study of slots on a Clark Y airfoil. The results of this investigation are summarized in Fig. 134. For the unflapped section, the most effective position for a single slot is near the leading edge, and the effectiveness decreases as the slot is moved aft. Multiple slots are relatively ineffective on the plain airfoil unless they include a slot near the

FIG.

133. Special retractable slat on Clark Y wing section.

leading edge, in which case a total of three slots, all located well forward, is optimum. For the flapped section, the slot located near the leading edge was effective. A single or double slot at the flap changed the type of flap from plain to slotted with- a corresponding increase of the maximum lift coefficient. Load data for the leading-edge slot are given by Harris and Lowry." These data show resultant-force coefficients as large as about 7.5 for that portion of the wing section ahead of a slot near the leading edge. If slots are considered as a fixed high-lift device, the profile drag in the high-speed. flight attitude is an important characteristic. Figure 134 shows that any of the slots investigated cause large increments in the minimum profile drag. This increment increases with the number of slots and de­ creases with rearward movement of the slots. Attempts have been made to maintain low drags with slots open by locating the slots so that there would be no flow through them in the high-speed. condition. lOS Such con­ figurations have failed to improve the ratio of maximum lift to minimum drag over that for the plain wing section. c. Leading-edge Flap8. A leading-edge flap may be formed by bending down the forward portion of the wing section in a manner similar to that in which the trailing edge is deflected in the case of plain flaps. Other types of leading-edge flap are formed by extending a surface downward and forward from the vicinity of the leading edge. As shown in Fig. 135, such flaps may extend smoothly from the upper surface near the leading edge, may be hinged at the center of the leading-edge radius, or may be hinged on the lower surface somewhat aft of the leading edge. Although

THEORY OF WING SECTIONS

228

Slot combination

CLma x

CDmi D

CLma'!C.

a

CDm iD

degrees

CL

max

«:

=:=::--.-....

1.291

0.0152

85.0

15

rr:

:::--=----..

1.772

0.0240

73.8

24

c:7~

1.596

0.0199

SO.3

21

C

7c:=::::..-..

1.548

0.0188

82.3

19

c:

:Jc::::,...

1.440

0.0164

87.8

17

r~~

1.902

0.0278

68.3

24

7~

1.881

0.0270

69.7

24

:J~

1.813

0.0243

74.6

23

~L7C?c:::=::--..-

1.930

rr: ­ rr: -

rL7c=---r~

I i

0.0340

I

I

56.8

I

25

I

1.885

0.0319

59.2

24

rC7C7oc:::::::::. -z:':70~ c:7C?c:=::::..-..

1.885

0.0363

51.9

25

1.850

0.0298

62.1

24

1.692

0.0228

74.2

22

c:7t:==7c::::,...

1.672

0.0214

c: -

1.510

0.0208

7a~

c:7C7oc::::,...

II I

I

78.2 72.6

I

1.662

I

I

0.0258

r

I

64.4

I i

Ii

I II

22 19

I

22

I

(a) Multiple fixed slots.

FIG.

134. Aerodynamic characteristics of a Clark Y wing with slots and flaps.

HIGH-LIFT DEVICES

Slot combination

CD mi u*

CLms x

229

II

-----------1----1----1----11----1 I

1.950

0.0152

2.182

!

128.2

12

0.0240

91.0

19

2.235

0.0278

80.3

20

2.200

0.0340

64.7

21

2.210

0.0270

81.8

20

1.980

0.0164

120.5

12

1.770

0.0164

IOS.O

14

0.0208

117.5

16

2.500

0.0258

96.8

18

2.185

0.0214

102.0

18

0.0243

93.2

19

2.320

0.0319

72.7

20

2.535

0.0363

69.8

20

0.0298

87.3

20

0.0298

68.3

21

I

2.-1-12

I

l

2.261

I

2.600 t

2.035

I

II i

• CD' with flap neutral. man (b) Multiple fixed slots and a slotted flap deflected 45 degrees. FIG.

134. (Concluded)

230

THEORY OF WING SECTIONS

none of these devices is as powerful 88 trailing-edge flaps, they may be used full span without mechanical interference with lateral-control de­ vices and they are effective when combined with trailing-edge high-lift devices. Leading-edge flaps reduce the severity of the pressure peak ordinarily associated with high angles of attack and thereby delay separa­

(0) DROOPED LEADING EDGE

] (b) UPPER SURFACE LEADING EDGE FLAP

(c)LOWER Sl,eqfACE LEAOINGEDGEFLAP

(d) FLAP HINGED ABOUT LEADING EDGE RADIUS FIG.

135. Various types of 1eading-edge flaps.

tion. Leading-edge flaps received little consideration until German in­ vestigators became interested in them during the Second '\\"'orld War. Krueger,.,60 Lemme,l3· M, 65 and Koster" showed increments of the maximum lift coefficient of 88 much as 0.7. These increments were, how­ ever, applied to maximum lift coefficients for the plain wings of the order of 0.72. These low maximum lift coefficients resulted from the small leading-edge radii of the wing sections usually used for these investigations and the very low Reynolds numbers of the tests. These investigations in­ dicated that the effectiveness of leading-edge Haps increased with decreasing leading-edge radius. Typical rcsultsH are shown in Fig. 136. The inere­

HIGH-LIFT DEVICES

231

moot of maximum lift coefficient, deLmasJ is plotted against the leading­ edge-radius parameter {r/c)/{t/c)2 where r is the leading-edge radius. The value of this parameter for NACA four- and five-digit wing sections is 1.1. FullmerU investigated t\VO types of leading-edge flap on the NACA 641-012 section at a Reynolds number of 6 million. The chord of the flap was 10 per cent of the section chord, and the configurations corresponded to

........

.6

o Withou/ split fbp

.~

.8

~

I

<, """,-re

?J

-R=0.57x 10'

a Corf(c~fig.135";lh split flop

~

rI

~~ rr R=2.I Xd <,

"',

r-,

n Corf(bJ,fig.l35

-

<>

-

<,

lei

~

o

Ccd(bJ,"~r.135wifh splitfbp

r-,

r-, ~

.2

=60

cf Corf(c~fig.135

I

\

.2

s

~~

itR=O.72x 106

-

~=a20

(UBM3067) I

~

-

Withsplit flop

r-R=2.2 x KJS

t- r-, ~

/11/

r:,.

.6

.8

'" <,

1.0

~

'" <,

"-

£4

~

~

(~;2 FlO. 136. Effeetiveneea 01 leadi11ll-4M1ge flaps for various wing sections. Increment of lift resulting from leading-edae flap, c, 88 a function of the leading-edge radius coefficient. (r/c}/{t/c)t. .al:

b and c of Fig. 135. The increments of maximum lift coefficient and of the angle of attack for maximum lift are shown in Fig. 137. The leading-edge radius parameter for the NAC...\ 641-012 section is 0.72. The maximum increments of Fig. 137 are shown plotted on Fig. 136 for comparison with the German results. The maximum lift coefficient of the plain wing sec­ tion for these tests was 1.42. 8.7. Boundary-layer Cpntrol. The idea of removing the low-energy air of the boundary layer, or of adding kinetic energy to the boundary layer, as a means for increasing the maximum lift has been obvious since the basic mechanism for separation was first understood." The kinetic energy of the layers of air close to the surface may hi increased by removing low­ energy air through suction slots or a porous surface. Another common method is to blow high-energy air through backward-directed slots. The

232

THEORY OF lYING SECTIONS

air handled through either the suction or blowing slots may be carried through the interior of the wing and the necessary energy supplied by a blower. Alternately, the pressure difference required may be obtained from the variation of pressure about the wing section itself. This method

I!!«JJ:...

l.J:Mer 5lIfoce~. '-e4e flop } o t.ower:5lIfoce flop willi a ftrJilhg-edoe flqp lJppei-sufoce leodirJg -e4e flop } £fiper-Sllfoce Ieoditig-edge fbp with Imiling-e4e ftp

Fig./35,

calle)

A- /.'71: /{/. ..,~ conl{b)

8 A

-.

4

I

-

- ----

~

~

~

/

-./'"

'¥

~

rA

8

/ "'" -,

Q,.-

.4

~

Do..

A

lJ()

~

~

/

I

~.

y" /00

/20

140

/60

l.I!odtY;-edge f/qJ deflecfionl 6,LE!deg.

FIG. 137. Variation of the increment of maximum section lift coefficient and the increment of section angle of attack for maximum section lift coefficient with leading-edge flap deflec­ tion. NACA 64t..Q12 airfoil section with leading- and trailing-edge split flaps. R, 6.0 X U)6.

has been used successfully only in the case of blowing slots on the upper surface using air taken from the lower surface. Examples of such arrange­ ments are the previously discussed slotted flaps, slots, and leading-edge slats. It is obvious that, if boundary-layer control is applied at sufficiently close intervals along the upper surface of a wing section, separation can be avoided up to very high values of the lift coefficient. The increments of maximum lift coefficient are obtained as an extension of the lift curve to

HIGH-LIFT DEVICES

233

higher angles of attack as contrasted with the displacement of the lift curve resulting from deflection of trailing-edge flaps. The problem con­ fronting the designer is to apply sufficient boundary-layer control to obtain the desired values of the maximum lift coefficient without increasing the weight and complexity of the airplane to such an extent as to nullify the gains in performance expected from the higher lift coefficients. These considerations of weight and complexity have prevented extensive use of 6.0

~

5.0

/

~4.0 ......"

i

Theory.. . "

.~

.~

~

)

§ .10

.~ ~

(j)

f

/r

/ /

v

l(

/'

~.

)C

/

~/

~.

J

~

I

2.0

V» '--- Experiment _

r--­

CO=Q038

/

V)'­

1/

1.0

a

I

V·

JV .-20.

-/0

0

10

20

Angled ottoclt1t%, deg. 138. Variation of lift coefficient with angle ot attack for a thick wing section with boundary-layer control.

FlO.

any except the simple types of boundary-layer control such as slotted flaps and leading-edge slots and slats. Successful application of boundary-layer control must delay both tur­ bulent separation over the aft portion of the wing and permanent laminar separation over the forward portion (see Sec. 5.3). The prevention of permanent laminar separation for thin or even moderately thick sections by means of suction slots is a difficult problem involving critically located slots in a region of greatly reduced pressure that requires high-pressure blowers. Nearly all the early investigations avoided this problem by using very thick sections with large leading-edge radii. 102. 101, 104 Using such sec­ tions, Schrenk103 obtained lift coefficients of over 5.0 using a single suction slot (Fig. 138) with a volume-flow coefficient Co of 0.038. The high drag coefficients of such thick wing sections in the high-speed flight condition led to investigations of boundary-layer control on thinner

THEORY OF WING BECTIONS

sections.II,u,WI Knight and Bamber" obtained maximum lift coefficients-­ of about 3.0 (Fig. 139) for the NACA 84-M wing section with a single backward-opening blowing slot.

NACA 84-Mwingsecfion. SloIwidfh=Q667%c Slotof 53. 9 ~ c from leading edge. Inlemo/ pressure /2 q higher than free- sllfJOlTl sialic

pressure

I

~

,

I

I

I

I

c, s/olled Ming sediJn

JV

2.4

L'

/

V

7 If

I

7

\ \ y

\

./

.~ ,V

8/

FIG.

.rl.

I) ~ ~

~

0

-:

/

Il

o

.L

~ ~c, pbiI wing section I

I · ••

I

II'-lfdstnIt/'wing ssdion ~

II .I

7.

~J I

/ / 1/ Jif 1/

..4

~

~"",. -.-...... ~

~

8

K:.~~~~_

~ -~ ~ ~

16

24

32

40

48

~ofolfodr,f4deg.

139. Effect of boundary-layer control by means of a baekward-opening blo'triDc 1101.

Much of the more recent work'" lion boundary-layer control has com­ bined suction slots with flaps and leading-edge slots. Quinn" obtained maximum lift coefficients exceeding 4.0 for the NACA 65a-418 section with a double-elotted flap and a single suction slot (Fig. 140). The maximum tift coefficient for the unfIapped section increased rapidly with small

HIGH-LIFT DEVICES

235

~-----.45c----~-a

44

4.0

-V "'--V

0, =65°

~

-"

L-----' ~ ~ ~

/ l-

"""'"

~.6

~ ~~

~

~

~

~

"6f=4S~

/

0,=0°"",,,,V~

~

~

~

10-"""

----

~

~

~ 1.6 I(

/ ......

1.20

.008

••016

.024

.032

.040

Flow coefficient, CQ FIG.I40. Configuration and maximum lift characteristics of the NACA 65,-418 wing aection with double-elotted flap and boundary-layer control by suction. R, 6.0 X 10'.

236

THEORY OF WING 8ECTIONS

amounts of flow through the slot, but the variation of maximum. lift coeffi­ cient for the Happed section was nearly linear with the flow coefficient up to a value of 0.040. Quinn93 has concluded that the maximum effectiveness of suction slots is nearly reached when the quantity of air removed is equal to that which would pass through the displacement thickness of the boundary layer at the slot location with the local velocity outside the boundary layer. Quinn found that the maximum lift was limited by leading-edge stalling for the 18 per cent thick sections. He used a leading­

L..,....o-" ~

~V

~

V

.01

).---"

~

.02

.03

.04

FkM coefficienf,CQ FIo. 141. Configuration and maximum lift characteristics for the NACA M 1A212 wiug section with leading-edge slat, double-slotted flap and boundary-layer control.

edge slatt 5 to control the leading-edge stalling of the NACA 641A212 section with a suction slot and a double-slotted flap. This combination of high-lift devices gave a maximum lift coefficient of nearly 4.0 for this thin section95 (Fig. 141). 8.8. The Chcrdwise Load Distribution over Flapped W'mg SectiODS. Allen' obtained empirical load distributions that permit calculation of reasonable approximations to the load distributions of "ring sections with deflected plain or split flaps. These load distributions are analogous to those associated with changes of camber in that the ratio Pb&/c~ is inde­ pendent of the angle of attack where Pili is the incremental basic load distribution associated with flap deflection and en., is the basic normal­ force coefficient associated 'lith flap deflection. The total load distribu­ tion is the sum of the incremental basic distribution P"" the incremental additional load distribution Po" and the load distributionof the unflapped section at the same angle of attack. The incremental additional load

HIGH-LIFT DEVICE8

237

distribution P a & is taken to be the same as the additional load distribution associated with angle of attack. That is, for a normal additional force coefficient of unity, P = 4 dV a -! GI. V V where the values of dVa/V and vIV are tabulated in Appendix I for various wing sections. Allen also found empirically that the values of P",/~ at any point (x/c)/(I - E) (points ahead of the flap hinge) and [I - (x/c»)/E (points aft of the hinge) were independent of the flap-chord ratio E for any given angular deflection of the flap. Values of P"./c.., are given in Table 6 for both plain and split flaps. The incremental load distribution associated with flap deflection is thus defined in terms of two known types of distribu­ tions. The remaining problem is to determine the magnitudes of these two types of distributions associated with a given flap deflection. The magnitudes of the load distributions are determined from force test data obtained at the desired angle of attack with the flap neutral and deflected. The additional load distribution contributes no moment about the quarter-chord point. The incremental basic distribution is therefore selected to produce the measured increment of moment coefficient between the flap-neutral and flap-deflected conditions. The remainder of the differ­ ence between the normal-force coefficients with the flap neutral and de­ flected is equal to the incremental additional load distribution. The normal-force distribution is assumed to act as shown in Fig. 142. From force tests of the wing section with flap neutral, C1ft1 (quarter-chord pitching-moment coefficient) and ella (normal-force coefficient) correspond­ ing to the normal-force distribution shown in Fig. 142a are obtained. From force tests of the wing section at the same angle of attack with the flap deflected, ems and CftS corresponding to the normal-force distribution shown in Fig. 142b are obtained. Let

aem =

Cm: -

em l , }

=

en: -

C ra.,

deft

(8.1)

where em.' and Cfta' are the pitching-moment and normal-force coefficients corresponding to the normal-force distribution for the section with flap neutral when plotted normal to the chord of the section with flap deflected as shown in Fig. 142c. Then ~Cm and .1c,. are the pitching-moment and normal-force coefficients of the incremental normal-force distribution when the incremental distribution is plotted normal to the chord of the section with the flap deflected as ShO"l1 in Fig. 142d. In many cases, the approximation (8.2)

THEORY OF WING SBCTIONS

238

is sufficiently exact except when the flap-chord ratio E and the flap deflec­ tion 8 are simultaneously large or when the shape of the mean line is such that the load is large near the trailing edge. In such cases, values of Crat' and Cat' may be obtained graphically from the theoretical load distribution for the unflapped section adjusted to agree with the experimental data. Let Ac.' and Ac.' be the pitching-moment and the normal-force coeffi­ cients of the incremental normal-force distribution plotted normal to the flap-neutral chord as shown in Fig. 142e. Because the incremental basic

em,'

h=~n=~/====-__

~_cn_/.....;:' ;;;;:;::::==--..

(a) NORMAL-FORCE DISTRBI1lON

(e) DISTRI8UTION SHOWN IN(a)

(b) NORMAL-FORCE DISTRIBUTION FOR AIRR)IL WITH FLAPNEUTRAL. FOR AIRFOL WITH FLAPDEF'l.B:T. ED~

f'::

~~

6Cn

"

WITH FLAPNORMAL FORCE 0ISTR8UT1ON PLDTTED NORMAL

lOFLAP DEFLECTED CHORD.

/l(j

(d) INCREMENlJ'L NORMAL -FORCE DISTRBITION CMJSED BYFLAP

AC~

~ (e) DISTRI8UTIDN SHOWN'IN (d)

PLDnED NORMAL TOFLAP-NElITRAL DEFLECTION. CHORD. FIG. 1.(2. Normal-force distribution and incremental normal-force diBtribution for flaps neutral and deflected.

normal-force distribution is responsible for the increment of the quarter­ chord pitching moment, then, if G is the moment arm in terms of the chord of the basic normal force about the quarter-chord point, ac.' == Gc..,

t:.e,., == e.., + en.,

wliere en _, = additional normal-force coefficient associated with flap de­ flection

c...,=~' c,,~.

}

LAc.'

=

~c",

-

(8.3)

G

The value of G is a function of E and 8. Values of G are given in Table 7. The correlation between the fictitious values of ~c",,' and ACta, and the measured values of de,. and ~CR must be established in order to determine c... and cfteI from force tests. The incremental flap normal-force coefficient is

HIGH-LIFT DEVICES

I, -

nl,

239

(8.4)

Ca -gEc ­

where nl, is the incremental flap normal force per unit span. Then J1c.' =z a~ + Ec"" (1 - cos 8) } and &:." = &:., - Ee. ' (1 - cos 6)(% - E)

'I,

(8.5)

The incremental1lap normal force may be considered as a combination of two components due to the incremental additional and the incremental basic normal-force distributions. Let 1'0, and "rll, be the ratios of the flap normal force to the section normal force for the incremental additional and the incremental basic normal forces, respectively. Then (8.6)

or (8.7)

The contribution to the flap load of the additional normal-force dis­ tribution is small compared with the basic contribution; and, for the purpose of determining &:." and J1Ca', the following approximation may therefore be used:

and Eqs. (8.5) become 4c.' = J1c.

&:."

/ii;w + En, G (1 -

= &:., -

E-y", ~' (1 - cos 6)(~ - E)

so that At;., ~c.'

where

cos a)

= 4c. + T" ~c.} = 'T'" ~c.

(8.8)

E(1 - cos 8)("Yb,/G) cos 8)(~ - E)('YbtJ/G)

T"

= 1 + E(l -

T.

= 1

+ E(l -

1 cos 8)(% - E)('Y6,/G)

The values of'T. and 'T.. as determined by Allen are given in Tables 8 and 9. In using this method, the values of J1c. and tic", are obtained from force tests of the flapped and unflapped section at the same angle of attack. From Eq. (8.8)~ values of 11c"" and &:." are obtained using Tables 8 and 9.

~ TABLE

6.-P",/C"6,

DISTRIBUTION b. Plain flap at 8 • 20 degrees

a. .Plain flap at 8 - S, 10, and 16 degreel 0.05 0.10 0.15 0.20 0.25 0.30 0.31 0.40

B

o.~

0.&0 0.55 0.80 0.65 0.70 0.05 0.10 0.16 0.20 0.25 0.30 0.36 0.40 0.46 0.50 0.66 0.60

--0 -0 -0 -0 -0 -0 --0 -0 ---0 -0 --0 -0 0 0 0 0 0 0

-- -- 0 0.18 0.26 0.39 0.50 0.62

0 0.19 0.27 0.40 0.62 0.64

0.15 0.23 0.34 0.44 0.55

0.16 0.24 0.36 0.46 O.M 0.67

0.17 0.25 0.36 0.47 0.68

0.17 0.25 0.38 0.49 0.60

0.18 0.26 0.39 0.&0 0.62

0.20 0.21 0.22 0.23 0.29 0.30 0.32 0.34 0.42 0." 0.46 0.49 O.M 0.67 0.60 0.63 O.M 0.67 0.70 0.73 0.77

0.25 0.36 0.52 0.67 0.82

0.27 0.39 0.56 0.72 0.88

0.15 0.23 0.34 0.44 O.M 0.55

0.16 0.23 0.34 0.46 0.66

0.16 0.24 0.35 0.46 0.67

0.17 0.26 0.36 0.47 0.68

0 0.17 0.26 0.38 0.49 0.60

0.66 0.80 0.98 1.24 1.74

0.67 0.82 1.00 1.26 1.75

0.69 0.83 1.01 1.27 1.7J

0.71 0.8& 1.03 1.29 1.77

0.73 0.88 1.06 1.3l 1.79

0.75 0.90 1.08 1.34 1.81

0.77 0.93 1.11 1.38 1.84

0.92 1.10 1.30 1.59 2.06

0.98 1.17 1.39 1.67 2.16

1.05 1.25 1.48 1.78 2.29

0.66 0.79 0.97 1.23 1.73

0.66 0.80 0.98 1.24 1.74

0.67 0.82 1.00 1.26 1.75

0.69 0.83 1.01 1.27 1.76

0.71 0.85 1.03 1.29 1.77

0.73 0.88 1.05 1.31 1.79

0.76 0.90 1.08 1.34 1.81

0.70 0.60 0.60

8.74 6.45 4.96 3.86 3.09 2.48

6.04 4.49 3.51 2.72 2.18 1.72

4.83 8.78 2.92 2.23 1.78 1.42

4.40 3.32 2.67 1.97 1.57 1.M

4.01 2. • 2.31 1.77 1.41 1.11

3.71 2.77 2.12 1.84 1.30 1.02

3.&0 3.35 3.23 3.15 3.11 3.06 3.04 2.80 2." 2.39 2.32 2.26 2.22 2.19 2.00 1.90 1.81 1.74 1.69 1.84 1.60 1.63 1.44 1.37 1.32 1.27 1.22 1.19

3.02 2.16 1.57 1.16 1.21 1.14 1.08 1.04 1.00 0.96 0.93 0.91 0.96 0.89 0.86 0.81 0.77 0.7& 0.72 0.70

5.S3 1.08 4.23 3.71 3.33 2.98

4.05 3.&3 2.99 2.63 2.3& 2.00

3.38 2.98 2.60 2.16 1.92 1.73

3.02 2.61 2.19 1.90 1.89 1.60

2.83 2.36 1.97 1.71 1.62 1.35

2.70 2.18 J.81 1.68 1.40 1.24

0.40 (back of 0.30 hinge) 0.20 0.10 0.05 0

1.90 1.40 0.92 0.48 0.32 0

1.34 0.09 0.63 0.34 0.19 0

1.10 0.81 0.&3 0.27 0.16 0

0.98 0.70 0.46 0.24 0.14 0

0.88 0.63 0.41 0.21 0.12 0

0.78 0.67 0.38 0.20 0.11 0

0.73 0.63 0.35 0.18 0.10 0

2.85 2.32 1.93 1.39 1.16 0

1.87 1.63 1.3& 0.98 0.70 0

I.M 1.34 1.11 0.79 0.68 0

1.31 1.18 0.96 0.69 0.&0

1.19 1.04 0.86 0.62 0.4& 0

1.09 0.9& 0.79 0.67 0.40 0

~/e

0 0.05 0.10 0.20 0.30 0.40

0 0.16 0.22 0.33 0.43 0.54

1-8 (ahead 01 0.60 0.66 hinge) 0.60 0.79 0.70 0.97 0.80 1.23 0.90 1.73

1.00 --- 0.90

0.80 1 - ~/c

B

0.16 0.23 0.34 0.45

0.19 0.27 0.40 0.&2

0.89 0.60 0.33 0.17 0.10 0

0.80 0.96 1.15 1.42 1.88

0.81 0.47 0.31 0.18 0.09 0

0.84 1.00 1.19 1.46 1.92

0.62 0.46 0.29 0.13 0.08 0

0.88 1.0j 1.24 1.52 1.98

0.69 0.43 0.28 0.16 0.08 0

0.57 0.42 0.27 0.14 0.08 0

0.&6 0.40 0.26 0.13 0.07 0

0.&3 0.39 0.25 0.13 0.07 0

0.16 0.22 0.33 0.43

a

- --0 -0 0 0 0.20 0.29 0.42 0.14 0.67

0.21 0.30 0.44 0.57 0.70

0.22 0.32 0.46 0.60 0.73

0.23 0.34 0.49 0.63 0.77

~ ~ ~

0.77 0.93 1.11 1.38 1.84

0.80 0.96 1.15 1.42 1.88

0.8J 1.00 1.19 1.46 1.92

0.88 1.05 1.24 1.52 1.98

0.92 1.10 1.30 1.59 2.06

~

2.63 2.05 1.70 1.48 1.30 J.15

2.58 1.96 1.62 1.39 1.22 1.08

2.56 1.88 1.65 1.32 1.17 1.03

2.56 1.83 1.49 1.27 1.12 0.98

2.58 1.78 1.44 1.22 1.08 0.94

2.62 1.75 1.40 1.18 1.04 0.91

1.02 0.88 0.73 0.53 0.38 0

0.96 0.82 0.68 0.50 0.35 0

0.91 0.78 0.65 0.47 0.33 0

0.88 0.73 0.62 0.45 0.31 0

0.83 0.72 0.69 0.43 0.30 0

0.80 0.89 0.56 0.41 0.28 0

~

s ~

~

C':S

~

~

d. Plain or .pUt flap at a- 40 degrees

c. Plain flap at' .. 30 deereee

-

I

0.05

E

z/c 1-8 (ahead of hinge)

--­ !-.=. ~/c B (bank of hinge)

-­

0 0.05 0.10 0.20 0.30 0.40

0 0.15 0.22 0.33 0.43

O.M

0.&0 0.60 0.70 0.80 0.00

0.66 0.70 OJ.7 1.23 1.67

1.00 O.DO

".rJO

0.80 0.70 0.60

O.SO 0.40 0.30 0.20 0.10 0.05 0

4.61 4.34 4.08 3.82 3.56 3.25 2.89 2.44 1.81

1.53 0

0.16

0.20

0.26

0.30

0.36

0 0.18 0.23 0.34 0.4& 0.56

0 0.16 0.24 0.3& 0.46 0.67

0 0.11

0 0.t7

0.6H

O.US

0.67 0.82 I.OU

1.24

1.26

1.68

1.69

:4.12 3.20

2.60 2.70 2.56 2.37 2.20

0.10

-­ 0 0.16 0.23 0.34 0.44 0.&5 0.66 O.RO

3.07 2.89 2.60 2.4.9

2.29 2.0' 1.72 1.27 0.92 0

0.40

0.45

0.60

0.06

0.10

0.16

0.20

0.25

0.30

0.35

0.40

0.19 0.27 0.40 0.52 0.64

0 0.20 0.29 0.42 0.54 0.67

0 0.21 0.30 0.44 0.67 0.70

0.16 0.22 0.33 0.43

0 0.16 0.23 0.34 0.44 0.66

0.16 0.23 0.35 0.46 0.56

0 0.16 0.24 0.36 0.46 0.67

0 0.11 0.26 0.38 0.47 0.38

0.17 0.23 0.39 0.49 0.60

0 0.18 0.26 0.40 0.61 0.82

0 0.19 0.27 0.42 0.62

O.M

tt:

0.77 O.!l:l 1.11 1.38 1.74

0.80 O.!J(\

0.84 1.00

r.rs

1.IH

1.42 1.77

0.89 0.83 1.01 1.26 1.61

0.71 0.86 1.0:1 1.28 1.61

0.73 0.88 1.05 1.30 1.62

0.7& 0.00 1.08 1.33 1.62

0.77 0.93 1.11 1.36 1.63

:;;~

2.01) 1.8.'1 1.66 1.62 1.40 1.29

2.01

1.17

1.11 0.98 0.82 0.61 0.44 0

- ­ - ­ -­ -­ - ­ -­ - ­ -0 ­ - ­ -­ -­ - ­ -­ -­ -­ -­ 0 0 0

0.1':4 1.01 1.27 1.70

2.36 2.:~7

2.0~

2.20& 2.0tJ 1.94 1.79

1.88

1.64

1.GS

1.45

1.41 1.03 0.76 0

O.~

0.25

0.38 0.47 0.68

0.38 0.49 0.60

0 0.18 0.26 0.39 0.60 0.62

0.71 0.145

0.73

0.75

n.AA

o.no

1.05

1.0M 1.34 1.73

I.oa 1.2U

r.ai

1.71

1.72

2.22 2.15 2.02 1.88 1.74 1.61

2.13 2.01 1.86 1.73 1.60 1.47

2.08 1.90

1.46 1.29

1.34 1.18 1.00 0.74

1.25 1.10 0.93 0.60 0.49 0

1.22

1.09

0.90

0.81

0.86

o

I 0O,6'J

0.53 0

l.i6 1.62 1.49 1.37

1.03 0.87 0.65 0.48

0

1.77 1.59 1.45 1.34 1.22

O.M

0.66 0.80

1.46 1.80

0.88 0.79 O.n7 1.22 1.68

1.23 1.&9

0.68 0.82 1.00 1.2& 1.60

1.UO 1.73 1.S2 1.40 1.28 1.17

4.10 4.24 •. 29 4.27 4.14 3.92

2.90 3.01 3.04 3.02 2.91 2.76

2.42 2.52 2.&3 2.48 2.39 2.27

2.12 2.23 2.22 2.18 2.10 1.98

1.90 2.02 2.00 1.96 1.89 1.77

1.88 1.88 1.84 1.81 1.74 1.62

1.81 1.78 1.73 1.70 1.62 1.61

1.76 1.72 1.66 1.69 1.62 1.42

1.06

3.64 3.28 2.82 2.11 1.79 0

2.&6 2.31 1.98 1.48 1.08 0

2.11 1.90 1.62 1.20 0.89

1.83 1.04 1.40 1.0& 0.77 0

1.64 1.47 1.26 0.96 0.89 0

1.50 1.34 1.15 0.87 0.63 0

1.40 1.24 1.07 0.81 0.58 0

1.31 1.17 1.00 0.76

0.93 0.78 0.68 0.41

0

O.OM

0

:::~

-s ~ tlj

~

~ t'I.)

O.M 0

~

~

~ TABLE

8.-P.,Ie"",

DISTRmUTIoN.-(Conlinued)

I. Plaln or epUt Sap at' -

,. Plain or 8PUtSap at' - 80 deere-

•

0.08

----~

ale 1-8 [ahead of billie)

--1 -ale

B (back of billie)

0 0.05 0.10 0.20 0.80

0.10

0.16

0.20

0.26

0.30

0.36

0.40

0.06

0.10

0.15

0.20

0.26

eo d..-. 0.30

--- - --- -- - - - - - - - - - - - - - - - - - - - 0 0.18 0.22 0.33 0.43

0 0.18 0.23 0.36 0.46 O.M

0 0.18 0.24 0.34 0.48 0.67

0 0.11 0.25 0.38 0.47 0.&8

0 0.17 0.26 0.38 0.49

O.~

O.M

0 0.18 0.23 0.34 0.44 0.66

0.60 0.80 0.70 0.80 0.90

0.88 0.'19 0.97 1.21 1.M

0.86 0.80 0.98 1.22 1.66

0.68 0.82 1:00 1.23 I .•

0.89 0.83 1.03 1.26 1.67

0.71

1.00 0.90 0.80 0.70 0.80 0.50

3.81 4.18 4.26 4.27 4.21 4.07

2.88 2.90 8.00 3.02 2.97 2.88

2.21 2.46 2.48 2.43 2.38

2.00 2.16 2.20 2.18 2.14 2.06

1.88 1.96 1.98 1.98 1."

I.M

1.81 1.77 1.89

0.40 0.30 0.20 0.10 0.05 0

3.88 3.62 3.08 2.34 2.02 0

2.71 2.49 2.16 1.64 1.22 0

2.24 2.M 1.78 1.33 1.00 0

1.96 1.78 1.63 1.18 0.87 0

1.73 1.68 1.37 1.06 0.78 0

1.59 1.44 1.28 0.06 0.70 0

2.80

O.U 1.OS 1.28 1.61

0.60 0.73 0.88 1.08

1.29 1.67 1.77 1.81

1.82

0 0.18 0.28 0.39 0.61 0.82

0 0.19 0.21 0.40 0.62 0.84

0 0.13 0.22 0.33 0.43

0 0.16 0.23 0.34 0.44

O.M

0.73 0.90 1.11 1.32 1.67

0.77 0.93 1.14 1.36 1.67

1.71 1.72 1.72 1.70 1.83 1.37 1.48 1.34 1.18 0.90 0.66 0

0.36 ~

O.~

-0 -

O.M

0 0.18 0.23 0.33 0.43 0.68

0 0.18 0.24 0.36 0.48 0.67

0 0.17 0.26 0.38 0.47 0.68

0 0.17 0.23 0.38 0.49 0.80

0 0.18' 0.28 0.39 0.61 0.82

0.88 0.79 0.97 1.20 1.62

0.87 0.80 0.98 1.21 1.63

0.68 0.82 0.99 1.22 1.M

0.89 0.83 1.01 1.24 1.66

0.71 0.86 1.02 1.26 1.M

0.73 0.88 1.05 1.28 1.63

0.76 0.90 1.07 1.29 1.61

0.77 0.93 1.10 1.32 1.80

1.87 1.68 1.M 1.69 1.M 1.48

3.8l 4.M 4.18

2.22 2.38 2.43 2.48 2.46 2.42

1.97 2.08 2.13 2.18 2.17 2.11

1.81 1.88 1.94 1.98 1.96 1.89

1.71 1.78 1.78 1.81 1.79 1.73

1.88 1.88 1.88 1.70 1.87 1.82

1.69

4.27 4.18

2.69 2.80 2.M 3.02 3.01 2.93

1.82 1.69 1.69 1.87 1.62

1.39 1.26 1.09 0.84 0.81 0

4.01 3.71 3.27 2.&1 2.18 0

2.82 2.82 2.29 1.78 1.31 0

2.33 2.11S 1.88 1.43 1.08 0

2.02 1".88 1.83 1.26 0.93 0

1.80 1.86 1.48

1.66 1.62 1.33 1.03 0.78 0

1.M 1.41 . 1.24 0.98 0.71 0

1.46 1.32 1.18 0.90 0.88 0

4.27

1.1~

0.84 0

0.19 0.27 0.40 0.62

O.M

~

~ ~ ~

~

~ ~ ~ ~

~

~

1/.

-

%/c

1-E

(ahead or hiDe.)

--­ 1 - %/c

B (baok of hinge)

..

0.06

E

A. Split Sap at a - 30 d. . . .

Split flap at 3 • 20 del....

-_. -­ - ­ - ­ -­ -­ -­-­ -­ -­ -­ -­ -­ -­ --­ -­ -­ -­ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.10

0.16

0.20

0.24

0.30

0.36

0.40

0.46

0.60

0.0&

0.10

0.16

0.20

0.21

0.30

0.3&

0.40

0.46

0.80

0.16 0.23 0.34 0." 0.&&

0.18 0.23 0.36 0.46 0.66

0.18 0.24 0.36 0.46 0.67

0.17 0.21

0.17 0.21

0.36 0.47 0.68

0.38 0.49 0.80

0.18 0.28 0.39 0.61 0.82

0.19 0.27 0.40 0.62 O.H

0.20 0.29 0.42 0.66 0.87

0 0.21 0.30 0.'" 0.&7 0.70

O.lS 0.22 0.33 0.'3 0.14.'

0.16 0.23 0.34 0.44 0.11

0.18 0.23 0.81 0.41 0.68

0.18 0.24 0.3& 0.48 0.67

0.17 0.26 0.38 0.47 0.18

0.17 0.28 0.38 0.49 0.80

0.18 0.26 0.39 0.61 0.82

0.19 0.27 0.40 0.62 0.84

0.20 0.29 0.42 0.&1 0.87

0.21 0.30 0." 0.67 0.70

0.71 0.86 1.03 1.29 1.73

0.73 0.88 1.05 1.31 1.73

0.76 0.00 1.08 1.34 1.73

0.77 0.93 1.11 1.38 1.73

0.80 0.90 1.16 1.42 1.74

0.84 1.00 1.19 1.46 1.74

0.68 0.79 0.97 1.23 1.M

0.66 0.80 0.98 1.24 1.86

0.88 0.82 1.00 1.28 1.86

0.69 0.83 1.01 1.27 1.67

0.71 0.86 1.03 1.29 1.88

0.73 0.88 1.06 1.31 1.88

0.76 0.90 1.08 1.34 1.88

0.17 0.93 1.11 1.38 1.89

0.80 0.96 1.16 1.42 1.70

O.M

1.90

1.88

1.90 1.78 1.81 1.46 1.28

1.83

1.84 1.79 1.63 1.48 1.32 1.16

4.&& 4.60 4.49 <&.21 •. 02 3.71

3.15 3.19 3.18 3.02 2.83 2.80

2.83 2.69 2.16

2.14 2.14 2.09 1.91 1.83 1.88

1.98 1.99

1.89 1.88 1.81 1.70 1.57 1.43

1.83 1.80 1.72 1.59 1.48 1.36,

1.78 1.73

2.14

2.32 2.38 2.32 a.19 2.04 1.87

1.73 1.68 1.18 1.48 1.36 1.22

0.99

3.36 2.96

2.38 I.M 2.08. 1.71

1.89 1.48

2.88

1.88 1.84 0.98 0

1.14

1.88 0.9& 0.70 0

1.29 1.12 1.01 0.73 0.13 0

1.21' 1.15 1.05 1.00 0.96 I 0.90

1.08

1.11 1.32 1.19 0. • 0.88 0

0 0.05 0.10 0.20 0.30 0.40

0 0.16 0.22 0.33 0.43 0.14.

O.SO 0.60 0.70 0.80 0.90

0.66 0.79 0.97 1.23 1.69

0.88 0.80 0.98 1.24 1.70

0.88 0.82

1.28 1.71

0.89 0.83 1.01 1.27 1.72

1.00 0.90 0.80 0.70 0.60 0.60

4.79 4.84 4.65 4.32 3.04 3.61

3.33 3.36 3.29 3.06 2.77 2.46

2.79 2.83 2.74 2.11 2.27 2.03

2.• 4 2.69 2.•0 2.21 2.00 1.77

2.22 2.28 2.18 1.99 1.80 1.69

2.08 2.09 1.99 1.84 1.86 1.<&8

1.97 1.98 1.81 1.12 1.64 1.36

0.40 0.30 0.20 0.10 0.06 0

3.06 2.80 2.13 1.14. 1.29 0

2.16

1.77 ,1.14. 1.61 1.30 1.23 \ 1.08 0.88 0.77 O.M 0.66 0 0

1.37

1.26

1.18

1.06

0.96 0.69 0.50 0

0.87 0.83 0.t5 0

1.17 0.98 0.81 0.69 0.42

1.83 1.&0 1.08 0.78 0

1.00

0

1.10 0.92 0.76 0.11 0.89 0

1.70 1.M 1.38 1.21 1.04 0.88 0.72 0.52 0.37

0

O.M 0.68 0.49 0.36 0

1.91 1.63 0

2." 2.32

0.81 0

1.92 1.82 1.69 1.14

J.M 1.12 1.41 1.28

1.00 1.19 1.46 1.70

~

~

~

~

-s

o~ ~

~ C';J

I

1.38 1.21 1.09 0.79 0.17

0

O.es 0.49

o

I

O.M 0.47

0

1.09 0.96 0.81 0.81 0.44 0

~

THEORY OF WING SECTIONS

244

The incremental basic eft" and additional en., normal-force coefficients are obtained from Eqs. (8.3) and Table 7. The appropriate values of the TABLE 7.-VALUES OF

~Eaeg ~~I

5,10,15 I

30

20 4.

-0.474

0.05 0.10 0.15 0.20 0.25

-0.448 -0.423 -0.397 -0.372 -0.347

0.30 0.35 0.40 0.45 0.50

-0.294 -0.268 -0.242

0.55 0.60 0.65 0.70

-0.215 -0.189 -0.163 -0.136

-0.320

G 40

50

60

Plain flaps

-0.476 -0.451 -0.428 -0.404 -0.380

-0.477 -0.453 -0.431 -0.408 -0.387

-0.478 -0.455 -0.434 -0.411 -0.392

-0.479 -0.456 -0.435 -0.414 -0.395

-0.4i9 -0.456 -0.435 -0.415 -0.396

-0.357 -0.334 -0.311 -0.288 -0.265

-0.366 -0.346 -0.325 -0.304 -0.283

-0.372 -0.352 -0.332

-0.375 -0.357 -0.339

-0.377 -0.360 -0.342

-0.242 -0.220

b. Split flaps

...... . . . ... ......

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

~

...... ...... ......

I

I

......

......

...... ......

incremental basic Table 6 as follows:

I P",

-0.476 -0.452 -0.430 -0.407 -0.385

-0.477 -0.454 -0.432 -0.409 -0.388

-0:478 -0.455 -0.434 -0.411 -0.392

-0.479 -0.456 -0.435 -0.414 ":'0.395

-0.479 -0.456 -0.436 -0.415 -0.396

-0.363 -0.342 -0.320 -0.299 -0.278

-0.367 -0.347 -0.327

-0..372 -0.352

-0.375 -0.357 -0.339

-0.3i7 -0.360 -0.342

-0.30i -0.287

-0.333

I

normal-force distribution are obtained by use of

The appropriate value of the incremental additional Paa normal-force dis­ ttibution is obtained from the data of Appendix I as follows: (8.10)

HIGH-LIFT DEVICES

245

TABLE 8.-VALUES OF T'.

~

10

0 0.05 0.10 0.15 0.20

0 -0.00 -0.01 -0.01 -0.01

0.25 0.30 0.35 0.40 0.45

-0.01 -0.02 -0.02 -0.03

0.50 0.55 0.60 0.65 0.70

-0.03 -0.04 -0.05 -0.06 -0.07

~I

40

30

50

60

~--_.:...--_-_..:.-.._---=------~--_---:.._---

o 0.05 0.10 0.15 0.20

4.

~0.02

0 -0.02 -0.03 -0.04 -0.05 -0.06 -0.07 -0.08 -0.09 -0.11 -0.12 -0.14 -0.17

I

I

Plain Haps 0 -0.05 -0.0; -0.09 -0.11

0 -0.09 -0.14 -0.18

0 -0.14 -0.23 -0.30

-0.23

-0.38

-0.14 -0.16 -0.19 -0.21 -0.25

-0.27 -0.32 -0.37 -0.42

-0.46 -0.54 -0.63 -0.71

I

0 -0.21 -0.34 -0.47 -0.59

I

I

-0.72

-0.85 -0.98 -1.11

-0.28

, I

-..

----------··--0

b. Split

--r----O-

-0.02

f1~lPS

-0.04 -0.05

-0.05 -0.07 -0.10 -0.12

0.25 0.30 0.35 0.40 0.45

-0.06 -0.07 -0.08 -0.10 -0.11

-0.15 -O.li -0.20 -0.23 -0.26

0.•50

-0.13

-O.2H

-0.03

­

-~.09 1--~.14···

-0.14 -0.18 -0.23

-0.2. -0.32 -0.37 -0.42

o

;

-0.23 -0.30 -0.38

-0.21 -0.34 -0.47 -0.59

I

-0.46 -0.54 -0.63 -0.71

-0.72 -0.85 -0.98 -1.11

t t

i1 !

I

i

The entire incremental normal-force distribution is obtained by adding the incremental basic and additional distributions (8.11)

The final load distribution is obtained by adding this incremental dis­ tribution to the load distribution for the unflapped lying section at the same angle of attack (see Sec. 4.5). Approximate hinge-moment coefficients may be obtained by an exten­ sion of this analysis." Such hinge-moment coefficients are nol considered to be reliable because the discrepancies between the predicted and actual

THEORY OF WING SECTIONS

246

TABLB

~IIO

9.-VALUB8

20

01' 'F'.

30

40

50

60

1.00 1.06 1.09 1.11 1.12

1.00 1.10 1.15 1.18 1.21

1.00 1.15 1.22 1.28 1.32

1.14 1.14 1.15 1.15

1.23

1.36 1.38 1.39 1.39

a. Plain flaps 0 0.05 0.10 0.15

0.20 0.25

0.30 0.35 0.40 0.45

0.50 0.55 0.60 0.65 0.70

1.00 1.00 1.00 1.01 1.01

1.80 1.01 1.02 1.02

1.03

1.00 1.03 1.05 1.06 1.06

1.01 1.01 1.01 1.01 1.01

1.03 1.03 1.03 1.03 1.03

1.07 1.07 1.07 1.08 1.07

1.01 1.01 1.01 1.01 1.00

1.03 1.03 1.03

1.07

1.24 1.25

1.25

b. Split flaps

0 0.05 0.10 0.15 0.20

....

0.25 0.30 0.35 0.40

.....

....

.... .....

....

.... ....

....

0.45

....

0.50

....

1.00 1.01 1.02 1.02 1.03

1.00 1.03 1.05 1.06 1.07

1.00 1.06 1.09 1.11 1.12

1.00 1.10 1.15 1.18 1.21

1.03 1.03 1.03

1.14 1.14 1.15 1.15

1.23 1.24 1.25 1.25

1.03

1.07 1.08 1.08 1.08 1.08

1.03

1.07

1.03

1.00

1.15 1.22 1.28 1.32 1.36

1.38 1.39 1.39

load distributions tend to be appreciable at the trailing edge where the contribution to the hinge-moment coefficient is the largest. In applying this method, it should be remembered that, although the tabulated characteristics are differentiated by flap deflection, the important variable is the degree to which the flap is stalled. The tabulated charac­ teristics for plain flap deflections up to 15 degrees are for unstalled condi­ tions, whereas those for larger flap deflections represent progressively increased separation. Consideration should be given to the selection of the characteristics to be used in order to represent the actual flow condi­ tions properly.

CHAPTER 9 EFFECTS OF COMPRESSmILITY AT SUBSONIC SPEEDS 9.1. Symbols. CII pressure coefficient, (p - PflO)/ Jip 1f1 CIIe critical pressure coefficient corresponding to a loeal Maeh number of unity B

to~ energy

K

constant

per unit ID888

M }'{ach Dumber, V fa R universal gas constant, e" -

c~

8

croes-eeetional area of a stream tube T absolute temperature

V velocity of the free stream

component of velocity along the span V. component of velocity normal to span

(J speed of sound

a lift-curve slope

v.

b span

c, section lift coefficient

Cp specific heat at constant pJ'e8SU1'e

specific beat at eoostant volume

Cr e base of Naperian loprith~ 2.71828

In logarithm to the base e

m Jn888 flow per unit area

loeal static pressure

Po total pressure

PflO static pressure of the free stream

t time

t thickness ratio

u component of velocity parallel to the % axis

v volume

component of velocity parallel to the y axis

V component of velocity parallel to the z axis to z; y, Z Cartesian coordinates a angle of attack fJ angle of sweep .., ratio of the specific beats, c./c.

.,

P. 11"

P

1/ v''! - l\ft ratio of the circumference of a eirele to its diameter

density of the fluid

ID888

9.1. Introduction. The wing-section theory and experimental data presented in the preceding chapters are applicable to conditions where the variation of pressure lR small compared with the absolute pressure. This 247

248

THEORY OF WING SECTIONS

condition is well satisfied when the speed is low compared with the speed of sound. At a flight speed of 100 knots at sea level, the impact pressure is only about 34 pounds per square foot as compared with an ambient static pressure of 2,116 pounds per square foot. At 300 knots, the impact pressure increases to approximately 321 pounds per square foot; and, at 600 knots, the impact pressure is 1,494 pounds per square foot. At the higher speeds, the pressure and corresponding 'volume changes are obviously not neg­ ligible. It is to be expected, therefore, that wing-section characteristics at high speeds will not agree with those predicted by incompressible Bow theory or with experimental data obtained at low speeds. Examination of the complete equations of motion89 shows that the parameter determining the effect of speed is the ratio of the speed to the speed of sound. This ratio is called the "Mach number." The physical significance of the speed of sound becomes evident in considering the difference between subsonic and sup­ ersonic flows, The speed of sound is the speed at which pressure im­ pulses are transmitted through the FlO. 143. Significance of Mach angle. air. At SIOlV flight speeds, the pres­ sure impulses caused by motion ~t the wing are transmitted at relatively high speed in all directions and cause the air approaching the wing to change its pressure and velocity gradually. Slow speed flows are accord­ ingly free of discontinuities of pressure and velocity. At supersonic speeds, no pressure impulses can be transmitted ahead of the wing, and the pres­ sure snd velocity of the air remain unaltered until it reaches the immedi­ ate vicinity of the wing. Supersonic ft.ow is thus characterized by discontinuities of pressure and velocity. As shown in Fig. 143, the pressure impulses are propagated in all directions at the speed of sound a while the 'ring moves through the air at the velocity V. The envelope of the pressure impulses is a straight line (for small impulses) which makes a slope sin-1a/V or sin-t M with the direction of motion, where M is the Mach number. This line is called the" Mach line." The air ahead of the Mach line is unaffected by the approaching wing. The present discussion of the characteristics of wing sections in com­ pressible flow is limited to subsonic speeds. The pressure-velocity relations for a stream. tube will be developed, and a brief summary will be presented of compressible flow theory as applied to wing sections. The theory of wing sections in subsonic flow generally consists of the determination of

EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS

249

the first-order compressibility effects on the incompressible flow solutions previously developed. The theory is generally limited to speeds at which the velocity of sound is not exceeded in the field of flow. A brief discussion of experimental data will, however, include data in the lower transonic region where the speed of sound is exceeded locally. 9.3. Steady Flow through a Stream Tube. a. Adiabatic Law. The relation among pressure, density, and temperature for a perfect gas is p = pRT where (in consistent units) p = pressure p = density, moles per unit volume T ~ absolute temperature R = universal gas constant = Cp - c. Cp = specific heat per mole at constant pressure Ct, = specific heat per mole at constant volume This equation may be written

(9.1)

or PIl't

r;=

112"2 T2

where the subscripts 1 and 2 refer to any two conditions of/the gas. When the temperature is constant (isothermal changes), the relation between the pressure and the density is obviously P1 PI -=­

'P2 P2 Except for this special case, pressure and volume changes are accompanied by temperature changes. In the high-speed ftO\VS of primary interest, these changes occur so quickly that individual elements of the gas generally do not lose or gain any appreciable heat. Changes in state of the gas are thus assumed to be adiabatic. In this case, (9.2)

where v = ratio of specific heats, cp/cv From Eqs, (9.1) and (9.2), the corresponding temperature relation is

E! =

(Tl)"Y/{Y-l)

(9.3) T2 b. Velocity of Sound. Expressions for the velocity of sound may be derived from consideration of the equations of motion and of continuity. The equation of continuity for a compressible fluid may be developed in a manner parallel to that for Eq. (2.1) and is P2

THEORY OF WING SECTIONS

250

a(pu)

ax

+ iJ(pv) + a(pw) == _ ap ay az at

(9.4)

For motion in one dimension, this equation reduces to iJ(pu)

ap

ax== - at

or ap

au iJp -+p-+uat ax ax ==0

dividing 1;ly p

lap + au+~ap = 0 pat

(9.5)

pax

a2:

In the concept of a velocity of sound, the disturbances are considered to be small 80 that the changes in density are small compared with the density and the corresponding velocities are small. Retaining only the first powers of small quantities, Eq. (9.5) becomes

! ap + au.... 0 = a In p + au p at ax at ax

(9.6)

For one-dimensional flow, Eq. (2.3) becomes _ 8p == p a~+ pu8u

ax

at

ax

Again retaining only the first power of small quantities and dividing by p, webave

au lap at -=-j;8x

(9.7)

It is known from the equation of state that p is a function of (9.7) can then be written

au at == -

Idp8p

; dp az ==

dp

a In

p.

Equation

p

- dp ax

(9.8)

Difierentiating Eq. (9.6) with respect to z and Eq. (9.8) with respect to t, and combining, we have h dpatu (9.9) at' == dp8r The solution of this equation is u

=f{~ -

Ji, t) + ft(X+ Ji,t)

These functions represent disturbances traveling in opposite directions with the veloc~ty V dp/dp. This velocity is termed the velocity of sound a. As stated previously, the changes in pressure occur so rapidly that no heat is considered to be conducted to or away from the elements of gas.

EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS

251

The relation between the pressure and density is therefore considered to be adiabatic [Eq. (9.2)]. Then

fiiP = FYP vdP v-;

a=

(9.10)

For a perfect gas,

a == K-v'i'

(9.11)

where T is the absolute temperature, and the value of K is given in the following table for various English units, and for dry air, -y == 1.4. Units of temperature Units of velocity Feet per second Miles per hour Knots

OF. absolute

°C. absolute

49.02 33.42 29.02

65.76

. . .

44.84 38.94

In applying the concept of the velocity of sound to various problems, it should be kept in mind that the concept as derived is valid only for small disturbances. Very strong shock waves and blasts should not be expected to propagate at speeds corresponding to the local velocity of sound. c. Bernoulli's Eqootion for Comprem.1Jle Flow. In deriving Bernoulli's equation for incompressible flow [Eq. (2.5)]J the followingexpression was obtained: \

Writing this equation in integral form, we have

!2 JT2 + fdpP = E

(9.12)

Relating the pressure and density by the adiabatic law of Eq. (9.2) and per­ forming the indicated integration,

!

2

V2 + _'1_. ~ 'Y-1p

zr

E

=

! V! + c T 2

"

(9.13)

This equation can also be derived from thermodynamic reasoning. The thermodynamic reasoning indicates that the quantity E in Eq. (9.13) is a constant along any stream tube for adiabatic changes whether reversible or not. For example, E is constant throughout & stream tube containing a normal shock. For reversible changes, Eqs. (9.2) and (9.13) together with a knowledge of the initial conditions permit the calculation of the variation of pressure and density with velocity.

252

THEORY OF WING SECTIONS

d. Cro8s-sectional Areas and PreBSUTe8 in a Stream Tube.

Along a

stream tube, the equation of continuity for steady flow is

pSV = constant where S = cross-sectional area of tube It is interesting to contrast the manner in which the cross-sectional area varies with velocity for compressible and incompressible flow, For the latter case, the area is seen to vary inversely with the velocity. The determination of the variation of area with velocity for the compressible case requires more extensive analysis. Differentiating the equation of con­ tinuity, and dividing by pSV, we obtain

! dB +! dp +.!.. 8dV pdV V

=0

(9.14)

Differentiating Bernoulli's equation with respect -to the velocity, V

or

+ _'Y_ (.! dp _ 1!.) dp = 0 'Y - 1 p dp

v + _1_ (1. a

2 _

'Y - 1 p

or

p'- dV 2

a ) dp = 0 p dV

a2 dp

V+ pdV =0

(9.15)

From Eq. (9.14),

dS=_~(l+Vdp)

dV

V

pdV

Substituting the value of dp/dV from Eq. (9.15), we obtain

dB dV

=_

§.(lV2) = V a2

~(1- W) V

(9.16)

This relation shows that the stream. tube contracts as the velocity increases for Mach numbers less than unity. The area of the stream tube is a minimum for a Mach number of unity and increases with Mach number for supersonic flow. From consideration of Eqs, (9.2), (9.3), and (9.11), it may be seen that the pressures, densities, temperatures, and velocities of sound are con­ nected by the equations (9.17)

where the subscript m may denote the condition at any point but, in the follo"ring development, the subscript will denote the condition at the minimum cross-sectional area of the stream tube where the local Mach number equals unity.

EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS

253

In order to obtain an expression for the cross-sectional area of the stream tube in terms of the local Mach number M, it is convenient to express Bernoulli's equation (9.13) as

!

2

l\I2 + _1_

'Y - 1

= ~+

2

1 am 2( 'Y - 1) a2

(9.18)

where a and am represent, respectively, the velocities of sound at any point along the stream tube and at the point of minimum cross-sectional area. From continuity, we can write

s;

pV

8 = Pma.. 1.0

J

.8 I

'fI"

,/

~

<,

r-, ~

/

. J

.6

s"'~

/

.4

-,

/

r-,

r-,

"'r-;

J .2

/ I

f

~ <,

-....... ---....

~

o

.4

.8

1.2

2.4

2.0

1.6

2.8

3.2

II

FIo. 144. Variation of etream-tube area with local Mach number.

where Sand S. are, respectively, the cross-sectional areas of the stream tube at any point and at the point of minimum area. Substituting the value of piPa from Eq. (9.17),

~=

(:j2/(Y-U

(a9

=

M

(0:)

(7+U/(y-U

and, from Eq. (9.18),

s. _. ( S -

l\{

~

+ .1

) ('Y+U/2

(~- 1)M2 + 2

(9.19)

A plot of 8 m/S against M is given in Fig. 144. The pressure relations along the stream tube may be obtained by sub­ stituting the value of am/a from Eq. (9.18) into Eq. (9.17)

p ( "y + 1 pm = (-y - l)M2

)y/(y-t>

+2

(9.20)

254

THEORY OF WING SECTIONS

where p == pressure at any point along stream tube.

It is often more convenient to express the local pressure in terms of the

pressure at the stagnation condition,

p ( 2 )-r/<..,-I.) Po == ('Y - 1)M2+ 2

(9.21)

where Po is the pressure of the fluid at rest. The temperature relations may be obtained from Eqs. (9.17) and (9.18) as T 'Y+ 1 T,. == (y - I)M2 + 2 or To == 1 + y - 1 MI (9.22) T 2

It is interesting to note that there is a limiting velocity which is reached when the gas is expanded indefinitely into a vacuum. Bernoulli's equation (9.13) may be written 111-+

20.2

1 'Y- 1

(a)' a..

'Y+l

==2(-y-l)

With infinite expansion, the temperature approaches zero and the local

velocity of sound a approaches zero. Accordingly

V ....... =

a.

~'Y + 1 'Y-l

(9.23)

The local Mach number M, of course, continues to increase indefinitely as the expansion is increased. e. Relations for a N ormol8hock. In deriving the relations for the cross­ sectional areas, pressures, and temperatures existing in a stream tube, the entire process was considered to be adiabatic and reversible. Experience has shown that this condition is satisfied whenever the velocity is increasing in the direction of flow. However, when an attempt is made to decelerate supersonic flows, discontinuities in velocity, pressure, and temperature generally occur. These discontinuities occur in a very short distance along the direction of motion and are termed U shock waves." The velocity and total pressure of the air decrease and the temperature and static pressure increase in going through the shock. The process, although adiabatic, is irreversible and characterized by an increase of entropy. The shock is stationary for a tube of fixed configuration with constant pressures at the inlet and exit. Since the velocity on the upstream face is supersonic, the shock can hardly be considered to propagate at the velocity of sound. Similar discontinuities occur forward of the nose of a blunt body traveling at supersonic speeds. In the case of a simple tube or of the shock im­

EFFECTS OF COMPRES8IBILITY AT SUBSONIC SPEEDS

255

mediately forward of the nose of a blunt body, the plane of the shock is normal to the direction of flow. This type of shock is a fundamental one and is termed a "normal shock," as distinguished from the oblique shocks shown in Fig. 143. Relations for determining the conditions across a normal shock wave from the conditions on one side are derived'" from considerations of mass, momentum, and energy. If the subscripts 1 and 2 denote, respectively, the conditions on opposite sides of the shock and if m is the mass flow per unit area and p is the static pressure, the condition for continuity for a shock stationary with respect to the observer is

m The equation of conservation of momentum is PIUl

=

P2~

(9.24)

=

P2 - PI = m(ul - U2)

(9.25)

The equation for conservation of energy is PIUl -

PtU2

_!2 m (u~! -

Ut!)

=~

"Y- 1

(l!: _PI) [.J2

PI

(9.26)

The solution" for these conditions is P2

P; =

(P2/PI) (1' + 1) / (1' - 1) + 1 (p,./Pl) + (1' + 1)/ ()' - 1)

(9.27)

This equation relates the density and pressure ratios across the shock when This equation corresponds to a change in entropy except in the case where PI equals P2. It can be shown" that P2 must be greater than PI for the change in entropy to be positive. It would be convenient to relate the pressure and density rat.ios across the normal shock to the upstream Mach number. Such an expression may be obtained by expressing Eqs.. (9.24) and (9.25) in terms of the local Mach number and combining to give PI and PI are the upstream conditions.

P2 - 1 = PI

Substituting the value of 1\'112 = ,.

Mt~'Y(l -~)

PI/ P2 from

'Y [1 -

\

P2

Eq. (9.27) gives

" (P2/Pl) -:- 1 , _
(9.28)

+

From Eq. (9.24), we can write

M~ = ~~ ~ l\I 1

P2 P2

(9.29)

Values of the ratio Pot/Po. may be calculated from Eqs. (\J.23) , (9.27), (9,,28), and (9.29) where po is the total pressure. This ratio is an index of

THEORY OF WING SBCTIONS

256

the useful energy remaining in the air downstream of the shock. Values of Mt/M1, p,./pl, PI/PI, and PoJPo. are plotted against M in Fig. 145. 9.4. First-order Compressibility Effects. a. Glauert-Prandtl rule. The Glauert-Prandtl rule36 relates the lift coefficient or slope of the lift curve of a wing section in compressible flow with that for incompressible flo,v. This relation was derived for the case of small disturbance velocities and

20 5 1.0

16 4

~

~

\"

.8

<,

M2

J2 3

r-, "-

~

.6

I PD2

r-,

,/

<, ~~

............

./

8

2

.4

/t,I" 4

I

/' ./

~

/

~

~

V ~

"

:>< ~

./ ~

.2

~ PDI

V

?~ V

~

~

/

~

'" V

~

~

V

7 ./

."".,.,.,...,

~~

.c~2 P,

/'

---r-, ~

'-.

~

~

-

~ -..............

v~

r-----.. ~

r----.-

~~

L4

1.8

26

3.4

4.2

M, FlO.

145. Pressure, density, and Mach-number relations for a normal shock.

low Mach numbers. These conditions are approximated for thin wing sections with small amounts of camber at low lift coefficients at speeds well below the speed of sound. The Glauert-Prandtl relation is Cl c

at;

ci,

a;

1 vI-M2

-=-=-==~

(9.30)

where the subscripts c and i denote, respectively, the compressible and incompressible eases. In deriving this formula, Glauert considered all the velocities over the wing section to be increased by the same factor. Equa­ tion (9.30) may therefore be applied equally well to the moment coefficient. The Glauert-Prandtl rule agrees remarkably well with experimental data, considering the assumptions made in its derivation. A comparison of this rule as applied to the slope of the lift curve for three wing sections of 6, 9, and 12 per cent thickness is given" in Fig. 146. KapianM obtained a first-step improvement of the Glauert-Prandtl rule

EFFECTS OF COMPRESSIBILITY AT ~UBSONIC SPEEDS

257

for the lift of an elliptical cylinder and extended it to arbitrary symmetrical profiles, The Kaplan rule is

~~ = I' + 1 ~ t [1'(1' - 1) + ~('Y + 1)(1'2 - 1YJ where

Il

(9.31)

1 vl-M2

= -_""'/=== ;

t = thickness ratio

.24.-----..-----or----or----..,.--....--....

.201-----t-------t------+------1,........--+----1

.'61-----+-----+------I----I-oI-~---&.--_t

dCt

da, .12....------4-----+-­

~-~~-+-_+__-t----_t

.081-----'-------1-----+------;------1

.2

.6

.4

.8

·ID

N FlO. 146. Lift-eurve slope variation with Mach number for the NACA 0012-03. 0009-63. and 0006-63 airfoils.

Equation (9.31) approaches the Glauert-Prandtl rule as the thickness ratio approaches zero. It will also be noted that the Kaplan rule shows a "peak dependence on the ratio of the specific heats 1'. The Kaplan rule for wing sections 5, 10, 15, and 20 per cent thick in air together with the Glauert­ Prandtl rule are plotted in Fig. 147. b. Effect oj j{ach number on tile Pressure CoejJicicnt. The Glauert­ Prandtl rule supplies a first approximation to the variation of pressure coefficient with Mach number. C~l

Cp ,

1

=

vI - ]\{2

THBORY OF WING SECTIONS

258

Numerous attempts have been made to obtain more accurate expressions for the variation of pressure with Mach number, notably by Chaplygin,15 Temple and Yarwood, 119 and von Karman and Tsien. 129,14O Garrick and Kaplan" succeeded in unifying these results 88 approximate solutions of

It',l 0.20

~

J5

2.5

rule

./0 1.05

I

I I ~Kaplon

J

I I

1.Pra1dt1-G!aJerf I I

Ir' III

rule

JI

2.0

flI '1 / J

1.5

~

1·°0

-----ae

~ 0.4

0.6

0.8

1.0

M1

FIG. 147. Ratio of lifts for compressible and incompressible fluids as function of stream Mach number.

the general problem and presented two other approximations. No simple general solution of the problem is known. The I{arman-Tsien relation is widely used in the United States. Ex­ perimental evidence appears to indicate that this relation is as applicable as any of the solutions. This relation is CPM 1

CPt = viI - M2 + [M2/(I + viI - M2)J(C p J 2) (9.32)

EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS

259

A plot of the pressure coefficient as a function of the free stream Mach num­ ber AI is given in Fig. 148 for a number of values of the incompressible

/.0

.9

.8

1 .~

i· 7 ~

t~

~

~

.6

~

u~

.5 .4 .3

.2

.­

~

./

°0

./

.4 .5 .6 .7 Free-stream Mach 1III11ber,Mo

148. Variation of local preesure coefficient and local Mach number with free stream Mach number according to KArm8.n-Tsien.

FIG.

pressure coefficient. Also shown in Fig. 148 are curves of the local Mach number ML. A special use for relations such as Eq. (9.32) is the prediction of the stream Mach number at which the velocity of sound is reached locally

THEORY OF WING 8ECTIONS

260

over the wing section. Jacobs" applied the Glauert-Prandtl rule to this problem. It is now customary in the United States to use the Karman­ 3.5

./

.4

,,5

.6

.7

Free -stream MochnumberlMo FIG.

.8

.9

148. (Concluded)

Tsien relation [Eq. (9.32)]. This problem may be solved from Fig. 148 by following the curve for the highest incompressible pressure coefficient on the wing section to the line ML -= 1 and reading the corresponding stream

EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS

261

Mach number. Figure 148 may be used conveniently when the maximum local pressure coefficient is known experimentally at some subcritical Mach number. If the incompressible pressure coefficient is known theoretically, it is more convenient to use Fig. 149, which gives the corresponding criti­ cal Mach number directly. 1.0

.9

.8

1

\

\

I

\

II

~

-,

r-,

<,

~

~:--

~

.4

I I

.~

.4

I

l6 2.0 LDw-spetJd presstn coefficient, Cp

.8

/2

2.4

2.8

Flo. 149. Critical Mach number chart from KarmAn-Tsien relat.ion.

9.6. Flow about Wing Sections at High Speed. Consideration of the effects of compressibility on the flow over wing sections may be conveniently divided into two parts" The first part deals with the characteristics in the range of subcritieal Mach numbers. 111C second part deals with the characteristics at supercritical Maeh numbers. The critical Mach number dividing these t\VO ranges is defined us the stream Mach number at which the local velocity of sound is just attained at any point in the field of flow. a. Flow at Subcritical Moen Numbers. The first-order effects of com­ pressibility on the potential flow at suberitical Mach numbers are well described by the theoretical relations of Sec. 9.4. An indication of the accuracy of the Karman-Tsien relation is given!" in Fig. 15Ob. This figure shows the experimental pressure distribution for the NACA 4412 section at a Mach number of 0.512 and a pressure distribution predicted from the experimental data obtained at a Mach number of 0.191 (Fig.

THEORY OF WING SECTIONS

262 -1.6 -1.2

~-

-.8

-A

c'P

0

-- ---­

~

..........

/

........... ~

.

r

~

--

~

-.......

~

~~

...

r

~

........

~ ....... ~~

-­

~ ~

(b) M=0.5/2

(a) M=0./9/

>

~~

(

.8 --/.6

-­ ~

-1.2

~

.......~

./

'

I

-.8

f ,.. r 4

-.4 Cp 0

--

""'­

.......

.......

-I. 6 ~

-I. 2

\

\

.... ~,

I

r----."",,-: ~

1',

--.

rr

-=:-....

~~

817 ~

<,~......

'i~

~

-

\

"\

/

J

"<, ......

4y-. ~~

""lr

""'-..... ......

°T

....

:

4

Iii ~

~

~

"<, \

\

Cpc

~

-'i' ......1'.

~...... I'-o..

~~

(f)

(e) M=Q664

8

~

(d) M=0.640

"'

Cp -.

~,

-­I tp;

(c) M=0.596

.8

,

~

M:Q690 ;

I. 2 -I.2

-. 8".... -.4

t

i

I.........

~

~.-

~

-._~

, \

.'"

1Jc

~

~~

OJ

~-

, .......

'~ ......

..

Dotted line in(b)represents application of Kormon-Tsien

.......~

.4

-

-~

-~

(g) M=O.735

8

..

reloliJn 10 doloin tal x t,pper surface o Lower surface

~

I. 0

20

40

60

Chor~percenf FIG.

80

100 0

20

40

60

80

Chord,percefl'

150. Pressure distribution for the NACA 4412 airfoil. a=: 1°52.5'.

100

EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS

263

15Oa). The accuracy appears to be satisfactory for such applications. The larger area of the pressure diagram at the higher Mach number corre­ sponds to an increased lift coefficient at the same angle of attack. The resulting change of lift-curve slope has been shown (Fig. 146) to agree fairly well with the Glauert-Prandtl rule. The applicability of these theoretical relations is limited to cases where the effects of separation are not marked. If separation is present, the increased pressure gradients associated with higher Mach numbers tend to increase the separation and thus change the whole field of flow, Moreover, the absence of appreciable separation at low speeds is not necessarily a sufficient criterion, because the increased pressure gradients at the higher Mach numbers may cause separation in marginal cases. The theoretical relations may therefore be applied with confidence only to wing sections of normal shape at low lift coefficients. b. Flmo at Supercritical Mach Numbers. The flow at supercritical Mach numbers is partly subsonic and partly supersonic and is characterized by the presence of shocks. No theory dealing with these mixed subsonic and supersonic flows has been developed to the extent of being useful in the prediction of wing characteristics. The almost complete lack of theoret­ ical treatment requires reliance on experimental data at supercriticaI Mach numbers. For the typical case of a wing section operating at a small positive lift coefficient, the velocity of sound is reached first on the upper surface, as is shown in Fig. 150 by the pressure coefficient increasing negatively beyond the critical value CPc_ Figure 150c shows no drastic change in the pressure distribution when the local velocity' of sound is exceeded by a small amount. Many experimental data indicate that drastic changes in the forces on wing sections do not occur until the critical Mach number is exceeded by a small but appreciable margin. There is some doubt as to whether a shock necessarily occurs when the velocity of sound is locally exceeded by a small margin. DC In any case, the losses associated with a shock from very low supersonic velocities are very small, as shown b)" Fig. 145. Such small losses would not be expected to produce drastic changes in the field of flow. A shock occurs when the critical Mach number is exceeded appreciably, as shown by Fig. 150d to g. The shocks act, at least qualitatively, like

a normal shock to reduce the velocity to subsonic values. The resulting rather sudden changes in the pressure distribution are shown dotted in Fig. 150. Schlieren pictures of the shocks corresponding to the diagrams of Figs. 150! and g are shown-" in Fig. 151. The positions of the shocks as shown by the photographs correspond closely with the dotted regions of the pressure diagrams. Figure 152 presents another series of schlieren photographs showing the shocks on a NACA 23015 wing section at an

THEORY OF WING SECTION8

264

angle of attack of 3 degrees. These pictures.!" which were taken in the NACA rectangular high-speed wind tunnel, show the progressive intensi-

FIG.

151. Schlieren photograph of flow for the NACA 4412 f!,irfoil. a

=

1°52.5'.

fication and rearward movement of the shock with increasing Mach num­ ber. The character of the flow with shock is greatly affected by the inter­ action of the shock and the boundary layer. It is apparent that the shock cannot extend to the surface through the region of Iow velocity in the

EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS

265

boundary layer. If the plane of the shock is perpendicular to the flow, the increase of static pressure through the shock will correspond to that shown in Fig. 145. The relatively high pressure downstream of the shock will tend to propagate upstream through the boundary layer. The pres­

M:O.40

M=O.60

M=O.550

M:O.650

FlO. 152. Schlieren photographs of flow. N.:\CA 2:J015 airfoil. NACA rectangular high­ speed wind tunnel. AiI"foil chord, 2 inches. Angle of attack. 3 degrees.

sure distribution measured at the surface cannot , therefore, be expected to show a discontinuity. Figure 150 shows that the sharp pressure rise is, in fact, shown as a steep gradient at the surface. Such propagation of the pressure along the boundary layer necessarily leads to pressure gradients across the boundary layer in the vicinity of the shock. This situation is quite different from the usual boundary-layer conditions at low l\Iach numbers. The large adverse pressure gradients associated with the shock always produce a large increase of boundary-layer thickness and frequently cause

266

THEORY OF WING SECTIONS

separation, as shown U 4 in Figs. 152 and 153. This separation is an im­ portant or even predominant factor in causing the large force changes occurring at supercritical speeds. The relative importance of the shock losses and the effects of separation on the drag of a typical wing section

M=O.750

M:O.70

M:O.821

M=O.80 FIG.

152. (Concluded)

are shown-" by Fig. 154. This figure shows the total pressure loss across the wake of the section. These losses are intimately associated with the drag, and, to the first order, the drag coefficient is proportional to the area under the curve. Intense shock has not occurred at the lower Mach num­ ber, and the diagram has the typical shape obtained at lo,v speeds. Intense shoek has occurred at the higher Mach number, and the drag coefficient has obviously increased. This increase may be considered to be of t\VO parts. The very large losses of pressure in the center of the diagram are similar in shape to the lower speed diagram and are attributed to the usual type of loss associated with skin friction and sep RXat,i on. The smaller

267

EFFECTS OF COMPRE88IBILITY AT 8UBSONIC SPEEDS

pressure losses extending far out into the stream are attributed to the losses across the shock. It is apparent that the drag rise associated with the usual type of loss is as important as that directly associated with the losses across the shocks. The pressure rise measured'P on the surface of the wing section in the vicinity of the shock is smaller than that corresponding to a normal shock with the losses of total pressure indicated by the wake surveys. The pressure rise correspond­ ing to a normal shock may also be obtained by means of Fig. 145 from the local Mach num­ ber ahead of the shock as indicated by pres­ sure measurements on the surface. In this case, also, the computed pressure rise is greater than that shown by the surface pressures. The com­ plete explanation of this discrepancy is not known, but two factors could tend to reduce l\>1 0.691 the pressure rise on the surface below that FIG. 153. Schlieren photograph of separated flow for rear portion corresponding to a normal shock. of NACA 23015 airfoil. XACA The first factor is the pressure gradient rectangular biJ!h-slx-t'd wind through the boundary layer previously dis­ tunnel. Airfoil ehord, [; inehes. Angle of ut tar-k, (. dt~~rN'~' cussed. The importance of this factor may he inferred from Figs. 152 and 153. The lines in the schlieren photographs correspond to density gradients. Dark Jines are 8ho''''11 in the vicinity of 81:

()

~.4-

~ ~

.s-: M Q810

t1

~ .2­

15

~

.708

./­

.~

Col)

§

0 240 200

/60

/20

80

40

0

40

80

/20

160

200

240

Distance across KOte.l perceri chord FIG. 154. '''''ake shape and total pressure defect as influenced by Mach number, 0012 airfoil, a = 0 degrees; NACA rectangular high-speed wind tunnel.

~ACl~.

the shock making a small angle with the surface. Although these density gradients may be partly associated with temperature gradients, the sharp­ ness and intensity of these lines are thought to indicate the presence of appreciable pressure gradients. The second factor is the possible reaction of the boundary layer on the shock. The thickening or separation of the boundary' layer tends to pro­ duce oblique shocks, Such shocks would produce some pressure rise ahead

THEORY OF WING SECTIONS

268

of the normal shock and would accordingly reduce the intensity of the normal shock, especially close to the surface. The extent to which the normal shock is locally softened in this manner is uncertain, but Figs. 152 and 153 suggest that this mechanism may be of considerable importance. Investigations made by the NACA, by Liepman,· and by Ackeret4 and his associates emphasize the complexity of the interaction of the shock and the boundary layer. These investigations indicate important differences between the shock phenomena for laminar and for turbulent boundary layers. Although exceptions have been noted, laminar boundary layers .4

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tend to produce multiple shocks of lower intensity, or U lambda-type" shocks, while those associated with turbulent boundary layers tend to resemble intense normal shocks more closely. The understanding of these phenomena awaits further theoretical and experimental investigations. The typical changes of forces experienced at supercritical speeds by a wing section of moderate thickness are shown!" in Fig. 155. The Mach number for force break is seen to be appreciably higher than the critical Mach number. At" Mach numbers higher than that for force break, the lift coefficient decreases, the drag coefficient increases, and the moment coefficient usually increases negatively. 9.6. Experimental Wing Characteristics at High Speeds. The existing three-dimensional lying theory, which is the basis for the concept of section characteristics, is not valid for supercritical speeds where part of the flow is supersonic. Consequently section data cannot be applied quantitatively to the prediction of the characteristics of wings at such speeds. Moreover, present trends for wings designed for efficient operation at supercritical speeds are toward low aspect ratios and large amounts of sweep. As

EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS

269

pointed out previously, section data are of doubtful quantitative signif­ icance for such plan forms at any speed. Under these circumstances, the detailed consideration of section characteristics as design data is not warranted. Nevertheless, the characteristics of wings at supercritical speeds de­ pend to some extent on the lying sections employed. The variation of the wing characteristics with wing section appears to be qualitatively similar to the variation of the wing-section characteristics- if the lying plan form is such as to permit any semblance to two-dimensional flow, Consequently, the purpose of this section is to present typical data showing qualitatively the major effects of variation of the wing profile at high subsonic Mach numbers. a. Lift Characteristic«; One of the first and most severe effects of com­ pressibility encountered in flight was the tendency of many airplanes to "tuck under" in high-speed dives. This U tucking under" tendency con­ sisted of a large negative shift in the angle of trim tbgether with a large increase of stability that resisted the efforts of the pilot to trim at the desired positive lift coefficients required for recovery from the dive. The resulting large elevator motions required to recover from the dive, or even to prevent the dive from becoming steeper, corresponded to excessive con­ trol forces, thus leading to the impression that the "~t.ick became frozen." These changes in longitudinal stability limited the maximum safe flying speed for many airplanes. These effects are intimately associated with tile changes in lift charac­ teristics of wing sections at Mach numbers above that for force break. The relatively thick cambered wing sections commonly used during the Second ,Vorld ar experienced a positive shift of the angle of zero lift and a reduction of lift-curve slope. Although ID3ny effects associated with other characteristics of the airplane are present, the change in angle of zero lift directly affects the angle of trim and the reduction of lift-curve slope directly increases the longitudinal stability. The shift in the angle of zero lift is associated with the camber of the wing section. ..~ symmetrical section shows, of course, no change in the angle of zero lift 116 (Fig. 156). The rather large positive shift in the angle of zero lift shown!" in Fig. 157 for the X.~(~.:\ 2409-34 section is typical for cambered sections without reflex. In this case, the change amounts to about 2 degrees for a change of Mach number from 0.80 to only 0.83. The effect of increasing the camber is showu'" by Fig. 158. Doubling the amount of camber causes a similar shift to occur at a lower Mach number. The effects of thickness on the lift characteristics of"symmetrical sec­ tions are shown" in Figs. 159 to 161 for the N..~CA 0006-34, 0008-34, and 0012-34 sections. The 6 per cent thick section shows a large increase of lift coefficient at a given angle of attack with increasing Mach number at

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THEORY OF WING SECTIONS

270

values below about 0.85. At Mach numbers above the force break, the lift coefficient decreases rapidly, but it never goes below the low-speed values, at least at Mach numbers up to 0.95. The 12 per cent thick section, however, fails to show much increase of lift coefficient at subcritical Mach numbers, but it shows a large decrease of lift coefficient at Mach numbers just above the force break. At a Mach number of 0.85, the lift coefficient,

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and accordingly the lift-curve slope, are only about two-thirds of the low­ speed value. At higher Mach numbers, the lift coefficients again increase. The effects of Mach number on the lift characteristics of the 8 per cent thick section are intermediate. These data show that the thin sections experience force breaks at higher Mach numbers than the thicker sections and that the lift force breaks are much less severe for the thin than for the thicker sections. The effects of thickness for a series of cambered sections are shown" in Figs. 162 to 165. These data indicate to an even greater degree the adverse effect of thickness on the character of the lift force break. Comparison of these data with those for the symmetrical sections (Figs. 159 to 161) shows the adverse effects of camber on the Mach number for lift force break anti on the shift in the angle of zero lift.

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THEORY OF WING SECTIONS

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Comparatively few data are available to show the effect of thickness distribution on the characteristics above the force break. The trends indicated by Stack and von Doenhoff116 cannot be considered representa­ tive for properly designed families of "ring sections because of the arbitrary

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geometrical manner in which the sections were varied. If the thickness distributions are varied in such a manner as to avoid peaks in the low­ speed pressure distributions, as in the case of the N ACA 6-series sections, the indications are that the ~ffect of thickness distribution is minor com­ pared with the effects of thickness ratio and camber. There is some indica­ tion 1l 6 that the trailing-edge angle should be kept small to avoid adverse effects on the lift characteristics similar to those for thick sections. Figures 166 to 168 show the17b lift and drag characteristics of the NAC..~ 66-210 wing section. The most outstanding feature of these data as com­

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THEORY OF WING SECTIONS

276

pared with th~ for the NACA

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coefficient of only about 0.55 and almost no increase of lift for a change of angle of attack from 4 to 5 degrees. Figure 169 ShO\VS376 the predicted critical Mach numbers and the Mach numbers for force divergence as functions of the lift coefficient for the NACA 66-210 section. The outstanding characteristic of these data is the wide range of lift coefficients over which high Mach numbers for force divergence are realized as compared with the range for high critical Mach numbers. The predicted critical Mach numbers are lower than those for force divergence but approximate the latter values over the range of lift coefficients where the speed of sound is first reached near the location of minimum pressure at the design lift coefficient. It is apparent that critical Mach numbers predicted on the basis of the attainment of the velocity of

EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS

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sound near the leading edge are useless as an indication of the force charac­ teristics of NACA 6-series sections. The general effects of Mach number on the maximum lift coefficients of wing sections may be inferred from the data'19 of Fig. 170. These data were ....- Symbol a 1.2 I - 10-

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obtained from tests in the NACA Langley Hi-foot high-speed wind tunnel of a lying having an aspect ratio of Gand having X.•\CA 23016 sections at the root and NACA 23009 sections at the tip. At Mach numbers above 0.30, the maximum lift coefficient decreases rapidly with Increasing Mach number up to values of about 0.55 where the maximum lift coefficient is approximately 1.0. At higher Mach numbers, the actual maximum lift coefficient decreases more slowly; but the angle of attack, or lift coefficient, at which the lift curve departs radically from its normal slope continues to

THEORY OF WING SECTIONS

278

decrease to the limits of the tests. The resulting variation of maximum lift coefficient with Mach number is shown" in Fig. 171. ' The increase of maximum lift coefficient with Mach number at values below 0.30 is associated with the variation of Reynolds number for these .20

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tests. This effect was studied more extensively in the NACA 19-foot­ pressure wind tunnel where the effects of Mach and Reynolds numbers could be separated" for low Mach numbers. These data indicated, at least qualitatively, that the peak maximum lift coefficient is determined by the critical Mach number at maximum lift which, of course, is attained at relatively low free-stream Mach numbers. At subcritical Mach numbers, the maximum lift coefficient is primarily a function of the Reynolds num­ ber.&8 at very low Mach numbers, although the effect of Mach number is

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not negligible. At supercritical Mach numbers, the value of the maximum lift coefficient decreases rapidly with increasing speed, and the effect of Reynolds number is secondary. For the wing tested, the critical Mach 1.6 I

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number of the order of 0.25 to 0.30 for the flaps-retracted condition and of 0.17 to 0.20 with O.2Oc split flaps deflected 60 degrees. It should be noted that the maximum usable lift coefficient for an air­ plane at high Mach numbers may not be the same as the actual maximum lift coefficient. Referring to Fig. 170, the airplane may be unable to fly on the relatively flat portions of the lift curves above the force breaks, be­ cause of the serious buffeting associated with the partly separated flow and, perhaps also, because of stability and control difficulties.

EFFECTS OF COJIPRESSIBILITY AT SUBSONIC SPEEDS

281

At supercritical speeds, changing the camber near the trailing edge, as by small deflections of a flap or aileron, cannot affect the flow over the front part of the section because pressures cannot propagate upstream through the supersonic region. Consequently, the flap effectiveness must be greatly reduced at supercritical Mach numbers. This effect is clearly shown!" by Fig. 172. This loss of flap effectiveness is related to the positive 1.6

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shift of the angle of zero lift for cambered sections. The rearward port.ion of a cambered section rna)" be considered to be a special case of a deflected flap. The loss of flap effectiveness at the force break leaves the forward portion of the section at an effectively lower angle of attack, and the resulting change in lift coefficient corresponds to a positive shift of the angle of zero lift. Conversely, a strongly reflexed section would show the opposite effect. It follows that the change in lift characteristics of cam­ bered sections may be reduced by incorporation of reflex or of an upwardly deflected flap. b. Drag Characteristics. The effect of thickness ratio on the drag characteristics of symmetrical wing sections is illustrated" by Figs. 173 to 175. These data show the typical rapid increase of drag coefficient at high Mach numbers. This rapid rise (force hreak) occurs at smaller Mach numbers as the angle of attack and thickness are increased. Similar data

THEORY OF n'ING SECTIONS

282 6,0·

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Mach number~M FIG.

173. Effect of compressibility on the drag of the NACA 0006-34 airfoil.

./0 .09

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FIG. 174. Etfe~t of Mmpre..."Sibility on the drag of the NACA 0008-34 airfoil.

283

THEORY OF WIJ.VG SECTIONS

284

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Mach ntmber, M 175. Effect of compressibility on the drag of the NACA 0012-34 airfoil.

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Mach fNJmber,M Flo. 176. Effect of compressibility on the drag of the NACA 2306 airfoil.

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177, F.Jrect of compressibility on the drag of the NACA 2309 airfoil.

~

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THEORY OF Ti'ING SECTIONS

286

for the cambered sections, NACA 2306, 2309, 2312, and 2315, are shown lO in Figs. 176 to 179. The effect of camber is to reduce the Mach number at which the drag force break occurs as compared with the data of Figs. 173 to 175. Although the thickness distribution is not the same for these

1

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symmetrical and cambered sections, the differences attributable to this fact are very small, at least for the lower thickness ratios. Comparisons of the predicted critical Mach numbers and the Mach numbers for drag force break are presentedf" in Fig. 169 for the NACA 66-210 wing section. For the range of lift coefficients corresponding to high predicted critical speeds, the Mach number for drag force break is about 0.01 to 0.03 greater than the critical. The Mach number for drag force break is intennediate between that for lift force break and the critical

EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS

287

value. The slope of the boundary for drag force break is similar to that for the critical Mach number, but high Mach numbers without force break

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may be realized over a much wider range of lift coefficient than is indicated by the theoretically predicted critical speed. c. M omen: Characteristics. Figures 157 and 158 illustrate116 the nega­ tive shift of the moment coefficient with increasing Mach number that is

THEORY OF WING SECTIONS

288

typical for cambered wing sections. In addition to this shift, there is a tendency at the highest Mach numbers shown for the moment coefficient to increase negatively with increasing lift coefficient as is shown37b more .3

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clearly for the N ACA 66-210 section in Fig. 180. This rearward shift of the aerodynamic center at high Mach numbers is stabilizing, and the more negative values of the moment coefficient tend to make the airplane trim at lower angles of attack. These moment changes thus add to the pre­ viously discussed effects of the lift coefficient in producing the "tucking under" tendency of airplanes with thick cambered wing sections.

EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS

289

The moment characteristics of the NACA 0006-34, 0008-34, 0012-34, 2306, 2309, 2312, and 2315 wing sections are shown" in Figs. 181 to 187. These data indicate that the changes in moment characteristics with Mach /

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Mach number, M Effect of compressibility on the pitching moment of the N ACA 0006-34 airfoil.

number are minimized for symmetrical sections, although a rearward shift of the aerodynamic center at high Mach numbers persists, 9.7. Wings for High-speed Applications. As discussed in the previous section, all wing sections experience undesirable variations in their charac­ teristics at high subsonic Mach numbers. Although these undesirable variations can be minimized by the use of thin symmetrical sections, it does not appear to be possible to eliminate them in two-dimensional flow. In three-dimensional flow, however, it is obvious that the shocks cannot extend tothe wing tips without modification. The presence of the shock at

THEORY OF WING 8ECTIONS

290 ./

o Ct =QI

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0

1

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EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS

291

./

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Mach rvmber,M 183. Etlect of compreeaibility on the pitching moment of the NACA 0012-34 airfoil

THEORY OF WING SECTIONS

292

o -.1

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EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS

o

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FIG.

."

ct =0.2

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293

Ct

./

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=Q4

.,...

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.J

THEORY OF WING SECTIONS

294 o

I I I I 1 I

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---.~

Ct=O

o

I I

1 I

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FIG.

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c,,=Q4 .6 .5 Ma:h number. M

.,

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rw

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186. Effect of compressibility on the pitching moment of the NACA 2312 airfoil.

EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS·

o

295

". ~

~

ll( ~ ~

t ;,0

-J

C

-

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Mach number, M FlO.

187. Effect of compressibility on the pitching moment of the NACA 2315 airfoil.

296

THEORY OF WING SECTIONS

the lying tip with the accompanying sudden pressure rise would induce a three-dimensional flow that would tend to reduce the severity of the shock. The existence of such H tip relief" became apparent in early tests of propellers at high tip speeds. The data showed that losses of efficiency did not OCcur until tip Mach numbers were reached that were well in excess of those at which the sections used at the tips suffered large increases of drag in two­ dimensional flow, Corresponding advantages for "rings may be expected if the aspect ratio is sufficiently low, Experimental data have confirmed the advantages of low aspect ratio at high Mach numbers. The theoretical approach for wings of very low aspect ratio is entirely different from that for normal wings of high aspect ratio, and the concept of wing-section characteristics is not applicable. Jones" showed that, for wings with aspect ratios approaching zero, the lift depends on the angle of attack and on the positive rate of increase of span in the direction of the air flow, or db C, = ret dx where C, = local lift coefficient of section perpendicular to direction of air flow a = angle of attack b = span x = distance in the direction of air flow db/dx is positive This simplified theory indicates that C, = 0 for those portions of the wing where db/d,x is zero or negative. This theory shows that the center of lift for a triangular wing with the apex forward is at the center of area. The theory indicates that, for the particular ease of a slender triangular plan form, the lift coefficient and center of pressure are independent of Mach number at both subsonic and supersonic speeds so long as the wing lies well within the Mach cone. The applicability of the theory at a Mach number of 1.75 is shown by Fig. 188. Low-aspect-ratio wings of triangular plan form thus appear to provide one solution to the stability difficulties in the transonic speed range. Another solution to the problems of the transonic speed range is the use of large amounts of sweep. The manner in which sweep is effective is clearly seen from consideration of a wing of infinite span with sideslip (Fig. 189). It is obvious that the spanwise component of velocity V. will produce no forces on the wing, neglecting viscosity, and that the force on the wing will be determined by the normal component of velocity V". The- resulting velocity l' may be transonic or even supersonic, while the normal component V ft is less than the critical speed if the angle of sideslip, or sweep, fJ is sufficiently large. This effect of sweep in avoiding compres­

EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS

297

I

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«,dig 188. Test of triangular airfoil in Langley model supersonic tunnel 1\lach number, 1.75; Reynolds number, 1,600,000.

FIG.

veIOCi/,ry component normal ( 10 wingtVcosjJ)

--,----~

117

streon veJocily..... V

Angleof

sweea-«

fi

Spamvise velocity component (VsinjJ)

FIG.

189. Velocity components on a swept wing.

THEORY OF WING SECTIONS

298

10------------ 80mm - - - - - - - - - - -..

-k::

~---------------

Profile R-4009

I

I

1 /

1/'

/

~ol rnetJSUI'1!fTI(

400

<,,/1 (.,'-1'

-..tso~ FIG.

190. Wing placed obliquely across a two-dimensional wind tunnel.

-qs

x D,S

~O

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FIo. 191. Comparison of pressure distribution over normal and oblique wing.

EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS

299

sibility troubles was independently discovered by German investigators and by JonesU during the Second World War. Application of this principle results in wings with large amounts of sweepback or sweepforward. Such plan forms obviously violate the simple two-dimensional considerations at the root and tip even if the wing has no taper. These violations, however, do not appear to be sufficient to destroy the value of sweep if the wing is of reasonably high aspect ratio, as in­ dicated by tests'" of a wing placed obliquely across a two-dimensional wing tunnel (Fig. 190). Figure 191 shows that the pressure distribution over the center section of the oblique wing is reasonably close to that expected from the simple theory, despite the fact that the tunnel w8JIs violate the assumed end conditions. Experimental studies have shown that swept wings of all useful aspect ratios are effective in postponing and alleviating compressibility troubles. Consequently, the theory of swept wings of arbitrary plan forms is being intensively investigated. 1I • u , lG. U7 It thus appears that the wings of the high-speed airplanes of the future may be quite different from those of comparatively slow subsonic air­ planes. These highly swept low-aspect-ratio "rings, however, have serious disadvantages for the low-speed flight necessary for landing and take-off. Some of these disadvantages are lo\v lift-curve slopes, high angles of stall, and poor stability characteristics. These disadvantages seriously com­ promise the design of airplanes using such plan forms and may force the use of more conventional plan forms with very thin supersonic sections for some applications. In anycase, these disadvantages at slow speeds are expected to dictate the use of conventional plan forms for airplanes de­ signed for such speeds that it is possible toavoid serious compressibility difficulties by the use of thin wing sections.

REFERENCES 1. ABBOTr, FRANK T., JR., and TURNER, HAROLD R., JR.: The Effects of Roughness at High Reynolds Numbers on the Lift and Drag Characteristics of Three Thick Airfoils. NACA ACR No. IAH21, 1944 (Wartime Rept. No. L-46). 2. ABBOTT, IRA H., and GREENBERG, llARBy: Tests in the Variable-Density Wind Tunnel of the NACA 23012 Airfoil with Plain and Split Flaps. NACA Rept. No. 661, 1939. 3. ABBOTr, IRA H., VON DOENHOFF, ALBERT E., and STIVERS, LoUIS S.: Summary of Airfoil Data.. NACA Rept. No. 824, 1945. 4. ACKERET, J., FELDMAN, F., and RoTr, N.: Investigations of Compression Shocks and Boundary Layers in Gases Moving at High Speed. NACA TM No. 1113, 1947. 5. AutEN, WILLIAM, JR.: Standard Nomenclature for Airspeeds with Tables and Charts for Use in Calculation of Airspeed. NACA TN No. 1120, 1946. 6. ALLEN, H. JULIAN: Calculation of the Chordwise Load Distribution over Airfoil Sections with Plain, Split or Serially-hinged Trailing Edge Flaps. NACA Rept. No. 634, 1938. 7. ALLEN, H. JULIAN: A Simplified Method for the Calculation of Airfoil Pressure Distribution. NACA TN No. 708, 1939. 8..-\LLEN, H. JULIAN: General Theory of Airfoil Sections Having Arbitrary Shape or Pressure Distribution. NACA ACR No. 3029, 1943. 9..~LLEN, H. JULIAN: Notes on the Efieet of Surface Distortion on the Drag and Critical Mach Number of Airfoils. NACA ACR No. 3129, 1943. 10. ANDERSON, RAYMOND F.: Determination of the Characteristics of Tapered Wings. NACA Rept. No. 572, 1936. 12. BAIBSTOW, L.: Skin Friction. J. Roy. Aeronaut. Soe., January, 1925, pp. 3-23. 13. BAMBER. ~fILLABD J.: Wmd-tunnel Tests on Airfoil Boundary Layer Control Using a Backward-opening Slot. NACA Rept. No. 385, 1931. 14. BLASIUS, H.: Grenzschichten in FlOssigkeiten mit kleiner Reibung. Z. J.fath. PhyBik, vol 66, pp. 1-37, 1908. 15. BL.~SIUS, H.: Ver. deut. lng. ForachungSMft 131, 1911. 16. BOSR.Ul, JOHN: The Determination of Span Load Distribution at High Speeds by the Use of High-speed Wind Tunnel Section Data. NACA. ACR No. 4B22, 1944 (Wartime Rept. No. L-436). 17. BRASLOW, .ALBERT L.: Investigation of Effects of Various Camouflage Paints and Painting Procedures on the Drag Characteristics of an NAC.A 65(.al)-420 a = 1.0 Airfoil Section. NACA CB No. UG17, 1944 (Wartime Rept. No. L-141). 18. BROWN, Cu~nroN E.: Theoretical Lift and Drag of Thin Triangular Wings at Supersonic Speeds. N.ACA TN No. 1183, 1946. 19. BURGERS~ J. 1.1.: The Motion of a Fluid in the Boundary Layer along a Plane Smooth Surface. Proc. First Intern. Congr. AppL. Mech., Delft, 1924, pp. 113-128. 20. CAHILL, JONES F.: Two-dimensional Wind-tunnel Investigation of Four Types of High-lift Flap on an NACA 65-210 Airfoil Section. NACA TN No. 1191, 1947. 21. CAHILL, JONES F., and RACIsz, STANLEY: Wmd-tunnel Development of Optimum Double-elotted-flap Configurations for Seven Thin NAC.o\. Airfoil Sections. NACA RM No. L7B17, 1947, also TN No. 1545. 300

REFERENCES 22.

301

F.: Aerodynamic Data for a Wing Section of the Republic XF-12 Airplane Equipped with a Double Slotted Flap. NACA MR No. L6AOSa, 1946 (Wartime Rept. No. L-544). 23. CAHILL, JONES F.: Summary of Section Data on Trailing-edge High-lift Devices. NACA RM No. LSD09, 1948. 25. CHAPLYGIN, SERGEI: Gas Jets. NACA TM No. 1063, 1944 (from Scientific Mem­ oirs, Moscow University, 1902, pp. 1-121). 26. CIlABTEBS, ALEX C.: Transition between Laminar and Turbulent Flow by Trans­ verse Contamination. NACA TN No. 891, 1943. 27. DRYDEN, H. L., and KUETHE, A. l\{.: Effect of Turbulence in Wind Tunnell\feasure­ menta. NACA Rept. No. 342, 1929. 28. DURAND, W. F.: Aerodynamic Theory, vol. 3, Div. G. "The Mechanics of Viscous Fluids" by L. Prandtl. Verlag Julius Springer, Berlin, or Durand Reprinting Comm., Cal. Inst. of Tech. 29. DURAND, W.F.: Aerodynamic Theory, vol. 3, Div. H. "The Mechanics of Com­ pressible Fluids" by G. I. Taylor and J. W. Maceoll, Verlag Julius Springer, Berlin, or Durand Reprinting Comm., Cal. lost. of Tech. 30. FERRI, ANTONIO: Completed Tabulation in the United States of Tests of 24 Airfoils at High Mach Numbers. (Derived from interrupted work at Guidonia, Italy, in the 1.31- by 1.74-foot High-speed Tunnel) N.ACA ACR No. LSE21, 1945 (War­ time Rept. No. L-143). 31. FISCHEL, JACK, and RIEBE, JOHS 1\1.: "~'ind-tunnel Investigation of a NACA 23021 Airfoil with a O.32-airfoil-chord Double Slotted Flap. N ACA ARR No. IAJ05, 1944 ("·artime Rept. No. L-7). 32. FULLMER, FELlCIEN F., Ja.: Two-dimensional "-:-ind-tunnel Investigation of the NACA 641-012 Airfoil Equipped with Two Types of Leading-edge Flap. NACA TN No. 1277, 1947. 33. FURLONG, G. CHESTER, and FrrzPATRlCK, J.~:MES E.: Effects of Mach Number and Reynolds Number on the Maximum Lift Coefficient of a Wing of NACA. 230­ series Airfoil Sections.. NACA l\IR No. L6F04, 1946, also TN No. 1299. 34. GARRICK, I. E., and KAPLAN, CARL: On the Flow of a Compressible Fluid by the Hodograph ?\Iethod. I. Unification and Extension of Present-day Results. N..4.CA ACR No. 1AC24. 1944, also Rept. No. 789. 35. GLAUERT, H.: Theoretical Relationships for an Aerofoil with Hinged Flap. R. &. AI. No. 1095, British ARC, 1927. 36. GLAUERT,H.: The Effect of Compressibility on the Lift of an Aerofoil. R. & M. No. 1135, British ARC, 1927. 37. GLAUERT, H.: "'The Elements of Aerofoil and Airscrew Theory," Cambridge University Press, London, 1926. 374. GOLDSTEIN, SIDNEY: Low-drag and Suction Airfoils, Eleventh \Vright Brothers Lecture, J. Imt. Aeronaut.. Sei., '·01. 15, 1\0. 4, pp. 189-214, April, 1948. 37b. GRAHAM, DONALD J.: High-speed Tests of an Airfoil Section Cambered to Have Criticall\fach Xumbers Higher than Those A.tt:lmnble with a Uniform-load Mean Line. NAC.~ TX No. 1396, 1947. 38. GJlUSCHWrrz. E.: Die turbulente Reibungsschicht in ebener Stromung bei Druckab­ fall und Druckanstieg, Ing. Arehi», Bd. 11. Heft 3, PP. 321-346, September, 1931. 39. H ..\ BRIS, THOMAS A.: .'\\"ind-tunnel Investigation of an K.t\CA Airfoil with Two Arrangements of a Wid~hord SIQtted Flap. N.ACA TN No. 715, 1939. 40. HARRIS, TIIOM.~S .~., and RECAST, ISIDORE G.: Wind-tunnel Investigation of NACA. 23012, 23021, and 23030 Airfoils Equipped with 4Q-percenkhord Double Slotted Flaps. !\.-\CA Rept. No. 723. 1941. CAHILL, JONES

302

THEORY OF WING SECTIONS

41. HARRIS, "rHOKAS A., and LoWRY, JOHN G.: Pressure Distribution over an NACA 23012 Airfoil with a Fixed Slot and a Slotted Flap. NACA Rept. No. 732, 1942. 42. HOOD, MANLEY J.: The Effects of Some Common Surface Irregularities on Wing Drag. NACA TN No. 695, 1939. 43. HOOD, MANLEY J., and GAYDOS, M. EDWARD: Effects of Propellers and of Vibra­ tion on the Extent of Laminar Flow on the NACA 27-212 Airfoil. NACA ACR, October 1939, (V-lartime Rept. No. L-784). 44. JACOBS, EASTMAN N.: Methods Employed in America for the Experimental Investi­ gation of Aerodynamic Phenomena at High Speeds. NACA Misc. Paper No. 42, 1936. Paper presented at Volta meeting in Italy, Sept. 30 to Oct. 6, 1935. 45. JACOBS, EASTMAN N.: Preliminary Report on Laminar Flow Airfoils and New Methods Adopted for Airfoil and Boundary-layer Investigations. NACA ACR, June, 1939, (Wartime Rept. No. L-345). 400. JACOBS, EASTMAN N., and ABBOTT, IRA H.: Airfoil Section Data Obtained in the N.A.CA Variable-density Tunnel as Affected by Support Interference and Other Corrections. NAC.~ Rept. No. 669, 1939. 46. JACOBS, EASTMAN N., and PINKERTON, RoBERT M.: Tests in the Variable-density Wind Tunnel of Related Airfoils Having the Maximum Camber Unusually Far Forward. NACA Rept. No. 537, 1935. 47. JACOBS, EAsTMAN N., and PINKERTON, RoBERT M.: PressureDistribution over a Symmetrical Airfoil Section with Trailing Edge Flap. NACA Rept. No. 360, 1930. 48. JACOBS, EASTMAN Nc, PINKERTON, ROBERT M., and GREENBERG, HARRY: Tests of Related Forward-camber Airfoils in the Variable-density \Vind Tunnel. NACA Rept. No. 610, 1937. 49. JACOBS, EAsTMAN N., WARD, ]{ENNETH E., and !)INKERTON, RoBERT ~f.: The Characteristics of 78 Related Airfoil Sections from Tests in the Variable-density Wind Tunnel. NACA Rept. No. 460,1932. 50. JONES, RoBERT T.: Correction of the Lifting-line 1"heory for the Effect of the Chord. NACA TN No. 617, 1941. 51. JONES, RoBERT T.: Properties of Low-aspect Ratio Pointed \Vinga at Speeds below and above the Speed of Sound. NACA TN No. 1032, 1946. 52. JONES, RoBERT T.: Wing Plan Forms for High-speed Flight. NACA TN No. 1033, 1946. 53. JONES, RoBERT T.: Thin Oblique Airfoils at Supersonic Speed. NACA TN No. 1107, 1946. 54. KAPLAN, CARL: Effect of Compressibility at High Subsonic Velocities on the Lifting Force Acting on an Elliptic Cylinder. NACA TN No. 1118, 1946. 55. KEENAN, JOSEPH H., and NEUMANN, ERNEST P.: Friction in Pipes at Supersonic and Subsonic Velocities. NACA TN No. 963, 1945. 56. KENNARD, EARLE H.: "Kinetic Theory of Gases," l\lcGraw-Hill Book Company, Inc., New York, 1938. 57. KNIGHT, MONTGOMERY, and BAMBER, MILLARD J.: Wind Tunnel Tests on Airfoil Boundary Layer Control Using a Backward Opening Slot. NACA TN No. 323­ 1929. 58. KOSTER, H.: Messungen an Profile OJ)() 12-0, 5545 mit Spreitz- und Nasanspreiz­ klappe, Deutsche VerSuchsanstalt fm Luftfahrt, UM 1317, July 29, 1944. 59. KaeGER, W.: Systematische Windkanalmessungen an einem LaminarftUgeI mit Nasanklappe, Aerodynamische Versucbsanstalt GOttingen, Fb Nr. 1948, June 13, 1944. 60. KROGER, W.: Wind-tunnel Investigation on a Changed Mustang Profile with

REFERENCES Nose Flap. 1177,1947.

Force and Pressure Distribution Measurements.

303 NACA. Tl\1 No,

61. LAMB, HORACE: "Hydrodynamica," Cambridge University Press, London, 1932. 62. LEES, LESTER: The Stability of a Laminar Boundary Layer in a Compressible Fluid. NACA TN No. 1115, 1946. 63. LEMME, H. G.: Kraftmessungen und Drukverteilungmessungen an einem Fliigel mit Knicknase, VorflOgel, Wolbungs- und Spreizklappe, Aerodynamische Ver­ suchsanstalt GOttingen, Fb. Nr. 1676, Oct. 15, 1942. 64. LEMME, H. G.: Kraftmessungen und Druckverteilungmessungen an einem Rech­ teckfliigel mit Spaltknicknase, Wolbungs- und Spreizklappe oder Rollklappe, Aerodynamische Versuchsanstalt Gottingen, Fb. Nr. 1676/2, l\fay 14, 1943 (also NACA Tl\{ No. 1108, 1947). 65. LEMME, H. G.: Kraftmessungen und Druckverteilungmessungen an einem Rech­ teckftiigel mit Doppelknicknase, Aerodynamische Versuchsanstalt Gottingen, Fb. Nr. 1676/3, June 2, 1944. 66. LIEPMAN, HANS WOLFG~"G: The Interaction between Boundary Layer and Shock Waves in Transonic Flow, J. Aeronaut. Sci., vol. 13, No. 12, pp.623-637, Decem­ ber,I946. 6&. LIN, C. C.: On the Stability of Two-dimensional Parallel Flows, Part I, QU4rt. Appl. jfath., vol 3, No.2, pp. 117-142, July, 1945; Part II, voL 3, No.3, pp, 218-234, October 1945; and Part III, vol. 3, No.4, PP. 277-301, Januar.)", 1946. 67. LIPPISCH, A., and BEUSCHAUSEN, 'V.: Pressure Distribution Measurements at High Speed and Oblique Incidence of Flo,,·. NACA Tl\1 No. 1115, 1947. 68. LoFTIN, LAURENCE K., JR.: Effects of Specific Types of Surface Roughness on Boundary-layer Transition. NACA. ~\CR L5J~ 1945 (Wartime Rept. No. L-48). 6&. LoFTIN, LAURENCE K., JR., and COllEN, I{ENNETR B.: Aerodynamic Characteristics of a Number of Modified NACA Four-digit-series Airfoil Sections. NACA RM No. L7122, 1947. 69. LoWRY, JOHN G.: "rind-tunnel Investigation of an NACA 23012 Airfoil with Several Arrangements of Slotted Flap with Extended Lips. NACA TN No. 808, 1941. 70. MUNK, l\IAX !\'1.: The Determination of the Angles of Attack of Zero Lift and Zero Moment, Based on Munk's Integrals. NACA TN No. 122, 1923. 71. l\IUNK, !\{AX M.: Elements of the \Ving Section Theory and of the Wing Theory. NACA Rept. No. 191, 1924. 72. :l\{UNJt, Max 1\1.: Calculation of Span Lift Distribution (Part 2), Aero Digest, vol. 48, No.3, p. 84, Feb. 1, 1945. 73. NAlMA.N, IRVEN: Numerical Evaluation of the e-Integral Occurring in the Theo­ dorsen Arbitrary Airfoil Potential Theory. NACA ARR No. lAD27n., April, 1944 (Wartime Rept. No. L-136). 73a. NAD1AN, IRVEN: Numerical Evaluation by Harmonic Analysis of the e-Funetion of the Theodorsen Arbitrary Potential Theory. NACA ARR No. LSRI8, Septem­ ber, 1945 (Wartime Rept. No. L-153). 14. NEELY, RoBERT H., BOLLECH, THOMAS V., WESTRICK, GERTRUDE C., and GRAHAM, RoBERT R.: Experimental and Calculated Characteristics of Several NACA 44­ series Wings with Aspect Ratios of 8, 10, and 12, and Taper Ratios of 2.5 and 3.5. NACA TN No. 1270, 1947. 75. NIItURADSE, J.: Gesetzmissigkeiten der turbulenten Stromung ill glatten Rohren, ForBCh. Gebiete Ingenieunc., 1932, FOfSChungwjt 356. 700. NIItURADSE, J.: VOrtrige aus dem Gebiete der Aerodynamik, p, 63, Aachen, 1929. 76. NITZBERG, GERALD E.: A Concise Theoretical Method for Profile-drag Calculation. NACA ACR No. 4005,1944.

304

THEORY OF WING SECTIONS

77. NOYES, RICHARD W.: Wind-tunnel Tests of a Wing with a Trailing-edge Auxiliary Airfoil Used as a Flap. NACA TN No. 524, 1935. 78. PANKHURST, R. C.: A Method for the Rapid Evaluation of Glauert's Expressions for the Angle of Zero Lift and the Moment at Zero Lift. R. &; 1\1. No. 1914, British ARC, 1944. 79. PEARSON, E. 0., JR., EVANS, A. J., and 'VEST, F. E.: Effects of Compressibility on the Maximum Lift Characteristics, and Spanwise Load Distribution of a 12-foot­ span Fighter-type Wing of NACA 23O-series Airfoil Sections. NACA ACR No. L5GI0, 1945 (\Y'artime Rept, No. L-51). 80. PEIRCE, B. 0.: U A Short Table of Integrals," Ginn &: Company, Boston, 1929. 80a. PIERCE, II.: On the Dynamic Response of Airplane Wings Due to Gusts. NACA TN No. 1320, 1947. 81. PINKERTON, RoBERT ~'I.: Analytical Determination of the Load on a Trailing Edge Flap. NAC.A TN No. 353, 1930. 82. PINKERTON, ROBERT 1\1.: Calculated and Measured Pressure Distributions over the Midspan Section of the NA.C..A\ 4412 Airfoil. NACA Rept. No. 563, 1936. 83. PLATT, ROBERT C.: Aerodynamic Characteristics of Wings with Cambered Ex­ ternal-airfoil Flaps, I neluding Lateral Control with a Full-span Flap. N ACA ltept. No. 541, 1935. 84:. PLATr, ROBERT C., and ASBOTr, IRA H.: Aerodynamic Characteristics of NACA 23012 and 23021 Airfoils with 2O-percent-chord Extemal-airfoil Flaps of NACA 23012 Section. NACA Rept, No. 573, 1936. 85. PLATT, H.OBERT e., and SHORTAL, JOSEPH A.: Wind-tunnel Investigation of "rings with Ordinary Ailerons and Full-span External Airfoil Flaps. NACA Rept, No. 603, 1937. 86. POHLIIAUSEN, 1\:.: Zur niiherungsweisen Integration der Differentialglcichung der laminaren 'Grenzschicht, Abhandl. aero. Inst. Aachen, 1. Lieferung, 1921. See also z. angelO. ,,"tath. !alech., Btl. 1, Heft 4, pp. 252-258, 1921. 87. PRA.~DTL, 1.1.: Motion of Fluids with Very Little Viscosity. NACA TN No. 452, 1928 (originally presented to the Third International Mathematical Congress, Heidelberg, 19(4). 88. PRANDTL, L.: Applications of Modern Hydrodynamics to Aeronautics. NACA Rept. No. 116, 1921. 89. PRANDTL, L. t and TIETJENS, O. G.: "Applied Hydro-and Aeromechanics,"l\{cGraw­ Hill Book Company, Inc., New York, 1934. 90. PUCKETT, Allen E.: Supersonic Wave Drag of Thin Airfoils, J. Aeronaut. Sci., vol. 13, No.9, pp. 475-484, September, 1946. 91. PURSER, PAUL E., FISCHJ<;L, JACK, and RIEBE, JOHN l.f.: Wind-tunnel Investigation of an N.ACA 23012 Airfoil with a O.30-airfoil-chord Double Slotted Flap, NACA ARR No. 3LI0, 1943 (Wartime Rept. No. L-469). 92. PURSER, PAUL E., and JOHNSON, IIAROLD S.: Effects of Trailing-edge Modifications on Pitching-moment Characteristics of Airfoils. NACA cn No. UI30, 1944 (Wartime Rept. No. L-664). 93. QUINN, JOHN II., JR.: Tests of the NACA. 65,-018 Airfoil Section with Boundary­ layer Control by Suction. NACA CB No. IAHIO, 1944 (Wartime Rept. No. L-2(9). 94. Qt:IN~, JOHN H., JR.: Wind-tunnel Investigation of Boundary-layer Control by Suction on the ~.A.CA 65,-418, a = 1.0 Airfoil Section with a 0.29 Airfoil-chord Double Slotted Flap. NACA TN No. 1071, 1946. 95. ,QUINN, JOHN H .• JR.: Tests of the NACA 641A212 Airfoil Section with a Slat, a Double Slotted Flap, and Boundary-layer Control by Suction. NACA TN No. 1293, 1947.

REFERENCES

305

96. RECANT, I. G.: Wind-tunnel Investigation of an NACA 23030 Airfoil with Various Arrangements of Slotted Flap. NACA TN No. 755, 1940. 97. REID, E. G., and BAMBER, M. J.: Preliminary Investigation on Boundary Layer Control by Means of Suction and Pressure with the U.S.A. 27 Airfoil. NACA TN No. 286, 1928. 98. ScHILLER, L.: Die Entwicklung der laminaren Geschwindigkeitsverteilung und ihr Bedeutung fUr Zihigkeitsmessungen, Z. angew. ltfath. ~'/ech., Bd. 2, Heft 2, pp. 96-106, 1922. See also ForBCh. Gebiete lngenieurw., Heft 248, Aachen III, 3, 1922. 99. ScHLICHTING, H.: Amplitudenverteilung und Energiebilanz der kleinen StOrungen bei der Plattenstromung, Nachr. Ga. Wi8s. GiiUingen, Moth, physik. Klaeee, Bd. 1, 1935. 100 ScHLICHTING, H.: Zur Entstehung der Turbulenz bei der Plattenstromung, Nachr. Ge«: Wiaa. Gottingen, Alath. physik. Klasse, 1933, pp. 181-209. 101. ScHLICHTING, H., and ULRICH, A.: Zur Berechnung des Umschlages Laminar Turbulent, Jahrbuch 1942 der deutschen Luftfahrtforschung, pp. 18-135, R. Olden­ bourg, Munich. 102. ScHRENK, OSKAR: Experiments with a \Ving Model from Which the Boundary Layer Is Removed by Suction. NACA T~I No. 534, 1929. 103. ScHRENK, OSKAR: Experiments with a Wing from \Vhich the Boundary Layer Is Removed by Suction. NACA Tl\1 No. 634, 1931. 104. SCHRENK, OSKAR: Experiments with Suction-type Wings, NACA TM No. 773, 1935. 105. ScHUBAUR, G. B.: Air Flow in the Boundary Layer of an Elliptical Cylinder. NACA Rept. No. 652, 1939. 106. ScHUBAUR, G. B., and SKRAMSTAD, If. 1\:.: Laminar-boundury-layer Oscillations and Transition on a Flat Plate. N.AC.A ACIt, April, 1943 (\Vartinle Rept, No.8). 107. ScIIULDENFllEI, l\{ARVIN J.: \Vind-tunnel Investigation of an NACA 23012 Air­ foil with a Handley Page Slot and Two Flap Arrangements. NAC1\ ARR, Febru­ ary, 1942 (Wartime Rept. No. L-261). lOS. SHERMAN, ALBERT, and HARRIS, ''1''. A.: The Effects of Equal Pressure Fixed Slots on the Characteristics of a Clark-Y Airfoil. :KACA rrN No. 507, 1934. 109. SHElWAN, ALBERT: A Simple l\lcthod of Obtaining Span Load Distribution. NACA TN No. 732, 1939. 109a. SILVERSTEIN, ABE, KATZOFF, SAMUEL, and lIooTMAN, JAMKS: Comparative Flight ar.J Full-scale Wind-tunnel Meusureruents of the :\In.xinluDl Lift of an Airplane. NACA Rept. Ko. 618, 1938. 110. SIVELLS, JAMES C.: Experimental Characteristics of Three \Yinb"S of NACA 64-210 and 65-210 Airfoil Sections with and without 2° \Yashout. !\.~Ci\ 'l"X No. 1422, 1947. Ill. SIVELLS, JAMES C., and KEELY, ROBERT II.: Method of Calculating Wing Char­ acteristics by Lifting-line Theory Using Xonlinear Section Lift Data, K.ACA 1'N No. 1269, 1947. 112. SQUIRE, H. B., and YOUNG, A. D.: The Calculation of the Profile Drag of Aero­ foils. R. & l\{. No. 1838, British AIlC, 1938. 113. STACK, JOHN: Tests of Airfoils Designed to Delay the Compressibility Burble. NACA TN No. 976, 1944. 114. STACK, JOHN: Compressible Flow's in Aeronautics, Eighth Wright Brothers Lec­ ture, J. Aeronaut. s«; vol. 12, No.2, April, 1945. 115. STACK, JOHN, LINDSAY, \V. F., and LI'M'ELL, ROBI-.:RT E.: The Compressibility Burble and the Effects of Compressibility on Pressures and Forces Acting on an Airfoil. NACA Rept. No. 646, .1938.

306

THEORY OF WING SECTIONS

116. STACK, JOHN, and VON DOENHOFF, ALBERT E.: Tests of 16 Related Airfoils at High Speeds. NACA Rept. No. 492, 1934. 117. STEWART, H. J.: Lift of a Delta. Wing at Supersonic Speeds, Quart. Appl. Math., vol. IV, No.3, pp. 246-254, October, 1946. 118. TAm, ITIBO: A Simple Method. of Calculating the Induced Velocity of a Mono­ plane Wing, Aeronaut. Research Inst. Tokyo Imp. Uniu. &pt. 111, vol. IX, p. 3, August, 1934­ 119. TEMPLE, G., and YARWOOD, J.: The Approximate Solution of the Hodograph Equations for Compressible Flow. Rept. No. S.M.E. 3201, RAE, June, 1942. 120. TETERVIN, NEAL: A Method. for the Rapid Estimation of Turbulent Boundary­ layer Thick~ess for Calculating Profile Drag. NACA ACR No. UGl4, July, 1944 (Wartime Rept. No. L-16). 121. THEODORSEN, THEODORE: On the Theory of Wing Sections with Particular Refer­ ence to the Lift Distribution. NACA Rept. No. 383, 1931. 122. THEODORSEN, THEODORE: Theory of Wing Sections of Arbitrary Shape. N.ACA Rept. No. 411, 1931. 123. TuEODORSEN, THEODORE: Airfoil Contour Modification Based 011 e-Curve Method of Calculating Pressure Distribution. NACA ARR No. IAG05, 1944 (Wartime Rept. No. L-135). 124. THEODORSEN, THEODORE: ..~ Condition on the Initial Shock. NACA TN No. 1029, 1946. 125. THEODORSEN, THEODORE, and GARRICK, 1. E.: General Potential Theory of Arbi­ trary 'Ving Sections. NACA Rept. No. 452, 1933. 126. THEODORSEN, THEODORE, and REGIER, ARTHUR: Experiments on Drag of Re­ volving Disks, Cylinders and Streamline Rods at High Speeds. NACA .A.CR No. IAFI6, 1944 (Rept. No. 793). 127. TOLLMIEN, "rIO: The Production of Turbulence. NACA T1"1 No. 609, 1931. 128. TOLLMIEN, W.: General Instability Criterion of Laminar Velocity Distributions, NACA T!\l No. 792, 1936. 129. TSIEN, Hsue-Shen: Two-dimensional Subsonic Flow of Compressible Fluids, J. Aeronaut. s«, vol. 6, No. 10, pp. 399-407, August, 1939. 132. VON DoENHOFF, ALBERT E.: A Method of Rapidly Estimating the Position of the Laminar Separation Point. NACA TN No. 671, 1938. 133. VON DOENHOFF, ALBERT E., and ABBOTr, FRANK T., Ja.: The Langley Two­ dimensional Low.- turbulence Pressure Tunnel. NACA TN No. 1283, 1947. 134. VON DoENHOFFJ ALBERT E., and TETERVIN, NEAL: Investigation of the "aria­ tion of Lift Coefficient with Reynolds Number at a Moderate Angle of Attack on a Low-drag Airfoil. NACA CB, 1942 (Wartime Rept. No. L-661). 135. VON DOENHOFF, A. E., and TETEBVIN, NEAL: Determination of General Relations for the Behavior of Turbulent Boundary Layers. NACA ACR No. 3G13, 1943 (Rept. No. 772). 136. VON KARMAN, T.: tJber laminare und turbulente Reibung, Z. angew. Alath. It/ech., Bd. 1, Heft 4. pp. 233-252, 1921. 137. VON KAmIAN, T.: Gastheoretische Deutung der Reynoldssehen Kennzahl, Abhandl. Aero. Imt. Aachen, Heft 4, 1925, Verlag Julius Springer, Berlin. 138.. VON K.AB.MAN, T.: Mechanical Similitude and Turbulence. NACA Tl\{ No. 611, 1931. 139. VON KA.mUN, T.: Turbulence and Skin Friction, J. Aeronaut. Sci., vol. 1, No.1, pp. 1-20, January, 1934. 140. VON K!RM:!N, T.: Compressibility Effects in Aerodynamics, J. Aeronaut. Sci., vol. 8, No.9, pp, 337-356, July, 1941.

REFERENCES

307

141. VON KAmfIN, T., and MILLIKAN, C. B.: On the Theory of Laminar Boundary Layers Involving Separation. NACA Rept. No. 504, 1934. 142. VON KAmfAN, T., and TSIEN, H. B.: Boundary Layer in Compressible Fluids, J. Aeronaut. s«, vol. 5, No.6, April, 1938. 143. WARnELD, CALVIN N. for the NACA Special Subcommittee on the Upper Atmosphere: Tentative Tables for the Upper Atmosphere. NACA TN No. 1200, 1947. 144. WEICK, FRED E., and ILuuus, THOMAS A.: The Aerodynamic Characteristics of a Model Wing Having a Split Flap Deflected Downward and Moved to the Rear. NACA TN No. 422,1932. 145. WEICK, FRED E., and PLATr, RoBERT C.: Wind-tunnel Tests on a Model Wing with Fowler Flap and Specially Developed Leading-edge Slot. NACA TN No. 459, 1933. 146. WEICK, FRED E., and SANDERS, ROBERT: Wind-tunnel Tests of a Wing with Fixed Auxiliary Airfoils Having Various Chords and Profiles. NACA. Rept. No. 472, 1933. 147. 'WEICK, FRED E., and SHORTAL, JOSEPH 1\.: The Effect of Multiple Fixed Slots and a Trailing-edge Flap on the Lift and Drag of a Clark Y.Airfoil. NACA Rept. No. 427, 1932. 148. WENZINGER, CARL J.: Wind-tunnel Investigation of Ordinary and Split Flaps on Airfoils of Different Profile. NACA Rept. No. 554, 1936. 149. WENZINGER, CARL J.: Pressure Distribution over an Airfoil Section with a Flap and Tab. NAC.A Rept. No. 574, 1936. 150. WENZINGER, CARL J.: "rind-tunnel Tests of a Clark Y \Ving Having Split Flaps with Gaps. NACA TN No. 650, 1938. 151. WENZINGER, CARL J.: Pressure Distribution over an NAC.A 23012 Airfoil with an NACA 23012 External-airfoil Flap. NAC.A Rept. No. 614, 1938. 152. WENZINGER, CARL J., and DELANO, JAMES B.: Pressure Distribution over an NACA 23012 Airfoil with a Slotted and a Plain Flap. NACA Rept. No. 633, 1938. 153. 'VENZINGEB, CAllL J., and HARRIS, THOMAS A.: Wind-tunnel Investigation of an NACA. 23012 Airfoil with Various Arrangements of Slotted Flaps. NACA. Rept. No. 664,1939. 154. WENZINOER, CARLJ., and HAlUlIS, THOMAS l\.: Wind-tunnel Investigation of NACA 23012, 23021, and 23030 Airfoils with Various Sizes of Split Flap. NACA Rept. No. 668, 1939. 155. WEN ZINGER, CARL J., and H.O\RRIS, THOMAS A.: Wind-tunnel Investigation of an NACA 23021 Airfoil with Various Arrangements of Slotted Flaps. NAC..~ Rept. No. 677, 1939. 156. WENZINGER, CARL J., and RoGALLO, FRANCIS M.: Resume of Airload Data on Slats and Flaps. NACA TN No. 690, 1939. 1564. WRAGG, C. A.: Wragg Compound Aerofoil. U. S. Air Service, March, 1921, PP4 32-34. 157. YOUNG, A. D.: Note on the Effect of Slipstream on Boundary Layer Flow. Rept. No. B.A. 1404. British R.~E, May, 1937. 158. YOUNG, A. D.: Further Note on the Effect of Slipstream on Boundary Layer Flow. Rept. No. B.A. 1404a, British RAE, October, 1938. 159. YOUNG, A. D., and ~{ORJUS, D. E.: Further Note on the Eff~t of Slipstream on Boundary Layer Flow. Rept. No. B.A. 1404h, British RAE, September, 1939. 160. ZALOVICIt, JOHN A.: Profile Drag Coefficients of Conventional and Low-drag Airfoils as Obtained in Flight. NACA ACR No. 1AE31, 1944 (Wartime Rept. No. L-139).

APPENDIXES APPENDIX I

BASIC THICKNESS FORMS CoNTEN)'S

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16-021 .

333

63-006 .

334

63-009 .

335

63-010 . 63r.Q12. 63r015. 63r018. 6&-021.

336

337

338

339

340

341

63AOO6

342

63A008 63AOIO 631A012

344

~A015

345

343

346

64-006 . 309

THEORY 0// WING SECTIONS

310

Page 347 348

N ACA Designation

64-008 .

M-009 . . . . · . 64-010 . 641-012. 64:r01S . . • · • • • • . . 64:r018. . . . . . . . .

64.-021. .

349

350 351 352 353

. ....

64AOO6 64AOOS

354

65-009 . .

355 356 357 358 359 360 361

66-010 . 65t-012.

363

65r015. .

3M

MAOI0 64tAOI2 MsA015 .

66-006 . . 65-008 . .

382

65r018. .

65A21..

365

. .....

366

65AOO6

867 368

65AOO8 65AOIO 65 1A012 .

369

370 371

65tA015 .

66-006 . 66-008 . . 66-009 66-010 . . ~~12. . . . . . . .

66r015 . . 66r018. .

372

.

373 374 375

376

66r021 ..

377 378 379

67,1-015 . 747A015 .

380 381

APPENDIX I

311

I••

,_ -----......

~

r--- '---

r---

h

.8

0006

-MICA

\

-

.4 .",.-

~

D

.4

I

x

I

y

(per cent c) __ (pe __ r _OO_Il_t_c}_.

o

o

I

(t' /1')2

.

1.0

.8

.6

%C

1'/l'

0.5 1.25 2.5 5.0

..... 0.947 1.307 1.777

0 0.880 1.117 1.186 1.217

0 0.938

7.5

2.100 2.341 2.673 2.869 2.971

,_._

I

.1J.'./ V

\----1

3.992 2.015

1.057

1.364

1.089 1.103

0.984 0.696

1.225 1.212 1.206 1.190 1. t 79

1.107 1.101 1.098 1.091 1.086

0.562 0.478 0.378 0.316 0.272

50 60 70

3.001 2.902 2.647 2.282 1.832

1.162 1.136 1.109 1.086 1.057

1.078 1.066 1.053 1.042 1.028

0.239 0.189 0.152 0.123 O.(l97

80

1.312

1.026

1.013

0.073

~:~

;::~

10 15

20 25 30

40

~. 90 95 100

__.. _ ;:~_l_J: I~.E.

radius: 0.40 per cent c

7'HEORY OF lVING SECTIONS

312 1.6

1.2

r:

I

-- t---

100....

---.. r---.

r--- r---. . . .

.8 /MCA OOOB .4 L,-.---

I'---

-

-",

L.--- ~

o

.4,

.8

.6

1.0

S,C

z

11

(V/V)t

V/V

At'./V

1.263 1.743 2.369

0 0.792 1.103 1.221 1.272

0 0.890 1.050 1.105 1.128

2.900 1.795 1.310 0.971 0.694

7.5 10 15 20 25

2.800 3.121 3.564 3.825 3.961

1.284 1.717 1.272 1.259 1.241

1.133 1.130 1.128 1.122 1.114

0.561 0.479 0.379 0.318 0.273

30

4.001 3.869 3.529 3.043 2.443

1.223 1.186 1.149 1.111 1.080

1.106 1.089 1.072 1.054 1.039

0.239 0.188 0.152 0.121 0.096

1.749 0.965 0.537 0.084

1.034 0.968 0.939

1.017 0.984 0.969

0.071 0.047 0.031 0

(per cent c) (per cent c)

0 0.5 1.25 2.5 5.0

40

50 60

70

80 90

95 100

-_

.._---.

0

.....

.....

. ....

L.E. radius: 0.70 per cent e NACA 0008 Basic Thieknees Form

313

.APPENDIX I I. 6

2/_

----

I.

---..,~

J

"""--

(~)' o.

1\,

I

o.

~

i

,.---

I

,

'---

o

o

,

0.2

0.4

I

~ ~

f 0.6

08

a

1.0

C

% 11 (per cent c) (per cent c)

(v/V)!

,,/1"

~'./V

2..089

0 0.917 1.023 1.092 1.137

0 0.958 1.011 1.045 1.066

4.839 1.338 0.966 0.691 0.564

1.162 1.188 1.206 1.217 1.202

1.078 1.090 1.098 1.103

40

2.436 2.996 3.396 3.867 4.000

0.485 0.387 0.326 0.248 0.197

50 60 70 80 90

3..884 3.547 2.987 2.213 1.244

1.185 1.163 1.127 1.067 0.993

95

0.684: 0.080

0.932 0

_.­

0 1.25

2.5 5.0 7.5 10 15 20

30

100

0 0.756 1.120 1.662

1.096

1.089

1..079 t

L062 1.033 0.996 0.965

0

L.E. radius: 0.174 per cent c NACA 0008-34 Basic Thickness Form

0.157 0.128 0.100 0.074 0.047 0.031 0

THEORY OF "K'ING BECTION.r;

314

I.'./I

_r _....

r--­ ~

-,

I.

-r----. ---......

r----. r--......

'\

--

\

IMCA 0009

1

..­ ~

~ ......

< :,

II

%

Y

(peI: cent c) (per cent c) 0 0.5 1.25 2.5 5.0

0

.....

1.420 1.961 2.666

.8

.6

%,C

(V/V)2

tJ/V

lJJJ./V

0 0.750 1.083 1.229 1.299

0 0.866 1.041 1.109 1.140

0.595 1.700 1.283 0.963 0.692 0.560 0.479 0.380 0.318 0.273

i

7.5 10 15 20 25

3.150 3.512 4.009 4.303 4.456

1.310 1.309 1.304 1.275

1.145 1.144 1.142 1.137 1.129

30

4.501 4..352 3.971 3.423 2.748

1.252 1.209 1.170 1.126 1.087

1.119 1.100 1.082 1.061 1.043

0.239 0.188 0.151 0.120 0.095

1.967 1.086 0.605 0.095

1.037 0.984 0.933 0

1.018 0.982 0.966 0

0.070 0.046 0.030 0

40 50

60 70 80

90 95 100

~~,

..

I...E. radius: 0.89 per cent r. NACA 0009 Basic Thieknees Form

_---­

315

APPENDIX I 2.D

.­

1.2

(v)'

rf ----

:---.­

,

r---­

r--...

---... r---­

~ .......

.8 NACA 0010

--"\ 1

- ---­ -­

~~

'--.. "--­

.4, ste X

i

Y

I I

0 0.5 1.25 2.5 5.0

0

0

.....

0.712 1..061 1.237 1.325

0.844 1.030 1.112 1.151

2.962

1.341 1.341 1.341 1.329 1.309

3..500

3.902 4.455 4.782 4.952

25

~D./lr

0

1.578 2.178

7.5 10 15 20

l?/V'

(r,/V)2

(per cent c) (per eeut c)

1.0

.8

.6

~

2.372 1.618 1.255 0.955 0.690

0.. 559 0.479 iO.380 0.318 0.273

1..158

1.158 1.158 1.153 1.144 I"

30 40

5.002 4.837 4.412 3.803 3.053

50 60 70 80

90 95 100

I I

2.187 1.207 0.672 0.105

1.284 1~237

1.190 LI38 1.094 I

!

II

I

1.040

0.960 0.925

.....

f

I I t

l

I

1.133 1..112 1.091 1.067 1.046 1.020 0.980 0.962

. ....

L.E. radius: 1.10 per cent c NACA 0010 Basic Thickness Form

0.239

0.188 0.150 0..119 0.094

0.069

0.045-'

I

0=1

0.030

THEORY OF WING SECTIONS

316

i6

L2

-

~

f

""----­

............

~

(vY.8 ~

r-,

,

['\ NACA 00/0--.14

.4 ~

--­----­

--­-

r---.

o

.4

~

.8

.6

~

~~

1.0

x/c

--x y (per cent c) (per cent c)

(u/V)t

vv

~Va/V

0 0.944 1.400 2.078 2.611

0 0.892 1.011 1.113 1.167

0 0.944 IJ)05 1.055 1.080

3.857 1.282 0.950 0.688 0.564

40

3.044 3.744 4.244 4.&13 5.000

1.200 1.238 1.256 1.265 .1.253

1.095 1.113 1.121 1.124 1.119

0.486 0.389 0.327 0.249 0.197

50 60 70 80 90

4.856 4.433 3.733 2.767 1.556

1.235 1.205 1.157 1.089 0.990

. 1.111 1.098 1.076 1.044 0.995

0.856 0.100

0.910

0

0.954

I

0

I

0 1.25 2.5 5.0 7.5

10 15 20 30

95 100

i

I

I

0.159 0.127 0.100 0.073 0.045 I

L.E. radius: 0.272 per cent c NACA 0010-34 Basic Thickness Form

0.030 0

317

APPENDIX I

1.6

l2

~

,r

""".-­

r-,

\ NACA 0010"J5

l----"'

~

-

r---. ---.....

o

.4

.2

xlc

r---. ............. ~

.8

6

v/V ---~.

,

---­ 10

bv./V

I

__.- -_.i-

-~.068-

o

0

o

1.25 2.5 5.0 7.5

0.878 1.267 1.844 2.289

0.954 1.032 1.087

0.977 1.016 1.043

1.122

1..059

0.679 0.. 555

10 15 20 30 40

2.667 3.289 3.789 4.478 4.878

1.141 1.172 1.194 1.214 1.229

1.068 1.083 1.093 1.. 102 1.109

0.476 0.382 0.323 0.247 0 . 198

50

5.000

60 70

4.867 4.389

80 90

I

3.500 2.100

1.235 1.240 1.227 1.176 1.046

1.111 1.114 1.108 1..084 1.023

0.. 162 0.131 0.104 0.076 0.048

1.178 0.100

0.920

0.959

j

0.030 0

95 100

I

I-----~---

._._ .. ... ~

I j

o

o

1.309 0..952

_---~_._-~--_.~._._-

L.E. radius: 0.272 per cent c

NACA 0010-35 Basic Thickness Form

THEORY OF WING SECTIONS

318

1.6 ___ L

, ,NA1CAoh,o

r- .............

1""Ia......

r

~I .........

.....--...... ......

::::::

~

R

NACA 0010-64-

r--- r---.

~

-

~-

o

:&

.~

10 15 20

30 40

50 60

70 80 90 95 100

~

to

.6

x/c

(v/V)t

»rv

~../V

0 1.511 2.044 2.722 3.178

0 1.108 1.245 1.286 1.277

0 1.053 1.116 1.134 1.130

2.324 1.286 0.966 0.690 0.556

3.533 4.056 4.411 4.856 5.000

1.269 1.261 1.248 1.244 1.242

1.127 1.123 1.117 1.116 1.115

0.475 0.377 0.316 0.241 0.193

4.856 4.433 3.733

1.556

1.231 1.211 1.155 1.089 0.980

1.110 1.101 1.074 1.043 0.990

0.155 0.126 0.098 0.072 0.045

0.856 0.100

0.912 0

0.955

0

0.030 0

Y

(per cent c) (per cent c)

0 1.25 2.5 5.0 7.5

.4

.....--

2..767

L.E. radius: 1.10 per cent r. NACA 0010-64 Basic Thickness Form

I

--

APPENDIX I

319

1.6

12

(v!

~~

.........

f

.8

WACA 0010-65

.4

- ---....

~ """"""""

---­

'-- .......

o

"',­\

.2

~ ~

.8

.6

A-

,

LO

x-/c

II

x

y

(lIIV)!

(per cent c) (per cent c)

,

0

1.25 2.5 5.0 7.5

I i

10 .15 20

30 40

50

60 70 80

_

..

I I

I I

0 1.068

2.584 1.295 0.970 0.684 0.551

0

1.140 1.273 1.271 1.252

3.300 3.756 4.089 4.578 4.889

1.236 1.213 1.200 1.196 1.212

1.112 1.101 1.095 1.094 1.101

0.470 0.372 0.312 0.239 0.193

1.229 1.234 1.226 1.173 1.049

1.109 1.111 1.107 1.083 1.024

0.158 0.128 0.103 0.076 0.046

0.915 0

0.957 0

5.000 4.867 4.389 3.500

I

2.100

I

1.178 0.100

---- l

41Ja / V

1.467 1.967 2.589 2.989

90

_-~.,

»rv

0

95 100

t

I

... _.

t

I

1

tI j

__

!

._...._.­

1.128 1.127 1.119

I

I

I

II I

'-'

L. E. radius: 1.10 pet" cent r

NACA 0010-65 Basic Thickness Form

0.029 0 ,~

..

,­

-..

THEORY OF WING SECTIONS

320

l6

l2

r­ ...............

~

f

---­ '"

.....

,

,

\ NACA0010-66

.4 ~

r---... o

--..... ...............

-

--­ ,.,,--­

.2

.4

x/c z 11 (per cent c) (per cent c)

.6

lO

(V/V)I

v/V

4v./V

0 1.489 2.011 2.656 3.089

0 1.130 1.246 1.286 1.282

0 1.063 1.116 1.134 1.132

2.434 1.289 0.959 0.687 0.554

40

3.400 3.856 4.178 4.678 4.822

1.258 1.225 1.209 1.189 1.178

1.122 1.107 1.100 1.090 1.085

0.471 0.372 0.310 0.236 0.190

SO 60 70 80 90

4.956 0.000 4.889 4.300 2.833

1.184 1.214 1.265 1.278 1.135

1.088 1.102 1.125 1.130 1.065

0.153 0.129 0.104 0.080 0.049

95 100 -------

1.656 0.100

0.960 0

0.980 0

0.030 0

0 1.25 2.5 5.0 7.5 10 15 20 30

L.E. radius: 1.10 per cent c

NACA OOlO-M Basic Thickness Form

APPENDIX 1

321

1.6

1.2

-

(

..............

I

~

~-

--~

~ ......

..........

.8

,

'\

MACA 0012

~

i'--­

-

-r--- r--­

-

~~

.8

D

% 11 (per cent c) (per cent c)

0 0.5 1.25 2.5 5.0 7.5 10 15 20 25

~ ~

-

..­ 1.0

(fJ/V)1

v/V

Avo/V

0 0.640 1.010 1.241 1.378

0 0.800 1.005 1.114 1.174

1.988 1.475 1.199 0.685

4.200 4.683 5.345 5.737 5.941

1.402 1.411 1.411 1.399 1.378

1.184 1.188 1.188 1.183 1.174

0.558 0.479 0.381 0.319 0.273

1.350 1.288 1.228 1.166 1.109

1.162 1.135 1.1OS 1.080 1.053

0.239 0.187 0.149 0.118 0.092

1.022 0.978 0.952 0

0

..... 1.894 2.615 3.555

30

6.002

40 50 60 70

5.803 5.294

80 90 95 100

2.623

1.044

1.448 0.807

0.956 0.906 0

4.563 3.664

0.126 I~E.

radius: 1.58 per cent c

NACA 0012 Basic Thickness Form

0.934

I

I

0.068 0.044 0.029 0

THEORY OF WING BECTIONS

322

L6 -'--­

V

- r---.

7

~ ..........

~

<,

f

~

,

~

r---­ r---..

!'--.. t--.........

o

.2

% 11 (per cent c) (per cent c)

0 1.25 2.5 0.0 7.5

10 15 20 30 40

50 60

70 80 90 ~5

100

,

r\ NACA 0012-34

~

.4

-x./c

.8

.6

I'--.......

~~

1.0

(v/V)t

fJ/V

I:.fJ./V

0 1.133 1.680 2.493 3.133

0 0.865 0.997 1.122 1.186

0 0.930 0.999 1.069 1.089

3.154 1.251 0.933 0.683 0.560

3.653 4.493 5.093 5.800 6.000

1.229 1.282 1.310 1.329 1.311

1.109 1.132 1.145 1.153 1.146

0.484 0.389 0.329 0.250 0.198

5.827 5.320 4.480 3.320 1.867

1.284 1.249 1.192 1.112 0.985

1.133 1.11'8 1.092 1.055 0.992

0.158 0.128 0.098 0.071 0.045

1.027 0.120

0.894 0

0.946

0.029 0

0

L.E. radius: 0.391 per cent c

NACA 0012-34 Basic Thic1mess Form

APPENDIX I

323

L6

,r:

r----. ~ ~

.8

r-,

1\ \1

NACA 00/2-64­

.4

-

----. ---­ -

~ ~ lo-.........

~

l---­ ~

o

.4-

% 11 (per cent c) (per cent c)

.8

.6

xle

J,O

(V/V)2

»rv

~v./V

0 1.813 2.453 3.267 3.813

0 1.072 1.270 1.330 1.325

0 1.035 1.127 1.153 1.151

2.019 0.952 0.685 0.554

1.322 1.313

40

4.240 4.867 5.293 5.827 6.000

1.150 1.146 1.141 1.139 1.140

0.474 0.372 0.315 0.241 0.199

SO 60 70 80 90

5.827 5.320 4.480 3.320 1.867

1.280 1.244 1.189 1.102

1.131 1.115 1.090

0.993

0.996

0.154 0.126 0.096 0.070 0.044

95

1.027 0.120

0.889 0

0.943

0 1.26 2.0 S.O 7.5 10 15 20

30

100

1.303 1.297 1.300

1.050

0

L.E. radius: 1.582 per cent c NACA 0012-64 Basic Thickness Form

1.236

0.028 0

-

THEORY OF WING SECTIONS

324

1.6

I.Z

r I

~

--......... i"'-<,

~

I

..............

~ --;

''\

.8

\,

NACA 0015

<, o

-

.--

r

---

---

:---~

----

---

0 0.5 1.25 2.5 5.0

I--'

1.0

I

Av./V

-

2.367 3.268 4.443

0 0.546 0.933 1.237 1.450

0 0.739 0.966 1.112 1.204

1.600 1.312 1.112 0.900 0.675

7.5 10 15 20 25

5.250 5.853 6.682 7.172 7.427

1.498 1.520 1.520 1.510 1.484

1.224 1.233 1.233 1.229 1.218

0.557 0.479 0.381 0.320 0.274

30

7.502 7.254 6.617 5.704 4.580

1.450 1.369 1.279 1.206 1.132

1.204 1.170 1.131 1.098 1.064

0.239 0.185 0.146 0.115 0.090

3.279 1.810 1.008 0.158

1.049 0.945 0.872 0

1.024 0.972 0.934 0

0.065 0.041 0.027 0

40

50 60 70 80 90 95

}OO

0

~V

(V/V)t

~

.s

z/c

I

X Y (per cent c) (per cent c)

r---. ~ .

.....

.....

L.E. radius: 2.48 per cent c NACA 0015 Basic Thickness Form

_.-

_ ..

_~

...­

325

APPENDIX I

1.6

1.2

~

(

..........

I I

~~

<,

~ .........

.........

~ "­

<,

1\

\,

NACA DOl.

.4

-: <,

-

~

..----

I"--.

.4

()

7.5 10 15 20 25

30 40

50 60

70

80 90 95

100

--­ ---­

~

~

.6

r--­ ~

.8

~ ~

1.0

(,'1V)2

__v~1

4v./IT

0 0.465 0.857 1.217 1.507

0 0.682 0.926 1.103 1.228

1.342 1.178 1.028 0.861 0.662

6.300 7.024 8..018 8.606 8.912

1.598 1.628 1..633 1.625

0.555

1.592

1.264 1.276 1.278 1.275 1..262

9.003 8.705 7.941 6.845 5.496

1.556 1.453 1.331 1.246 1.153

1.247 1.205 1.154 1.116 1.074

0.238 0.184 0.144 0.113 0.087

1.051 0.933 0.836 0

1..025 0.966 0.914 0

0.063 0.039 0.025 0

v

x

t

(per cent c) (per cent c) 0 0.5 1.25 2.5 5.0

~

0

.....

2.841 3.922 6.332

3.935 2.172 1.210 0.189

I

I

L.E. radius: 3..56 per cent c NACA 0018 Basic '£hickness Form

0.479 . 0..381 0.320 0.274

THEORY OF 'A,'IING SECTIONS

326

I

~

r-, <,

7

1.2

"' r-,

1

-"

11

(per cent c) (per cent c) 0 0.5 1.25 2.5 5.0

0

r--­ ---..

(V/V)t

~

I

\,

r--- r---. ~

.8

.6

X/C

-,

\

L.---­ ~

.4

<,

(J(J2/

---

~

V r-, ...-...

()

X

r-,

~~ MeA

.4

r-,

~

1.0

fJ/V

li.v./V

0 0.630 0.887 1.087 1.242

1.167 1.065 0.946 0.818 0.648

6.221

0 0.397 0.787 1.182 1.543

7.5 10 15 20 25

7.350 8.195 9.354 10.040 10.397

1.682 1.734 1.756 1.742 1.706

1.297 1.317 1.325 1.320 1.306

0.550 0.478 0.381 0.320 0.274

30

10.504 10.156 9.265 7.986 6.412

1.664 1.538 1.388 1.284 1.177

1.290 1.240 1.178 1.133 1.085

0.238 0.183 0.142 0.111 0.084

4.591 2.534 1.412 0.221

1.055 0.916 0.801 0

1.027 0.957 0.895 0

0.061 0.037 0.023 0

40

50 60

70 80 90

95 100

...... 3.315 4~576

L.E. radius: 4.85 per cent c NACA 0021 Basic Thickness Form

327

APPENDIX I

rr-«:

7 I

r-.

,,~

" r-, <,

I

r-,

'"

.s M4CA 0024

. .4

o

V --r-, .j.

y x (per cent c) (per cent c)

-

r---.

r----.. r----.. ~

x/c

L..--- ~ .8 ..8

~

\

\

t---.

1

~

I. D

(V/V)!

V/V

4D./V

3.788 5.229 7.109

0 0.335 0.719 1.130 1.548

0 0.579 0.848 1.063 1.244

1..050 0.964 0.870 0.771 0.632

7.5 10 15 20 25

8.400 9.365 10.691 11.475 11.883

1.748 1.833 1.888 1.871 1.822

1.322 1.354 1.374 1.368 1.350

0.542 0.476 0.383 0.321 0.274

30

50 60 70

12J)04 11.607 10.588 9.127 7.328

1.777 1.631 1.450 1.203

I

1.333 1.277 1..204 1.151 1.097

0.238 0.181 0.140 0.109 0.082

80 90 95 100

5.247 2.896 1.613 0.252

1.065

I I

1.032 0.944 0.879 0

0..059 0.035 0.022 0

0 0.5 1.25 2.5 5.0

0 .........

40

I

1.325

0.891 0.773 0

j

L.E. radius: 6.33 per eent c NACA 0024 Basic 'I'hickness Form

THEORY OF WING SECTIONS

328 2.0

1.6

1.2

---.

r:

~

\

.B NACA 16-006

-­

e---'

11

10 15 20 30 40 lit)

60

10 80 90

95

100

!

(fl/V)t

v/V

0 0.646 0.903 1.255 1.516

0 1.059 1.085 1.097 1.105

0 1.029 1.042 1.047 1.051

5.471 1.376 0.980 0.689 0.557

1.729 2.067 2.332 2.709 2.927

1.108 1.112 1.116 1.123 1.132

1.053 1.055 1.057 1.060 1.064

0.476 0.379 0.319 0.244 0.196

3.000 2.917 2.635 2.099 1.259

1.137 1.141 1.132 1.104 1.035

1.066 1.068 1.064 1.051 1.017

0.160 0.130 0.104 0.077 0.049

0.707 0.060

0.962 0

0.981 0

0.032 0

(per cent c) (per cent c) 0 1.25 2.5 5.0 7.5

,

/.0

.8

%

\ \

L.E. radius: 0.176 per cent c NACA 16-006 Basic Thickness Form

l1v./V

APPENDIX I

329

..

,

1.2

,­

...........

~~

.8

\

NM:A 16-00g

1

.4

---

~-

r---­

-

~

.4

(J

%

11

Z/c

\

.8

.6

"""""""­

~

/.0

(V/V)2

V/V

J1v./V

0 0.969 1..354 1.882 2.274

0 1.042 1.109 1.139 1.152

0 1.021 1.053 1.067 1.073

3.644 1.330 0.964 0.684 0.554

10 15 20 30 40

2.593 3.101 3.498 4.063 4.391

·1.158 1.168 1.177 1.190 1.202

1.076 1.081 1.085 1.091 1.096

0.475 0.378 0.319 0.245 0.197

50 60 70 80 90

4.500 4.376 3.952 3.149 1.888

1.211 1.214 1.206 1.156 1.043

1.100 1.099 1.075 1.022

0.160 0.. 131 0.103 0.076 0.047

95

1.061 0.090

0.939 0

0.969 0

0.030 0

(per cent c) (per cent c) 0 1.25 2.5 5.0 7.5

tOO

I

1~106

L.E. radius: 0.396 per cent c NACA 16-009 Basic Thickness Form

THEORY OF WING SECTIONS

330

1.6

I-Z

T ----

to--"'"

~

~

""r\.\

.8 NACA 16-DI2

.4 ~

I'---

-

~ 100-0-..

D

!.........-.

.t

.4

\

t--- t--....... L--- ~ I.D

.6

SIC.

x

y

(v/V)t

V/V

I1v./V

0 1.292 1.805 2.509 3.032

0 1.002 1.109 1.173 1.197

0 1.001

1.053 1.083 1.094

2.624 1.268 0.942 0.677 0.561

10 15 20 30 40

3.457 4.135 4.664 5.417 5.855

1.208 1.223 1.237 1.257 1.271

1.099 1.106 1.112 1.121 1.128

0.473 0.378 0.319 0.245 0.197

50 60 70 80 90

6.000 5.836 5.269 4.199 2.517

1.286 1.293 1.275 1.203 1.051

1.134 1.137 1.129 1.097 1.025

0.161 0.131 0.102 0.075 0.045

95 100

1.415 0.120

0.908 0

0.953 0

0.027 0

(per cent c) (per cent c) 0 1.25 2.5 5.0 7.5

L.E. radius: 0.703 per cent c

NACA 16-012 Basic Thickness Form

331

APPENDIX I

1.6

:",.--­ ~

/

~

-

--'" -, "'"

{

\ M4CA /6-0/5

~

i'---.. D

x

~

---­

-

10 15 20

--­ ---­

...---­

.8

/.0

0 1.615 2.257 3.137 3.790

0 0.956 1.105 1.200 1.239

0 0.978 1.051 1.095 1.113

0.916 0.668 0.547

4.322

1.256 1.278 1.297 1.327 1.349

1.121 1.130 1.139 1.152 1.161

0.471 0.377 0.318 0.245 0.197 0.161 0.131 0.102 0.074 0.043

40

SO 60

7.500 7.293

1.364

70

6.. 587

80 90

5.248 3.147

1.348 1.254 1.053

1.168 1.172 1.161 1.120 1.026

95

1.768 0.150

0.875 0

0.935 0

100

~

v/V

6.168 5.830 6.772 7.318

30

...............

(V/V)2

(per cent c) (per cent c)

0 1.25 2.5 5.0 7.5

r--­

~

v

\,

1.374

L.E. radius: 1.100 per cent c NACA 16-015 Basic Thickness Form

I

~v./V

2.041 1.209

I

0.025 0

THEORY OF WING SECTIONS

332

-

.w

-

/'

~

-

- --­

..........

<,

-7

t-,

\

I

\

.8

V

~

~ t----

- r--­

~

.6

I--­

(J

% 'Y (per cent c) (per cent c)

,

\

NACA /S-OI8 .4

~

~

L.---­ ~ .8

~ ~

/.0

(V/V)I

fJ/V

~fJ./V

0 1.938 2.708 3.764 4.548

0 0.903 1.092 1.217 1.271

0 0.950 1.045 1.103 1.128

1.744 1.140 0.883 0.657 0.541

5.186 6.202 6.996 8.126 8.782

1.302 1.332 1.357 1.399 1.426

1.141 1.154 1.165 1.183 1.194

0.468 0.376 0.318 0.245 0.198

70 80 90

9.000 8.752 7.904 6.298 3.776

1.447 1.452 1.421 1..306 1.051

1.203 1.205 1.192 1.143 1.025

0.162 0.131 0.102 0.073 0.042

95 100

2.122 0.180

0.837 0

0.915 0

0.024 0

0 1.25 2.5 5.0

7.5 10 15 20 30 40

SO 60

f

L.E. radius: 1.584 per cent c NACA 16-018 Basic Thickness Form

APPENDIX 1

.8 ~

V

- I -/ ,

---- --­

--.... <,

~

I.

\

~

NACA /S-(J21 .4

V

---­

~ r--.-.

...-­

-~

. ---.--.... ~

_l.---' ~

~

D

.4

z 11 (per cent c) (per cent c)

1\ '\ \

zjc

.8

B

~

to

(V/V)t

tJ/V

!lv.IV

0 2.261 3.159 4.391 5.306

0 0.826 1.062 1.221 1.295

0 0.909 1.031 1.105 1.138

1.574 1.069 0.828 0.640 0.534

10 15 20 30 40

6.050 7.236 8.162 9.480 10.246

1.342 1.391 1.419 1.474 1.506

1.159 1.179 1.191 1.214 1.227

0.463 0.374 0.317 0.245 0.198

50 60 70

10.500 10.211 9.221 7.348 4.405

1.535 1.536 1.495 1.361 1.039

1.239

1.166 1.019

0.162 0.131 0.102 0.072 0.041

2.476 0.210

0.801

0.895 0

0.023 0

0 1.25 2.5 5.0 7.5

80 90 95

100

I

0

1.239 1.223

ThE. radius: 2.156 per cent c NACA 16-021 Basic Thickness Form

THEORY OF WING SECTIONS

334 1.8

,0 1.2

(f)~

I

t

.....

I

~CI,,03 ,..,.,.. $Ur"~

~03 L~ ?? J

."foce

r-

r--..

~ ~

~

.8 NACA t13-D06

~

...............

D

.4

x y (per cent c) (per cent c)

'/. S,C

1.0

.8

.6

(f)/V)2

V/V

J1v./V

0 0.503 0.609 0.771 1.057

0 0.973 1.050 1.080 1.110

0 0.986 1.025 1.039 1.054

4.483 2.110 1.778 1.399 0.981

5.0 7.5 10 15 20

1.462 1.766 2.010 2.386 2.656

1.130 1.142 1.149 1.159 1.165

1.063 1.069 1.072 1.077 1.079

0.692 0.562 0.484 0.384 0.321

25 30

2.841 2.954 3.000 2.971 2.877

1.170 1.174 1.170 1.164 1.151

1.082 1.084 1.082 1.079 1.073

0.279 0.245 0.218 0.196 0.176

2.723 2.517 2.267 1.982 1.670

1.137 1.118 1.096 1.074 1.046

1.066 1.057 1.047 1.036 1.023

0.158 0.141 0.125 0.111 0.098

1.342 1.008 0.683 0.383 0.138

1.020 0.994 0.965 0.936 0.910

1.010 0.997 0.982 0.967 0.954

0.085 0.073 0.060 0.047 0.032

0

0.886

0.941

0

0 0.5 0.75 1.25 2.5

35 40 -is 50

55 60

65 70

75 80 85 90 95 100

LE. radius: 0.297 per cent c -NACA 63-006 Basic Thickness Form

APPENDIX.] I.IJ

ro.. 1.2

f/<

.8

,

c, •.

/

33~ I

~ ~ slrfaet!J

~

~

~ I ~ r--.08 Lower .•ur'f«:e

~~

V

~

~

MACA 63-otJ!J

.4 [,--­

~

r--­ " - ­ _L--- ~ -

"'""-­

.z

a %

.4

Y

{11/V)2

(per cent c) (per cent c)

0 0.5 0.75 1.25

2.5 5.0 7.5 10 15 20

0 0.749 0.906 1.151 1.582

2.196

%/c

0

0.885 i

I I

1.002 l.OSI 1.130

lO

.6

fJ/V ,

J1tJ./V

0 0.941 1.001 IJ)25

3.058 1.889

1.063

1.647

1.339 0.961

1.180 1.205 1.221 1.241 1.255

1.08B

1.105 1.114 1.120

0.689 0.560 0.484 0.386 0.324

4.275

1.264

1.124

0.281

4.442 4.500 4.447 4.296

1.269 1.265 1.255 1.235

1.126 1.120 1.111

0.248 0.220 0.196

1.208

1.099

1.175

65

4.056 3.739 3.358 2.928

70

2.458

1..084 1.068 1.051 1.032

0.156 0.140 0.124 0.109 0.095

75 80

1.966 1.471

85 90

0.550

95

0.196

0.984 0.942 0.903 0.868

1.012 0.992 0.971 0.950 0.932

0.082 0.069 0.057 0.044 0.030

0

0.838

0.915

0

25 30 35 40

45 50 55 60

100

2.655

3.024 3.591

3.997

0.990

!

1.141 1.104 1.065

1.025

1.098

1.125

L.E. radius: 0.631 per cent c

N.~CA

63-009 Basic Thickness Form

0.175

THEORY OF WING SECTIONS

336 ttl

/'

I.e

.8

c, ..

I

10 , . . , . ..,-'!ot»

ro;: ~~~

/r V

v

I

1'--.10 LOWf1r

~

I

.".,~

~~

~~

~

NACA 63-010 .4

-r --­ r---­ t---..­

~

L----­ I"-­ "'-­

-­

o

.4

I

~-

/.0

.8

.6

%/C

x JI (per cent c) (per cent c)

0 0.5 0.75 1.25

2.5 5.0 7.5 10 15 20 25

30 35 40 45 50

55 60

65 70

(,,/V)2

fJ/V

11v./V

0.841 0.978 1.037 1.131

0 0.917 0.989 1.018 1.063

2.775 1.825 1.603 1.316 0.952

1.193 1.223 1.245 1.270 1.285

1.092 1.106 1.116 1.127 1.134

0.687 0.560 0.484 0.386 0.325

4.753 4.938 5.000 4.938 4.766

1.296 1.302

1.299 1.286 1.262

1.138 1.141 1.140 1.134 1.123

0.282 0.248 0.220 0.196 0.175

4.496 4.140 3.715 3.234 2.712

1.231 1.193 1.154 1.113 1.069

1.110 1.092 1.074 1.055 1.034

0.156 0.139 0.123 O.lOS 0.094

1.025

1.012 0.989 0.967 0.945 0.924

0.081 0.069 0.056 0.043 0.030

0.907

0

0

0 0.829 1.004 1.275 1.756 2.440 2.950 3.362

3.994 4.445

I

75 80 85

2.166 1.618

90 95

0.604 0.214

0.979 0.935 0.893 0.853

0

0.822

100

1.088

L.E. radius: 0.770 per cent c NACA *':>.-1)10 Basic Thickness Form

APPENDIX I l8

,

~

_C,..

337

14 Lt:Per surfQCfl

.............

I

loy ~ ~ £2 <.J (/ .14 Lo",.,- surfoct!!I """ ~ ~ ~

---­

L.--.

I{

.8

~

,

'-

~

NACA 63J-a2

.4

~

"--.....

~

- r---

~

-

a

.P

.4

I

~

.6

-----­ --­--

v

(vIV)2

"/1'

0 0.985 1.194 1.519 2.102

0 0.750 0.925 1.005 1.129

0 0.866 0.962 1.003

5.0 7.5 10 15 20

2.925 3.542 4.039 4.799 5.342

1.217 1.261 1.294 1.330 1.349

1.103 1.123 1.138

25

5.712 5.930 6.000 5.920 5.704

1.362 1.370

5.370 4.935 4.420 3.840 3.210 2.556 1.902 1.274 0.707 0.250

1.023 0.969 0.920 0.871 0.826

x

(per cent c) (per cent c) 0 0.5 0.75 1.25 2.5

30 35 40 45 50

55 60 65

70

75 80 85

90 95

100

---­

I i

0

1.063

I

I

Jlv.IV

2.336 1.695 1.513 1.266 0.933

1.161

0.682 0.559 0.484 0.387 0.326

1.317

1.167 1.170 1.169 1.161 1.148

0.283 0.249 0.221 0.196 0.174

1.276 1.229 1.181 1.131 1.076

1.130 1.109 1.087 1.063 1.037

0.155 0.137 0.121 0.106 0.091

1.011 0.984 0.959 0.933 0.909

0.079 0.067 0.055 0.042 0.029

0.889

0

1.153

1.366 1.348

II

1.0

.8

%C

0.791

I

I

L.E. radius: 1.087 per cent c NACA 631-012 Basic Thickness Form

THEORY OF WING SECTIONS ~ ,•.22 ~ !IUrlI:It»

.........

(

'.I

r

III I - I

(v)'

--........ r-,

.J1I'"

~

/

~~~~ r-,

~~22

~

Loww-

"'oce . . ~ ~

~

A

~~............

NACA tJ3z-tJl5

.4­

V

~

-r---- r---

<, t---.

L.---- J---­

r--- t--­ ~

a

~,...

.8

%

11

1.0

(vlv)!

V/V

4v./V

0 1.204 1.462 1.878 2.610

0 0.600 0.822 0.938 1.105

0 0.775 0.907 0.969 1.051

1.918 1.513 1.379 1.182 0.903

5.0 7.5 10 15 20

3.648 4.427 5.055 6.011 6.693

1.244 1.315 1.360 1.415 1.446

1.115 1.147 1.166 1.190 1.202

0.674 0.557 0.484 0.388 0.330

25 30

7.155 7.421 7.500 7.386 7.099

1.467 1.481 1.475 1.446 1.401

1.211 1.217 1.214 1.202 1.184

0.286 0.251 0.222 0.196 0.174

6.665 6.108 5.453 4.721 3.934

1.345 1.281 1.220 1.155 1.085

1.160 1.132 1.105 1.075 1.042

0.153 0.135 0.118 0.102

3.119 2.310 1.541 0.852 0.300

1.019 0.953 0.894

1.009

0.789

0.976 0.946 0.916 0.888

0.076 0.063 0.051 0.039 0.026

0

0.750

0.866

0

(per cent c) (per cent c) 0 0.5 0.75 1.25 2.5

35 40 45 50 55

60 65 70 75 80 85 90

95 100

0.83~

L.E. radius: 1.594 per cent c NACA 63r015 Basic Thickness Form

0.088­

APPENI)[.\'- I eo

~,-.32

L /.6

0,.

1.2

/"

/ / I,l V' J

~

"

r-,

~~

<,

~~

~

I

I

~

~

NfCA 633-018

--

~

-

r--­

o

%

<,

~32 LOflllf!lr ""acll

I

,

..............

V

I

V

lIppe". surfloc.

<,

(

339

'"

I

I---­ ~

ZIt:

~

r--­

~

1.0

.B

6

(v/1T ) 2

y

(per cent e) .(per rent c)

r--.. r---

"IV

/i,v./V 1.639

0.848

0 0.664 0.837 0.921

i

0

!

i

0.5 0.75

I

1.25

f

2.5

f

5.0 7.5 10 15 20

25 30

35 40 45 50 55 60

65 70 75 80 85 90

95 100

I I

I I I I

0

0

1.404 1.713 2.217 3.104

0.441 0 ..700 1.065

1.032­

4.362 5.308 6.068 7.225

1.260 1.360 1.424

1.122 1.166 1.193 1.225 1.244

1.500

8.048

1.547

8.600 8.913 9.000 8.845 8.482

1.579 1.598 1.585

1.550 1.490

1.361

1.258 1.105 0.871 0.663 0.553 0.484 0.390

0.333

1.257

0.289

1.264

0.253

1.259 1.245 1.221

0.223 0.197 0.173

1.188 1.153 1.119 1.082

0.152 0.133 0.115 0.099

1.043

0.084­

1.004

7.942 7.256 6.455 5.567 4.622

1.252 1.170 1J187

3.650 2.691 1.787 0.985 0.348

1.009 0.933 0.868 0.807 0.753

0.966 0.932 0.898

0.868

0.072 0.059 0.048 0.036 0.024

0

0.712

0.844

0

1.411 1.330

I

~--

L.E. radius: 2.120 per cent c N.~CA

63...018 Basic Thickness Form

THEORY OF WING SECTIONS

340

a

.z

z 71 (per cent C) (per cent C)

.4

Z/c

.6

(V/V)2

vlV

Av.IV

0 1.583 1.937 2.527 3.577

0 0.275 0.564 0.7'5 1.010

0 0.524 0.751 0.851 1.005

1.439 1.236 1.156 1.034 0.842

5.065 6.182 7.080 8.441

1.260 1.394 1.487 1.592

20

9.~O

1.655

1.122 1.181 1.219 1.262 1.286

0.653 0.550 0.484 0.392 0.335

25

10.053 10.412 10.&00 298 9.854

1.698

1.578

1.303 1.312 1.307 1.286 1.256

0.291 0.255 0.225 0.198 0.173

50

9.206

55 60 65

8.390

1.479 1.380 1.281 1.180 1.084

1.216 1.175 1.132 1.086 1.041

0.150 0.130 0.112 0.096 0.081

0.997 0.954 0.916 0.880 0.849

0.068 0.057

2.021 1.113 0.392

0.994 0.911 0.839 0.774 0.721

0..046

0

0.676

0.822

0

0 0.5 0.75 1.25 2.5 5.0

7.5 10

15

30 35 40 45

to.

70

7.441 6.396 &.290

75

4..160

80 85

90 96 100

a.OM

1.721

1.709 1.654

. L.E. radius: 2.6liO per cent c r~ACA 63M}21

Basic 'Thickness Form

0.035 0.023

APPENDIX I

0 1.2.

r-l

IlL"

r:

341

c/ =.01Uppersurface

."

I-+-- r---.

"".01 lower surrace I

.8

I

-- r-.

NACA 6JAOO6

.4 .....-

r--- "---

o %

.2

11

1.25

2.5

x/c

.6

vlV

J1v./I··

0 0.495 0.595 0.7M 1.045

0 0.900 1.063 1.086 1.112

0 0.949 1.031 1.042 1.055

4.560

1.447

1.134

1.747 1.989

1.142 1.1SO 1.159 1.165

1.065 1.069 1.072 1.077

5.0 7.5 10 15

2.362

20

2.631

25 30 35 4{)

2.820 2.942 2.996 2.985

45

2.914

50 55 60

2.788

1.168

1.079

1.370 0.976

0.693 0.563 0.485 0.383 0.321

1.162 1.151

1.067 1.058 1.049 1.039 1.028

0.. 140 0.126 0.112 0.085 0.072 0.060 0..047 0.033 0

1.169

1.859

75

1.556

1.035

80

1.248

1.010

85

0.939

90 95

0.. 630

0.986 0.96-1

0.322

0.939

1.017 1.005 0.993 0.982 0.969

100

0.013

0

0

2.613 2.396 2.143

2.079 1.794

1.081 1.082 1.081 1.078 1.073

1.170

1..138 1.120 1.100 1.079 1.057

65 70

to

.8

(fllV)t

(per cent c) (per cent c) 0 0.5 0.75

A-

L.E. radius: 0.265 per cent c T.E. radius: 0.014 per cent c NACA 63AOO6 Basic Thickness Forol

0.278 0.24:4 0.217 0.195 0.175 0.158

0.098

THEORY OF WING SECTIONS.

342

l6 , c/

,

=.05Upper surface

I

0

t

~

?(

-r- ~

~

~

'.OS lower surface

.............

~

NACA6.1AOO8

~

..............

o

--

.2.

.4

.6

lO

.8

X/I! %

(V/V)2

»rv

AIl./V

0 0.658 0.791

0 0.850 1.034

0 0.922 1.017

11

(per cent c) (per cent c)

1.003

1.080

1.039

1.391

1.132

1.064

3.465 1.961 1.674 1.344 0.967

1.930 2.332 2.656 3.155 3.51&

1.168 1.185 1.198 1.212 1.221

1.081 1.089 1.095 1.101 1.105

0.689 0.562 0.484 0.383 0.322

3.766 3.926 3.995 3.978 3.878

1.227 1.230 1.228 1.219 1.204

1.108 1.109 1.108 1.104 1.097

0.279 0.246 0.218 0.195 0.174

3.705 3.468 3.176 2.837 2.457

1.183 1.159 1.132 1.104 1.073

1.088

1.077 1.064 1.051 1.036

0.156 0.138 0.123 0.109 0.096

90 95

2.055 1.647 1.240 0.833 0.425

1.042 1.010 0.980 0.951 0.919

1.021 1.005 0.990 0.975 0.959

0.083 0.070 0.058 0.045 0.030

100

0.018

0

0

0

0 0.5 0.75 1.25 2.5 5.0 7.5 10 15 20 25

30 35 40

45 50

55 60

65 70 75 80 85

L.E. radius: 0.473 per cent c T.E. radius: 0.020 per cent c N ACA 63AOOS Basic Thickness Form

APPENDI..Y. I

343

l6 .~". cl =.09 Upper surface

.... r/L ~

1.2

(vY 1fA

--..;~

,...-

~

~

~~

""~

'.09 lower surface

.8

NACA6JAOIO

~

-

~

---..;

'--...........

o

, .6

.8

x/c x

11 (per cent c) (per cent c) 0 0.5 0.75

o

~

2.5

0.816 0.983 1.250 1.737

5.0 7.5 10

15

20

2.412 2.917 3.324 3.950 4.400

25

4.714

30

4.913

35

40

4.995 4.968 4.837

!I

1.25

45

50 55 60

65 70 75

80 85 90 95

100

I

I

4.613

i 4.311 3.943

I II

I

i

I

I

I

,

t

1

!

3.517 3.044 2.545 2.040 1.535 1.030 0.525 0.021

I

(V/V)2

I

i

1.043

i

1.140

1.068

I

1.200

1.095 1.107 1.116 1.126 1.132

0

i

0.774 0.985

! I

I

I

I

I

I

I

i

II I

1.225 1.245

1.268 1.282 1.290 1.294 1.291 1.279 1.258

LO

&./"..

'II V 0 0.880 0.992 1.021

~

,

-

1.136

1.138 1.136 1.131 1.122

I

I I

I

2.805 1.833 1.594 1.307 0.957 0.684 0.560

0.483

I 0.383 0.324

I I

I

0.280

0.247 0.218 0.195 0.174

0.155 0.137

0.122 O.lOS 0.09-1

1.230 1.196 1.162

1.125 1.086

1.109

1.048 1.009

1.02-1 1.004

I

!

0.972 0.938

O.9CiO

0.969 0.949

I

j

0

0

I 0

I

i I

1.094 1.078

1.061 1.042

i

I

I

, 1

0.986

L.E. radius: 0.742 per cent c T.E. radius: 0.023 per cent c

NACA 63AOI0 Basic Thickness Form

I

I

0.081 0.068 0.057 0.044 0.030

THEORY OF WING SECTIONS

344

16

IJI' cl :./2 Vpp~r surface

rl)Lt

~ ~

~/~

(fit rr .B ,

~

"'-o

~

~

............

'.It Lower surface

~

NACA6~AOI2

--

~

~

..4

~

~Iooo..:.

t - ~-

.6

.8

x/c

z 11 (per cent c) (per cent c)

LO

(vlv)'J

plV

i1v./V 2.361 1.701 1.515

0

0

0.5

0.973 1.173 1.492 2.078

0 0.686 0.924 0.985 1.136

0 0.828 0.961 0.992 1.066

0.935

2.895 3.504 3.994 4.747 5.287

1.229 1.265 1.291 1.324 1.344

1.109 1.125 1.136 1.151 1.159

0.679 0.559 0.482 0.384 0.325

25 30 35

5.664:

40 45

5.957

5.792

1.355 1.360 1.357 1.340 1.312

1.164 1.166 1.165 1.158 1.145

0.281 0.248 0.219 0.196 0.174

50

5.517 5.148 4.700 4.186 3.621

1.275 1.234 1.191 1.145 1.098

1.129 1.111 1.091 1.070

0.154 0.136 0.120 0.106 0.092

85 90

1.826

1.051 1.007 0.964

1.025

80

3.026 2.426 1.225

0.925

9S

0.625

100

0.025

0.75 1.25

2.5 5.0 7.5 10

15 20

55 60 65

70 75

"-­

~ ............ ~

jIlII""""

5.901

5.995

1.048

1.258

0.880

1.003 0.982 0.962 0.938

0.079 0.066 0.055 0.042 0.029

0

0

0

L.E. radius: 1.071 per cent c T.E. radius: 0.028 per cent c

NACA 6&A012 Basic Thickness Form

APPENDIX I

345

,~t:l =.18 Upper surface

1.6

i2

...............

f

O~ ~- .................. ~

I

't"

~

V

f----

II ~ ,

r--:::: ~ e-,

~~

\

.8

r/

J8 lower $ur'Fact!

I

~

r--... ~

NACA 6:12 AO/5

- r---

~"""""

<, r---..

o

.2

.4

I'--- ~

~~

.,.....- "---~

.6

J.()

x/c z 11 (per cent c) (per cent c)

(V/V)2

"IV

&J./V

0 0.550 0.825 0.882 1.120

0 0.742 0.908 0.939 1.058

1.504 1.370 1.176 0.005

1.257

1.121 1.150 1.167 1.187 1.199

0.669 0.555 0.482 0.384 0.326

1.396

1.206 1.210 1..207 1.198 1•.182

0.282 0.250 0.220 0.196 0.174

6..858 6.387 5.820 5.173 4.468

1.349 1.296 1.237 1.175 1.115

1.161 1.138 1.112 1.084 1..056

0.152 0.134 0.118 0.104 0.090

90 95

3.731 2.991 2.252 1.512 0.772

1.055 1.000 0.950 0.900 0.850

1.027 1.000 0.975 0.949 0.922

0.077 0.064 0.052 0.040 0.028

100

0.032

0

0

0

0 0.5 0.75 1.25 2.5

5.0 7..5 10 15 20

25 30 35 40 45 50

55 60 65

70

75 80 85

0 1.203

1.448 1.844 2.579 3.618 4.382 4.997 5.942 6.619 7.091

7.384 7.496 7.435 7.215

1.323

1.361 1.408 1.437

1.455 1.464 1.458 1.435

1.930

I

L.E. radius: 1.630 per cent c T.E. radius: 0.037 per cent c

NACA ~015 Basic Thickness Form

THEORY OF WING SECTIONS

346

1.2

ke,..oe ~ - * ­

,D ,r..,.

~

I~

~oe Lower

"""ot:e

r----... r----.

..............

~

.B

!MeA M-008 .4 L..­ ~

o

.2

.4

,I

.s-

.11

1.0

%,C

'II

%

(per cent c) (per cent c)

(vlV)t

fJIV

&D./1'

0 0.5 0.75 1.25 2.5

0 0.494 0.596 0.754 1.024

0 0.995 1.058 1.085 1.1OS

0 0.997 1.029 1.042 1.053

4.623 2.175 1.780 1.418 0.982

5.0

1.405 1.692 1.928 2.298 2.572

1.119 1.128 1.134 1.146 1.154

1.058 1.062 1.065 1.071 1.074

0.692 0.560 0.483 0.385 0.321

2.772 2.907 2.981 2.995 2.919

1.160 1.1M 1.168 1.171 1.160

1.077 1.079 1.081 1.082 1.077

0.279 0.246 0.220 0.198 0.178

2.775 2.575 2.331 2.050 1.740

1.143 1.124 1.102 1.079 1.054

1.069 1.060 1.050 1.039 1.027

0.1fi8 0.142 0.126 0.112 0.098

1.412 1.072 0.737 0.423 0.157

1.028 1.000 0.970 0.939 0.908

1.014 1.000 0.985 0.969 0.953

0.085 0.072 0.060 0.047 0.031

0

0.876

0.936

0

7.5 10 15 20 25

30 35

40 45

50 55 60

85 70 75 80

85 90 95

100

L.E. radius: 0.256 per cent c NACA 64-006 Basic ThicJmess Form

347

APPENDIX 1 .6

(

/'

~

~C',

•.04

Up~

$urfoce

I ~~ ~ ~IU Lower surlOCtfI ~ ~ r r---.. ~

~

~

~

NACA 84-008

.4 ~

r--o

x

-

.... 10--.

.2

I

.4

y

(V/V)2

I

0

0

II

0.658 0.794 1.005 1.365 1.875

/.0

.8

.6

xjc

I--­

V/V

fJ.V./V

0.912 1.016 1.084 1.127

0 0.955 1.008 1.041 1.062

3.544 l.994 1.686 1.367 0.969

1.152 1.167 1.179 1.195 1.208

1.073 1.080 1.086 1..093 1.099

0.688 0.560 0.480 0.385 0.323

1.217 1.225 1.230 1.235

45

3.704 3.884 3.979 3.992 3.883

1.103 1.107 1.109 1.111 1.105

0.279 0.246 0.220 0.198 0.176

50 55 60 65 70

3.684 3.411 3.081 2.i04 2.291

1.191 1.163 1.133 1.102 1.069

1.091 1.078 1.050 1.034

0.158 0.141 0.125 0.110 0.096

75 80 85 90 95

1.854 1.404 0.961 0.550 0.206

1.033 0.995 0.957 0.918 0.878

I.Q16 0.997 0.978 0.958 0.937

0.083 0.071 0.059 0.046 0.031

0

0.839

0.916

0

(per cent c) . (per cent c)

0 0.5 0.75 1.25

2.5 5.0 7.5 10 15 20

25 30 35

40

I

2..259 2..574

3.069 3.437

1.220

1.064 I

I

t

100

I

L.E. radius: 0.455 per cent c NACA 64-008 Basic Thickness Form

THEORY OF WING SECTIONS

348 /.6

{

--V ~

",.-

K

~

~

c,•.

06 UpptIr surftXVI

-.l

t--""_ ~

~

~

T

r--:otJ Lower .urf'DCfI ~

T

~

~

.8

~

HACA 1U-IJ09

~

--

,...-

r-----. ......... o

.2

x 'Y (per cent c) (per cent c)

0 0.5 0.75 1.25 2.5 5.0

7.5 10 15 20

25 30

35 40

45

50 55 60 65

70 75

80 85 90 95

100

.4

%/c

II--..

I - - to--"

.6

.8

to

(V/V)2

V/V

~./V

0 0.739 0.892 1.128 1.533

0 0.872 0.990 1.131

0 0.934 0.995 1.037 1.063

3.130 1.905 1.637 1.340 0.963

2.109 2.543 2.898 3.455 3.868

1.166 1.186 1.200 1.221 1.236

1.080 1.089 1.095 1.105 1.112

0.686 0.560 0.479 0.383 0.323

4.170 4.373 4.479 4.490 4.364

1.246 1.255 1.262 1.267 1.246

1.116 1.120 1.123 1.126 1.116

0.281 0.248 0.221 0.198 0.176

4.136 3.826 3.452 3.026 2.561

1.217 1.. 183 1.149 1.112 1.073

1.103 1.088 1.072 1.055 1.036

0.158 0.140 0.125 0.109 0.095

2.069 1.664 1.069 0.611 0.227

1.033 0.992 0.950 0.907 0.865

1.016 0.996 0.975 0.952 0,930

0

0.822

0.907

0.082 0.070 0.057 0.044 0.030 ()

1.075~

L.E. radius: 0.579 per cent c

NACA M-009 Basic Thickness Form

APPENDIX I I. 6

349

I

I

ve, -.08 ~

I

I surf.:.

L

.--­ ~ /. .. rti: ~

-~

~

r--.D9

I

-~

~

I~

L...-

""oc. ~ ~

8

r-, r-,

NACA 64-010

"

~

r - ­ ~ I"'---­

~

-

I'---­ _ ()

z

5.0 7.5 10 15 20

25 30

~

I.D

.8

.6

1/

(v/l')2

v/l"

l1v./V

0 0.820 0.989 1.250 1.701

0 0.834 0.962 1.061 1.130

0 0.913 0.981 1.030 1.063

2.815 1.817 1.586 1.313 0.957

2.343 2.826 3.221 3.842

1.181 1.206 1.221 1.245 1.262

1.087 1.098 1.105 1.116 1.123

0.684 0.559 0.480 0.386 0.325

1.129 1.134 1.138 1.140 1.131

0.280 0.246 0.220 0.199 0.176

(per cent c) (per cent c) 0 0.5 0.75 1.25 2.5

~

4.302

4.639 4..864: 4.980 4.988

45

4.843

1.275 1.286 1.295 1.300 1.279

50

4.586 4.238 3.820 3.345 2.827

1.241 1.201 1.161 1.120 1.080

1.114 1.096 1.077 1.058 1.039

0.158 O.lS9 0.124 0.109 0.095

2.281 1.722 1.176 0.671 0.248

1.036 0.990 0.944 0.900 0.850

1.018 0.995 0.972 0.949 0.922

0.081 0.069 0.057 0.044 0.030

0

0.805

0.897

0

as

40

55 60 65 70

75 80

85 90 95

100

L.E. radius: 0.720 per cent c NACA 64-010 Basic Thickness Form

THEORY OF WING SECTIONS

350

I

-

-~

1.2

.8

1 I I ,-J2 Upper .8Urface

~~ r~ ~ ~ ~ ~ ~

1(/ ~2 Lower .".r«11

r/

~~

l'.

"­

MCA 841-tJ12

t->

- r-­

~

i'--­ t - - .

-

~ """"'-­

~~

~

1.0

.8

(J

y

x

II

(V/V)2

»rv

0 0.978 1.179 1.490 2;035

0 0.750 0.885 1.020 1.129

0 0.866 0.941 1.010 1.063

5.0 7.5 10 15 20

2.810 3.394 3.871 4.620 5.173

1.204 1.240 1.264 1.296 1.320

1.097 1.114 1.124 1.139 1.149

0.685 0.559 0.482 0.388 0.328

25 30 35 40 45

5.576 5.844 5.978 5..981 5.798

1.338 1.351 1.362 1.372 1.335

1.156 1.162 1.167 1.171 1.156

0.281 0.247 0.221 0.199 0.177

50 55 60

5.480 5.056 4.548 3.974 3.350

1.289 1.243 1.195 1.144 1.091

1.136 1.115 1.093 1.070 1.044

0.158 0.138 0.122 0.103 0.088

2.695 2.029 1.382 0.786 0.288

1.037 0.981 0.928 0.874 0.825

1.018 0.990 0.963 0.935 0.908

0.074 0.063 0.052 0.045 0.028

0

0.775

0.880

(per cent c) (per cent c) 0 0.5 0.75 1.25 2.5

65 70 75 80

85 90 95 100

L.E. radius: 1.040 per ce~t c NACA 041-Oi2 Buic Thickness Form

I

~Vtl./V

2.379 1.663 1.508 1.271 0.943

II I

0' .--

~

~"

APPENDIX I

351

.22 LtlWt!r ."(Dce

.2

.4

.6

xjc

.8

1.0

I

X

Y

1---_

(v/l'F

I

tJ/V -----I-----!-----:­

(per cent c) (per cent c)

~o/l' I

o

o

o

o

0.5 0.75

1.208 1.456 1.842 2.528

0.670 0.762 0.896 1.113

0.819 0.873 0.947 IJJ55

1.231

1.109

1.284

15

3.504 4.240 4.842 5.785

20

6.480

1.323 1.375 1.410

1.133 1.150 1.172 1.187

0.670 0.559 0.482 0..389 0.326

25

6.985 7.319 7.482 7.473 7.224

1.454 1.470 1.485 1.426

1.198 1.206 1.213 1.218 1.195

0.285 0.250 0.225 0.202 0.179

1.25

2.5 5.0 7.5 10

30

35 40

45 50

70

6.810 6.266 5.620 4.895 4.113

75 80

3.296 2.472

55 60 65

85

1.677

90 95

0.950 0.346

1.434

1.365 1.300

1.233 1.167 1.101

1.033 0.967 0.902 0.841 0.785

1.168 1.140 1.110 1.080 1.049

1.016 0.983 0.950 0.917 0.886

1.939 1.476 1.354 1.188 0.916

i

I

I

I i I i

I "

0.158 0.135 0.121 0.105 0.090 0.078 0.065 0.054 0.041 0.031

100 o 0.730 o -----~-----_...:..-----_I~~._I_ -----1 L.E. radius: 1.590 per cent c

NACA 64 r015 Basic Thickness Form

THEORY OF WING SECTIONS

352

()

.4

y x (per cent c) (per cent c)

0 0.5 0.75 1.25 2.5 5.0 7.5 10 15 20

25 30 35

40 45 50 55

60 65

70 75 80 85

90

95 100

0 1.428 1.720 2.177 3.005 4.186 5.076 5.803 6.942 7.782

8.391 8.789 8.979 8.952 8.630

(V/V)2 0

0.54.6 0.705 0.862 1.079

I

I

1.244 1.327 1.380

1.450 1.497 1.535 1.562 1.585

vii'

1.518

J1fJ./l'

0.739 0.840 0.920 1.039

1.646 1.360 1.269 1.128 0.904

1.115 1.152 1.175 1.204 1.224

0.669 0.558 0.486 0.391 0.331

1.239 1.259 1.265 1.232

0.288 0.255 0.228 0.200 0.177

0

1.250

1.600

1.0

.8

.6

Z/c

8.114 7.445 6.658 5.782 4.842

1.354 1.272 1.190 1.109

1.198 1.164 1.128 1.091 1.053

0.154 0.134 0.117 0.102 0.088

3.866 2.888 1.951 1.101 0.400

1.028 0.952 0.879 0.812 0.747

1.014 0.976 0.937 0.901 0.864

0.074 0.063 0.051 0.039 0.027

0

0.695

0.834

0

1.436

i

L.E. radius: 2.208 per cent c

NAC.A 64r018 Basic Thickness Form

APPENDIX I

353

(~)z

.4

-I.

1.0

.8

.6

%,C

I I

(per ::nt C) (per

0 0.5 0.75 1.25

2.5 5.0 7.5

10

15

20

25

30

35

40

45

50

55

60

65

70

75

80

85

90

95

100

~nt C)

I

()

j

1.646

I

\

{V!V)2_

3.485

0 0.462 C.603 0.759 1.010

4.871 5.915 6.769 8.108 9.095

1.248 1.358 1.431 1.527 1.593

9.807 10.269 10.481 10.431 10.030

1.654

I

I

1.985

!

2.517

I

t

I

1.681 1.712 1.709 1.607

9.404 8.607 7.678 6.649 5.549 4.416 3.287 2.213 1.245 0.449 0

1.117 1.165 1.196 1.236 1.281 1.297 1.308 1.307 1.268 1.228

0.932 0.851 0.778 0.711

1.010 0.965 0.923 0.882 0.844

0.653

0.808

1.020

..

1.458 ;

1.274 1.203

1.084 0.878 I i

1.262

1.186 1.143 1.099 1.055

1.112

I I

0.680 0.776 0.871 1.005

~./V

_ _-­

0

1.507 1.307

!

----

1.406 1.209

I

v/lf"

I

I

I

1

I

I

i

!

1I

II

I

I

I

I

L.E. radius: 2.884 per cent c

NACA 64.-021 Basic Thickness Form

0.665 0.557 0.486 0.395 0.335 0.293 0.259 0.232 0.202 0.178 0.155 0.134 0.116 0.099 0.084 0.071 0.059 0.047 0.036 0.022

0

THEORY OF WING SECTIONS

354 0

l2

(vt

..-I P:-rJf

II

/ "I =.02 Upper surface

H--

~02 Lowersurface

...............

.8

...........

~

NACA 64AOO6

.--~

o

.2

y x (per cent c) (per cent c)

.4

lO

.6

x/c

(V/V)2

v/V

t.Vo/V

0 0.485 0.585 0.739 1.016

0 1.019 1.046 1.076 1.106

0 1.009 1.023

4.688 2.101 1.798

1.037 1.052

0.980

5.0 7.5 10 15 20

1.399 1.684 1.919 2.283 2.557

1.118 1.126 1.132 1.141 1.149

1.057 1.061 1.064 1.072

0.694 0.564 0.482 0.382 0.321

25

2.757 2.896 2.977 2.999 2.945

1.154 1.158 1.162 1.165 1.156

1.074 1.076 1.078 1.079 1.075

0.278 0.246 0.219 0.197 0.177

70

2.825 2.653 2.438 2.188 1.907

1.142 1.125 1.107 1.087 1.066

1.069 1.061 1.052 1.043 1.032

0.159 0.143 0.126 0.112 0.099

75 80 85 90 95

1.602 1.285 0.967 0.649 0.331

1.043 1.018 0.992 0.964 0.935

1.021 1.009 0.996 0.982 0.967

0.087 0.074 0.061 0.047 0.033

100

0.013

0

0

0

0 0.5 0.75 1.25 2.5

30 35

40 45 50 55 60 65

1.068

L.E. radius; 0.246 per cent c T.E. radius: 0.014 per cent c NACA 64A.OO6 Basic Thickness Form

1.422

355

APPENDIX I / Cl

= .046 Upper surface

/

~ l2

(vt K J

I

~r--

~

r---. ~ ...

'.tJ4.6 Lower sarfoce

.............

.8

~

NACA 64,4008

.4 ~

,........

-.-.,

r-­ ..........

o

.2

.6

.8

LO

xle x

y

(per cent c) (per cent c)

0 0.5 0.75 1.25

0 0.646 0.778 0.983

2.5

1.353

(V/V)2

v/V

lw./V

0 0.947 1.005 1.068 1.122

0 0.973 1.033 1.059

3.546 1.972 1.697 1.352 0.971 0.692 0.564 0.481 0.382 0.323

1.002

5.0

1.863

7.5

2.245 2.559 3.047 3.414

1.151 1.165 1.176 1.191 1.201

1.073 1..079 1.084 1.091 1.096

3.681 3.866 3.972 3.998 3.921

1.2\>9 1.217 1.221 1.225 1.211

1.100 1.103 1.105 1.107 1.100

3.757 3.524·

1.191 1.167 1.141 1.113 1.084

1.091 1.080 1.068 1.055 1.041

0.158 0.141 0.125 0.111 0.098

1.026

1.010 0.993 0.975 0.956

0.084 0.072 0.059 0.045 0.032

0

0

10 15 20

25 30 35 40 45 50

55 60 65

3.234

70

2.897 2.521

75

2.117

80 85

1.698

90 95

1.278 0.858 0.438

1.053 1.020 0.987 0.951 0.914

100

0.018

0

L.E. radius: 0.439 percent c T.E. radius: 0.020 per cent c NACA 64AOOS Basic Thickness Form

I I

I

0.279 0.247 0.221 0.198 0.177

THEORY OF WING SECTIONS

356

L6 /C,

0,

= .08 Upper surface

Ii

12

r t. ~ '-::=:

t;(

r

-

~loo...-

~

"""'iiillIII

~

~~

~

t', .08 Lower 6urface

~

NACA 64AOIO

A

--­

~-

r---- I--....

o

~

.6

.4

t - - - ..... ~

....-­

.B

LO

x/c %

11

(V/V)2

vtV

&J./V

0 0.804 0.969 1.225 1.688

0 0.868 0.952 1.042 1.130

0 0.932 0.976 1.021 1.063

2.868 1.845 1.603 1.300 0.957

2.327 2.905 3.199 3.813 4.272

1.178 1.201 1.217 1.238 1.254

1.085 1.096 1.103 1.113 1.120

0.280 0.248 0.221 0.199 0.177

(per cent c) (per cent c)

0 0.5 0.75 1.25 2.5

5.0 7.5 10 15 20

I I

II

0.688 0.562 0.480 0.382 0.324

25

4.606

1.266

30

4.&17

25

4.968 4.995 4.894

1.275 1.282 1.268

1.125 1.129 1.132 1.135 1.126

4.684 4.388 4.021 3.597 3.127

1.240 1.208 1.174 1.139 1.102

1.114 1.099 1.084 1.067 1.050

0.158 0.140 0.124 0.109 0.096

1.063

95

2.623 2.103 1.582 1.062 0.541

0.981 0.938 0.893

1.031 1.011 0.990 0.969 0.945

0.083 0.070 0.058 0.044 0.031

100

0.021

0

0

40

45 50 55 60 65

70 75 80

85 90

1.288

1.023

L.E. radius: 0.687 per cent c T .E. radius: O'()?-3 per cent c

I

II

0

APPENDIX I

l6

,

0

#

357

/ c1 =./3 Upper surFQ(Y

- ~........ ~ ~~~ ~~~ ~

1.2.

L

{~

(vt V .8

~

........

~

"J~ lower surFace

~~

~

I

~

NACA 64,AO/2

~

-r--- !---

I'"""""""

f'-.-. ~

o

.2

.4

.6

x/c z

I

r--. .--­ ........

L.-­ ~

LO

,._­ i

l7/ v

y

(V/V)2

I

0 0.792 0.893

1.25

0 0.961 1.158 1.464

1.006

2.5

I

2.018

1.127

l.Oti2

0.941

5.0 7.5 10 15

1.201

1.000

1.235

1.111

1.257

1.121

20

2.788 3.364 3.839 4.580 5.132

25 30

5.534 5.809

35 40

5.965 5.993

45

5.863

50 55 60

5.605

(per cent c) (per cent c) 0

0.5 0.75

65

70 75

80 85

5.244 4.801 4.289 3.721

I

1.135

1.144

1.324

1.151 1.156 1.160 1.164 1.152

0.281 0.249 0.221 0.199 0.177

1.135 1.118

1.099 1.079 1.057

0.157 0.139 0.123 0.108 0.094

0.873

1.035 1.011 O.98i 0.962 0.934

0.080 0.068 0.056 0.042 0.029

0

0

0

1.34fi

1.289 1.250 1.207 1.164 1.118

0.974

1.263

0.925

0.644

100

0.025

I.U71 1.023

i

1.254

1.308

3.118 2.500 1.882

90 95

!

2.408 1.720 1.515

1.2..~

1.354 1.326

I

0

0.890 0.945 1.003

0.681 0.560 0.478 0.383 0.325

1.336

I

i

~l'CI/V

L.E. radius: 0.994 per cent c "f.E. radius: 0.028 per eent c

NAC.A.. 64 1A012 Basic Thickness Form

THEORY OF U'ING SECTIONS

358

,

L6

/c, = .2/ Upper surface

.-- r----- ~~ ~ ~

,0

l2

/

~

If J K,

(vt '/ .8

~~

~ ~

~

I

~~

~

".2/ lower surface

r-,

NACA 64-2 A O/5

I

.4

- r--- r---

~

r '-- r--- -

o

.2

- .6

~~

.4

-------

,......

~

LO

.8

x/c x 11 (per cent c) (per cent c)

(VJV)2

vJV

.:1t·a / l'

0.5 0.75 1.25 2.5

0 1.193 1.436 1.815 2.508

0 0.678 0.789 0.936 1.110

0 0.823 0.888 0.967 1.054

1.956 1.552 1.404 1.189 0.912

5.0 7.5

3.477 4.202

1.226 1.280

4.799 5.732 6.423

1.314 1.390

1.107 1.131 1.146 1.166 1.179

0.671 0.552 0.478 0.384 0.326

40 45

6.926 1.270 7.463 7.481 7.313

1.413 1.430 1.445 1.458 1.414

1.189 1.196 1.202 1.207 1.189

0.283 0.249 0.222 0.201 0.177

50 55 60 65 70

6.978 6.517 5.956 5.311 4.600

1.364 1.311 1.255 1.198 1.139

1.168 1.145 1.120 1.095 1.067

0.156 0.137 0.121 0.106 0.091

75 80

1.079 1.020 0.961 0.901 0.843

1.039 1.010 0.980 0.949 0.918

0.078 0.065 0.053 0.041

95

3.847 3.084 2.321 1.558 0.795

100

0.032

0

0

0

0

10

15 20 25

30 35

85 90

.1.360

LE. radius: 1.561 per cent c T.E. radius: 0.037 per cent c NACA 64tA015 Basic Thickness Fonn

0.027

APPENDIX I ~,Cls.OJ

.I.e 10

359 I

I .............

~r:L

r--:Ol Lower""'or:e

.8

I

I

Lt;per surfoce

r----.. r-----...

-.......

r-.

I NACA 65-006

.4

--­

~

D

.4

x

y

5.0 7.5 10

15 20 25 30 35 40 45

50 55 60

65 70

75 80 85

90 95

100

/.0

.8

(V/V)2

v/V

~Va/V

0 0.476 0.574 0.717 0.956

0 1.044 1.055 1.063 1.081

0 1.022 1.027 1.031 1.040

4.815 2.110 1.780 1.390 0.965

1.310 1.589 1.824 2.197 2.482

1.100 1.112 1.120 1.134 1.143

1.049 1.055 1.058 1.065 1.069

0.695 0.560 0.474 0.381 0.322

2.697 2.852 2.952 2.998 2.983

1.149 1.155 1.159 1.163 1.166

1.072 1.075 1.077 1.078 1.080

0.281 0.247 0.220 0..198 0.178

2.900 2.741 2.518 2.246 1.935

1.165 1.145 1.124 1.100 1.073

1.079 1.070 1.060 1.049 1.036

0.160 0.144 0.128 0.114 0.100

1.594 1.233 0.865 0.510 0.195

1.044 1.013 0.981 0.944 0.902

1.022 1.006 0.990 0.972 0.950

0.086 0.074 0.060 0.046 0.031

0

0.858

0.926

0

(per cent c) (per cent c) 0 0.5 0.75 1.25 2.5

.6

L.E. radius: 0.240 per cent

c

NACA 65-006 Basic Thickness Form

f'HEORY OF WING SECTIONS

360

~ 1

,

1/' -.001 lJp~r sur'a~

0 1.2

~

I I """'ilIlIiiii

~IH LtItIWr ~1ICtI

~

"'" r-,

.1 NACA 85-008

·4 ~

~

-

r---........

.....

~

a

.4

x 11 (per cent c) (per cent c)

Z/c

1.0

.8

.11

(V/V)2

V/V

Avo/V

0 0.627 0.756 0.945 1.267

0 0.978 1.010 1.043 1.086

0 0.989 1.005 1.021 1.042

3.695 2.010 1.696 1.340 0.956

5.0 7.5 10 15 20

1.745 2.118 2.432 2.931 3.312

1.125 1.145 1.158 1.178 1.192

1.061 1.070 1.076 1.085 1.092

0.689 0.560 0.477 0.382 0.323

25 30 35 40 45

3.599 3.805 3.938 3.998 3.974

1.203 1.210 1.217 1.222 1.226

1.097 1.100 1.103 1.105 1.107

0.281 0.248 0.221 0.199 0.178

50 55 60 65 70

3.857 3.638 3.337 2.971 2.553

1.222 1.193 1.163 1.130 1.094

1.105 1.092 1.078 1.063 1.046

0.160 0.145 0.128 0.113 0.098

75 80 85

2.096 1.617 1.131 0.664 0.252

1.055 1.014 0.971

0.873

1.027 1.007 0.985 0.961 0.934

0.084 0.072 0.059 0.044 0.031

0

0.817

0.904

0

0 0.5 0.75 1.25 2.5

90

95 100

(i923

L.E. radius: 0.434 per cent c

NACA 65-008 Basic Thickness Form

APPENDIX I I.G

I /e I -.IM

1.2

361 r I I tP/HIr ",'act1

f

C\ ,...-r

~

~~

I'-

~.D6

I -Low.- .ur'ac.

f'

V

~

.8 /MeA

ts5-009

~

r----

I000o......

o

L---

--

r--- I....

-

~

1.0

.8

.2

x y (per cent c) (per cent c)

"' r-,

(vlV)2

vjV

I1v.IV

0 0.700 0.845 1.058 1.421

0 0.945 0.985 1.037 1.089

0 0.972 0.992 1.018 1.044

3.270 1.962 1.655 1.315 0.950

5.0 7.5 10 15 20

1.961 2.383 2.736 3.299 3.727

1.134 1.159 1.177 1.200 1.216

1.065 1.077 1.085 1.095 1.103

0.687 0.660 0.477 0.382 0..323

25 30 35 40

4.050 4.282 4.431 4.496 4.469

1.229 1.238 1.246 1.252 1.258

1.109 1.113 1.116 1.119 1.122

0.280 0.248 0.220 0.198 0.178

4.336 4.086 3.743 3.328 2.856

1.250 1.220 1.185 1.145 1.103

1.118 1.105 1.089 1.070 1.050

0.160 0.144 0.128 0.111 0.097

2.342 1.805 1.260 0.738 0.280

1.059 1.013 0.963 0.912 0.856

1.029 1.006

0.981 0.955 0.925

0.084 0.071 0.059 0.044 0.030

0

0.797

0.893

0 0.5 0.75 1.25 2.5

45

50 55 60 65 70 75

80 85 90

ss 100

L.E. radius: 0.552 per cent c

NACA 65-009 Basic Thickness Form

I

0

THEORY OF WING SECTIONS

362 1.4

1.2

.8

I

--­ --­ (/

)\

r

t

--::: :::-­

~ \08

I

J

I

/ ', •.08 IJpptIr 6Ur'ace

,

~

~

LDWf!r

Sur~DC4P

~

""'--,

NACA 65-010

,--.............

- r--­

­­ .z

(J

y

x

.4

--­

~

.s

I"'---­ ~~

.8

/.

(V/V)2

v/V

l:.VA/V

2.5

0 0.772 0.932 1.169 1.574

0 0.911 0.960 1.025 1.085

0 0.954 0.980 1.012 1.042

2.967 1.911 1.614 1.292 0.932

5.0 7.5 10 15 20

2.177 2.647 3.040 3.666 4.143

1.143 1.177 1.197 1.224 1.242

1.069 1.085 1.094 1.106 1.114

0.679 0.558 0.480 0.383 0.321

25

4.503 4.760 4.924 4.996 4.963

1.257 1.268 1.277 1.284 1.290

1.121 1.126 1.130 1.133 1.136

0.280 0.248 0.222 0.199 0.179

70

4.812 4.530 4.146 3.682 3.156

1.284 1.244 1.202 1.158 1.112

1.133 1.115 1.096 1.076 1.055

0.160 0.141 0..126 0.110 0.097

75 80 85 90 95

2.584 1.987 1.385 0.810 0.306

1.062 1.011 0.958 0.903 0.844

1.031 1.005 0.979 0.950 0.919

0.082 0.070 0.058 0.045 0.030

0

0.781

0.884

0

(per cent c) (per cent c) 0 0.5 0.75 1.25

30 35 40 45 50 55 60 65

100

L.E. radius: 0.687 per cent c NACA 66-010 Basic Thickness Form

APPENDIX I

It---+--t-----i~

363

NACA 6~-OI2

.41----+---+-~f--__+_-_+_____f-__+_-_+___t-"'"""1

o

.2"

x

y

.4

1.0

.8

.6

X/C

(t,/lr)~

v/1T

&toll"

0 0.923 1.109 1.387 1.875

0 0.848 0.935 1.000 1.082

0 0.921 0.967 ].000 1.040

2.444 1.776 1.465 1.200 0.931

5.0 7.5 10 15 20

2.606 3.172 3.647 4.402 4.975

1.162 1.201 1.232 1.268 1.295

25 30 35 40

5.406

1.316

5.;16

1.332

5.912 5.997 5.949

1.343 1.350 1.357

(per cent c) (per cent c)

0 0.5 0.75 1.25 2.5

45 50 55

iI

I

1.078 1.096 1.110 1.126 1.1.38

1.147 1.154 1.159 1.162 1.165

0.282 0.251 0.223 0.204 0.188

1.159 1.138 1.115

60 65

70

3.743

1.295 1.243 1.188 1.134

75 80 85 90 95

3.059 2.345 1.630 0.947 0.356

1.073 1.010 0.949 0.884 0.819

1.036 1.005 0.974 0.940 0.905

0

0.748

0.865

100

I

0.702 0.568 0.480 0.389 0.326

5.757 5.412 4.943 4.381

1.343

1

1.065

0.169 0.145 0.127 0.111 0.094 0.074 0.062 0.049 0.038 0.025

1.090

L.E. radius: 1.000 per cent c

NACA. 65 1-012 Basic Thickness Form

I

0

THEORY OF WING SECTIONS

364

c, •. 22 ~ surl'oce

o

.2

x

'Y

8

6

1.0

(V/V)2

V/V

Av./l

0 1.124 1.356 1.702 2.324

0 0.654 0.817 0.939 1.063

0 0.809 0.904 0.969 1.031

2.038 1.729 1.390 1.156 0.920

5.0 7.5 10 15 20

3.245 3.959 4.555 5.504 6.223

1.184 1.241 1.281 1.336 1.374

1.088 1.114 1.132 1.156 1.172

0.682 0.563 0.487 0.393 0.334

25

6.764 7.152 7.396 7.498 7.427

1.397 1.4:18 1.438 1.452 1.464

1.182 1.191 1.199 1.205 1.210

0.290 0.255 0.227 0.203 0.184

7.168 6.720 6.118 5.403 4.600

1.433 1.369 1.297 1.228 1.151

1.197 1.170 1.139 1.108 1.073

0.160 0.143 0.127 0.109 0.096

3.744 2.858 1.977 1.144 0.428

1.077 1.002 0.924 0.846 0.773

1.038 1.001 0.961 0.920 0.879

0.078 0.068 0.052 0.038 0.026

0

0.697

0.835

0

(per cent c) (per cent c) 0 0.5 0.75 1.25 2.5

30 35

40 45

50 55 60

65 70 75 80 85

90 95

100

L.E. radius: 1.505 per cent c X ACA. 65r015 Basic Thickness Form

Y

APPENDIX I

(J

.4

x

JI

(per cent c) (per cent c) 0 0.5 0.75 1.25 2.5

0

1.337 1.608 2.014 2.751

365

.s

z/c

14

(v/V)~

v/1T

j.r:./V

0 0.625 0.702 0.817 1..020

0 0.i91 0.838 0.. 904 1..010

1.746 1.437 1.302

1.123 0.858

5.0 7.5 10 15 20

3..866

4.733 5.457 6.606 7.476

1.192 1.275 1.329 1.402 1.452

1..092 1.129 1.153 1.184 1.205

0.650 0.542 0.474

25

1.488 1.515 1.539 1.. 561 1.578

1.220

45

8.129 8.595 8.886 8.999 8.901

0.285 0.251 0.225 0.203 0.182

50

8..568

55

1.526 1.440 1.353 1.262

1.235 1.200

65 70

8.008 7.267 6.395 5.426

75

4.396

80

30 35 40

60

85 90

95

100

1.170

0.. 385 0.327

1.231 1:241 1.249 1.256

0.157 0.137 0.118 0.104

1.163 1.123 1.082

3.338 2.295 1.319 0.490

1.076 0.985 0.896 0.813 0.730

1.037 0.992

0

0.657

0.811

0.087 0.074 0.062 0.050 0.039 0.026

0.947 0.902 0.854

L.E. radius: 1.96 per cent c

NACA 65r 0 18 Basic Thickness Form

I

0

THEORY OF ,"lING SECTIONS

366 2.0

1.2

.8

»:

'I I ...--- IV ~

x

I

----

[\ /~ ~

~ r--"\

-: or / v / / \.J/4 I IV

a

I

I

\ . / 1 :1 -.-I/. ~ svr'oce

I 1.6

I

I

r:

<,

LDWtlr

~

'" ~~~

NACA 65. -OZI

- --- ---.r---

-- ---

~

y

"~

$urf"~ <,

{U/V)2

l---'

~

L-----

1.0

.8

f'/l~

(~r~nt~ 1_(p_e_r_~_n_t_~_I_~~~_~~~~~I!~~~~1 o

o

o

o

0.5 0.75 1.25 2.5

1.522 1.838 2..301 3.154

0.514 0.607 0.740 0.960

0.. 717 0.779 0.860 0.980

5.0

4.472 5.498 6.352 7.700 8.720

1.186 1.293 1..371 1..469 1.533

1.137 1.171 1.212

9.487 10.036 10.375 10.499 10.366

1.580 1..621

9.952 9.277 8.390 7.360 6.224

1.633

5.024.

1.073

3.800 2.598 1.484 0.546

0.970 0.872 0.778 0.694

o

0.616

7..5 10

15 20 25

30 35

40 45

50.

55 60

65 70 75 80

85 90 95

100

1.654

1.680 1..700 1.508

1.397 1.286

1.177

1.531 1.333 1.215 1.062 0.838

1.238

0.649 0.544 0.478 0.388 0.330

1.257 1.273 1.286 1.296 1.304

0.289 0.255 0.229 0.206 0.184

1.278 1.228 1.182 1.134 1.085

0.158 0.139 0.120 0.101 0.087

1.036 0.985 0.882 0.833

0.073 0.058 tl.047 0.035 0.020

0.785

o

1.089

0.934

L.E. radius: 2.50 per cent c NACA 65.-021 Basie-Thickness Form

APPENDIX I

367

1.6 0 I

L2.

Cl

/

=.01 Upper surface

,!,

~

'""""IIIIl

~-

r----. __

r-,

"a/lower surface I

.8

NACA6SAOO6

~

r--..

.4-

()

LO

.8

.6

x/c

%

Y

!

(per cent c). (per cent c)

I

(v/1T )2

!

v/V

0

0

0 0.5 0.75 1.25 2.5

0.718 0.981

5.0

1.313

7.5

1.591

10

15 20 25

0..164 O.5ft3

40

45

.2.992

50

2.925 2.793 2.602 2.364 2.087

35

55 60

65 70 75 80 85

1.775

1.029

1.039

1.139 1.145 1.149 1.153 1.157 1.159

1.070 1.072 1.074 1.076 1.077

1.157 1.141

1.07(;

1.131

-

II

I

4.879 2.145 1.763 1.365 0.966 0.688 0.562 0.480 0.382 0.323

I

t I

!

ii !

0.278 0.246 0.219 0.198 0.178

o.isu

I

1.124

1.()(i8 1.06U

,I

1.106 1.083

1.052 1.041

0.143 0.127 0.112 0.099

i

1.059 1.032 0.973 0.936

1.029 1.016 1.001 0.986 0.967

0.076 0.061 0.047 0.033

0

0

0

I

95

1.437 l.OS3 0.727 0.370

100

0.013

I

90

0 1.017 1.021

1.049 1.055 1.058 1.063 1.067

1.101 1.112 1.120

1.824 2.194 2.474

2.687 2.842 2.945 2.996

30

1.034

1.043 1.058 1.080

4v./V I

-~_._---

i

1.003

L.E. radius: 0.229 per cent c T.E. radius: 0.014 per cent c NACA 65AOO6 Basic Thickness Form

OJl87

THEORY OF WING SECTIONS

368

L6

0\

,'" cl-.05 Upper surface I T .........

.8

t?~5 lower surface r

~

......... ...............

r-,

WACA65AOO8

A

---

~

r---- ............

o

.4

I I

.6

lO

~/c i

% Y (per cent c) (per cent c)

0 0.5 0.75

I---...

(V/V}2

v/V

I

Av./V

!

0 0.973 1.001 1.038 1.088

0 0.986

2.5

0 0.615 0.746 0.951 1.303

5.0 7.5 10 15 20

1.749 2.120 2.432 2.926 3.301

1.127 1.145 1.157 1.175 1.186

1.062 1.070 1.076 1.084 1.089

25

3.585 3.791 3.928 3.995 3.988

1.195 1.202 1.207 1.213 1.217

1.093 1.096 1.099 1.101 1.103

0.279 0.247 0.219 0.198 0.178

3.895 3.714 3.456 3.135 2.763

1.214 1.191 1.167 1.108

1.102 1.091 1.080 1.067 1.053

0.161 0.144 0.128 0.112 0.098

95

2.348 1.898 1.430 0.960 0.489

1.076 1.041 1.002 0.961 0.916

1.001 0.980 0.957

100

0.018

0

0

1.25

30 35 40

45 50

55 60 65

70

75 80

85 90

i.iss

1.000

~

1.019 1.043

1.037 1.020

L.E. radius: 0.408 per cent c T.E. radius: 0.020 per cent c

i I

I

3.698 2.010 1.693 1.333 0.954 0.685 0.561 0.479 0.382 0.322

0.086 0.073 0.060 0.046

C

NACA 6SAOOS Basic 'rhickness Form

APPENDIX I

369

L6 ,e, = ./0 Upper surface I

~

0

,

0.V-

r[

~

.-. """"""'"'

~

~

~ .......

"'lIIIIIIlI

"'JO lower surf"ace

~

INAcA 6SA010

-

~~

~ t--...

o %

.2

.4

x/c

II

y

(per cent c) (per cent c)

o

0

0.5 0.75 1.25 2.5

0.765 0.928 1.183 1.623

5.0 7.5 10

15

20

2.182 2.650 3.040 3.658 4.127

25

35

40

4.483 4.742 4.912 4.995

45

4.983

50

55

4.863 4.632 4.304 3.899 3.432

30

60

65

70

90

95

2.912 2.352 1.771 1.188 0.604

100

0.021

75

80

85

I I I

I I

,I I

I

i

I

i

I

I I

I

I

I

I I

r--- ~-

-

.6

.8

(V/V)2

v/V

0 0.897 0.948 1.010 1.089

0 0.947 0.974

1.148 1.176 1.194 1.218 1.234

1.071

1.247 1.257 1.265 1.272 1.277

1.117 1.121 1.125 1.128 1.130

1.271 1.241 1.208

1.127 1.114 1.099 1.083

1.172

J.O

,

Av./V

I

2.987 1.878 1.619 1.303 0.936

1.084 1.093

I

1.104 1.111

I

0.679 0.559 0.478 0.382 0.323

I

I

1.005

1.044

1.133

1.064

1.091 0.999 0.949 0.893

1.045 1.023 0.999 0.974 0.945

0

0

1.047

~~

L.E. radius: 0.639 per cent c T.E. radius: 0.023 per cent c

NACA 65AOIO Basic Thickness Form

I

!

0.281 0.249

0.222

0.198 0.178 0.161

0.144 0.127 0.111

0.097

0.084 I

I 0.071

1

0.058 OJM5

0.029

I

0

THEORY OF WING SECTIONS

370 L6

u

.-I C, ·./S Upper surface

- <. . (/

,...O~

r(

~

~

~

~

~~ ~

.8 I

~ ....

~

" ./5 lower surface

-,

NACA6~AOI2

~

r'----

o

~

.2

--- -------- ----------

-

I--""""'"

.6

A

1.0

.8

-x/a

I,

y

(V/V)2

v/V

AVa/I'

0 0.913 1.106 1.414 1.942

0 0.824 0.883 0.969 1.081

0 0.908 0.940 0.984 1.040

2.520 1.757 1.543

5.0 7.5 10 15 20

2.614 3.176 3.647 4.392 4.956

1.166 1.204 1.228 1.263

25 30

5.383 5.693 5.897 5.995 5.977

%

(per cent c) (per cent c) 0 0.5 0.75 1.25 2.5

i.ses

0.914

1.285

1.097 1.108 1.124 1.134

0.672 0.557 0.477 0.382 0.324

1.301 1.313 1.324 1.332 1.338

1.141 1.146 1.151 1.154 1.157

0.281 0.250 0.224 0.198 0.178

5.828 5.544 5.143 4.654 4.091

1.329 1.292 1.251 1.156

1.153 1.137 1.118 1.097 1.075

0.161 0.143 0.126 0.111 0.096

90 95

3.467 2.798 2.106 1.413 0.719

1.104 1.051 0.994 0.936 0.871

1.051 1.025 0.997 0.967 0.933

0.082 0.069 0.057 0.043 0.027

100

0.025

0

0

0

35 40 45 50 55 60 65

70 75 80 85

1.204

1.080

L.E. radius: 0.922 per cent c 1'.E. radius: 0.029 per cent c NACA 65 1A012 Basic Thickness Form

APPENDIX I _/~ Cl

L6

J,:

r ~

~

-r-,

~

l,...---­

~

7 i>

.4

~~

"" ~

1A fl ~

-r--- r---. L-.­ ~

.2

~

11

0 1.131 1..371 1.750

2.412

r---- "'"-­ ~

~

lO

.8

x/c

(per cent c) (per cent c) 0 0.5 0.75 1.25 2.5

~

WACA 652 .40 15 ~,....

o

~

' ... .22 lower surrac«

[,....­ ~ I"----........

%

=.22Uppersurface

'"--.

/

.8

371

(v/V)%

vlV

&J./V

0 0.714 0.781 0.891 1.059

0 0,,845 0.884 0.944 1.029

2.048 1.586 1.417 1.195 0.880

1.089 1.115 1.131 1.152 1.166

0.660 0.553 0.476

0.282 0.252 0.227

5.0 7.5 10 15 20

3.255 3.962 4.553

1.187 1.243

5.488

1.328 1.359

25

45

6.734 7.122 7.376 7.496 7.467

1.401 1.416 1.427 1.437

1.176 1.184 1.190 1.195 1.199

50 55 60 65 70

7.269 6..903 6.393 5.772 5.063

1.419 1.368 1.311 1.249 1.186

1.191 1.170 1.145 1.118 1.089

0.161 0.142 0.124 0.109

75

4.282

3.451 2.598

95

0.887

1.123 1..056 0.986 0.913 0.841

1.060

80 85 90

1.028 0.993 0.956 0.917

0.080 0.067 0.055 0.041 0.026

100

0.032

0

0

0

30 35 40

6.198

1.743

1.280

1.383

0.382 0.326

0.204­ 0.181

0,(194

L.E. radius: 1.446 per cent c T.E. radius: 0.038 per cent c ....,.. • ,....

,. . . . n . . . " ' ..

~

_

~

:_1

'T."

_

THEORY OF WING SECTIONS

372 /.6

/C'i 01 1.2

uprr

Surface

JO

~-

--~,

/ Lower .urface

~'"

.8 NACA 68-006

.4

--­

-~

o

.2

%

11

(per cent c) (per cent c) 0 0.5 0.75 1.25

2.5 5.0 7.5 10 15 20

,

25

30

35 40 45 50 55

60

65 70

75 80 85 90

95

tOO

I I I

I

.4

X/C

(V/V)2

1.0

8

6

v/IT

Av./V

0 0.461 0.554 0.693 0.918

0 1.052 1.057 1.062 1.071

0

1.257 1.524 1.752 2.119 2.401

1.086 1.098 1.107 1.119 1.128

1.042

2.618 2.782 2.899 2.971 3.000

1.133 1.138 1.142 1.145 1.148

1.064 1.067 1.069 1.070 1.071

2.985 2.925 2.815 2.611 2.316

1.151 1.153 1.155 1.154 1.118

1.073 1.074 1.075 1.074 1.057

0.161 0.145 0.130 0.116 0.102

1.953 1.543 1.107 0.665 0.262

1.081 1.040 0.996 0.948 0.890

1.040 1.020 0.998 0.974 0.943

0.089 0.075 0.061 0.047 0.030

0

0.822

0.907

0

1.026

1.028 1.031 1.035

I I

I

1.048

1.052 1.058

1.062

L.E. radius: 0.223 per cent c

NACA 66-006lJasic Thickness Form

I

I

I ! /

I

4.941 2.500 2.020 1.500 0.967 0.695 0.554 0.474 0.379 0.320 0.278 0.245 0.219 0.197 0.178

373

APPENDIX I

'41'!1

~~,.

/0

(;)' I(~

~

1~03 LO~

03 LPPer surfoce

.s:

"""lIIIlIil

~~

-=

"' <,

MACA 66-D08

--

l...-­

r--- ........ .4

%

11

X/C

r-­ ~

.....

.~

1.0

j

{V/V)2

»rv

0 0.610 0.735 0.919 1.219

0 0.968

0 0.984 1.011 1.023 1.038

6.0 7.5 10 15 20

1.673 2.031 2.335 2.826 3.201

1.107 1.128 1.141 1.1SS 1.171

1.052 1.062 1.076 1.082

0.552 0.474 0.379 0.321

25

3.490

30 36

3.709 3.865

40

3.962

46

4.000

1.178 LI86 1.191 1.196 1.201

1.085 1.089 1.091 1.094 1.096

0.278 0.246 0.220 0.198 0.178

1.098

0.161 0.145 0.130 0.115 0.101

(per cent c) (per eent e)

0 0.5 0.75 1.25 2.5

1.023

1.046 1.078

1.068

liO

3.978

55

3.896

80

3.740 3.459

1.205 1.208 1.213 1.202

3.062

1.156

1.075

2.574 2.027 1.447 0.864 0.338

1.103 1.048 0.989 0.926 0.855

1.050 1.024

0.994 0.962 0.925

0

0.768

0.876

6S 70

75 80 85

90 IS

100

i

Av./J-· 3.794 2.220 1.825 1.388 0.949

I I

1.099

1.101 1.096

0.689

0.087

I

-_1--- ______

L.E. radius: 0.411 per cent c

NACA 66-008 aasic Thickness Form

0.073 0.058 0.045 0.029

0

THEORY OF WING SECTIONS

374

zo

f.Z

¥,C, •• OS ttJper eurface

I'

~ (t)2 ~ V .8

~

I

~~

~05 tppM­ .ur'~

~

"" r-, f44C4 66-oot/

.4 -r----

l"...--'

-

r-­ '""--­

a

.4

x

y

~/C

~

--­---­

.8

.6

1.0

wv»

fJ/V

~./V

0 0.687 0.824 1.030 1.368

0 0.930 0.999 1.036 1.079

0 0.964 0.999 1.018 1.039

3.352 2.100 1.750 1.340 0.940

5.0 7.5 10 15' 20

1.880 2.283 2.626 3.178 3.601

1.119 1.142 1.159 1.178 1.190

1.058 1.069 1.077 1.085 1.091

0.686 0.552 0.473 0.379 0.323

2S 30 35 40 45

3.927 4.173 4.348 4.457 4.499

1.201 1.210 1.217 1.221 1.228

1.096 1.100 1.103 1.105 1.1OS

0.280 0.246 0.220 0.197 0.178

50

4.475 4.381 4.204 3.882 3.428

1.232 1.237 1.240 1.230 1.172

1.110 1.112 1.114 1.109 1.083

0.161 0.145 0.130 0.116 0.100

2..877 2.263 1.611 0.961 0.374

1.113 1.050 0.985 0.915 0.839

1.055 1.025 0.992 0.957 0.916

0.085 0.071 0.057 0.043 0.028

0

0.747

0.864

0

(per cent c) (per cent c)

0 0.5 0.75 1.25 2.5

55 60 65

70

75 80

85 90

95

100

L.E. radius: 0.530 per cent c NACA 66-009 Basic Thickness Form

APPENDIX 1

. .........'-~

I. ~

I.

I"

I

375

I

I

/~, -.07 ~ Arf~ ~

~

K

-­I

I

r-.o71.-wr " " c .

I

~

f'~

'" r-,""

,~

"-.­

M-OJ(}

.4

L.o-- ....-­

r--- r--.. ,........ .-- I - -

r---- .........

I.D

.B

%

11

(per cent c) (per cent.c)

(rJ/V)2

v/IT

~.IV

0 0.5 0.75 1.25 2.5

0 0.759 0.913 1..141 1.616

0 0.896 0.972 1.023 1.078

0 0.947 0.986 1.011 1.038

3.002 2.012

5.0 7.5

2.087

1.125 1.154 1.174 1.198 1.215

1.061 1.074 1.084: 1.095 1.102

0.682 0.551 0.473 0.379 0.322

4.363

1.226

4.6.36

1.236 1.243 1.249 1.255

1.107 1.112 1.115 1.118 1.120

0.279 0.246 0.220 0.198 0.178

1.261 1.265 1.270 1.250 1.190

1.123 1.125 1.127 1.118 1.091

0.161 0.146 0.130 0.114 0.099

to IS 20

25 30 35 40

45 SO 55 60 65

70 75 80 85

90 95

100

2.536 2.917 3.630 4.001

4.&12 4.953 S.OOO 4.971 4.865 4.665 4.302

3.787

1.686 1.296

0.931

3.176 2.494: 1.773 1.054 0.408

1.121

1.059

1.052

1.026

0.979 0.904 0.821

0.989 0.951 0.906

0.085 0.070 0.056 0.043 0.027

0

0.729

0.854

0

L.E. radius: 0.662 per cent c NACA 66--010 Basic Thickness Form

THEORY OF WING

376

I. 6

I. 2

8ECTIONl~

I I I _/~, -./Z ~psr l~urFac.

--­r: ---,(J

IF

~ V< 'r--:IZ ILow."I I ~

-~

,~

:wr,~

~

)1I

~

'"

/MCA tiIia-O/2

~

-

~

~ t---..L...

o

-4

x (per cent c)

11

5.0 7.5 10 15 20

I

I

25 30

35 40

45 50 55 60 65

70 75 80 85

90 95

I

t--.....

I

xc

1.0

.8

.6

(V/V)2

V/V

s1v./V

0 0.906 1.087 1.358 1.808

0 0.800 0.915 0.980 1.078

0 0.894 0.957 0.990 1.036

2.569 1.847 1.575 1.237 0.913

2.496 3.037 3.496 4:.234 4.801

1.138 1.177 1.204 1.237 1.259

1.067 1.097 1.112 1.122

0.674 0.549 0.473 0.380 0.323

5.238 5.568 5.803 5.947 6.000

1.275 1.287 1.297 1.303 1.311

1.129 1.134 1.139 1.142 1.145

5.965 5.836 5.588 5.139 4.515

1.318 1.323 1.331 1.302 1.221

1.148 1.150 1.154 1.141 1.105

3.767 2.944 2.083 1.234 0.474

1.139 1.053 0.968 0.879 0.788

1.067 1.026 0.984­ 0.938 0.888

0

0.687

0.829

(per

0 0.5 0.75 1.25 2.5

I'--- r---

_I----­ ~ ~

cent c)

1.085

f

!

i

0.280 0.246 0.221 0.197 0.176

0.162 0.147 0.132 0.113 0.098 0.084 0.069 0.053 0.040 0.031

I

100

L.E. radius: 0.952 per cent. c NACA 66t-012 Basic Thickness Form

I

0

APPENDIX I

377

-., /.~ M

l,...-C",•• '

I./pp#Ir :!IU"fOC.

1

~

1 ~

1.1

-

0/

~J

(

I

21#:Per~

~II NACA

~

~~ ~~

I

,

,~

-r/,

i->

--"-

661 -0 15

r--- to---

1----

t---

o

~

.2

I

.4

z c) (per cent Y c) (per cent

i

X/C

.6

----

~

r--- "--~

~

1.0

.8

I

("'IF):

to/I'

0 0.760 0.840 0.929 1.055

0

2.139

0.872 0.916 0.9641.027

1.652

j

..1"./1'

0 0.5 0.75 1.25 2.5

0 1.122 1.343 1.675 2.235

~.O

7.5 10 15

3.100 3.781 4.358 5.286

20

~.995

1.288 1.317

1.078 1.099 1.114 1.134 1.148

25

6.543 6.956 7.250 7.430 7.495

1.340 1.356 1.370 1.380 1.391

1.158 1.164 1.170 1.175 1.179

7..450 7.283 6.959 6.372 5.576

1.401 1.411 1.420 1.367 1.260

1.184 1.188 1.192 1.169 1.122

4.632 3.598 2.530 1..489 0.566

1.156 1.053 0.949 0.841 0.744

1.075 0.974 0.920 0.863

0.080 0.065 0.051 0.039 0.025

0.639

0.799

0

30 35 40

45 50

55 60

65 70 75 80 85

90 95

100

1.163 1.208

1.242

0 ~~---_

... -_._......-- ..... _- ­

1.026

L.E. radius: 1.435 per cent c

N ACA 66::-015 Basic Thickness Form

j I

I I I

I

!

t

I

i

1.431 1.172 0.895

0.663 0.047 0.473 0.381 0.322 0.280 0.248 0.222 0..200 0.180

0.163 0.146 0.131 0.113 0.096

7.'HBORY OF WINO .SECTIONS

378

% 11 (per oeot c) (per cent c)

(fJ!V)I

fJ/V

MJ./v

0 0.806 0.857

1.773 1.456 1.312 1.121

0 0.5 0.15 1.25

0 1.323 1.571 1.952

2.5

2.646

1.0 7.5 10

3.690 4:.613 5.210

16

20

8.333 7.las

1.393

1.134: 1.182 1.1SO

25 30 3& 40 4&

7.848 8.348 8.101 8.918 8.998

1.423

1.193

1.4:45

1.202

1.484 1.481

1.210 1.217 1.223

SO IS 60

8.942

8.733 8.323

65 70

7.580 6.197

1.S09 1.522 1.534 1.438 1.302

75 80

5.4:51

8S

90 95 100

0 0.650

0.735 0..850 1.005 1.114 1.234 1.285 1.350

1.496

0.897 1.002 1.074

1.111

1.228 1.234

1.238 LI99 1.141

0..858

0.649 0.645 0.472 0.381 0.323 0.282 0.250 0.223 0.201 0.181 0.163 0.147 0.131 0.114 0.095

0.848

1.172 1..046 0.922 0.803 0.692

0.896 0.832

0.077 0.061 0.048 0.037 0..022

0

0.S87

0.766

0

4.206 2.934 1.714

1.083 1.022 0.9S0

L.E. radius: 1.955 per cent c

NACA 66r018 Basic Thickness Form

APPENDIX I

.4 z JI (per cent c) (per cent c) 0

0

0.5 0.75 1.25 2.5

1.525 1.804

5.0 7.5 10 15 20 25

30 35 40 45

SO 55 60 65

70

75

2.240 3.045

4.269

Z/c

(V/V)J

0 0.580 0.635 0.755 0.952

0 0.~1

0.797 0.869 0.976

&1.1 V 1.547 1.314 1.218 1.054

0.828

6.052 7.369 8.376

1.148 1.185 1.208

9.153 9.738 10.1M 10.407 10.500

1.499 1.528 1.551 1.574 1.594

1.224 1.236 1.245 1.255 1.263

.0.251 0.224 0.202 0.183

1.611 1.629 1.648

1.269 1.276 1.284

10.434 10.186 9.692 8.793 7.610 6.251 4.796 3.324

1.069 1.116

0.283

1.508

1.228

1.335

1.155

0.165 0.148 0.132 0.114 0.093

1.084

0.073

1.015 0.944 0.873 0.805

0.058 0.046 0.034 0.020

0.734

0

90

1.924

0.717 0

0.539

100

_IV

0.635 0.542 0.472 0.381 0.324

95

85

1.0

.•

1.143 1.2-&6 1.318 1.405 1.459

5.233

1.176 1.031 0.891 0.763 0.648

80

379

L.E. radius: 2.550 per cent c NACA 66r021 Basic Thickness Form

THEORY OF WING

380

~ECTIONS

2.D I

/e, • .J2 1.6

1.2

(v)' 8

(/ e­

t.

/ ~

~

I---~

t=-­

~

-

~

~

~

,~

/MeA

t->

I

svrFac»

i"'"-./2 LorNtN'" .,ftl~

r/ I

.4

o

~

I Uppttr

-----­ --­ .z

x

'!J

(per cent c) (per cent c)

-,

67,1-015

r---~

., x/c' (V/V)2

-----.. r--..... ~

~

/.0

.8

6

»rv

~G/V I

0 0.5 0.75 1.25

2.5 5.0 7.5 10 15 ~ 25 30

35 40 45 50

55 60

65 70 75 80 85

90 95

100

I

0 1.167 1.394 1.764 2.395

0 0.650 0.970 1.059

0 0.806 0.985 1.029

1.140

1.068

2.042 1.560 1.370 1.152 0.906

3.245 3.900 4.433 5.283 5.940

1.209

1.100 1.113 1.122 1.134 1.142

0.667 0.548 0.470 0.370 0.312

6.404 6.854 7.155 7.359 7.475

1.318 1.330 1.341 1.851 1.360

1.148 1.153 1.158 1.162 1.166

0.276 0.248 0.221 0.201 0.180

7.497 7.421 7.231 6.905 6.402

1.368 1.375 1.381 1.388 1.390

1.170 1.173 1.175 1.178 1.179

0.160 0.142 0.124 0.111 0.108

5.621 4.540 3.327 2.021 0.788

1.321 1.176 1.018 0.864 0.712

1.149 1.084 0.930 0.844

0.094 0..071 0.060 0..045 0.025

0

0.570

0.755

0

1.239 1.259 1.285 1.304

1.009

L.E.. radius: 1.523 per cent c NACA 67,1-015 Basic Thickness Form

II

APPENDIX I 2.0

381

r--....,.-...,...-.....--r--.......­ .......

-....-~

~l

............_ ....

I aurtlce =.22 Upper

,r-oIo---+--I-L---­ ......K~

•8

~

II '(

.. . . .

I .IJ--;---t----ir--f--+---+--+--J----+_-I

----I--­

'-........r--­ r----I-_.... _ _I---,-­ .2

.4

xlc. 6

~

.8

1.0

j

%

11

i

(per cent c) (per cent r) i - - - - - -.._ -­ ~_._.---

(I'll')"! --..

-"--~"---

~t~CI/l!

1,'/1"

- _.._ ­

..

---

---

()

0

()

0.5 0.75

1.199

1.25

1.801 2.462

0.660 0.799 0.942 1.100

0 0.812 0.894 0.971 1.049

2.028 1.680 1.560 1.325 0.990

6.384

1.201 1.259 1.295 1.339 1.369

1.096 1.122 1.138 1.156 1.170

0.695 0.551 0.465 0.383 0.324

6.898 7.253 7.454 7.494 7.316

1.390 1.409 1.42"& 1.435 1.391

1.179 1.187 1.193 1.198 1.179

0.283 0.252 0.224 0.199 0.176

7.003 6.584 6.064 5.449 4.738

1.348 1.306 1.265 1.221 1.178

1.161 1.143 1.125 1.105 1.085

0.156 0.138 0.122 0.108

3.921 3.020 2.086 1.193 0.443

1.115 1.027 0.938 0.852 0.774

1.056 1.013 0.969 0.923 0.880

0

0.703

0.838

1.435

2.5 5.0 7.5 10 15 20 25 30 35 40

3.419 4.143

4.743

5.684

I I I

45 50

55 60 65 70 75 80 85 90

95

100

I

i

! i

I t

I

I I

---_._-

I

I

0.093 0.079 0.065 0.052

O.MO

I

I

L.E. radius: 1.544 per cent. c

NACA 747A015 Basic Thickness Form

0.028 0.018

APPENDIX II MEAN LINES CONTENTS

NACA Designation

62

63 . . . . .

Page

383

384

64

65

66

385

386

387

388

389

67

210 .

220 .

390

230 .

240 250 .

a91 .

392

393

394:

4==0

4

== 0.1

4

== 0.2

a == 0.3 a -= 0.4 a::s= 0.5 4 == 0.6

a == 0.7

a == 0.8 . . . . . . ....

a := 0.8 (modified for use with N ACA 6A-eeries sections)

a -= 0.9 a -= 1.0 .

382

395

396

397

398

399

400

401

4
.

403

404

405

APPENDIX II

)~

I.0

1

I

I~

383

~

i""---. r--­

r--- r--.....

-....

....

r-,

NACA 62

f1ItJt:1nJine

~

.2

D

Cl.

-

...-­

== 0.90

.4

(Xi

== 2.81

x/c 0

~

Ca e/

.

r - - to--... 1.0

.8

.6

=:I

-

0.113

I

%

I/e

(per cent c)

(per cent c)

0

0 0.726 1,.406 2.625 3.656

0.60000 0.56250 0.52500 0.45000 0.37500

0 0.682 1.031 1.314 1.503

0 0.171 0.258 0.328 0.376

10

4.500

15

5.625

1.651 1.802

20 25

6.000 5.977 5.906

0.30000 0.15000 0 - 0.00938 - 0.01875

1.113

0.413 0.451 0.383 0.318 0.279

5.625 5.156 4.500 3.656 2.625

-

0.03750 0.05625 0.07500 0.09375 0.11250

0.951 0.843 0.741 0.635 0.525

0.238 0.211 0.185 0.159 0.131

90

1.406

95 100

0.727 0

- 0.13125 - 0.14062 - 0.15000

0.377 0.261 0

1.25 2.5 5.0 7.5

30 40

50

60 70 80

I

dyc/d.c

Pa

t:J.vlV '"'" P./4

1.530 1.273

Data for NACA Mean Line 62

I I

0.094 0.065 0

THEORY all It·ING SECTIONS

384

1.0

o

I

v- -........... <,

r--­

~

---­ ----r---_ ...

~

NACA 63 ",.", line

~

~

o

.2

e'i

x

(pe~~ent c)

I

=

.4

0.80

ai

'/ X,C

= 1.60°

---

t--- r--­

.8

.6

Ca e/

4

~

1.0

= - 0.134 I

y.

PR

dYe/dx

(per cent c)

, .It'jV

=

PR/4

---~~~.-

0 0.489 0.958 1.833 2.625

0.40000 0.38333

0.36667 0.33333 0.30000

10 15

3.333

20

5.333 5.833 6.000

0.26667 0.20000 0.13333 0.06667 0

()

1.25 2.5 5.0 7.5

4.500

25 30

50 60 70 80

5.878 5.510 4.898 4.041 2.939

-

00 95 100

1".592 0.827 0

- 0.14694 - 0.15918 - 0.17143

40

I I I I

1

0.02449 0.04898 . 0.07347 0.09796 0.12245

0 0.389 0.553 0.788 0.940

0

0.097 0.138 0.197 0.235

,

1.259 1.233 1.160

0.267 0.305 0.315 0.308 0.290

0.949 0.850 0.762 0.673 0.560

0.237 0.213 0.191 0.168 0.140

0.406 0.291 0

0.102 0.073

1.066 1.220

i

I

I

I

i

I i

Data for NACA Mean Line 63

0

385

APPENDIX II

2.0

to

/

--

V

----r--­

r--­ --...........

............

( (J

r-,

NACA 64 mean line

---- ---

-

~

II

.2

e'i = 0.76 X ( per

cent c

o

)

.4

ai'

= 0.74°

I(per cent Yc ) c -------1------ ------- I I

d!l
I

I

0

1.25 0.369 2.5 0.726 5.0 1.406 7.52.039

I

10 15 20 25 30

2.625 3.656 4.500 5.156 5.625

40 50 60 70 80

6.000 5.833 5.333 4.500 3.333

90

1.833 0.958

I -

o

J

95

100

I %C

I

Data for

­

-

.6

C.r/ t

------

=-

I!

pR

I--.

1.0

0.157

,j--~V-/-V-=-P-R/--4 i

--.-- ----- ---.------ ;------_._­

O.3()()OO

0

.

0

0.29062 0.28126 0.26250 0.24375

0.257 0.391 0.546 0.668

_

I

0.064 0.098 0.137 0.167

0.22500 0.18750 0.15000 0.11250 0.07500

O.i48 0.871 0.966 1.030 1.040

I

0 0.03333 0.06667 0.10000 0.13333

0.999 0.910 0.827 0.750 0.635

0.16667 0.18333 0.20000

0.466 0.334 0

X.~C.:\

l\IC1Ul Line 64

'I

I I

I 1

0.187 0.218 0.242 0.258 0.260

0.250 0.. 228 0 ..207 0.188 0.159 0.117 0.084 0

THEORY OF WING SECTIONS

386 3D

2.0

1.0

~~ D

V

----- --

- ----..

~

'\ NACA 65

mean . .

o ----

.8

e'i == 0.75 'l/c

%

(per cent c) (per cent c) 0 1.25 2.5 5.0 7.5 10

16 20 25 30 40 50

60 70 80 90

95 100

r--- r---

L.--- ~

0

Qi

a:

00

d1lo/dz

~

1.0

c.c/4 == - 0.187

PB

AvIV

c:

Pa/4

0.296 0.S85 1.140 1.665

0.24000 0.23400 0.22800 0.21600 0.20400

0 0.205 0.294 0.413 0.502

0 0.051 0.074 0.103 0.126

2.160 3.060 3.840 4.500 5.040

0.19200 0.16800 0.14400 0.12000 0.09600

0.671 0.679 0.760 0.824 0.872

0.14.3 0.170 0.190 0.206 0.218

5.760 6.000 5.760 5.040 3.840

0.04800 0 - 0.04800 - 0.09600 - 0.14400

0.932 0.951 0.932 0.872 0.760

0.233 0.238 0.233 0.218 0.190

2.160 1.140 0

- 0.19200 - 0.21600 - 0.24000

0.571 0.413 0

0.143 0.103 0

Data for NACA Mean Line 65

APPENDIX 11

387

.0

I .IJ

tJ

~

V

...----

~

---- --

:-....

'"

r-,

\

NACA 66 I'I'N!'GJ . .

-

41!11

.--- ~ ~ .2

D

c'- == 0.76

----

~

.4

a, == -

'/ Z,t:

0.74°

t--- ~ 1.0

.8

.6

Ca e/ .

== - 0.222 ---~

%

1/1:

(per cent c)

(per cent c)

dy,:!dx

1

PR

4v/V = PR/4

~._-----_.-...-----

0 1.25 2.5 5.0 7.5

0 0.247 0.490 0.958 1.406

0.20000 0.19583 0.19167 0.18333 0.17500

0 0.135 0.244 0.334 0.408

0.034 0.061 0.084 0.102

20 25 30

1.833 2.625 3.333 3.958 4.500

0.16667 0.15000 0.13333 0.11667 0.10000

0.466 0.557 0.635 0.700 0.750

0.117 0.139 0.159 0.175 0.188

40 50 80 70 80

5.333 5.833 6.000 5.625 4.500

0.06667 0.03333 0 - 0.07500 - 0.15000

0.827 0.910 0.999 1.040 0.966

0.207 0.228 0.250 0.260 0.242

90

2.625

95

1.406

- 0.22500 - 0.26250 - 0.30000

0.748 0.546 0

0.187 0.137 0

10 15

100

0

,

Data for NACA Mean Line 66

0

I

THEORY OF WING SECTIONS

388

A .W'

I

-­...

./

oV

~

~

~

~

~

r-,

~~

\

,

NACA 67

nwon line

~

-

"

~~

x

..-­

r--...... ~ ./J

cu == 0.80 (per cent c)

~

(Xi

Yc (per cent c)

== -

1.60 0

dyc/dx

c.c/.

== - 0.266

PB

J11J/V == PB/4

0 0.212 0.421 0.827 1.217

0.17-143 0.16837 0.16531 0.15918 0.15306

0 0.137 0.195 0.291 0.356

0 0.034 0.049 0.073 0.089

10 15 20 25 30

1.592 2.296 2.939 3.520 4.041

0.14694 0.13469 0.12245 0.11020 0.09796

0.406 0.483 0.560 0.616 0.673

0.102 0.121 0.140 O.IM 0.168

40 50 60 70 80

4.898 5.510 5.878 6.000 5.333

0.07347 0.04898 0.02449 0 - 0.13333

0.762 0.850 0.949 1.160 1.259

0.191 0.213 0.237 0.290 0.315

90 95 100

3.333 1.833 0

- 0.26667 - 0.33333 - 0.40000

1.066 0.788 0

0 1.25 2.5 5.0 7.5

I

I

Data for NACA Mean Line 67

I

I

0.267 0.197 0

APPENDIX II

389

20

to

" ~

\ \ ...

...........

------.

o

--­ NACA2/0

mean/ina

2.e c o

_

....•..

.2

_----,eli

x ) (per rent c

__ _,--_.= 2.09°

= 0.30 .---.

X/c

.4

__.

a;

__ ....

.. ..

-

-

-_._.==__-

CwL~/. ...

_~

y~ e ) ! dlleld..r !I (per cent ll

1.0

.8

.6

O.OO() .~_.--;-_

Pit

..

_.-.

-_ ..... _-_ .._.-._.

i ~v/l/" == PR/4

-- - - - - ~'-_._---.:----....---- -_. 1-"·- ---1--------····­

o

t .25 2.5

5.0 7.5 10 15 20

25 30

! i

I !

-to

50 00 70 80 90 95 100

I f

0 0.596 0.928 1.114 1.087

.

II -

I

0.59613: 0.36236 0.1 S5O-l 0.00018

1.058 0.999 0.940

0.881 0.823

I I

0.705! 0.588 I Q4W 'I 0.353 0.235 0.118 0.059 0

I IJ

_ 0.01175

o

0 1.381 1.565 1.221 0.781

0.345 0.391 0.305 0.195

0.626 0.489 O.-lO8 0.348 0.302

0.156 0.122 0.102

0.242 0.198

0.061 0.049 0.040 0.032 0.025

'

Ql00 0.128 0.098 0.065

!

0.044 0

Data for N ACA :\lean Line 210

0.087 0.075

i

O.(}16 0.011

o

THEORY OF WING SECTIONS

300

0

1.0

T' r-, I'---.... " - ­

IJ NACA 220 ItI«JI7line

..

""

D

.4

Cl.

== 0.30

x 1/e (per cent c) (per cent c) 0

ai

,I sIC

== 1.86 0

dYe/dx

.8

.6

1.0

c.c/t -= - 0.010

P.

4v/V == Pa/4

0 0.442 0.793 1.257 1.479

0.39270 0.31541 0.24618 0.13192 0.04994

0 0.822 1.003 0.988 0.900

0 0.206 0.251 0.247 0.225

10 15 20 25 30

1.535 1.463 1.377 1.291 1.205

0.00024:

0.801 0.615 0.465 0.378 0.326

0.200 0.154 0.116 0.095 0.082

40

SO 60 70 80

1.033 0.861 0.689 0.516 0.344

0.253 0.205 0.169 0.135 0.100

0.063 0.051 0.042 0.034 0.025

90 95 100

0.172 0.086 0

0.064 0.040 0

0.016 0.010 0

1.25

2.5 5.0 7.5

- 0.01722

Data for NACA Mean Line 220

APPENDIX II

391

(J

I.D

1/r-, --

...............

r---­

~

NACA 23IJ lin.

IIWJt1n

,. ~

II

.

Cit

.2

== 0.30

z (per cent c) 0

(per

Yc cent c)

.4

a, ==

sIc

1.650

0.30508 0.26594 0.22929 0.16347 0.10762

10 15 20 25 30

1.701 1.838 1.767 1.656 1.546

0.06174 - 0.00009 - 0.02203

40 50 60

1.325 1.104 0.883 0.662 0.. 442

2.5 5.0 7.5

70 80

90 95

100

0.221 0.110 0

1.0

c.c/t == - 0.014

Pa

dyc/dx

0 0.357 0.666 1.155 1.492

1.25

.11

0

0.853

0 0.132 0.168 0.198 0.213

0.859 0.678 0.519 0.419 0.361

0.215 0.170 0.130 0.105 0.090

0.274 0.217 0.. 177 0.144 0.105

0.069 0.054 0.044 0.036 0.026

0.069 0.042 0

0.017 0.011 0

0.528 0.673 0.791

- 0.02208

I

AviV == PRI4

Data for NACA l\fean Line 230

THEORY OF WING SECTIONS

392

Zeo

-: ~

V o

.............

r---. ~

-­

P-............

~C;l240

m~

o

.2

ci, x

= 0.30

Ye (per cent c)

(per cent c)

.4

ai

=

'l"e

.1

%,C

1.45°

dyc/ch

.8

.6

Cr../.

=-

1.0

0.019

..lv/II' = PR/4

PR -

------­

0 0.301 0.572 1.035 1.397

0.25233 0.22877 0.20625 0.16432 0.12653

0 0.377 0.491 0.625 0.718

0 0.094 0.123 0.156 0.180

10 15 20 25 30

1.671 1.991 2.079 2.018 1.890

0.09290 0.03810 - 0.00010 - 0.02169

0.750 0.677 0.566 0.477 0.410

0.188 0.169 0.142 0.119 0.103

40 50 70 80

1.620 1.350 1.080 0.810 0.540

0.304 0.234 0.186 0.150 0.110

0.076 0.059 0.047 0.038 0.028

90 95 100

0.270 0.135 0

0.071 0.047 0

0.018 0.012 0

0 1.25 2.5 5.0 7.5

60

I

- 0.02700

Data for NA.CA Mean Line 240

393

APPENDIX II

2.0

1.0 .-1_

--..

r-............ r--­

V a

r-­

~

-~

NACA 250

",.an "".

.!.!...2 c o

-

.4,

.8

.6

1.0

SIC

--- ._..

cr,

_~~-_.-

== 0.30

at

=

1.26°

- _._--------- ----------.

..

X

Yc

(per cent c)

(per cent c) ----~

....

_---­

1

dYc/dx

_ _ _...._

.. _ .. __ -4-..--_ _,_

0 0.258 0.498 0.922 1.277

0.21472 0.19920 0.18416 0.15562 0.12909

10 15 20 25 30

1.570 1.982 2.199 2.263 2.212

40 50 60 70 80

1.931 1.609 1.287 0.965 0.644

0 1.25 2.5 5.0 7.5

90 95 100

I

I

0.322 0.161 0

c.. ~/.

=-

0.026

-------------.- f--------.- -­ --­ l~H i ~/::'1~ = P,,/4 I - ­ ­ -­ . --------- ._-­ -_.-~-

- _.-......

0 0.281 0.369 0.477 0.552

0 0.070 ().092 0.119 0.138

0.10458 0.06162 0.02674 - O.OOOOi - 0.01880

0.592 0.624 0.610 0.547 0.470

0.148 0.156 0.153 0.137 0.117

- 0..03218

0.346 0.255 0.197 0.154 0.119

0.087 0.064 0.049 0..038 0.030

0.076 0.051 0

0.019 0.013 0

! i

I i

:

Data for KAC.-\. Mean Line 250

!

THEORY OF WING SECTIONS

394

2.0

-..............

~'"

.........

<,

/.0

~

~

r-, ""­ ........

~

~

.-.

v

MACA 0-0 IIIt1t1I'I

'M

2

D

Cia sa

1.0

%

1/c

(per cent c)

(per cent c)

0

0 0.460 0.641 0.964 1.641

0.5 0.75 1.25

2.5

-

~

V

CZi

=­ 4.56 0

c.c/.

t--- t - -........

a:

-

0.083

p.

4v/V == PR/4

0.75867 0.69212 0.60715 0.48892

1.990 1.985 1.975 1.950

0.498 0.496 0.494 0.488

1.900

1.850 1.800 1.700 1.600

0.475 0.463 0.450 0.425 0.400

d1lc/d%

5.0

2.693

7.5 10

3.507 4.161 6.124 6.747

0.36561 0.29028 0.23515 0.15508 0.09693

6.114 6.277 6.273 6.130 5.871

0.05156 0.01482 - 0.01554 - 0.04086 - 0.06201

5.516 5.081 4.581 4.032 3.445

-

0.07958 0.09395 0.10539 0.11406 0.12003

0.900 0.800 0.700 0.600

0.250 0.225 0.200 0.175 0.150

75 80 85

2.836 2.217

90 95

1.013 0.467

-

0.12329 0.12371 0.12099 0.11455 0.10301

0.500 0.400 0.300 0.200 0.100

0.125 0.100 0.075 0.050 0.025

·0

- 0.07958

0

0

Ui 20

25 30

35 40

45 50

55 60 65

70

100

1.604

1.500 1.400

1.300 1.200 1.100 1.000

Data for NACA Mean Line a

=0

0.375 0.350 0.325 0.300 0.275

APPENDIX II

395

3JJ

-

20

r-; ........

~'" r-;

1.0

-..........

"'~ -.......

'"

r-;

o NACA a-al mtIQI7

IinII

.!.L.e c .2

eli -

z' (per cent c)

0 0.5

0.75 1.25 2.5 5.0

7.5 10 15 20

25 30

35 40

45 50

55 60 65 70 75 80 85 90 95 100

-r---- r--­

~

V ()

1.0

«i sa

1/e

4.43°

dYe/dx

(per cent c) 0 0.440 0.616 0.933

1.608 2.689 3.551 4.253 5.261 5.906

0.38235 0.31067 0.25057 0.16087 0.09981

6.282 6.449

0.05281 0.01498 - 0.01617 - 0.04210 - OJ)6373

5.664 5.218 4.706 4.142

c.c/.

,

a:

-

to

0.086

PB

AviV = Pll/4

1.818

0.455

1.717 1.616

0.429 0.404

1.515

0.379 0.354

I i I !

0.73441 0.67479 0.59896 0.49366

6.443 6.296 6.029

r--..~

.8

.6

3..541

-

0.08168 0.09637 0.10806 0.11694 0.12307

2.916 2.281 1.652 1.045 0.482

-

0.12644 0.12693 0.12425 0.11781 0.10620

0

- 0.08258

h 11 II

; I

I} 1

.j

jJ

lJ

;

! I

!

1.414 1.313 1.212 1.111

0.328 0.303 0.278

1.010 0.909 0.808 0.707 0.606

0..253 0.227 0.202 0.177 0.152

0.505 0.404 0.303 0.202 0.101 ..

0.126 0.101 0.076 0.050 0.025

0

0

Data for N ACA Mean Line a == 0.1

THEORY OF WING SECTIONS

396

~

~ 1.0

~

.............

o

""'

..........

~" r-;

/MeA G-D.2

'e C

mtH1f7/ine

.2

~

r - - t---

~

D

.2

.4

./.

r--- t - - -

t--

1.0

.8

.6

%,C

e'i -= 1.0

x (per cent c) 0

0.5 0.75

5.0 7.5 10 15 20

I I

I I

45

60

65 70

I I

80

85 90

95

100

I

=: -

0.094

PR

l!JJ/IT == PB/4

0

0.414 0.581 0.882

1

1.530

0.69492 0.64047 0.57135 0.47592

2.583 3.443 4.169 5.317 6.117

0.37661 0.31487 0.26803 0.19373 0.12405

1.667

0.417

6.572 6.777 6.789 6.646 6.373

0.06345 0.02030 - 0.01418 - 0.04246 - 0.06588

1.563 1.459 1.355 1.250 1.146

0.391 0.365 0.339 0.313 0.287

5.994 5.527 4.989 4.396 3.762

-

0.08522 0.10101 0.11359 0.12317 0.12985

1.042 0.938 0.834 0.729 0.625

0.260 0.234 0.208 0.182 0.156

3.102 2.431 1.764 1.119 0.518

-

0.13363 0.13440 0.13186 0.12541 0.11361

0.521 0.417 0.313 0.208 0.104

0.130 Q.I04 0.078 0.052 0.026

0

- 0.08941

0

0

I

75

c. e/ .

d1lc/dx

(per cent c)

25 30 35 40 50 55

== 4.17°

1/e

1.25

2.5

(Ii

Data for NACA Mean Line a

= 0.2

397

APPENDIX II

~~

<,

I.:

'"'~ .........

r-, <,

'"'

A

NACA 4-0.3

<,

mean line

.­ A

~

--

~

o

,l

.4

:-- r---

r---

.8

%/C

e'l

(per

== 1.0

3.84

(Ii -

0

o 0.5 0.75 1.25 2.5

5.0 7.5 10 15 20

25 30

35

40 45 50 55 60

65 70

75 80

85 90

95 100

(per

' __ - .

o

0.389

to

C.ri. == - 0.106

~~ I :nt_:~Ldy./~~~ c)

­

_~~~~~=

1'_/4

0.832 1.448

0.65536 0.60524 0.54158 0.45399

2458 3.293 4.008 5.172 6.052

O.363-t-l O.3078U O.26(i21 O.2tr24U 0.15068

6.685 7.072 7.175 7.074 6.816

0.10278 0'-»833 - 0.00205 - OJlfH92

1.319 1.200

6.433 5.949 5.383 4.753 4.076

- 0.()8746 - O.l()567

U.989

- 0.1201-4 - 0.13119 - 0.13901

0.879 0.769 O.fl59

U.275 0.2-1. 0.220 0.192• 0.165

3.368 2.645 1.924 1.224 0.570

-

0.14365 0.14500 0.14279

0.549 0.440 0.330 0.220 0.110

0.137 0.110 0.082 0.055 0.028

o

- 0.09907

0

o

0.5-16

-

O.0371()

1.538

0.385

1.429

0.357 0.330 fl. 302

1.099

O.1363R 0.12430 j

Data for NAC..A.. Mean Line a

=

0.3

THEORY OF WING SECTIONS

398

:uJ1_ _-.....-~--,--r---r-.-....,.-.,.--,

- r---- r---.

.r---",",",-­

o

.4

ct; -.:

1.0

3.46°

(J[i -

x fie (per cent c) (per cent c) 0 0.5 0.75 1.25 2.5

0 0.366 0.514 0.784 1.367

6.0 7.& 10 15 20

"'2.330

z/c

c..clt

1.0

.8

.6

-= - 0.121

p.

4v/V - P/f/4

1.429

0.357

dYc/tk

0.61759 0.57105 0.61210 0.43106

3.131 3.824 4.968 5.862

0.34764 0.29671 0.25892 0.20185 0.Ui682

6.&46 7.039 7.343 7.439 7.275

0.11733 0.07988 0.04136 - 0.00721 - 0.05321

1.310

0.327

70

6.929 6.449 5.864 6.199 4.475

-

0.08380 0.10734 0.12567 0.13962 0.14963

1.190 1.071 0.952 0.833 0.714

0.298 0.268 0.238 0.208 0.179

76 80 8& 90 95

3.709 2.922 2.132 1.361 0.636

-

0.15589 0.15837 0.15683 0.lli062 0.13816

0.695 0.476 0.357 0.238 0.119

0.149 0.119 0.089 0.060 0.030

0

- 0.11138

0

0

25

30 35 40 4S 50 86 60

65

100

Data for NACA Mean Line a == 0.4

APPENDIX II

399

r-. <,

r-, <,

o

r-,

NACA 1:1=0.5 ",.., lin.

~

~

r---- t---

~

()

(per

:ent

C) , (per

:nt

C)

I

d~:dz_1

o

o

0.5 0.75 1.25 2.5

0.345 0.485 0.735 1.295

0.58195 0.53855 0.48360 0.40815 0.33070 0.28365 0.24890 0.19690 0.15650

5.0 7.5 10

15

2.206 2.970 3.630 4.740

20

5.620

25

30

6.310 6..840

7.215

35

40

7.430

45

7.490

50

7.350 6.965 6.405 5.725 4.955

55

60

65

70

70

4.130

80

85

90

95

3.265

100

t--- t-....

.8

.6

1.0

4IJor -

p.

P./4

1

I

1.333

0.333

0.12180 0.09000 0.05930 0.02800 - 0.00630

- 0.05305 - 0.09765 - 0.12550 - 0.14570 - 0.16015

2.395 1.535 0.720

-

I -

0.16960 0.17435 0.17415 0.16850 0.15565

o

II -

0.12660

1.200

I

1.067 0.933 0..800

I

I

0.300 0.267 0.233 0.200

0.667 0.533 0.400 0.267 0.133

0.167 0.133 0.100 0.067 0.033

o

o

Data for NACA Mean Line a

=-:

0.5

rHEORY OF WING SECTIONS

400

zo

1"-

1.0

"',

<,

o

r-,

NACA 41:0.6 If'IINIn Ii,.

o

~

l.---­ ~

r-- r---­

I"""""""'"

.4

eli

-= 1.0

G, == 2.58

%

1/c

(per cent c)

(per cent c)

0

0

0.5 0.75

0.325 0.455 0.695

,I.

%,C

0

2.5

1.220

5.0 7.5 10 15 20

2.080 2.805 3.435 4.495 5.345

0.31325 0.26950 0.23730 0.18935 0.15250

25 30

6.035 6.570 6.965 7.235 7.370

0.12125 0.09310 0.06660 0.04060 0.01405

35 40 45

== -

t-­

1.0

0.158

p.

dllt:/dz

0.54825 0.50760 0.45615 0.38555

1.25

c.c/4

r---

.8

.4

~v/V·

= PH/4

1.250

0.312

70

7.370 7.220 6.880 6.275 5.505

-

0.01435 0.04700 0.09470 0.14015 0.16595

1.094 0.938

0.273 0.234

75 80 85 90 95

4.630 3.695 2.720 1.755 0.825

-

0.18270 0.19225 0.19515 0.19095 0.17790

0.781 0.625 0.469 0.312 0.156

0.195 0.156 0.117 0.078 0.039

0

0

50 55 60 65

100

0

I

- 0.14550

nata for NACA Mean Line 4 == 0.6

APPENDIX II

401

(Ji

20

"­

1-0

-...

'"' -, -,

NACA a-47 ",." /WM

~

r--

I

----- ---.

...--- ---­

Cia =-:

1.0

I

·--I~-d I~

I

yc

Ii

0.305 0.425 0.655

0.51620 0.47795 0.42960

2.5

1.160

0.36325

5.0 7.5

1.955 2.645

0.29545 0.25450

10 15 20

3.240 4.245 5.060

0.22445 0.17995 0.14595

25 30 35 40 45

5.715

0.11740 O.092()() O.On840 OJ)4570 0.02315

50

7.155 7.090

0 - 0.02455

6.900 6.565 6.030

- 0.05185 - 0.08475 - 0.13650

X

(per :nt c) 0.5 0.75 1.25

(pe: rent c)

y,

6.240 6.635 6.925 7.095

55 60 65 70 75

5.205

- 0.18510

80 85 90 95

4.215 3.140 2.035 0.965

-

100

I

0

/.IJ

c...j. -= - 0.179

2.09 0

ai -

~

0.20855 0.21955 0.21960 0.20725

, - 0.16985

Data for XACA

~ienn

t

I

} 1.176

0.294

i !

t I

1

II

I J

0.980 0.784 0.588 0.392 0.196

o

Line a -- 0.7

I I

I

0.245 0.196 0.147 0.098 0.049

o

THEORY OF WING 8ECTION8

402

..---­

....,

-

<,

"

(J

NACA -e/J.8 ",.., n,. .2

a

,-------I.----

.4

e'i

-= 1.0

1.64

eli -

fie

%

2.5

0 0.287 0.404 0.616 1.077

5.0 7.5 10 15 20

1.841 2.483 3.043 3.986 4.748

25 30

5.367 5.863 6.248 6.528 6.709

1.25

35 40

45 50

55 60 65

70 75

80 85

90 95

100

SIC 0

.8

.6

""""""--­

/.0

c../. - - 0.202

p.

dl/c/a

(per cent e) (per cent e) 0 0.5 0.75

--- r---....

to-"""

AviV

== PIl/4

0.48535 0.44925 0.40359 0.34104 0.27718

0.23868 0.21(}lj() 0.16892 0.13734 0.11101 0.08775 0.06634 0.04601 0.02613

6.790 6.770 6.644 6.405 6.037

-

0.00620 0.01433 0.03611 0.06010 0.08790

5.514 4.771 3.683 2.435 1.163

-

0.12311 0.18412 0.23921 0.25583 0.24904

0

- 0.20385

1.111

0.278

0.833 0.556 0.278

0.208 0.139 0.069

0

0

Data for NACA MeaD Line a .. 0.8

APPENDIX 11

403

2.0

1.0

""

,,~

o

\

WACA • - O.B (modlfl_d) Mean II".

2

o t-.--­ o

~

- r--- r----.

~

.2

.4

.6

~

.8

1 0

x/e

eli

-= 1.0

Gr, -=

'lie

%

1.25

2.5 5.0 7.5 10 15 20

25 30 35 40 45 50

55 60

65 70 75 80 85 90 95

100

I

c.ct.

0 0.281 0.396 0.603 1.055

0.47539 0.44004 0.39531 0.33404

1.803 2.432 2.981 3.903 4.651

0.27149 0.23378 0.20618 . 0.16546 0.13452

5.257 5.742 6.120 6.394 6.571

0.10873 0.08595 0.06498 0.04507 0.02559

6.651 6.631 6.508 6.274 5.913

-

0.00607 0.01404­ 0.03537 0.05887 0.08610

5.401 4.673 3.607 2.452 1.226

-

0.12058 0.18034 0.23430 0.24521 0.24521

0

- 0.24521

:II:

0.219

------

PIl

AfJ/V .. P./4

1.092

0.273

11.~

0.274

}1.100

0.276

} 1.104

0.276

1.108 1.108

0.277 O.Z17

1.112 1.112 0.840 0.588 0.368

0.278 0.278

0

0

dy./a

(per cent e) (per cent c) 0 0.5 0.75

1.400

Data for NACA Mean Line a -= 0.8 (modified)

0..210 0.147 0.092

THBoRY OF WING SECTIONS

404

D

I. D

--

.,

1\ \

- r--­

~

NACAa.O,9 meon/i/'ltl

L"....--­ ~

~

IJ

t(J

ci, == 1.0

x (per cent c) 0 0.5 0.75 1.25 2.5

Gi

== 0.90 0

Yc

0 0.269 0.379 0.577

1.008

5.0 7.5 10 15 20

1.720 2.316 2.835 3.707 4.410

0.25786 0.22153 0.19liOO 0.15595 0.12644

2S

4.980 5.435 5.787 6.M5 6.:212

0.10196 0.08047 0.06084: 0.04234 0.02447

40

45 50

55 60

65 70 75 80 85 90

95

100

6.290 6.279 6.178 5.981 5.681

-

0.00678 0.01111 0.02965 0.04938 0.07103

2.984: 1.503

-

0.09583 0.1260:) 0.16727 0.25204: 0.31463

0

- 0.26086

5.265 4.714 3.987

0.225

z= -

Pa

dyc/dz

(per cent c)

0.45482 0.42OM 0.37740 0.31821

30 35

c.c/.

~/V

=

PR/4

1

1.05:1

0.263

0.526

0.132

0

0

Data for NACA Mean Line (I

:II:

0.9

APPENDIX II

405

:J.O

.0

.0

D NACA ••/.0 mean /;n. ~

- r--- r--­

..",..-- ~ -

D

.4

e'i

1.25

2.5

0.930

5.0 7.5 10

1.580

i

15 20 25

30

35 40

45

65 70 75 80 85

90 95

100

00

c..,J.

I J

2.120 2.585 3.365 3.980 4.475 4.860 5.150 5.355 5.475

0.08745 0.00745 0.04925 0.(13225 0.01595

5.515 5.355 5.150 4.860 4.475 3.980 3.365 2.585

11&:

-

0.250

PR

j

0.23430 0.19995 0.17485 0.13805 0.11030

5.475

,

dyc/dx

0.42120 0.38875 0.34770 0.29155

I

Ii I

-

0 0.01595

I,II

1.580

!

- 0.23430

0

ti

i

II

I

I

II I If urn i

1(

t

f; !

I

II

II

- 0.08745 - 0.11030 - 0.13805

= PR/4

Ii

I'

II

I~V/V

II

:I OJ>3225 I: 0.04925 0.06745

UJ

.8

.6

0 !

0.250 0.350 0.535

50 55 60

a, =

Yc (per cent c)

X

(per cent c) 0 0.5 0.73

= 1.0

sIc

I I I

i

I

I j

I - 0.17485

I I

Data ior N ACA :\IcuD Line Il = 1.0

j

I

0.250

APPENDIX III

AIRFOIL ORDINATES

CoNTENTS

NACA Designation

Page 408

0010-34 a == 0.8 (modified) e'i II: 0.2 . 0012-64 a == 0.8 (modified) e'i == 0.2 . 1408

1410

1412

2408

2410 . . . . .

2412

2415 . . . . .

408

408

409

409

409

410

410

410

2418

2421 2424

411

.....

411

411

412

412

4412

4415

.....

4418

4421

4424

23012

23015

23018 .

23021

23024

63-206 .

63-209 .

63-210 .

63 1-212 63.-412.

63r215 . .

412

413

413

413

414

414

414

415

415

416

416

417

.

417

418

418

63r415. '. . . . . .

63t-615. . . . . .

63,-218. .

63r418.

419

419

63,-618. .

420

420

63r221 . .

63.-421.

421

421

63A21 0

64-108 . . . .

64-110 .

64-206 . .

422

423

422

423

424

64-208 . . . .

64.-209. . . . .

424

406

APPENDIX III NACA Designation

64-210 .

641-112 . . ....

641-212 .

641-412. .

642-215. .

64r415. . . . . . .

64,-218 . .

64r418.

64r618 . .

64 221

.

64.-421 ..

407

.

Page

425

425

426

426

427

427

428

428

429

429

. . . . . 430

430

431

431

432

64A210

64A410

64IA212 .

642.~215 .

65-206 . .

65-209 . . 65-210 .

65-410 .

651-212 . .

65.-212 a == 0.6 . .

65.-412 .

65r215. . . . . . .

65r415. . . . .

65r415a == 0.5 . .

65r 218 . . . . . . .

65,..418. . . . .

65r418 a - 0.5. ­ 65r618. . . . .

65,-618 a == 0.5 .

65...221 .

65r421 .

65.-421 a == 0.5 .

66-206 . .

66-209 . .

66-210 . .

66.-212.

432

433

433

434

434

435

435

436

436

437

437

438

438

439

439

440

440

441

441

442

442

443

66r215.

443

66r415.

444

66r218. 66r418. 66...221. 67,1-215 . 747A315 . 747A415 . . • • • • .

. ....

444

445

445

446

446

447

(Stations and ordinates given in

NACA 0012-64

a = 0.8 (modified) e'i = 0.2

(Stations and ordinates given in

per cent of airfoil chord)

per cent of airfoil chord)

NACA 0010-34

a = 0.8 (modified) c" == 0.2

_ .._.

Upper surface

Lower surface

Upper surface

Ordinate Station

Ordinate

0 1.928 2.659 3.623 4.295

0 1.393 2.664 5.177 7.678

-

0 1.686 2.237 2.901 3.323

4.832 6.221 6.974 7.279

10.175 15.161 20.142 30.100 40.054

-

3.640 4.083 4.361 4.678 4.721

49.993 60.038 70.077 80.120 90.091

7.157 6.622 5.662 4.253 2.355

50.007 59.962 69.923 70.880 89.909

-

4.497 4.018 3.296 2.383 1.375

95.050 100.000

1.271 0.120

04.950 100.000

Ordinate Station

Ordinate

Station

0 0.813 1.324 2.593 5.113

-

0 0.632 0.820 1.186 1.714

0 1.107 2.336 4.823

4.887

0 0.790 1.062 1.608 2.436

7.378 9.875 14.876 19.886 20.017

3.094 3.637 4.523 5.172 5.080

7.622 10.125 15.124 20.114 30.083

-

2.122 2.445 2.961 3.312 3.684

9.825 14.839

5.645

19.858 29.000 39.946

39.955 49.994 (lO.O31 70.ntH 80.100

6.279 6.186 5.735 3.700

40.045 50.006 59.960 OH.H3H 79.900

-

3.721 3.526 3.131 2.5"Ul 1.83U

00.076 95.042 100.000

2.044 1.100 0.100

89.92-1 94.958 100.000

- 1.004 - 0.610 - 0.100

Station 0 0.687 1.176 2.407

tj.015

L.E. radius: 0.272 Slope of rudius through L.E.: 0.095

7.322

(Stations and ordinates given in per cent of airfoil chord) Upper surface

Lower surface

- 0.781 - 0.120

L.E. radius: 1.582 Slope of radius through L.E.: 0.095

~ 00

NACA 1408

Station

Ordinate

Lower surface

Station Ordinate

0 1.189 2.418 4.896 7.386

0 1.324 1.862 2.602 3.138

0 1.311 2.582 5.104 7.614

9.883 14.889 19.904 24.926 29.950

3.558 4.171 4.574 4.819 4.939

10.117 15.111 20.096 25.074 30.050

-

2.682 2.953 3.074 3.101 3.063

40.000 50.020 60.034 70.041 SO.03D

4.869 4.502 3.931 3.193 2.305

40.000 49.980 59.966 69.959 79.961

-

2.869 2.556 2.153 1.693 1.193

90.027 95.016 100.000

1.271 0.698 0.084

89.973 94.984 100.000

0

- 1.200 - 1.620 - 2.134 - 2.458

- 0.659 - 0.378 - OJJ84

L.E. radius: 0.70 Slope of radi us through L. E.: 0.05

~ ~ ~ ~

~

~

~

~ ~

C

~

NACA 1410

NACA 1412

NACA 2408

(Stations and ordinates given in per cent of airfoil chord)

(Stations and ordinates given in

(Stations and ordinates given in

per cent of airfoil chord)

Upper surface Station ()

Station

0 1./i15

0 t.tuM 2.378 4.845

0 1.ll30

2.398 4.870 7.358

2.297 3.194 3.837

9.854

29.937

4.338 5.062 5.531 5.809 5.940

40.000 50.025 00.042 70.051 80.049

5.836 5.385 4.692 3.804 2.741

90.034 95.021 100.000

1.513 0.832 0.105

24.907

Upper surface

Lower surface

Upper surface

Lower surface

..

Ordinate Station Ordinate

1.174

14.861 19.880

_­

Lower surface

--

per cent of airfoil chord)

0 1.320 2.002 7.642

-

2.055 2.726 3.157

10.146 15.139 20.120 25.093 30.063

-

3.462 3.844 4.031 4.091 4.064

5.130

I

I

I 40.000 49.975 I

- 3.836 - 3.4:i9

59.058 69.949

- 2.914 - 2.30,1

79.951

- 1.629

89.966 0.901 94.979 , - 0.512 I

100.000

-

!-

0.105

L.E. radius: 1.10 Slope of radius through L.E.: 0.05

Ordinate Station

7.330

0

0

l.or)t1 2.733 3.786 4.537

1.:\tI2 2.U22

5.118 5.951 6.486 6.799 6.040

9.82'. 14.833

19.857 24.889 29.925 40.000 50.029 60.051 70.061 80.0SS ,

6.803

5.155 7.670

lO.l7G 15.167 20.143

ss.iu

30.075 40.000 49.071

0.267 5.453 4.413 3.178

60.930 79.942

1.753 90.040 1 0.966 95.025 0.126 100.000

89.960 94.975 100.000

I

5fi.940

Ordinate 0

Station

Ordinate

Station 0 1.372 2.663 5.206

7.273

0 1.3140 1.977 2.829 3.471

0

- 1.8..10

1,12R

- 2.491 - 3.318

4.794

- 3.857

2.337

7.727

Ordinate 0

- 1.134 - 1.493 - 1.891 - 2.111

-

4.242 4.733 4.986 5.081 5.064

9.768 14.778 19.809 24.852 29.900

3.987 4.776 5.320 5.677 5.875

10.232 15.222 20.191 25.148 30.100

-

2.237 2.338 2.320 2.239 2.125

-

4.803 4.321 3.675 2.913 2.066

40.000 50.039 70.081 80.078

5.869 5.473 4.820 3.942 2.858

40.000 49.961 59.932 69.919 79.922

-

1.869 1.585 1.264 0.942 0.636

- 1.141 - 0.646 - 0.126

90.054 95.033 100.000

1.575 0.855 0.084

89.946 94.967 100.000

L.'E. radius: 1.58 Slope of radius through L.E.: 0.05

no. 008

~

~

t:::J

""-c

~

""-c '"'-.. ""-c

- 0.353 - 0.217 - 0.084

L.E. radius: 0.70 Slope of radius through L.E.: 0.1

.... o

co

NACA 2410 (Stations and ordinates given in per cent of airfoil chord)

Upper eturface Station

Lower surface

Ordinate Station Ordinate

1.098 2.297 4.742 7.217

0 1.694 2.411 3.420 4.169

0 1.402 2.703 6.258 ' 7.783

9.710 14.722 19.761 24.814 29.875

4.766 5.665 6.276 6.668 6.875

10.290 15.278 20.239 26.186 30.126

40.000 lj().049 60.085 70.102 80.097 90.067 96.041 100.000

6.837 6.366 5.lS8O 4.651 3.296

0

1.816 0.990 0.105

-

0 1.448 1.927 2.482

2.809 - 3.016 - 3.227 - 3.276 - 3.230 - 3.123 - 2.837 - 2.468 - 2.024 -l.Ml - 1.074 - 0.594 - 0.352

40.000 49.951 59.915 69.898 79.903 89.933 94.959 100.000 - 0.105

L.E. radius: 1.10 Slope of radius through L.E.: 0.1

NACA 2412 (Stations and ordinates given in per cent of airfoil chord) Upper surface

Lower surface

Station Ordinate Station Ordinate 0 1.2a 2.5 6.0 7.5

.. .... 2.15 2.99 4.13 4.96

10 16 20 25 30

6.63 6.61 7.26 7.67 7.88

40

7.80 7.24 6.36 6.18 3.75

so

60 70 80 90

95

100 100

2.08 1.14 (0.13)

.. ....

2.5 5.0 7.6 10 15 20 25

30 40 50 60

70 80

90 95

100 100

Upper surface

0 1.66 2.27 3.01 3.46

0

......

1.25

2.71 3.71 5.07 6.06

-

3.76 4.10 4.23 4.22 4.12

10 15 20 25 30

6.83 7.97 8.70 9.17 9.38

-

3.80 3.34 2.76 2.14 l.l5O

40

- 0.82 - 0.48 (- 0.13) 0

L,E. radius: 1.58 Slope of rArdius through L.E.: 0.10 •

•.o....

Lower surface

Station Ordinate Station Ordinate

-

0 1.25

NACA 2415 (Stations and ordinates given in per cent of airfoil chord)

2.5 5.0 7.5

50 60 70 80 90 95 100 100

0 1.25 2.5 5.0 7.6

-

0 2.06 2.86 3.84 4.47

10 15 20 25 30

-

4.90 5.42 5.66 5.70 5.62

9.25 8.57 7.50 6.10 4.41

40 50 60 70 80

- 5.26 - 4.67 -3.90 - 3.05 - 2.15

2.45 1.34 (0.16)

90

- 1.17 - 0.68 (- 0.16) 0

......

95 100 100

L.E. radius: 2.48 Slope of radius through L.E.: 0.10

~

~ ~

e.c:

~

s ~ ~ ~

~

~ r,j

NACA 2418

NACA 2421

NACA 2424

(Stations and ordinates given in per cent of airfoil chord)

(Stations and ordinates given in per cent of airfoil chord)

(Stations and ordinates given in

Upper surface Station Ordinate 0 1.20

2.5 6.0 7.5 10

I

16 20 2S 30 40 50 60

70 80 90

95

100 100

••

I

••••

3.28 4.45 6.03 7.17 8.05 9.34 10.15 10.65 10.88 10.71 9.89 8.66 7.02 5.08 2.81 1.55 (0.19)

..... ...

Lower surface Station

Ordinate 0

0

1.25 2.5 5.0 7.5

Station Ordinate Station Ordinate

0 1.25 2.5 5.0 7.5

-

2.45 3.44 4.68 5.48

-

6.03 6.74 7.09 7.18 7.12

-

6.71 5.99 5.04 3.97 2.80

40

100

- 1.63 - 0.87 (- 0.19)

90 95 100

100

0

100

10 15

20

25 30 40 50 60

70 80

90 95

L.E. radius: 3.66 Slope of rndius through L.E.: O.to

Lower surface

Upper surface

10 15 20 25 30 50

60 70 80

••••

t

••

1.25 2.5 5.0

8.29

7.5

9.28 10.70 11.59 12.15 12.38

10

15 20

25

30

12.16 11.22 9.79 7.94

40

5.74

80

3.18 1.76 (0.22)

90 95 100 100

.......

50

60

70

Upper surface

Lower surface

Station Ordinate Station Ordinate 0 3.646 4.965 6.614 7.692

-

0 2.82 4.02 5.51 6.48

0 0.885 2.012 4.380 6.820

0 3.892 5.449 7.502 9.052

-

7.18 8.05 8.52 8.67 8.62

9.300 14.333 19.427 24.555 29.700

10.215 11.888 12.959 13.593 13.874

10.700 - 8.465 15.667 - 9.450 20.573 - 9.959 25.445 - 10.155 30.300 - 10.124

-

8.16 7.31 6.17 4.87 3.44

40.000 50.118 60.203 70.244 80.233

13.606 12.532 10.903 8.824 6.352

40.000 49.882 59.797 69.756 79.767

- 1.88

90.161 95.098 100.000

3.502 1.930

0

3.87 5.21 7.00

per cent of airfoil chord)

- 1.06 (- 0.22) 0

L.E. radius: 4.85 Slope of radius through I".E.: 0.10

.....

0 1.615 2.988 5.620 8.180

-

-

9.606 8.644 7.347 5.824 4.130

~

~S2

>< ~ ~

.....

89.839 - 2.280 94.902 - 1.292 0 100.000

L.E. radius: 6.33

Slope of radius through L.E.: 0.10 ~ ..... .....

....

~

NACA 4412 (Stations and ordinates given in per cent of airfoil chord)

Upper surface Station

Ordinate

0

0

1.25 2.5

2.44 3.39 4.73 5.76

5.0 7.5 10 15 20 25 30 40

50 60 70 80

90 95 100 100

Lower surface

Station Ordinate 0 1.25 2.5 5.0 7.5

NACA 4415 (Stations and ordinates given in per cent of airfoil chord)

Upper surface Station 0 1.25 2.5 5.0

-

0 1.43 1.95 2.49 2.74

9.41 9.76

10 15 20 25 30

-

2.86 2.88 2.74 2.50 2.26

9.80 9.19 8.14 6.69 4.89

40 50 60 70 80

-

1.80 1.40 1.00 0.65 0.39

60 70 80

2.71 1.47 (0.13)

90 95 100

- 0.22 - 0.16 (- 0.13) 0

90 95 100 100

6.59 7.89

8.80

..... .

100

IJ.E. radius: 1.58

Slope of radius through L.E.: 0.20

Ordinate Station Ordinate t

••••••

- 3.27

6.91

7.5

- .3.71

Station 0 1.25 2.5 5.0 7.5

....... 3.76

5.00 6.75 8.06

3.98 4.18 4.15 3.98 3.75

10 15 20 25 30

9.11 10.66 11.72 12.40 12.76

11.25 10.53 9.30 7.63 5.55

40

-

3.25 2.72 2.14 1.55 1.03

40 50 60 70 80

12.70 11.85 10.44 8.55 6.22

3.08

90 95 100 100

- 0.57 - 0.36 (- 0.16) 0

90 95 100 100

3.46

20

9.27

1.67

(0.16) II

•••••

50 60

70 80

L.E. radius: 2.48 Slope of radius through L.E.: 0.20

~

Lower surface

Ordinate Station Ordinate

-

10.25 10.92 11.25

50

Upper surface

10 15 20 25 30

7.84

40

0 - 1.79 - 2.48

5.74

10 15

30

0 1.25 2.5 5.0

3.07 4.17

7.5

25

Lower surface

NACA 4418 (Stations and ordinates given in per cent of airfoil chord)

1.89

(0.19) ••••••• I

0

0 2.11 - 2.99 - 4.06 - 4.67

~

10

- 5.06

15

-

~

1.25 2.5 5.0 7.5

20 25 30 40 50 60 70

~

5.49 5.56 5.49 5.26

- 4.70 - 4.02

-3.24

80

- 2.45 - 1.67

90 95 100 100

- 0.93 - 0.55 (- 0.19) 0

L.E. radius: 3.56 Slope of radius through L.E.: 0.20

~

~ ~

~ ~ ~ ~

~

C

~

NACA 4421 (Stations and ordinates given in per cent of airfoil chord) Upper surface

Lower surface

Ordinate Station Ordinate

Station

..... . -

0 1.25 2.5 5.0

4.45 5.84 7.82

9.24

7.5

0 1.25 2.5

5.0 7.5

0 - 2.42 - 3.48 - 4.78

- 5.62

15 20

10.35 12.04 13.17

10 15 20

- fi.t5 - (}.75

25

13.88

25

- 6.92

30

14.27

30

- 6.76

40 50 60

14.16 13.18 11.60 9.50 6.91

40

-

10

70 80 00 95

100 100

I

3.85 2.11 (0.22)

..... ..

50 60

70 80 90 95 100 100

- 6.98

6.16 5.34 4.40 3.35 2.31

- 1.27 - 0.74 (- 0.22) 0

L. I~. radius: 4.85 Slope of radius through 1.1. E.: 0.20

NACA 4424

NACA 23012

(Stations and ordinates given in

(Stations and ordinates given in

per cent of airfoil chord)

per cent of nirfoil chord)

I .ower surface

Upper surface

Station

1.536

0 3.964 5.624

3.775 6.153

7.942 9.651

8.611 13.674 18.858 24.111 29.401

11.012 13,<)45 14.416 15.287

0 0.530

I5.nOG

50.235

14.47·' 12.674 10.312

00.405 70.487 80.464

7.447

i

4.099 2.240 •

It

••

••

L.E. radius: 6.33

0

0 1.970

- 3.472

3.4M 6.225

8.847

.io.ooo

I

··JU.7G5 59.505 00.513

iO.536 80.680 94.804 lOOJ)(X)

I

I

Station 0 1.25

- 4.656

2.5

- 6.0f>6 - 6.931

5.0 7.5

11.389 - 7.512 16.326 - 8.I09 21.142 - 8.416 25.&~9 - 8.411 30.500 - 8.238

15.738

40.000

90.320 95.196 100.000

Station Ordinate

Ordinate

Upper surface

10 15 20 25 30

- 7.006 - 0.698

40

- 5.562

Ordinate Station Ordinate

...... 2.67 3.61 4.91

5.80 6.43

7.10 7.53 7.60 7.55

60

7.14 6.41 5.47

- 4.312

70

4.36

- 3.003

80

3.08

- 1.655

90 95

-

0.964 0

Slope or radius through L.E.: 0.20

50

100

100

Lower surface

1.68

0.92 (0.13)

......

0 1.25

2.5 5.0

7.5 10 15

-

0 1.23 1.71 2.26 2.61

20

- 2.92 - 3.50 - 3.97

25

- 4.28

30

- 4.46

40 50 60 70 80

-

4.48 4.17 3.67 3.00

~ ~ ~

~

~

.......

><

.......

.......

.......

- 2.16

90 95

- 1.23 - 0.70

100 100

(- 0.13) 0

L.E. radius: 1.58 Slope of radius through L. E.: 0.305

~ ~

~

NACA 23015 (Stations and ordinates given in per cent 01 airfoil chord) Upper surface

Lower surface

Station Ordinate Station Ordinate 0 1.20 2.5

5.0 7.5

.... ., 3.84 4.44 5.89 6.90

0 1.25 2.5 6.0

7.5

-

0 1.54 2.25 3.04 3.61

NACA 23018 (Stations and ordinates given in per cent of airfoil chord) Lower surface

Upper surface Station 0 1.25 2.5 5.0

7.5

Ordinate •••••

t

10

7.64

10

- 4.09

10

8.83

15

8.52

20 26 30

8.92

15 20 26

15 20 25

30

-

9.86 10.36 10.56 10.OS

40

- 5.92

50 60 70 80

-

40 50 60 70 80 90 95 100 100

9.08 9.06 8.59 7.74 6.61

5.25 3.73

2.04: 1.12 (0.16)

.. ....

90 95

100 100

4.84 6.41 6.78 5.96 5.SO 4.81 3.91 2.83

- 1.89 - 0.90 (- 0.16) 0

L.E. radius: 2.48 Slope of radius through L.E.: 0.305

30 40 50 60 70 80

10.04 9.05 7.76 6.18

90

2.39 1.32 (0.19)

ss 100 100

•

4.09 6.29 6.92 8.01

4.40

.... ...

Station

Ordinate

0 1.25 2.5 5.0

- 3.80

7.5

- 4.60

0 - 1.83 - 2.71

NACA 23021 (Stations and ordinates given in per cent of airfoil chord)

0 1.25 2.5 5.0 7.5

Ordinate •••

It

••

4.87 6.14 7.93 9.13

10 15 20 25 30

- 5.22 - 6.18 - 6.86 -7.27 -7.47

10 15 20 25 30

10.03 11.19 11.80 12.05 12.06

40 50 60 70 80

-7.37 - 6.81 - 5.94 - 4.82 - 3.48

40 50 60 70 SO.

11.49 10.40 8.90 7.09 5.05

- 1.94

90

-

1.09

95

(- 0.19)

100 100

2.76 1.53 (0.22)

00 95 100 100

0

L.E. radius: 3.56 Slope of radius through L.E.: 0.305

~

Lower surface

Upper surface Station

~ .....

.......

Station Ordinate 0 1.25 2.5 5.0 7.5 10 15

20 25 30 40 50 60 70

80 90 95 100 100

-

0 2.08 3.14 4.52 5.50

-

6.32 7.51 8.30 8.76 8.95

-

8.83 8.14 7.07 5.72 4.13

- 2.30 - 1.30 (- 0.22) 0

L.E. radius: 4.85 Slope of radius through L.E.: 0.305

~

~

~ ~

~ ~ ~

~

~ ~ ~

~r;,j

APPENDIX III NACA 63-206

(Stations and ordinates given in

percent of airfoil chord)

NACA23024

(StatiODS and ordinates given in per cent of airfoil chord) Upper surface

I

Lower surface

Station Ordinate Station Ordinate

o

415

Upper Surface Station Ordinate

Lower Surface

Station

Ordinate 1

0

0

0

.458 .703 1.197 2.438

.551 .677 .876 1.241

.542 .797 1.303 2.562

4.932

0 -.451

0.?:l7 1.331 3.853 6.601

0 4.017 5.764 8.172 9.844

9.423 15.001 20.253 25.262 30.265

11.049 12.528 13.237 13.535 13.646

10.577 - 7.647 14.999 - 8.852 19.747 - 9.703 24.738 - 10.223 29.735 - 10.454

7.429 9.930 14.934 19.941

1.776 2.189 2.526 3.058 3.451

5.068 7.571 10.070 15.066 20.059

-

1.144 1.341 1.492 1.712 1.859

40.256 50.235 60.202 70.162 80.116

12.928 11.690 10.008 7.988 5.687

39.744 - 10.278 49.766 - 9.482 59.798 - 8.242 69.838 - 6.664 79.884 - 4.803

24.950 29.960 34.970 39.981 44.991

3.736 3.926 4.030 4.042 3.972

25.050 30.040 35.030 40.019 45.009

-

1.946 1.982 1.970 1.900 1.7R2

90.064 95.036

3.115 1.724

89.936 94.964

50.000 55.008 60.015 65.020 70.023

3.826 3.612 3.338 3.012 2.642

50.000 54.992 59.985 64.980 69.977

- 1.620

75.023 80.022 85.019 90.013 95.006

2.237 1.804 1.356 .900 .454

74.927 79.978 84.981 89.987 94.994

100

......

0 2.223 3.669 6.147 8.399

100

-

0 3.303 4.432 5.862 6.860

- 2.673 - 1.504 0

L.E. radius: 6.33 Slope of radius through LE.: 0.305

100.000

0

100.000

- .537 - .662 - .869

-- 1.4221 1.196 - .952 - .698

- .447 - .212 - .010 .134 .178

0

L.E. radius: 0.297 Slope of radius through L.E.: 0.0842

THEORY OF WING SECTIONS

416

NACA 63-209 (Stat.ions and ordinates given in per cent of airfoil chord) Upper surface

Lower surface

Station Ordinate Station 0 0.436 0.680 1.170 2.408

0 0.796 0.973 1.255 1.765

0 0.563 0.820 1.330 2.592

4.897 7.394 9.894 14.901 19.912

2.510 3.077 3.539 4.263 4.792

24.925 29.940 34.956 39.971 44.986

Ordinate

NACA 63-210

(Stations and ordinates given in

per cent of airfoil chord)

Upper surface Station

Lower surface

Ordinate Station Ordinate

-

-

0 0.696 0:833 1.041 1.393

0 0.430 0.669 1.162 2.398

0 0.876 1.107 1.379 1.939

0 0.570 0.831 1.338 2.602

-

0 0.776 0.967 1.165 1.567

5.103 7.606 10.106 15.099 20.088

-

1.878 2.229 2.505 2.917 3.200

4.886 7.382 9.882 14.890 19.902

2.753 3.372 3.877 4.665 5.240

5.114 7.618 10.118 15.110 20.098

-

2.121 2.524 2.843 3.319 3.648

5.169 5.414 5.530 5.518 5.391

25.075 30.060 35.044 40.029 45.014

-

3.379 3.470 3.470 3.376 3.201

24.917 29.933 34.951 39.968 44.985

5.647 5.910 6.030 6.009 5.861

25.083 30.067 35.049 40.032 45.015

-

3.857 3.966 3.970 3.867

50.000 55.012 60.022 65.029 70.033

5.159 4.834 4.429 3.958 3.430

50.000 54.988 59.978 64.971 69.967

-

2.953 2.644 2.287 1.898 1.486

50.000 55.013 60.024 65.032 70.036

5.599 5.235 4.786 4.264 3.684

50.000 54.987 59.976 64.968 69.964

-

3.393 3.045 2.644 2.204 1.740

75.034 80.032 85.027 90.019 95.009

2.861 2.267 1.663 1.067 0.512

74.966 79.968 84.973 89.981 94.991

-

1.071 0.675 0.317 0.033 0.120

75.038 80.036 85.030 90.021 95.010

3.061 2.414 1.761 1.121 0.530

74.962 79.964 84.970 89.979 94.990

-

1.271 0.822 0.415 0.087 0.102

100.000

I

0

100.000

0

L.E. radius: 0.631 Slope of radius through L.E.: 0.0842

100.000

0

100.000

~.671

0

L.E. radius: 0.770 Slope of radius through L.E.: 0.0842

APPENDIX III NACA 63 1-212 (Stations and ordinates given in per cent of airfoil chord)

Upper surface Station

Upper surface Station 0 0.336 0.567 1.041

0 1.032 1.260 1.622 2.284

0 0.583 0.843 1.355 2.622

- - - -

0 0.932 1.120 1.408 1.912

4.863 7.358

3.238 3.963 4.554 5.470 6.137

5.137 7.642 10.141 15.132 20.118

- - - - -

2.606 3.115 3.520 4.124 4.545

6.606 6.901 7.030 6.991

- - - - -

4.816 4.957 4.970 4.849 4.609

9.859

N ACA 63 1-412

(Stations and ordinates given in

per cent of airfoil chord)

Ordinate Station Ordinate

0 0.417 0.657 1.145 2.378

14.868 19.882

Ordinate

2.257 4.727 7.218 9.718 14.735

:::1

3.544

5.273 7.782 10.282 15.265 20.235

- - - - -

25.200 30.160 35.118

- 3.919

4.379 5.063 6.138 6.929 7.499 7.872 8.059 8.062 7.894

50.000 55.016 60.029 65.038 70.043

6.473 6.030 5.491 4.870 4.182

50.000 54.984 59.971 64.962 69.957

- - - - -

4.267 3.840 3.349 2.810 2.238

7.576 50.000 55.031 1 7.125 6.562 60.057 65.076 5.899 70. 087 1 5.153

75.045

3.451 2.698

74.955 79.958 84.965 89.975 94.988

- - - -

1.661 1.106 0.601 0.190 0.066

75.089 80.084 i 85.0iO I

90.025

95.012 100.000

1.947 1.224 0.566 0

100.000

0

I

4.344 3.492 2.618 90. 049 1 1.739 0.881 95. 023 1 0 100.000 •• _ . _ _ •• _ _ _ _ ... _

L.E. radius: 1.087 Slope of radius through L.E.: 0.0842

Ordinate - - - -

6.799

I

Station 0 0.664 0.933 1.459 2.743

25.100 30.080 35.059 40.038 45.018

29.840 882 34. 39.924 1 44.9H4 .

Lower surface

0 1.071 1.320 1.719 2.460

24.900 29.920 34.941 39.962 44.982

80.042 85.035

-

Lower surface

417

...... _ _ •• r

0 0.871 1.040 1.291 1.716

2.280 2.685 2.995 3.446 3.745

- 3.984

45.036

- 3.939 - 3.778 - 3.514

50.000 54.969 59.943 64.9201 69.913

- 3.164 - 2.745 2.278 1 1.i79 I - 1.265

40.076

I

I-

I-

I 74.911 I - 0.764 79.916 t - 0.308 84.930 0.074 89.951 0.329 94.977 0.383

I I

ioo.ooo ._ ...

--

I __0 _..

.-

.._ ...­

I.,.E. radius: 1.087 Slope of radius through L.E.: 0.1 sss

THEORY OF WING SECTIONS

418

NACA63r215 (Stations and ordinates given in per cent of airfoil chord) Upper surface Station

Ordinate Station Ordinate

0 0.601 0.863 1..380

Upper surface Station

Lower surface

Ordinate Station

Ordinate

2..652

-

0 1.150 1..388 1..766 2.420

0 0.300 0.525 0.991 2.198

0 1.287 1.585 2.074 2.964

0 0.700 0.975 1.509 2.802

5.171 7.677 10.177 15.166 20.148

-

3.328 3.999 4.535 5.336 5.895

4.660 7.147 9.647 14.669 19.705

4.264 5.261 6.077 7.3'8 8.279

5.340 7.853 10.353 15.331 20.295

- 3.000 - 3.565 - 4.009

25..125 30.100 35.074 45.023

-

6.259 6.448 6.470 6.315 6.004

24.760 29.800 34.852 39.905 44.955

8.941 9.362 9.559 9.527 9.289

25.250 30.200 35.148 40.095 45.045

-

5.361 5.474 5.439 5.243 4.909

7.768 7.203 6.524 5.751 4.906

50.000 54.981 59.965 64.953 69..947

-

5.562 5.013 4.382 3.691 2.962

50.000­ 55.039 60.070 65.093 70.106

8.871 8.298 7.595 6.780 5.877

50.000 54.961 59.930 64.907 69.894

-

4.459 3.918 3.311 2.660 1.989

4.014 3.106 2..213 1.368 0.616

74.945 79.949 84.957 89.970 94..986

-

2.224 1.513 0.867 0.334 0.016

75.109 80.102 85.085 90.059 95.028

4.907 3.900 2.885 1.884 0.931

74.891 79.898 84.915 89.941 94.972

- 1.327 - 0.716 - 0.193 0.184 0.333

0 0.399 0.637 1.120 2.. 348

0 1.250 1.528 2.792

4..829 7.323 9.823 14.834 19..852

3.960 4.847 5.569 6.682 7.487

24.875 29.900 34.926 39.952 44.977

8.049

50.000 55.019 60.035 65.047 70..053 75.055 SO.051 85.043 90.030 95.014 100.000

Lower surface

NACA63r415

(Stations and ordinates given in

per cent of airfoil chord)

1.980

8.392

8.530 8.457 8.194

0

40..048

100.000

0

L.E. radius: 1.594 Slope of radius through L.E.: 0.0842

100.000

0

100.000

-

0 1.087 1.306 1.646 2.220

- 4.656 - 5.095

0

L.E. radius: 1.594 Slope of radius through L.E.: 0.1685

APPENDIX III NACA 63r615 (Stations and ordinates given in per cent of airfoil chord) Upper surface Station

Ordinate

Lower surface Station

419

NACA 63a-218

(Stations and ordinates given in

per cent of airfoil chord)

Upper surface

Lower surface

Station

Station

-

0 1.017 1.214 1.517 2.013

0 0.382 0.617 1.096 2.319

0 1.449 1.778 2.319 3.285

0 0.618 0.883 1.404 2.681

-

0 1.349 1.638 2.105 2.913

2.664 3.123 3.476 3.972 4.290

4.796 7.288 9.788 14.801 19.822

4.673 5.728 6.581 7.895 8.842

5.204 7.712 10.212 15.199 20.178

-

4.041 4.880 5.547 6.549 7.250

25.150 30..120 40.057 45.027

-

7.704 7.940 7.970 7.774 7.387

-

6.839 6.161 5.384 4.537 3.650

-

2.754 1..894 1.113 0.467 0.032

Ordinate

0 0.205 0.418 0.866 2.0li0

0 1.317 1.634 2.159 3.129

0 0.795 1.082 1.634 2.950

4.492 6.973 9.473 14.504 19.558

4.560 5.667 6.578 8.010 9.066

5.508 8.027 10.527 20.442

-

24.625 29.700 39.857 44.932

9.830 10.331 10.587 10.598 10.384

25.37S 30.300 35.222 40.143 45.068

-

4.460 4.499 4.407 4.172 3.814

24.850 29.880 34.911 39.943 44.973

9.494 9.884 10.030 9.916 9.577

50.000 55.058 60.105 65.139 70.159

9.974 9.393 8.665 7.809 6.847

50.000 04.942 59.895 64.861 69.841

- 3.356 - 2.823 - 2.239

9.045 8.351 7.526

- 1.015

50.000 55.023 60.042 65.055 70.062

5.594

50.000 64.977 59.958 64.945 69.938

75.163

5.800

74.837 79.847 84.873 89.911 94.958

- 0.430 0.083 0.483 0.704 0.651

75.064 80.059 85.049 90.034 95.016

4.544 3.486 2.459 1.501 0.664

74.936 79.941 84.951 89.966 94.984

34.778

so. 153

85.127 90.089 95.042 100.000

4.693

3.555 2.398 1.245 0

1~.496

100.000

Ordinate

Ordinate

-

1.629

0

100.000

6.597

0

35.()89

100.000

0 ~_._--

L.E. radius: 1.594 Slope of radius through L.E.: 0.2527

L.E. radius: 2.120 Slope of radius through L.E.: 0.0842

THEORY OF WING SECTIONS

420

NACA 63r418 (Stations and ordinates given in per cent of airfoil chord) Upper surface Station

Ordinate

Lower surface Station

NACA 63.-618

(Stations and ordinates given in

per cent of airfoil chord)

Upper surface

Ordinate

Station

Ordinate

Lower surface Station

Ordinate

0 0.267 0.487 0.945 2.140

0 1.484 1.833 2.410 3.455

0 0.733 1.013 1.555 2.860

-

0 1.284 1.553 1.982 2.711

0 0.156 0.361 0.797 1.965

0 1.511 1.878 2.491 3.616

0 0.844 1.139 1.703 3.035

-

0 1.211 1.458 1.849 2.500

4.593 7.077 9.577 14.602 19.645

4.975 6.139 7.087 8.560 9.632

5.407 7.923 10.423 15.398 20.355

.;.. -

3.711 4.443 5.019 5.868 6.448

4.393 6.868 9.367 14.404 19.469

5.268 6.542 7.586 9.219 10.418

5.607 8.132 10.633 15.596 20.531

-

3.372 3.998 4.484 5.181 5.642

24.699 29.760 34.823 39.886 44.949

10.385 10.854 11.058 10.986 10.672

25.301 30.240 35.177 40.114 45.054

-

6.805 6.966 6.938 6.702 6.292

24.549 29.640 34.734 39.829 44.919

11.273 11.822 12.086 12.056 11.767

25.451 30.360 35.266 40.171 45.081

-

5.903 5.990 5.906 5.630 5.197

50.000 55.046 60.083 65.110 70.125

10.148 9.446 8.596 7.626 6.564

50.000 54.954 59.917 64.890 69.875

- 5.736

-

5.()66 4.312 3.506 2.676

50.000 55.069 60.125 65.164 70.187

11.251 10.541 9.667 8.655 7.534

50.000 54.931 59.875 64.836 69.813

-

4.633 3.971 3.241 2.475 1.702

75.128 80.119 85.099 90.069 95.032

5.438 4.280 3.130 2.017· 0.978

74.872 79.881 84.901 89.931 94.968

- 1.858 - 1.096 - 0.438 0.051 0.286

75.191 80.178 85.147 90.103 95.048

6.330 5.073 3.800 2.531 1.293

74.809 79.822 84.853 89.897 94.952

- 0.960 - 0.297 0.238 0.571 0.603

100.000

I

0

100.000

0

L.E. radius: 2.120 Slope of radius through L.E.: 0.1685

100.000

0

100.000

0

L.E. radius: 2.120 Slope of radius through L.E.: 0.2527

421

APPENDIX III NACA 63r221 (Stations and ordinates given in per cent of airfoil chord) Upper surface Station

NACA 63.-421

(Stations and ordinates given in

per cent of airfoil chord)

Lower surface

Upper surface

Ordinate Station Ordinate

Station

1.425

0 - 1.527 - 1.861 - 2.414

0.237 0.452 0.902

2.708

- 3.385

o

Lower surface

Ordinate

Station

Ordinate

0 0.7fl3 1.048 1.598 2.914

0 - 1.461 - 1.774 - 2.289 3.181

0 0.367 0.600 1.075 2.292

0 1.627 2.001 2.628 3."57

0 0.633 0.900

4.763 7.253 9.753 14.767 19.792

5.375 6.601 7.593 9.111 10.204

5.237 7.747 10.24i 11).233 20.208

-

4.743 5.753 6.559 7.765 8.612

4.5271 5.675 5.473 7.007 I 7.010 7.993 9.506 I 8.097 10.494 14.535 1 9.774 t 15.465 19.5851 10.993 20.415

24.824 29.860 34.897 39.934­ 44.969

10.946 11.383 11.529 11.369 10.949

25.176 30.140 35.103 40.066 45.031

-

9.156 9.439 9.469 9.227 8.i59

24.6491 11.837 29.719 12.352 34.;93 J 12.558 39.8()7: 12.439 4-1.937; 12.044

50.000 55.027 60.048 65.063 70.071

10.309 9.485 8.512 7.426 6.262

8.103

50.000 55.054

11.412 10.580

60.096

9.582

65.126, 70.1-l3 i

8.455 7.232

75.073

5.054 3.849 2.693 1.629 0.708

75.1451 80.135 i

5.947 74.855 2.367 4.643 I 79.865 I - 1.459

85.1·11

3.364

so.067 85.056 90.039 95.018 100.000

0

50.000

54.9i3

59.95~

I-

I-

- ;.295

I- 6.370 5.366

64.931 i -

69.929\74.927 79.933 l 84.944 1 89.961 94.982 -

il

4.318 3.264 2.257 1.347 0.595 0.076

__

.100.000 L_.~

.

-.--~

L.E. radius: 2.650 Slope of radius through L. E.: 0.0842

I {i

2.086!

9(!.07~

i

I

0 1.661 2.054 2.717 3.925

-

4.U! 5.314 6.029 7.082 7.809

25.351 30.281 35.207 40.133 I 45.OG3 1 -

8.257 8.464 8.438 8.155 7.60-1

5O.0on

7.000 6.200

54.946

I 11

I'

J -

!-

I-

59.9t).J 5.298 6-l.8~: I - 4.335 69.851 3.344

II-

8-4.889 I - O.G72

2.144 , 89.922

I- O.01H

l;:;;L..~.l~ l;:~ L_.~·242

1..E. radius: 2.650 Slope of radius through L.E.: O.l6S5

THEORY OF WING SECTIONS

422

NACA 63A210 (Stations and ordinates given in per cent of airfoil chord)

Upper surface Station

Ordinate

Lower surface Station

NACA 64-108

(Stations and ordinates given in

per cent of airfoil chord)

Upper surface

Ordinate

Station

Lower surface

Ordinate Station Ordinate

0 0.423 0.664 1.151 2.384

0 0.868 1.058 1.367 1.944

0 0.577 0.836 1.349 2.616

-

0 0.756 0.900 l.l25 1.522

0 0.472 0.719 1.215 2.460

0 0.682 0.828 1.058 1.457

0 0.528 0.781 1.285 2.540

-

0 0.632 0.758 0.950 1.271

4.869 7.364 9.863 14.869 19.882

2.769 3.400 3.917 4.729 5.328

5.131 7.636 10.137 15.131 20.118

-

2.047 2.428 2.725 3.167 3.468

4.956 7.455 9.955 14.958 19.962

2.032 2.471 2.832 3.405 3.835

5.044 7.545 10.045 15.042 20.038

-

1.716 2.047 2.316 2.733 3.039

24.898 29.916 34.935 39.955 44.975

5.764 6.060 6.219 6.247 6.151

25.102 30.084 35.065 40.045 45.025

-

3.662 3.764 3.771 3.689 3.523

24.968 29.974 34.980 39.987 44.994

4.152 4.370 4.494 4.528 4.431

25.032 30.026 35.020 40.013 45.006

-

3.256 3.398 3.464 3.456 3.335

49.994 55.012 60.028 65.041 70.052

5.943 5.637 5.245 4.772 4.227

50.006· 54.988 59.972 64.959 69.948

-

3.283 2.985 2.641 2.262 1.861

50.000 55.005 60.010 65.013 70.015

4.236 3.959 3.617 3.219 2.777

50.000 54.995 59.990 64.987 69.985

-

3.132 2.863 2.545 2.189 1.805

75.061 80.074 85.072 90.050 95.026

3.624 2.974 2.254 1.519 0.769

74.939 79.926 84.928 89.950 94.974

-

1.464 1.104 0.812 0.539 0.279

75.016 80.015 85.013 90.010 95.005

2.302 1.802 1.297 0.808 0.364

74.984 79.985 84.987 89.990 94.995

-

1.406 1.006 0.625 0.292 0.048

100.000

0.021

100.000

- 0.021

100.000

L.E. radius: 0.742 T.F. radius: 0.023 Slope of radius through L.E.: 0.095

0

100.000

0

L.E. radius: 0.455 Slope of radius through L.E.: O.()4-2

APPENDIX III NACA 64-110 (Stations and ordinates given in per cent of airfoil chord) Upper surface

Station 0 0.465

Ordinate Station Ordinate

1.207 2.450

0 0.844 1.023 1.303 1.793

4.945 7.443 9.944 14.947 19.953

2.500 3.037 3.479 4.178 4.700

24.959 29.967 34.175 39.984 44.992

5.087

0.71~

50.000 55.007 60.012 65.016 70.019 75.020 80.019 85.016 90.012 95.006 100.000

Lower surface

0

o.sss 0.788 1.293 2.550

D.055 1.557 10.056

1"5.053 20.04:7

25.04:1 30.033

-

0.442 0.524 0.645 0.836

-

2.184 2.613 2.963 3.506 3.904

4.934 7.432 9.933 14.937 19.943

1.719 2.115 2.444 2..970 3.367

5.066 7.568 10.067 15.063 20.057

-

1.087 1.267 1.410 1.624 I.nS

4.191 4.378 4.465 4.452 4.295

24.952 29.961 34.971 39.981

25.048 30.039 35.029 40.019 45'(J09

- 1.877 - 1.935 - 1.951

44.991

3.667 3.879 4.011 4.066 4.014

4.034 3.690 3.284 2.830 2.341

50.000 55.008 ! 60.015 I 65.020 70.023 I

3.878 3.670 3.402 3.080 2.712

50.000 - 1.672 54.992 - 1.480 59.985 - - 1.260 64.980 - 1.020 69.977 - 0.768

50.000 54.993 59.988 64.984 69.981

-

74.980 79.981 84.984 89.988

- 1.833 - 1.324

I

94.994

100.000

- 0..840 - 0.413 I -

I

Ordinate

0 0.541 0.796 1.302 2.560

5.138 4.786 4.356

0

Ordinate Station 0 0.542 0.664 0.859 1.208

40.016 45.008

2.729 2.120 1.512 0.929 0.406

Station

Lower surface

0 0.459 0.704 1.198 2.440

35.025

3.313

Upper surface

0 0.794 0.953 1.195 1.607

5.495 5.524 5.391

3.860

NACA 64-206

(Stations and ordinates given in

per cent of airfoil chord)

-

-

5.350

423

0.090 0______

L.E. radius: 0.720 Slope of radius through L.E.: 0.042

I i

2.307 800024 1.868 85.020 1.410 90.015 1I 0.940 0.473 95.007 75.025

100.000

1

0

0

-

1.924

- 1.824

I 74.975 I

I

i

- 0.517 79.976 - 0.276 0.064 84.980 0.094 89.985 I 94.993l 0.159

Ii

100.000 t

0

LE. radius: 0.256

Slope of radius through L.E.: 0.084

THEORY OF WING SECTIONS

424

NACA 64-208 (Stations and ordinates given in per cent of airfoil chord)

Upper surface Station

Ordinate

Lower surface Station

NACA 64-209

(Stations and ordinates given in

per cent of airfoil chord)

Upper surface

Ordinate

·Station

Ordinate

Lower surface Station

Ordinate 0 - 0.686 - 0.819 - 1.018

0 0.445 0.688 1.180 2.421

0 0.706 0.862 1.110 1.549

0 0.555 0.812 1.320 2.579

-

0 0.606 0.722 0.896 1.177

0 0.438 0.680 1.172 2.411

0 0.786 0.959 1.232 1.716

0 0.562 0.820 1.328 2.589

4.912 7.410 9.909 14.915 19.924

2.189 2.681 3.089 3.741 4.232

5.088 7.590 10.091 15.085 20.076

-

1.557 1.833 2.055 2.395 2.640

4.901 7.398 9.899 14.905 ~9.915

2.423 2.965 3.413 4.127 4.663

24.935 29.948 39.974 44.988

4.598 4.856 5.009 5.063 4.978

25.065 30.052 35.039 40.026 45.012

-

2.808 2.912 2.949 2.921 2.788

24.927 29.941 34.956 39.971 44.986 ,

50.000 55.011 60.020 65.027 70.031

4.787 4.506 4.152 3.733 3.263

50.000 54.989 59.980 64.973 69.969

-

2.581 2.316 2.010 1.673 1.319

50.000 55.012 60.022 65.030 1 70.035 ,

75.032

2.749 2.200 1.634 1.067 0.522

74.968 79.969 84.973 89.981 94.990

-

0.959 0.608 0.288 0.033 0.110

34.961

~.031

85.027 90.019 95.010 100.000

0

100.000

0

I,.E. radius: 0.455 Slope of radius through L.E.: 0.084

-

1.344

5.099 7.602 10.101 15Jl95 20.085

-

1.791 2.117 2.379 2.781 3.071

5.064 5.345 5.509 5.561 5.459

25.073 30.059 35.044 40.029 45.014

-

3.274 3.401 3.449 3.419

5.239 4.921 4.523 4.056 3.533

50.000 54.988 59.978 64.970 69.965

-

3.033 2.731 2.381 . 1.996 1.589

75. 036 1 2.964 2.360 80.035 85.030 ; 1.742 90.021 i 1.128 0.543 95.011

74.964 79.965 84.970 89.979 94.989

-

1.174.

I I

I

100.000

0

100.000

- 3.269

- 0.;68 - 0.396 - 0.094 0.089

0

L.E. radius: 0.579 Slope of radius through L.E.: 0.084

t

APPEl·lDIX III NACA 64-210 (Stations and ordinates given in per cent of airfoil chord)

Upper surface Station

Lower surface

Ordinate Station Ordinate

425

NACA 64 1-112

(Stations and ordinates given in

per cent of airfoil chord)

t· pper surface Station

Ordinate

o

0 0.431 0.673 1.163 2.401

0 0.867 1.056 1.354 1.884

4.890 7.387 9.887 14.894 19.905

2.656 3.248 3.736 4.514 5.097

2.024 5.110 2.400 7.613 10.113 I - 2.702 15.106 - 3.1~ 20.095 1 - 3.50il

2.967 3.605 4.128 4.956 5.571

24.919 29.934 34.951 39.968 44.985

5.533 5.836 6.010 6.059 5.938

25.081 30.066 i 35.049140.032 45.015 -

3.743 3.Sg-1 3.950 3.917 3.748

6.024 6.330 6.493 6.517 6.346

50.000 55.014 60.025 65.033 70.038

5.689 5.333 4.891 4.375 3.799

50.000 54.987 1 59.975 I 64.967169.962 -

3.483 3.143 2.749 1.855

6.032 5.604 5.084 4.489 3.836

75.040 80.038 85.033 90.024 95.012

3.176 2.518 1.849 1.188 0.564

74.960 I 79.962 , 84.968 89.977 94.988

- 1.386 - O.!2? - O.D03 - 0.15-1 0.068

3.143 2.427 1.718 1.044 0.446

100.000

0

0 0.569 0.827 1.337 2.599

0 0.767 0.916 1.140 1.512

-

Ii-

I-

!

100.000

1

2.31~

0

----

L.E. radius: 0.720 Slope of radius through L.E.: 0.084

1.002 1.213 1.543

2.127

o L.E. radius: 1.040 Slope of radius through I ... I~.: 0.042

'l'HEORY OF WING SECTIONS

426

NACA 64 1-212 (Stations and ordinates given in per cent of airfoil chord)

Upper surface Station

Lower surface

Ordinate Station Ordinate

NACA 641-412

(Stations and ordinates given in

per cent of airfoil chord)

Upper surface

Lower surface

Station Ordinate Station

-

Ordinate

0.841 1.353 2.618

-

0 0.925 1.105 1.379 1.846

0 0.338 0.569 1.045 2.264:

0 1.064 1.305 1.690 2.393

0 0.662 0.931 1.455 2.736

3.123 3.815 4.386 5.291 5.968

5.132 7.636 10.135 15.128 20.114

-

2.491 2.967 3.352 3.945 4.376

4.738 7.229 9.730 14.745 19.772

3.430 4.231 4.896 5.959 6.760

5..262 7.771 10.270 15.255 20.228

- 2.166 - 2.535 -2.828 - 3.267 - 3.576

24..903 29.921 34 . 941 39.961 44.982

6..470 6.815 7.008 7.052" 6.893

25..()97 30.079 35.. 059 40.039 45.018

-

4.680 4.871 4.948 4.910 4.. 703

24.805 29.842 34.882 39.923 44.963

7.363 7.786 8.037 8.123 7.988

25.195 30.158 35.118 40.077 45.037

- 3.783

- 3.608

50.000 55.016 60.029 65.039 70.045

6.583 fi.151 5.619 5.004 4.322

50.000 54.. 984 59.971 64.961 69.955

-

4.377 3.961 3.477 2.944 2.378

50.000 55.032 60.059 65.078 70.090

7.686 7.246 6.690 6.033 5.293

50.000 54.968 59.941 64.922 69.910

-

75.047 80.045 85.038 90.027 95.013

3.590 2.825 2.054 1.303 0.604

74.953 79.955 84.962 89.973

-

1.800 1.233 0.708 0.269 0.028

75.094 80.089 85.076 90.055 95.027

4.483 3.619 2.722 1.818 0.919

74.906 79.911 84.924 89.945 94..973

- 0.903 - 0.435 - 0.038 1 0.250 0.345

0 0.418 0.659 1.147 2.382

0 1.025 1.245 1.593 2.218

4.868 7.364 9.865 14.872 19.886

100.000

0

0 0~582

94.987 100.000

0

L.E. radius: 1.040 Slope of radius through L.E.: 0.084

100.000

0

100.000

-

0 0.864 1.025 1.262 1.849

- 3..898 - 3.917 - 3.839 3.274 2.866 2.406 1.913 1.405

0

L.E. radius: 1.040 Slope of radius through L.E.: 0.168

APPENDIX III NACA 64t-215 (Stations and ordinates given in per cent of airfoil chord) Upper surface Station

Lower surface

Ordinate Station

427

NACA 64r415

(Stations and ordinates given in

per cent of airfoil chord)

Upper surface

Ordinate

Station

Ordinate

Lower surface Station

Ordinate

0 0.399 0.637 1.122 2.353

0 1.254 1.522 1.945 2.710

0 0.601 0.863 1.378 2.641

-

0 1.154 1.382 1.731 2.338

0 0.299 0.526 0.996 2.207

0 1.291 1.579 2.038 2.883

0 0.701 0.974 1.504 2.793

4.836 7.331 9.831 14.840 19.857

3.816 4.661 5.356 6.456 7.274

5.164 7.669 10.169 15.160 20.143

-

3.184 3.813 4.322 5.110 5.682

4.673 7.162 9.662 14.681 19.714

4.121 5.075 5.864 7.122 8.066

5.327 7.838 10.338 15.319 20.286

-

2.857 3.379 3.796 4.430 4.882

24.878 29.901 34.926 39.952 44.977

7.879 8.290 8.512 8.544 8.319

25.122 30.099 35.074 40.048 45.023

-

6.089 6.346 6.452 6.402 6.129

24.756 29.803 34.853 39.904 44.954

8.771 9.260 9.541 9.614 9.414

25.244 30.197 35.147 40.096 45.046

-

5.191 D.372 5.421 5.330 5.034

50.000 55.020 60.036 65.048 70.055

7.913 7.361 6.691 5.925 5.085

50.000 54.980 59.964 64.952 69.945

-

5.707 5.171 4.549 3.865 3.141

50.000

9.016 8.456 7..762 6.954 6.055

50.000 54.960 59.928 64.904 69.889

- 4..604 - 4.076 - 2.834 - 2.167

15.058

4.191 3.267 2.349 1.466 0.662

74.942 79.945 84.954 89.967 94.984

- 2.401 - 1.675

5.084 4.062 3.020 1.982 0.976

74.885 79.891 84.908 89.934 94.968

- 1.504 - 0.878 - 0.328 0.086 0.288

100.000

0

80.055 85.046 90.033 95.016

100.000

0

1 100 .000

- 1..003 - 0.432 - 0.030

0

L.E. radius: 1.590 Slope of radius through L.E.: 0.084

55.040 60.072 65.096 70.111 75.115

so. 109

85.092 90.066 95.032 100.000

0

0 -

1.()91

- 1.299 - 1.610 - 2.139

-

3.478

L.E. radius: 1.590 Slope of radius through L.E.: 0.168

THEORY OF VlING SECTIONS

428

NACA 64r218 (Stations and ordinates given in per cent of airfoil chord)

Upper surface Station

Ordinate

Lower surface Station

NACA 64,-418

(Stations and ordinates given in

per cent of airfoil chord)

Upper surface

Ordinate

Station

Lower surface

Ordinate

Station

-

Ordinate

0 0.380 0.617 1.099 2.325

0 1.473 1.785 2.279 3.186

0 0.620 0.883 1.401 2.675

-

0 1.373 1.645 2.065 2.814

0 0.263 0.486 0.950 2.152

0 1.508 1.840 2.370 3.357

0 0.737 1.014 1.550 2.848

4.804 7.297 9.797 14.808 19.828

4.497 5.496 6.316 7.612 8.576

5.196 7.703 10.203 15.192 20.172

-

3.865 4.648 5.282 6.266 6.984

4.609 7.095 9.595 14.617 19.657

4.800 5.908 6.823 8.277 9.366

5.391 7.905 10.405 15.383 20.343

-

3.536 4.212 4.755 5.585 6.182

24.853 29.881 34.912 39.942 44.972

9.285 9.760 10.009 10.023 9.725

25.147 30.119 35.088 40.058 45.028

-

7.495 7.816 7.949 7.881 7.535

24.707 29.763 34.823 39.885 44.9-15

10.176 10.730 11.037 11.093 10.820

25.293 30.237 35.177 40.115 45.055

-

6.596 . 6.842 6.917 6.809 6.440

50.000 55.024

9.217 8.540 7.729 6.812 5.814

50.000 54.976 59.957 64.943 69.935

-

7.011 6.350 5.587 4.752 3.870

50.000 55.047 60.086 65.114 70.131

10.320 9.635 8.799 7.841 6.784

50.000 54.953 59.914 64.886 69.869

-

5.908 5.255 4.515 3.;21 2.896

4.760 3.683 2.623 1.617 0.716

74.932 79.936 84:.946 89.962 94.981

-

2.970 2.091 1.277 0.583 0.084­

7~.135

5.654 4.477 3.294 2.132 1.030

74.865 79.873 84.892 89.923 94.963

-

2.074 1.293 0.602 0.064 0.234

60.043

65.057 70.065 75.068

so.064

85.054 90.038 95.019 100.000

0

100.000

0

L.E. radius: 2.208 Slope of radius through L.E.: 0.084

80.127 85.108

I

90.0771 95.037

100.000

I

0

,

100.000

0

- 1.308 - 1.560 - 1.942 - 2.613

0

L.E. radius: 2.208 Slope of radius through L.E.: 0.168

APPENDIX III NACA 64r618 (Stations and ordinates given in per cent of airfoil chord)

Upper surface Station

429

N ACA Mc-221

(Stations and ordinates given in

per cent of airfoil chord)

Lower surface

L pper surface

Ordinate Station Ordinate

Station

Lower surface

I

Ordinate Station Ordinate

0 0.150 0.359 0.805 1.982

0 1.534 1.885 2.452 3.518

0 0.850 1.141 1.695 3.018

-

0 1.234 1.465 1.810 2.402

0 0.362 0.596 1.075 2.297

0 1.690 2.049 2.618 3.665

0 0.638 0.904 1.425 2.703

4.417 6.895 9.395 14.427 19.486

5.093 6.312 7.322 8.937 10.153

5.583 8.105 10.605 15.573 20.514

-

3.197 3.768 4.220 4.899 5.377

4.772 7.264 9.763 14.776 19.799

5.182 6.334­ 7.282 8.778 9.889

5.228 7.736 10.237 15.224 20.201

-

24.560 29.645 34.735 39.827 44.917

11.065 11.698 12.065 12.163 11.915

25.440 30.355 35.265 40.173 45.083

-

5.695 5.866 5.885 5.737 5.345

24.829 29.861 34.897 39.933

10.701 11.240 11.510 11.502 11.125

25.171 30.139 35.103 40.067 45.032

-

50.000 55.071 60.129 65.171 70.196

11.423 10.730 9.870 8.870 7.754

50.000 54.929 59.871 64.829 69.804

-

4.805 4.160 3.444 2.690 1.922

50.000 55.027 60.050 65.065 70.0i5

10.507 9.702 8.749 7.679 6.521

54.9731-

50.000

59.950 64.935 69.925

- 8.301 7.512 - 6.607 - 5.619 - 4.577

75.203 80.191 85.161 90.115 95.056

6.544 5.270 3.963 2.646 1.344

74.797 79.809 84.839 89.885 94.944

- 1.174 - 0.494 0.075 0.456 0.552

75.077 80.073 85.001 90.044 95.021

74.923 79.927

- 3.520 - 2.490

M.939 . 89.956 94.979

- 0.727

100.000

0

100.000

!- _. __ 0 ..

.-._-­

L.E. radius: 2.208 Slope of radius through L.E.: 0.253

4-1.

008

100.000 --

--~

i 1

5.310 4J)82

2.885 1.761 0.765

I

0

Il~.OOO 1

L.E. radius: 2.884 Slope of radius through

0 - 1.590 -

1.909

- 2.404 - 3.293

4.550 5.486 6.248 7.432

-8.~7

8.911 9.296 9.450 9.360 8.935

- 1.539 - 0.133

0

I~.I~.: O.OSl

-

THEORY 0/;' WING SECTION8

430

NACA 64r421 (Stations and ordinates given in per cent of airfoil chord)

Upper surface Ordinate

Station

Lower surface

NACA 64A210

(Stations and ordinates given in

per cent of airfoil chord)

Upper surface

Station

Ordinate

Station

0 0.424 0.665 1.153 2.387

Ordinate

-

Lower surface

-

Station

Ordinate

0 0.856 1.044 1.342 1.895

0 0.576 0.835 1.347 2.613

0 - 0.744

-

0 0.227 0.445 0.903 2.096

0 1.723 2.101 2.707 3.834

0 0.773 1.055 1.597 2.904

-

0 1.523 1.821 2.279 3.090

4.545 7.028 9.528 14.553 19.599

5.482 6.744 7.786 9.442 10.678

5.455 7.972 10.472 15.447 20.401

-

4.218 5.048 5.718 6.750 7.494

4.874 7.369 9.868 14.874 19.885

. 2.685 3.288 3.792 4.592 5.200

5.126 7.631 10.132 15.126 20.115

- 1.963

24.657 29.723 34.794 39.865 44.936

11.591 12.209 12.539 12.572 12.220

25.343 30.277 35.206 40.135 45.064

-

8.011 8.321 8.419 8.288 7.840

24.900 29.917 34.935 39.955 44.975

5.656 5.984 6.192 6.274 6.208

25.100 30.083 35.065 40.045 45.025

- 3.554 - 3.688 - 3.744

11.610 10.797 9.819 8.708 7.491

50.000 54.945 59.901 64.869 69.850

-

7.198 6.417 5.535 4.588 3.603

49.994 55.012 60.028 65.042 70.054

6.014 5.714 5.323 4.852 4.310

50.006 54.988 59.972 64.958 69.946

6.203 4.876 3.556 2.276 1.079

74.846 79.855 84.878 89.913 94.958

-

2.623 1.692 0.864 0.208 0.185

75.063 80.076 85.074 90.052 95.027

3.702 3.037 2.301 1.551 0.785

74.937 - 1.542 1.167 79.924 84.926 I - 0.859 89.948 - 0.571 94.974 - 0.295

100.000

0.021

50.000 55.055 60.099 65.131 70.150 75.154 80.145 85.122 90.087 95.042 100.000

I

0

100.000

0

L.E. radius: 2.884 Slope of radius through L.E.: 0.168

- 0.886 - 1.100 - 1.473 - 2.316

- 2.600 - 3.030 - 3.340

- 3.716 - 3.080 I

354 1 -- a.3.062 - 2.719 - 2.342 - 1.944

I-

100.000 -

- 0.021

LE. radius: 0.687

T.E. radius: 0.023 Slope of radius through L.E.: 0.095

I

APPENDIX 111 NACA 64A4:10

(Stations and ordinates given in

per cent of airfoil chord)

Upper surface

I i

0 0.350 : 0.582 1.059 2.276

NACA 64 1A212

(Stations and ordinates given in

per cent of airfoil chord)

Lower surface

Station Ordinate Station Ordinate

431

Upper surface Station

Lower surface

Ordinate Station

Ordinate

0 0.902 1.112 1.451 2.095

0 0.650 0.918 1.441 2.724

-

0 0.678 0.796 0.969 1.251

3.034 3.865 4.380 5.366 6.126

5.251 7.770 10.263 1'5.252 20.230

-

1.592 1.919 1.996 2.244 2.406

4.849 7.343 9.842 14.849 19.862

3.145 3.846 4.432 5.358 6.060

5.151 7.657 10.158 15.151 20.138

6.705 7.131 7.414 7.552 7.522

25.200 30.166 35.129 40.090 45.050

-

2.499 2.537 2.518 2.436 2.266

24.880 29.900 34.922 39.946 44.970

6.584 6.956 7.189 7.272 7.177

25.120 / - 4.482 30.100 - 4..660 35.078 f - 4.741 40.054 4.714 45.030 4.549

49.989 . 7.344 55.025 ' 7.040 60.057 . 6.624 65.085 j a.l06 70.108 J 5.490

50.011

2.O'M

49.993 55.015 60.034 65.050 70.064

6.935 6.570 6.103 5.544 4.903

50.007154.985 59.966 i 64.950 69.936 -

75.075 80.090

74.925 I - 2.037 79.910 - 1.563 84.912 - 1.159 89.938 - 0.771 94.968 - 0.398

4.749 7.230 9.737 14.748 19.770 24.800 29.834 34.871 39.910 44.950

; ; : .

I

I 4.780

54.975 59.943 64.915 69.892

3.967 3.018 2.038 1.028

74.874 79.849 84.852 89.896 94.947

100.000 1 0.021

100.000

75.126 SO.151 85.148 90.104 95.053

J

i ; j j

1~

0 0.409 0.648 II 1.135 2.365

0 1.013 1.233 1.580 2.225

-

1.736 1.418 1.086 0.760

-

0.460 0.229 0.132 0.076 0.048

90.062 95.032

4.197 3.433 2.601 1.751 0.888'

- 0.021

100.000

0.025

L.E. radius: 0.687 T.E. radius: 0.023 Slope of radius through L.E.: 0.190

85.088

-

I

0 0.591 ·0.852 I 1.365 2.635 I -

I-

0 0.901 1.075 1.338 1.803

- 2.423 - 2.874 3.240 3.796 - 4.200

1I

­

II-

1-

I

IO?OOO

I-

4.275 3.918 3.499 3.034 2.537

0.025

L.E. radius: 0.994 r.s, radius: 0.028 Slope of radius through L.E.: 0.095

THEORY OF WING SECTIONS

432

NACA 64JA215 (Stations and ordinates given in per cent of airfoil chord)

L pper surface

Lower surface

NACA 65-206

(Stations and ordinates given in

per cent of airfoil chord)

Upper surface

Ordinate Station

Ordinate

Station

0 1.243 1.509 1.930 2.713

0 0.612 0.876 1.393 2.667

0 1.131 1.351 1.688 2.291

0 0.460 0.706 1.200 2.444

4.811 7.304 9.802 14.811 19.827

3.833 4.683 5.391 6.510 7.351

5.189 7.696 10.198 15.189 20.173

- 3.111 - 3.711 - 4.199

24.849 29.875 34.903 39.933 44.963

7.975 8.417 8.686 8.766 8.627

25.151 30.125 35.()97 40.067 45.037

49.992 55.018 60.042 65.063 70.079

8.308 7.843 7.258 6.566 5.782

50.008 54.982 59.958 64.937 69.921

75'()93 80.111 85.109 90.076 95.039

4.926 4.017 3.039 2.046 1.039

74.907 79.889 84.891 89.924 94.961

100.000

0.032

100.000

Station 0 0.388 0.624 1.107 2.333

I

Lower surface

-

Ordinate Station

Ordinate

0 0.524 0.642 0.822 1.140

0 0.540 0.794 1.300 2.556

0 0.424 0.502 0.608 0.768

- 5.491

4.939 7.437 9.936 14.939 19.945

1..625 2.012 2.340 2.869 3.277

5.061 7.563 10.064 15.061 20.055

-

5.873 6.121 6.238 6.208 5.999

24.953 29.962 34.971 39.981 44.990

3.592 3.824 3.982 4.069 4.078

25.047 30.038 35.029 40.019 45.010

-

5.648 5.191 4.654 4.056 3.416

50.000 55.009 60.016 65.022 70.-6

4.003 3.836 3.589 3.276 2.907

50.000 54.991 59.984 64.978 69.974

-

-

2.766 2.147 1.597 1.066 0.549

75.028 80.027 850024 90.018

2.489 2.029 1.538 1.027 1 95'(x)9 ! 0.511

74.972 79.973 84.976

- 0.699 - 0.437 - 0.192 0.007 0.121

-

-

4.948

i

- 0.032

L.E. radius: 1.561 T.E. radius: 0.037 Slope of radius through L.E.: 0.095

100.000 t

0

89.982

94.991

I 100.()()()

-

-

- 0.993 - 1.164

- i.306 - 1.523 - 1.685 - 1.802

- 1.880 - 1.922 - 1.927 - 1.888 1.797 1.646 1.447 1.216 0.963

0

L.E. radius: 0.240 Slope of radius through l..E.: 0.084

433

APPENDIX III NACA 65-209 (Stations and ordinates given in per cent of airfoil chord) Upper surface Ordinate

Station

Lower surface Station

Ordinate

0 0.441 0.684 1.177 2.417

0 0.748 0.912 1.162 1.605

0 0.559 0.816 1.323 2.583

-

0 0.648 0.772 0.948 1.233

4.908 7.405

°2.275 2.805 3.251 3.971

5.092 7.595 10.096 15.091 20.082

-

25.071 30.058 35.044 40.029 45.014

-

~.904

14.909 19.918

4.522

24.929

4.944

29.942

5.254 5.4:61

34.956 39.971 44.986

I

5.567 5.564

I

5.439 50.000 5.181 013 55. 1 4.814 60.024 4.358 65.033 3..828 70.039 3.237 75.041 2.601 80.040 1.933 035 85. 90.026 1 1.255 0.596 95.013 100.000

I

0

Upper surface Station

Lower surface

I

Ordinate Station Ordinate

o 0.435 0.678 1.169 2.408

0 0.819 0.999 1.273 1.757

-

0 0.719 0.859 1.059 1.385

1.643 1.957 2.217 2.625 2.930

4.898 7.394 9.894 14.899 19.909

2.491 3.069 3.555 4.338 4.938

5.102 7.606 10.106 15.101 20.091

-

1.859 2.221 2.521 2.992 3.346

3.154 3.310 3.401 3.425 3.374

24.921

30.0641- 3.788

25.079

- 3.607

34.951

5.397 5.732 5.954

39.9()s

6.067

40.032 t - 3.~

6.058

45.016

50.000 I - 3.233 I 54.9871 - 2.991 ~9.976 I - 2.672 ti4 .967 - 2.298 69.961 - 1.884 74.959 79.960 84.965 89.974 987 94. 1 100.000

NACA 65-210 (Stations and ordinates given in pel" cent of airfoil chord)

1.447 1.009 0.587 0.221 0.036

0

L.E. radius: 0.552 Slope of radius through L.E.: O.OSl

29.936

44.984 I

i

5.915 5.625 5.217 4.712 4wl28

75.045 8l!.044 , 85 90.028 95.01-l,

3.479 2.783 2.057 1.327 0.622

1\

Jl38I'

II-

35.~~

-

I-

3.894

- 3.868 I

50.000 i 55.014 OO.tl27 j (\5.03() 70.043

I

0 i 0.565 0.822 1.331 I 2.592 I

50.000 I 54.986 SU.973 I 64.96! 69.951 i -

3.709 3.435 3.075 2.652 2.184

74.955

Ii -

79.9~6

I -

1.689 1.191 0.711 0.293 0.010

I-

1°

I II!

!-

1-

1 8-1.062 I 89. 972 94.986

I!

I_~:~~J_~ i 100.000

-

0__

1..E. radius: 0.687 Slope of radius through L.E.: 0.084­

THEORY OF WING SECTIONS

434

NACA 65-410 (Stations and ordinates given in per cent of airfoil chord) Upper surface Station

Lower surface

Ordinate Station

NACA 651-212

(Stations and ordinates given in

per cent of airfoil chord)

Upper surface

Lower surface

Ordinate Station Ordinate

Ordinate

Station

0 0.661 0.781 0.944 1.191

0 0.423 0.664 1.154 2.391

0 0.970 1.176 1.491 2.058

0 0.577 0.836 1.346 2.609

0 0.372 0.607 1.089 2.318

0 0.861 1.061 1.372 1.935

0 0.628 0.893 1.411 2.682

4.797 7.289 9.788 14.798 19.817

2.800 3.487 4.067 5.006 5.731

5.203 7.711 10.212 15.202 20.183

- 1.536 -1.791 - 1.999 - 2.314 - 2.547

4.878 7.373 9.873 14.879 19.890

2.919 3.593 4.162 5.073 5.770

5.122 7.627 10.127 15.121 20.110

-

24.843 29.872

6.290 6.702 6.983 7.138 7.153

25.157 30.128 35.()97

2.710 2.814 2.863 2.854 2.773

24.906

29.923 34.942 39.961 44.981

6.300 6.687 6.942 7.068 7.044

25.094 30.077 35.058 40.039 45.019

- 4.510 - 4.743

45.032

-

7.018 6.720

50.000 54.971 59.947 64.927 69.915

-

2.606 2.340 2.004 1.621 1.211

50.000 55.017 60.032 65.043 70.050

6.860 6.507 6.014 5.411 4.715

50.000 54.983 59.968 64.957 69.950

74.910 79.912 84.924 89.943 94.971

- 0..792

75.053 80.052 85.045 90.033 95.017

3.954 3.140 2.302 1.463 0.672

74.947 79.948 84.955 89.967 94.983

34.903

39.936 44.968

50.000 55.029 60.053 65.073 70.085 75.090 80.088 85.076 90.057 95.029 100.000

6.288 5.741 5.099 4.372 3.577

2.129 1.842 0.937 0

40.064

100.000

-

- 0.393 - 0.037 0.226 0.327

0

L.E. radius: 0.687 Slope of radius through L.E.: 0.168

100.000

I

0

100.000

0 - 0.870 - 1.036 - 1.277 - 1.686

2.287 2.745 3.128 3.727 4.178

- 4.882 - 4.926

- 4.854 - 4.654

- 4.317 I1 - 3. 872 - 3.351 - 2.771

-

2.164 1.548 0.956 0.429

- 0.040 0

L.E. radius: 1.000

Slope of radius through L.E.: 0.084

1

APPENDIX III NACA 65.-212 (J == 0.6 (Stations and ordinates given in per cent of airfoil chord)

Upper surface

Lower surface

435

NACA 65t-412 (Stations and ordinates given in per cent of airfoil chord)

Upper surface

Lower surface

Station Ordinate Station Ordinate

Station Ordinate Station Ordinate 0 0.399 0.638 1.124 2.356

0 0.982 1.194 1.520 2.113

4.837 7.329 9.827 14.833 19.848

0 0.601 0.862 1.376 2.644

-

0 0.852 1.012 1.242 1.625

3.017 3.728 4.330 5.298 6.042

5.163 7.671 10.173 15.167 20.152

-

2.185 2.606 2.956 3.500 3.904

24.869 29.894 34.921 39.951 44.983

6.611 7.029 7.304 7.444 7.423

25.131 30.106 35.079 40.049 45.017

-

4.197 4.401 4.518 4.550 4.475

50.017 65.051 60.094 65.123 70.124

7.231 6.856 6.318 5.634 4.842

49.983 54.949 59.906 64.877 69.876

-

4.283 3.968 3.566 3.124 2.640

75.112 80.090 85.064 90.036 95.013

3.983 3.082 2.173 1.297 0.521

74.888 79.910 84.936 89.964 94.987

-

2.131 1.604 1.085 0.595 0.191

100.000

0

1100.000

0

L.E.. radius: 1.000 Slope of radius through L.E.: 0.110

0 0.347 0.580 1.059

2.28.1

0 1.010 1.236 1.588 2.234

0 0.653 0.920 1.441 2.717

-

0 0.810 0.956 1.160 1.490

4.757 7.247 9.746 14.757 19.781

3.227 4.010 4.672 5.741 6.562

5.243 7.753 10.254 15.243 20.219

-

1.963 2.314 2.604 3.049 3.378

24.811 29.846 34.884 39.923 44.962

7.~93

7.658 7.971 8.139 8.139

25.189 30.154 35.116 40.077 45.038

-

3.613 3.770 3.851 3.855 3.759

50.000 55.035 60.064 65.086 70.101

I

7.963 7.602 7.085 6.440 5.686

50.000 54.965 59.936 64.914 69.899

-

3.. 551 3.222 2.801 2.320 1.798

75.107 I 80.103 85.090 , 90.066 95.033

4.847 3.935 2.974 1.979 0.986

74.893 79.897 84.. 910 89.934 94.967

- 1.267 - 0.751 - 0.282 0.089 0.278

I

100.000

0

100.000 _.

--

0 .-.-._.­

I~.E. radius: 1.000 Slope of radius through L, E.. : 0.168

THEORY OF WING SECTIONS

436

NACA 65r215 (Stations and ordinates given in per cent of airfoil chord) Upper surface

Station

Ordinate

Lower surface Station

NACA 65r415

(Stations and ordinates given in

per cent of airfoil chord)

Upper surface

Ordinate

Station

Lower surface

Ordinate Stati~n Ordinate

0 0.406 0.645 1.132 2.365

0 1.170 1.422 1.805 2..506

0 0.594 0.855 1.368 2.635

-

0 1.070 1.282 1.591 2.134

0 0.313 0.542 1.016 2.231

0 1.208 1.480 1.900 2.680

0 0.687 0.958 1.484 2.769

4.848 7.342 9.841 14.848 19.863

3.557 4.380 5.069 6.175 7.018

5.152 7.658 10.159 . 15.152 20.137

-

2.925 3.532 4.035 4.829 5.426

4.697 7.184 9.682 14.697 19.726

3.863 4.794 5.578 6.842 7.809

5.303 7.816 10.318 15.303 20.274

- 3.510 - 4.150 - 4.625

24.882 29.904 34.927 39.952 44.976

7.658 8.123 8.426 8.569 8.522

25.118 30.096 35.073 40.048 45.024

-

5.868 6.179 6.366 6.427 6.332

24.764 29.807 34.854 39.903 44.953

8.550 9.093 9.455 9.639 9.617

25.236 30.193 35.146 40.097 45.047

-

4.970 5.205 5.335 5.355 5.237

50.000 55.021 60.039 65.053 70.062

8.271 7.815 7.189 6.433 5.572

50.000 54.979 59.961 64.947 69.938

-

6.065 5.625 5.047 4.373 3.628

50.000 55.043 , 60.079 I 65.106 ~ 70.124 i

9.374 8.910 8.260 7.. 462 6.542

50.000 I 54.957 59.921 64.894 69.876

-

4.962 4.530 3.976 3.342 2.654

75.065 80.063 85.055 90.040 95.020

4.638 3.653 2.649 1.660 0.744

74.935 79.937 84.945 89.960 94.980

-

2.848 2.()61 1.303 0.626 0.112

75.131 II 80.126 1 85.109\ 90.080 j 95.040 I

5.532 4.447 2.175 1.058

74.869 79.874 84.891 89.920 ( 94.960

-

1.952 1.263 0.628 0.107 0.206

0

1100.000 I

100.000

0

100.000

0

L.E. radius: 1.505 Slope of radius through L.E.: 0.084

I

I

100.000

I

3.320

I

-

0 1.008 1.200 1.472 1.936

- 2.599 - 3.098

0

L.E. radius: 1.505 Slope of radius through L.E.: 0.168

APPENDIX III NACA 65r415 (J == 0.5 (Stations and ordinates given in per cent of airfoil chord)

NACA 65.-218 (Stations and ordinates given in per cent of airfoil chord) Upper surface

Lower surface

Lower surface

Ordinate Station Ordinate

Station

Station Ordinate Station Ordinate

4.099 5.122 5.985 7.383 8.459

5.426 7.946 10.451 15.432 20.389

- 2.335 - 2.746 -.3.081 - 3.591 - 3.963

24.671 29.743 34.825 39.916 45.019

9.280 9.883 10.280 10.470 10.423

25.329 30.257 35.175 40.084 44.981

- 4.232 - 4.411 I- - 4.508 - 4.526 - 4.431

50.152 10.106 55.262 9.501 60.307 8.672 65.314 7.684 70.294 , 6.573

49.848 54.738 59.693 64.686 69.706

- 4.226 - 3.929 - 3.548

75.253 80.199

74.747 79.801 84.863 89.923 94.973

-

90.077 95.027 100.000 ,

-_._­

),.~~.

0

100.000

- 2.609

1

radius: 1.505 Slope of radius through L.E.: 0.233

0 0.612 0.875 1.390 2.660

4.819 7.311 9.809 14.818 19.835

4.178 5.153 5.971 7.276 8.270

5.181 7.689 10.191 15.182 20.165

-

24.858

9.023 9.566 9.916 10.070 9.996

25.142 30.116

I-

40.058 45.028

!-

I I J

75.077 1 80.074 I 85.(163 f 90. 046 1 95.023 I

2.083 1.545 1.014 0.527 0.139

0

0 1.382 1.673 2.116 2.932

50.000 55.026 60.047 65.063 70.073

-3.1~

-

0 0.388 0.625 1.110 2.340

29.884_ 34.912 39.942 44.972

100.000 ; ...

~

85.13i

5.387 4.157 2.930 1.755 0.715

-

-

I

0.755 1.036 1.573 2.874

0 0.957 1.132 1.3i7 1.776

0

- 1.282 - 1.533 -

I

I

I

1.902

- 2.560

- g.546

-

35.0881 -

4.305 4.937 5.930 6.676

7.233 7.622 7.856 7.928 7.806

!-

9.671 9.103 8.338 7.425 6.398

50.000 7.4ti5 54.974 f - 6.913 59.953 j - 6.196 U4.937 : - 5.365

69.927

5.290 4.133 2.967 1.835 0.805

I

0

1

I

I

- 4.454

74.923 : 79.920 84.93i 89.95·1 94.977 100 .000 .

_ _ _· " _ " ' ·_ _ .W __ ". _ _ _ _ ' -

4.574 7.054 9.549 14.568 19.611

0

-

0 1.233 1.520 1.965 2.812

~

0 0.245 0.464 0.927 2.126

~

-

Upper surface

437

3.500 2.Ml 1..621 0.801 0.173

0 . . . . . . _ _ _ - . . . , _... _

L.E. radius: 1.96 Slope of radius through L.E.: 0.084

THEORY OF WING SECTIONS

438

NACA 65a-418 (Stations and ordinates given in per cent of airfoil chord)

Lower surface

Upper surface Station

Ordinate Station

Ordinate

NACA 65,-418 a == 0.5 (Stations and ordinates given in per cent of airfoil chord)

Upper surface Station

0 0.278 0.503 0.973 2.181

0 1.418 1.729 2.209 3.104

0 0.722 0.997 1.527 2.819

4.639 7.123 9.619 14.636 19.671

4.481 5.566 6.478 7.942 9.061

5.361 7.877 10.381 15.364 20.329

-

3.217 3.870 4.410 5.250 5.877

24.716 29.768 34.825 39.884 44.943

9.914 10.536 10.944.__ 11.140 11.091

25.284 30.232 35.175 40.116 45.057

-

6.334 6.648 6.824 6.856 6.711

50.000 55.051 60.094 65.126 70.146

10.774 10.198 9.408 8.454 7.368

50.000 M.949 59.906 64.874 69.854

-

6.362, 5.818 5.124 4.334 3.480

75.154 80.147 85.127 90.092 95.046

6.183 4.927 3.638 2.3liO 1.120

74.846 79.853 84.873 89.908 94.954

-

2.803 1.743 0.946 0.282 0.144

100.000 J~.E.

0

100.~

__

0 - 1.218 - 1.449 -1.781 - 2.360

0

radius: 1.96 Slope of radius through L.E.: 0.168

Lower surface

Ordinate Station

Ordinate

0 0.197 0.411 0.868 2.057

0 1.440 1.766 2.271 3.233

0 0.803 1.089 1.632 2.943

-

0 1.164 1.378 1.683 2.197

4.493 6.966 9.459 14.481 19.533

4.715 5.891 6.882 8.482 9.709

5.507 8.034 10.5041 15.519 20.467

-

2.951 3.515 3.978 4.690 5.213

24.604 29.691 34.789 39.899 45.022

10.643 11.325 11.770 11.970 11.897

25.396 30.309 35.211 40.101 44.978

-

5.595 5.853 5.998 6.026 5.905

50.182 55.313 60.364 65.372 70.347

11.506 10.788 9.820 8.674 7.397

49.818 54.687 59.636 64.628 69.653

-

S.626 5.216 4.696 4.094 3.433

75.298 80.232 85.159 90.089 95.030

6.038 4.636 3.247 1.930 0.777

74.702 79.768 84.841 89.911 94.970

100.000

0

- 1.331 - 0.702 - 0.201

_-_

100.000 -_...

- 2.734

- 2.024

0

... -

L.E. radius: 1.96 Slope of radius through Js; E.: 0.233

APPENDIX III NACA 65.-618 (Stations and ordinates given in per cent of airfoil chord) Upper surface Station

Ordinate

Lower surface Station

Ordinate 0 1.146 1.356 1.651 2.152

NACA 65,-618 a = 0.5 (Stations and ordinates given in per cent of airfoil chord) Upper surface Station

0 0.172 0.385 0.839 2.026

0 1.446 1.776 2.293 3.268

0 0.828 1.115 1.661 2.974

4.462 6.936 9.431 14.455 19.506

4.776 5.971 6.978 8.602 9.848

5.538 8.064 10.569 15.545 20.494

24.574 29.652 34.738 39.826 44.915

10.803 11.504 11.972 12.210 12.186

25.426 30.348 35.262 40.174 45.085

-

5.433 5.672 5.792 5.784 5.616

50.000 65.077 60.141 65.189 70.219

11.877 11.293 10.479 9.482 8.338

50.000 54.923 59.859 64.811 69.781

-

5.259 4.723 4.063 3.302 2.506

75.230 80.220 85.189 90.138

7.075 5.719 4.306 2.863 1.433

74.770 79.780 84.811 89.862 94.932

- 1.705 - 0.943 - 0.268 0.239 0.463

95..068

100.000

0

100.000

-

- 2.880 - 3.427 - 3.876

- 4.564 - 5.072

0

L.E. radius: 1.96 Slope of radius through L.E.: 0.253

439

Lower surface

Ordinate Station

Ordinate

II 0 0 0.941 . - 1.055 1.244 - 1.239 1.811 - 1.493 3.154 - 1.895

0 0.059 0.256 0.689 1.846

0 1.469 1.821 2.375 3.449

4.248 6.706 9.194 14.225 19.301

5.115 6.448 7.575 9.404 10.815

5.752 8.294 10.806 15.775 20.699

24.407 29.537

11.893 12.687 13.209 13.456 13.395

25.593 - 4.321 30.463 . - 4.. 479 35.3161 - 4.551 40.151 - 4.540 44.966 - 4.407

12.974 12.173 11.090 9.806 8.374

49.727 54.532 59.454 64.443 69.481

34.684 39.849 45.034

50.273 55.468 60..546

65.557 70.519

75..445 so.347 85.239 90.133 95. 046 1 100.000 I

i !

I

I

- 2..469

- 2.884 - 3.219 - 3.716 - 4.071

- 4.154 - 3.815

- 3.404 - 2.936 - 2.428

6.851 i 74.555 . - 1.895 5.279 I 79.653 t - 1.361 3.720 84.761 1 - 0.846 2..233 89.867 - 0.. 391 0.920 94.954 - 0.055

I

1

0

I lOO.()~:lO________~ _~ ___

L.E. radius: 1.96 Slope of radius through L. e.: 0.349

THEORY OF WING SECTIONS

440

NACA 65r 221 (Stations and ordinates given in per cent of airfoil chord) Upper surface

Station

Ordinate

Lower surface Station

Station 0 0.247 0.468 0.933 2.135

0 1.567 1.902 2.402 3.335

0 0.628 0.892 1.410 2.684

-

0 1.467 1.762 2.188 2.963

4.791 7.280 9.778 14.787 19.808

4.783 5.918 6.865 8.370 9.514

5.209 7.720 10.222 15.213 20.192

-

24.834 29.865 34.898 39.932 44.967

10.381 11.007 11.404 11.570 11.461

25.166 30.135 35.102 40.068 45.033

-

50.000 55.030 60.054 65.072 70.084

11.055 10.372 9.461 8.390 7.195

75.088 80.084 85.072 90.052 95.026

5.918 4.595 3.270 2.000 0.861

I

0

Upper surface

Ordinate

0 0.372 0.608 1.090 2.314

100.000

NACA 65r421

(Stations and ordinates given in

per cent of airfoil chord)

Ordinate

Lower surface

Station

Ordinate

0 1.601 1.956 2.493 3.505

0 0.753 1.032 1.567 2.865

-

0 1.401 1.676 2.065 2.761

4.151 5.070 5.831 7.024 7.922

4. 582 1 5.085 7.062 6.329 9.557 7.371 14.575 9.034 19.616 10.304

5.417 7.938 10.443 15.425 20.384

-

3.821 4.633 5.303 6.342 7.120

8.591 9.063 9.344 9.428 9.271

24.668 29.729 34.796 39.865 44.934

11.271 11.976 12.433 12.640 12.556

25.332 30.271 35.204 40.135 45.066

-

7.691 8.088 8.313 · 8.356 8.176

50.000 ~54.970 ~ 59.946 64.928 69.916 -

8.849 8.182 7.319 6.330 5.251

50.000 55.059 60.108 65.145 70.168

12.158 11.467 10.531 9.419 8.166

50.000 54.941 59.892 64.855 69.832

-

7.746 7.087 6.247 5.299 4.278

74.912 79.916 84.928 89.948 94.974

4.128 3.003 1.924 0.966 0.229

75.176 80.167 85.143 90.104 95.051

6.811 5.388 3.940 2.514 1.176

74.824 79.833 84.857

- 3.231 - 2.204 - 1.248

100.000

-

0

L.E. radius: 2.50 Slope of radius through L.E.: 0.084

I

100.000

0

89.896\- 0.446 1 94.949

100.000 I.

0.088 ,

0

L.E. radius: 2.50 Slope of radius through L.E.: 0.168

1

APPENDIX III NACA65r421 a = 0.5 (Stations and ordinates given in per cent of airfoil chord) Upper surface

Station

Lower surface

0

o

0.155 0.363 0.813 1.992

1.620 1.991 2.553 3.631

0.845 1.137 1.687 3.008

-

1.344 1.603 1.965 2.595

4.414 6.880 9.371 14.395 19.455

5.315 6.651 7.773 9.572 10.951

5.586 8.120 10.629 15.605 20.545

-

3.551 4.275 4.869 5.780 6.455

24.538 29.639 34.754 39.882 45.026

12.000 25.462 12.765 30.361 13.258 1 35.246 40.118 13.470 13.362 44.974

-

6.952 7.293 7.486 7.526 7.370

50.211 55.362 60.421 65.428 70.398

12.890 12.056 10.942 9.637 8.193

49.789 54.638 59.579 64.572 69.602

-

7.010 6.484 5.818 5.057 4.229

75.340 80.264 85.181

6.664 5.09i 3.550

74.660 79.736 84.819 89.900 94.966

-

3.360 2.485 1.634 0.867 0.257

95.034

100.000

NACA66-206 (Stations and ordinates given in per cent of airfoil chord) Upper surface

Station

Lower surface

Ordinate Station

Ordinate

Ordinate Station Ordinate

o

90.IOO

441

I

2.095

0.833 0

1 1100.000:

0

0

L.E. radius: 2.50 Slope of radius through L.E.: 0.233

0 0.461 0.707 2.447

0 0.509 0.622 0.798 1.102

0 0.539 0.793 1.298 2.553

-

0 0.409 0.482 0.584 0.730

4.941 7.439 9.939 14.942 19.947

1.572 1.947 2.268 2.791 3.196

5.059 7.561 10.061 15.058 20.053

-

0.940 1.099 1.234 1.445 1.604

3.513 3.754 3.929 4.042 4.095

25.046 30.038

1.202

24.954 29.962 :H.971 39.981 44.990 1

I

I

50.000 t

- 1.723 - 1.810 35.029 - 1.869 40.019 f - 1.900 1.905 45.010

1-

4.088 55. 009 1 4.020 60.018 I 3.886 65.026 I 3.641 70.031 , 3.288

I

75.034

I

I

SOo0341

85.031 90.023 t 95.012 I

100.000

I

2.848 2.339 1.780 1.182 0.578 l)

50.000 54.991 59.982 64.974 69.969

, - 1.882 1.830 t - 1.744­ ! - 1.581 t - 1.344

Ii

74.966

i - 1.058

84.969 89.977 I 94.988

- 0.434 - 0.148 0.054

I 7909661 - 0.747 ! 1 100 .000

0

L.E. radius: 0.223 Slope of radius through L.E.: 0.084

THEORY OF WING SECTIONS NACA 66-209 (Stations and ordinates given in per cent of airfoil chord) Upper surface Station

Ordinate

Lower surface Station

Ordinate

Upper surface Station

Lower surface

-

Ordinate Station Ordinate

0

0

0

0.442 0.686 1.179 2.420

0.735 0.892 1.135 1.552

0.558 0.814 1.321 2.580

-

0.635 0.752 0.921 1.180

0 0.436 0.679 1.171 2.412

0 0.806 0.980 1.245 1.699

4.912 7.409 14.912 19.921

2.194 2.705 3.141 3.850 4.396

5.088 7.591 10.092 15.088 20.079

-

1.562 1.857 2.107 2.504 2.804

4.902 7.399 9.898 14.903 19.912

2.401 2.958 3.432 4.202 4.796

588 2. 5.098 7.601 10.102 15.097 20.088

24.931 29.944 34.957 39.971 44.986

4.821 5.145 5.378 5.528 5.594

25.069 30.056 35.043 40.029 45.014

-

3.031 3.201 3.318 3.386 3.404

24.924 29.937 34.952 39.968

5.257 5.608 5.862 6.024 6.095

25.076 30.063 35.048 40.032 45.016

50.000 55.014 60.027 65.038 70.046

5.578 5.476 5.275 4.912 4.400

50.000 54.986 59.973 64.962 69.954

-

3.372 3.286 3.133 2.852 2.456

50.000 55.016

60.030 65.042 70.051

6.074 5.960 5.736 5.332 4.759

50.000 54.984 59.970 64.958 69.949

75.050

3.772 3.058 2.283 1.477 0.690

74.950 79.950 84.9&6 89.965 94.982

-

1.982 1.466 0.937 0.443 0.058

75.056 80.055 85.049 90.037 95.019

4.071 3.289 2.445 1.570 0.724

74.944 79.945 84.951 89.963 94.981

9.~

SO.05O 85.044 90.034 95.018 100.000

0

0

NACA 66-210 (Stations and ordinates given in per cent of airfoil chord)

100.000

L.E. radius: 0.530 Slope of radius through L.E.:

0

0.084

44.984­

100.000

0

0 0.564 0.821 1.329

100.000

0 - 0.706 - 0.840

1-

- 1.031

1.327

- 1.769 - 2.110

- 2.389 - 2.856 - 3.204

1

- 310467! - 3.664

- 3.802 - 3.882

=~:I

- 3.770 - 3.594 1 - 3.272

- 2.815 1 - 2.281 -

1.697 1.099 0.536 0.092

0

L.E. radius: 0.662

Slope of radius through L.E.: O.~

I

APPENDIX III NACA~-212

(Stations and ordinates given in per cent of airfoil chord) Upper surface Station

Lower surface

Ordinate Station Ordinate

NACA66r215

(Stations and ordinates given in

per cent of airfoil chord)

Upper surface Station

0 0.424 0.666 1.156 2.395

0 0.953 1.154 1.462 1.991

0 0.576 0.834 1.344 2.605

-

0 0.853 1.014 1.248 1.619

0 0.406 0.646 1.134 2.370

4.883 7.379 9.878 14.883 19.894

2.809 3.459 4.011 4.905 5.596

5.117 7.621 10.122 15.117 20.106

-

2.177 2.611 2.977 3.559 4.004

4.855 7.349 90848 14.854 190868

24.908 29.925 34.943 39.962

44.981

6.132 6.539 6.833 7.018 7.095

25.092 30.075 35.057 40.038 45.019

-

50.000 &5.019 60.036 65.051 70.061

7.068 6.931 6.659 6.169 5.487

50.000 54-.981 59.964 64.949 69.939

75.066 80.065 85.057 90.043 95.022

4.661 3.739 2.755 1.750 0.189

74.934 79.935 84.943 89.957 94.978

100.000

0

100.000

443

Ordinate

Lower surface

Station Ordinate

0 1.168 1.409 1.778 2.417

0 0.594 0.854 1.366 2.630

3.413 4.202 4.872 5.957 6.790

5.145 7.651 10.152 15.146 20.132

4.342 4.595 4:.773 4.876 4.905

7.437 24.886 29.906 1 7.9Z1 34.929 ( 8.280 39.952 I 8.501 44.976 8.590

25.114 30.094 35.071 40.048

-

4.862 4.741 4.517 4.109 3.543

50.000 55.023 6O.0f5 65.063 70.075

8.553 8.378 8.030 7.402 6.547

50.000 54.977 59.955

-

2.871 2.147 1.409 0.716 0.157

75.081 80.079 85.070 90.052 95.026

5.526 4.393 3.202 2.005 0.881

0

L.E. radius: 0.952 Slo~ of radius through L.E.: 0.084

f

1

j

100.000

I

0

-

f -

2.781 3.354 3.838 4.611 5.198

-

5.647 5.983 6.220 6.359 6.400

69.925

-

6.347 6.188 5.888 5.342 4.603

74.919 79.921 84.930 89.948 94.974

-

3.736 2.801 1.856 0.971 0.249

45.024

64.937

100.000

-

0 1.068 1.269 1.564 2.045

0

L.E. radius: 1.435 Slope of radius through L.E.: 0.084

THEORY OF WING SECTIONS

444

NACA 66r415 (Stations and ordinates given in per cent of airfoil chord)

Upper surface

Ordinate Station Ordinate

Station 0 0.314 0.544 1.019 2.241

I

4.711 7.199 9.696 14.709 19.736 24.771 29.812 34.SS7 39.904 44.952

Lower surface

I

Upper surface Station

Lower surface

Ordinate Station

Ordinate

0 1.206 1.467 1.873 2.592

0 0.686 0.956 1.481 2.759

-

0 1.006 1.187 1.445 1.848

0 0.389 0.628 1..115 2.. 346

0 1.368 1.636 2.054 2.828

0 0.611 0.872 1.385 2.654

3.718 4.617 5.381 6.624 7.581

5.289 7.801 10.304 15.291 20.264

-

2.454 2.921 3.313 3.. 932 4.397

4.827 7.320 9.818 14.825 19.841

4.002 4.933 5.724 7.004 7.982

5.173 7.680 10.182 15.175 20.159

-

3.370 4.085 4.690 5.658 6.390

8.329 8.897 9.309 9.571 9.685

25.229 30.188

-

4.749 5.009 5.189 5.287 5.305

24.863 29.887 34.914 39.942 44.971

8.742 9.31,' 9.731 9.989 10.093

25.137 30.113 35.086 40.058 45.. 029

-

6.952 7.373 7.671 7.847 7.903

50.000 55.. 028 60.054 65.075 70.()89

10.045 9.828 9.394 8.610 7.568

50.000 54.972 59..946 64.925 69.911

-

7.839 7.638 7.252 6.550 5.624

75.095

6.345 5.001 3.606 2.230 0.961

74.905 79.907 84.919 89.940 94.970

- 4.555 - 3.409 - 2.260 .

35.143 40.096 45.048

50.000 55.046 60.090 65.126 70.150

9.656 9.473 9.100 8.431 7.518

50.000

- 5.244

54.954 59.910 64..874 69.850

-

5.093 4.816 4.311 3.630

75. 162 1 80.159 I 85.139 t 90.104 95.. ~'i3

6.419 5.187 3.872 2.519 1.196

74.838 79.841 84.861 89.896 94.947

-

2.839 2.003 1.180 0.451 0.068

100.000

NAC.A 66.-218

(Stations and ordinates given in

per cent of airfoil chord)

0

100.000

0

LE. radius: 1.. 435 Slope of radius through L.E.: 0.168

80.093

85.. 081 90.060 95.030 100.. 000

0

100.000

0 - 1.268 -

1.496

- 1.840 - 2.. 456

i

96 - 1.1 1 - 0.329

0

L.E. radius: 1.955 Slope of radius through L.E.: 0.084

I

THEORY OF WING SECTIONS

446

NACA 67,1-215 (Stations and ordinates given in per cent of airfoil chord)

Upper surface

Station

Lower surface

NACA 747A315

(Stations and ordinates given in

per cent of airfoil chord)

Upper surface

Ordiilate Station Ordinate

Station

Lower surface

-

Ordinate Station Ordinate

0 0.402 0.642 1.128 2.361

0 1.213 1.460 1.867 2.577

0 0.598 0.858 1.372 2.639

-

0 1.113 i.320 1.663 2.205

0 0.. 229 0.449 0.911 2.109

0 1.305 1.599 2..065 2..935

0.771 1.051 1.589 2.891

-

1.927

4.848 7.344 9.845 14.854 19.869

3.557 4.321 4.947 5.954 6.735

5.152 7.656 10.155 15.146 20.131

-

2.925 3.473 3.913 4.808 5.143

4.564 7.053 9.558 14.599 19.668

4.264 5.286 6.140 7.497 8.503

5.436 7.947 10.442 15.401 20.332

-

2.518 2.952 3.304 1 3.843 4.247

24.887 29.908 34.930 39.953 44.976

7.348 7.825 8.185 8.430 8.570

25.113 30.092 35.070 40.047. 45.024

-

5.558 5.881 6.125 6.288 6.380

24.758 29.867 35.001 40.200 45.375

9.242 9.731 9.982 9.962 9.572

25.242 30.133 34.999 39.800 44.625

-

4.546 4.773 4.926 5.020 5.040

50.000 55.024 60.047 65..068 70.086

8.600 8.516 8.302 7.935 7.373

50.000 54.976 59.953 64.932 69.914

-

6.394: 6.326 6.160 5.875 5.429

50.447 55.463 60.435 65.366 70.241

8.964 8.206 7.324 6.365 5.354

49.553 54.537 59.565 64.634 69.759

-

5.014 I 4.930 I 4.772 1 4.509

75.098 80.100 85.092 90.071 95.037

6.515 5.335

74.902 79.900 84 ..908 89.929 94.963

-

4.725 3..743 2.653 1.. 503 0.471

75.130 80.073 85.038 90.016

4.336 3.295 2.257 1.289

95.004:

0.481

74.870 79.927 84.962 89.984 94.996

100.000

3..999

2.537 1.103 0

100.000

0

100.000

0

0

I 100.000 -­

L.E. radius: 1.523 Slope of radius through L.E.: 0.084

0 - 1.031 - 1.207 - 1.473

- 4.110 I - 3.502 - 2.743

- 1.91sl - 1.097 - 0.405 0

- .._---,_.­

L.E. radius: 1.544 Slope of radius through L.E.: 0.232

445

APPENDIX III NACA 66,-418

(Stations and ordinates given in per cent of airfoil chord) Upper surface Station

0 1.205 1.412 1.719 2.256

0 0.372 0.610 1.095 2.323

0 1.570 1.869 2.342 3.226

0 0.628 0.890 1.405 2.677

-

3.042 3.651 4.163 4.977 5.589

4.800 7.291 9.788 14.797 19.815

4.580 5.653 6.565 8.039 9.170

5.200 7.709 10.212 15.203 20.185

- 6.053 - 6.399

24..840

4.306 5.347 6.231 7.669 8.773

9.f133 10.287 34.829 ! 10.759 I 885 11.059 39. 1 11.188 44.943 I

I

75.191 80.185 85.162 90.120 95.000

100.000 I

-

Ordinate Station Ordinate

-

4.656 7.140 9.636 14.651 19.683

I

Lower surface

Station

0 1.405 1.692 2.147 3.000

50.000 55.056 ftO.l07 65.149 70.178

Upper surface

Ordinate Station Ordinate

0 0.280 0.509 0.981 2.194

24.726 29.775

Lower surface

NACA66r221

(Stations and ordinates given in

per cent of airfoil chord)

11.148 10.923 10.464 9.639 8.539 7.238 5.794 4.276 2.744 1.275

0

0 ~720

0.991 1.519 2.806

5.344

7.860 10.364: 15.349 20.317

40.11~ - 6.i;~ f 45.051 1- 6.80~

10.047 10..709 29.869 11.183 34.900 11..478 39.933 44.967. 11.595

50.000 54.944 59.893 M.SSI 69.822

50.000 55.032 60.063 65.087 70.103

25.274 30.225 35.171

- 6.6.39

-

I-

6.736 6.543 6.183 5.519 4.651

74.809 . - 3.658 79.8151 - 2.610 84.838 - 1.584 89.880 - O.6i6 94.940 - 0.011 100.000

0

L.E. radius: 1.955 Slope of radius through L.E.: 0.168

J

75.109

so. 106

85.092 90.067 95. 034 1 100.000 t

i

11.537 11.281 10.763 9.823 8.581 7.145 5.591 3.996 2.440 1.032

0

-

0 1.470 1.729 2.128 2.8M

- 3.948 - 4.805 - 5.531 6.693 - 7.578

I-

1-

25.160 f - 8.257 8.765 30.131 35.100 I - 9.123

40.06719.~ 45..033 - 9.405 5O.0~

I-

9.331

54.968

! -

9.091

59.937l - 8.621 64.913 69.897

- 7.763

I - 6.637

1-

74.891 5.355 79.894 ; - 3.999 84.908 2.f>50

I-

89.933

i-

1.406

94.966 , - 0.400

lOO.ooo 1

0

L.E. radius: 2.550 Slope of radius through L.E.: 0.084

APPENDIX III

447

NACA 747A415

(Stations and ordinates given in

per cent of airfoil chord)

Lower surface

Upper surface

Station

Station Ordinate

Ordinate

0 0.183 0.398 0.852 2.041

0 1.318 1.622 2.106 3.016

0 0.817 1.102 1.648 2.959

4.487 6.972 9.476 14.521 19.598

4.411 5.488 6.390 7.827 8.897

5.513 8.028 10.524 15.479 20.402

- 3.501 - 3.845

24.698 29.818 34.964 40.176 45.364

9.687 10.216 10.497 10.499 10.121

25.302 30.182 35.036 39.824 44.636

-

50.447 55.474 60.454 65.393 70.273

9.516 8.753 7.859 6.878 5.838

49.553 54.526 59.546 64.607 69.727

- 4.462 4.381 4.235 3.992 - 3.622

75.164 80.107

4.783 3.692 2.592 1.546 0.639

85..066 90.037 95.015

ioo.ooo

0

f t

-

0 0.994 1.160 1.406 1.822

- 2.349 - 2.730

- 3.038

4.095 4.286 4.411 4.485 4.~93

II

I-

74.836 79.893 84.934 89.963 94.985\100.000

3.053 2.344 1.578 0.838 0.247 0

L.E. radius: 1.544 Slope of radius through L.E.: 0.274

APPENDIX IV

AERODYNAMIC CHARACTERISTICS OF WING SECTIONS C.aNTENTS

NACA Designation 0006 . ()()()9

.

.

•

.

.

Page

452

.

0016-34:

.....

0010-35

. . . . ..

0016-34: a == 0.8 (modified) eli = 0.2 . . 0012 . . . . . . . . . . . . . .

0012-64 0012-64 a 1408

.

== 0.8

1410

(modified) C'l, == 0.2 . . .

462

464

466

468

470

472

1412 2408

474

476

478

2410 2412

2415 2418 2421

454

456

458

460

480

482

. .

484

2424 4412

486

488

4415

490

4418 4421 4424 23012 23015 23018 23021 23024 63-006

492

494

496

498

500

502

504

506

508

510

. . .. .

. 63-009 . 63-206 . 63-209 . . 63-210 . 631-012 . . 631-212

.

520

6&.-412

.

522

524

63r015.. 63r215 . . . 63r415. 63z-615. .

512

.514

516

518

.

526

528

530

THEORY OF WING SECTIONS

450

Page 532 534

NACA Designation

63r018. 63r218. 63r418. 63,-618.

536

538 540

63.-021 . .

542 544 546

634-221 . .

6&-421. 63AOI0 63A210

64-108 64-110 64-206 64:-208 64:-209

548

.....

64-()(M). . • • • • . • 64-()()9 • . .

. . .

.. . . . ..

558 560 562 liM

64-210 . . 64.~12. . . . . . . .

566 568 570

64t-112. .

Mt-212 . . 64.-412 . . 64r015. 64r215 . 64r415 . 64r018 .

572 574 576 578 580 582 584 586 588 590 592

64r218 . . 64r418. 64r618 . .

64r021 . . 64.c-221.

64r421. MAOIO

594

596 li98 600 602 604 606 608 610 612 614 616

64A210 MA410 64 1A212 . .

64,A215 . 85-()()6.

. .....

. . . . .

65--009 . 66-206 . . . 65-209 . 65-210 . . 65-410 . 6&-012. 65t-212 65s-212 II == 0.6 . ~-412.

550 652 554 556

. . . . .

6Sr015. . . . . . . . 65r215. . . . . . . . 66r415. . . . . . 65.t-410 II :II: 0.5 • .

.

618 620 622

624

626 628

630

APPENDIX IV NACA Designation 65r018 . 65,-218 . 65r418. . . . . 65r418 a - 0.5 . 65.-618. . . . . 65r618 a :=II 0.5 .

65.-021

.

65.-221 . 65r421 . 65r421 a - 0.5 . 66-006 . . . 66-009 . . ~206

•..

66-209 . . 66-210 . .

66r012. 66,-212. 66:-015. 602-215.

66r415. 66,-018. . 66,-218. . 66,.418.

66c-021 .. 66r221. 67,1-215 . . 747A315.

147A415.

451

Page

632

634

636

638

640

642

644

646

648

650

652

8M

656

6li8

660

662

664

666

668

670

672

674

676

678

680

682

684

686

THEORY OF WING SECTIONS

452 3. 6

,­

3. "!:II

S

2. 4­

2J)

I,..'" a; ., ~J~

I .6

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Section D!'19. NACA 0006 Wing

0

8

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Dffack, «0, dsq .

SectiOI'

32

453

APPENDIX IV

. ~

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·c,

N ~C.A 0006 Win« Section (Continued)

.8

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~~ ~I-

1.6

THEORY OF WING SECTIONS

454 3.8

3.2

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-

24

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J

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24

455

APPENDIX IV .IJ3,

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R->--,. *~~I o 30-'0_ .250 0 I I a 6.01 I .2500 I I

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250 .005 6 6.0 ~ Standard roughness . O.zOe simotatea split flop deflected 60· v 6.0

4

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p

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Section liff coefficient, e,

NACA 0009 ·,\ing Section (Conlinued)

.8

1.2

/4

THEORY OF WING SECTIONS

456

2.4

2.0

1.8 I

I.. 1.2

,

0.1

..

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APPENDIX IV

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SectioD lift coefficieDt. NACA 0010-34 Wing Section (Continued)

i:~: :' I. a.c. pos1 tloD

R

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APPE~VDIX

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APPENDIX IV

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APPEl\~DIX

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APPENDIX IV

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APPENDIX IV

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APPE1VDIX IV

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24

32

APPENDIX IV

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1.6

THEORY OF WING SECTIONS

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.12

APPENDIX IV

527

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32

APPENDIX IV

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APPENDIX IV

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535

A.PPENDIX IV

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APPENDIX IV

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APPENDIX IV

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APPENDIX IV

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APPENDIX IV

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APPENDIX IV

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584

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24

APPENDIX IV

585

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1.2

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APPENDIX IV

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APPENDIX IV

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THEORY OF WING ljECTION8

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32

APPENDIX IV

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NACA 65z-415 Wing Section a == 0.5

24

631

APPENDIX IV

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24

32

633

APPENDIX IV

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THEORY OF WING SECTIONS

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635

APPENDIX IV

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THEORY OF WING SECTIONS

636 36

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24

637

APPENDIX IV .1)16

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THEORY OF WING SECTIONS

638 3.6

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32

APPENDIX IV .2

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640 3.6

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24

32

641

APPENDIX IV .2

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.8

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1.6

THEORY OF WING SECTIONS

642

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24

32

643

APPENDIX IV

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644 3.6

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24

32

645

APPENDIX IV .21

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NACA 65.-021 \Ving Section (Continued)

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THEORY OF WING SECTIONS 3.6

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24

APPENDIX IV

687

INDEX

A Ackeret, J., 268

Adiabatic law, 249

Aerodynamic center, experimental, 182

theoretical, 69

wing, 5, 19

Aerodynamic characteristics, experimen­ tal, 449

Airfoils (see Wing sections)

Allen, H. Julian, 65, 164, 194, 202, 236

Anderson, Raymond F., 9

Angle of attack, 4

effective, 22, 25

ideal, 70

induced, 20

wing, 8,11

Angle of zero lift, experimental, 128

theoretical, 66, 69

approximate methods, 72

variation with flap deflection, 192

wing, 16

Aspect ratio, 6

at high speeds, 296

B

Bamber, Millard J., 234

Bernoulli's equation, 35, 251

Birnbaum, Walter, f~

Blasius, H., 87, 96

Boshar, John, 20

Bound~layer, 80

compressibility effects, 109

concept, 81

crossftows, 28

ht.minar,85

Blasius solution, 87

characteristics, 85

separation, 83, 86, 93

skin friction, 87

thickness, 89, 92

transition, 105, 157

velocity distribution, 87

Boundary layer, momentum relation, 90

slip, 83

thickness, displacement, 89

momentum, 92

turbulent, 95

pipe flow, 95

separation, 83, 103

skin friction, 97, 174

thickness, 102

velocity distribution, 97, 100

Boundary-layer control, 231

c Cahill, Jones F., 206, 221

Camber, 65

application to thickness forms, 75, 112

design of mean lines, 73

effect of, on aerodynamic center, 182

on angle of zero lift, 128

on lift curve, 132

on maximum lift, 136

on minimum drag, 148

on pitching moment, 179

on profile drag, 151, 178

induced, 28

shapes and theoreti~l characteristics,

382

Chaplygin, Sergei, 258

Chord, 113

mean aerodynamic, 21

Circular cylinder in uniform stream, 41,

49

with circulation, 44, 49

Circulation, 37

lift, 45

thin wing sections, 65

vortex, 43

Complex number, 47

Complex variable, 47

Compressibility effects, 247

boundary layer, 109

drag characteristics, 281

first order, 256

689

7'REORY OF lVING SECTIONS

690

Compressibility effects, flap effectiveness,

281

lift characteristics, 269

pitching-moment characteristics, 287

pressure coefficient, 257

wing characteristics, 268

Conformal mapping, 47, 49

Conformal transformations, 49

Continuity, 32, 38, 249

D Design of wing sections, 62

Designation of wing sections, III

German designations of NACA sec-

tioDS,118

KACA four-digit sections, 114

XACA five-digit sections, 116

X.ACA modified four- and five-digit

sections, 116

XACA L-series sections, 119

X A.CA 6-series sections, 120

K.-\CA. 7-series sections, 122

Doublet, 41, 49

Downwash, 9, 20

Drag characteristics, 148

calculation, 107

compressibility etIects, 281

minimum, 148

roughness effects, 157, 175

section induced, 25

variation with lift, 149

wing induced, 16, 27

E Experimental techniques, 125

Flow around wing sections, 83

at high speeds, 261

Fluid, perfect, 32

Fullmer, Felicien F., 231

G

Garrick, I. E., 63, 258

Glauert, H., 9, 65, 67, 192

Glauert-Prandtl rule, 256

Goldstein, Sidney, 63

Gruschwitz, E., 103

H

Harris, Thomas A., 218, 227

High-lift devices, 188

I

Irrotational motion, 37

J Jacobs, Eastman N., 194, 260

JQDes, Robert T., 296, 299

Jones edge-velocity correction, 11, 25

l~

Kaplan, Carl, 256, 258

Kaplan rule, 257

}{&nnaD integral relation, 90

}{arman-Tsien relation, 258

Keenan, Joseph H., 109

}\:night, Montgomery, 234

Koster, H., 230

Krueger, W., 230

Kutta-Joukowsky condition, 52

F

L

Families of wing sections (see Wing sec­

tions)

Fischel, Jack, 218

Flaps, 188

external airfoil, 215

leading-edge,7Zl

plain trailing-edge, 190

split trailing-edge, 197

slotted trailing-edge, 203

Laminar flow (see Boundary layer)

Lanchester-Prandtl wing theory, 7

Leading edge, 113

Lees, Lester, 109

Lemme, H. G., 230

Liepman, n. \V., 268

Lift characteristics, compressibility effects,

269

design lift, 70

691

INDEX

Lift characteristics, experimental, 128

ideal lift, 70

maximum lift, 134

theoretical, 53

wing, 3, 8

maximum, 19

Lift-curve slope, effective, 11

experimental, 129

theoretical, 53, 69, 256

wing, 11

Limiting speed, 254

Loads, chordwise, 53, 60, 73, 75

flaps, 236

plain trailing-edge, 192, 236

slotted trailing-edge, 215, 221

split trailing-edge, 202, 236

span\\?ise,9, 19, 25

Loftin, Laurence K., Jr., 157

Lowry, John G., 227

M Mach line, 248

Mach number, 81, 248

critical, 259, 261

(See also Compressibility effects)

Mean lines, 382

(See also Camber)

Millikan, C. B., 93

Moment (see Pitching-moment charac­

teristics)

Motion, equations of, 32, 85

irrotational, 37

Munk, Max ~I., 23, 65, 72

N

Naiman, Irven, 56, 74

Navier-Stokes equations, 85

Keely, Robert II., 20

Neumann, Ernest P., 109

Xikuradse, J., 99

Normal shock, 254

Noyes, Richard W., 304

o Ordinates of cambered wing sections, 406

p Pankhurst, R. C., 72

Pinkerton, Robert M., 61

Pitching-moment characteristics, 179

compressibility effects, 287

theoretical, 66, 69

approximate, 72

wing, 4, 16, 27

Platt, Robert C., 215, 226

Pohlhausen, K., 93, 106

Prandtl, L., 9, 20, 85, 96

Prandtl-Glauert rule, 256

Pressure coefficient, 42

Pressure distribution, theoretical, 53

approximate, 75

empirically modified, 60

Propeller slipstream, effect on drag, 170

Purser, Paul E., 218

Q Quinn, John H., Jr., 234, 236

R

References, 300

Regier, Arthur, 109

Reynolds number, 81

(See also Scale effect)

Reynolds, 0., 95

Rogallo, Francis l\{., 202

s Sanders, Robert, 225

Scale effect, 81

aerodynamic center, 182

lift-curve shape, 133

lift-curve slope, 129

maximum lift, 137

minimum drag, 148

profile drag, 151

Schlichting, H., 105

Schrenk, Oskar, 233

Schubauer, G. n., 105

Sections (see \Ving sections)

Sherman, Albert, 20

Shock, normal, 254·

on wing sections, 263

Shortal, Joseph A., 227

THEORY OF WING SECTIONS

692

Sinks, 39

Sivells, James Co, 20

Skin friction (Bee Boundary layer)

Skramstad, H. x., 105

Slats, leading-edge, 225

Slots, 2:1.7

Sound, velocity of, 249

Source, 39, 49

Speed, limiting, 254

Squire, H. s., 101, 109

Stack, John, 272

Stagnation points, 45

Stream function, 36

Stream tube, compressible flow in, 249, 252

Streamlines, 36

Superposition, 39, 75

Surface condition, 143, 157

effect of, on drag, 157, 175

on lift, 143

on transition, 157

standard leading-edge roughness, 143

SWL-ep, 296

T Taper ratio, 11

Tani, Itiro, 20

Techniques, experimental, 125

Temple, G., 258

Tetervin, Neal A., 102, 103

Theodorsen, Theodore, 54, 63, 65, 70, 109

Thickness distributions, 309

effect of, on aerodynamic center, 182

on angle of zero lift, 129

on lift-curve shape, 133

on lift-curve slope, 132

on maximum lift, 134

on minimum drag, 148

on pitching moment, 179

on profile drag, 148, 178

Thickness forms and theoretical data; 309

Thickness ratio, effect of, on aerodynamic

center, 182

on angle of zero lift, 129

on lift-curve shape, 133

on lift-curve slope, 53, 132

on maximum. lift, 134

on minimum drag, 148, 178

on pitching moment, 179

on profile drag, 151

(See also Wing sections)

Trailing edge, angle, 117

effect of, on aerodynamic center, 182

on lift-curve slope, 132

definition, 113

Transformation of circle in to wing sec­ tion, 50

Transient conditions, 133, 143

Tsien, H. S., 109, 258

Turbulent flow (Bee Boundary layer)

Twist, 11

u firich, A., 105

U nconservative sections, 175

Uniform stream, 39, 49

v

Velocity, limiting, 254

Velocity potential, 38

Vibration, effect on drag, 173

Viscosity (see Boundary layer)

von Kdrmdn, To, 82, 90, 93, 109, 175, 258

Vortex, 42, 49

system on wings, 9

Vorticity, 37

w Weick, Fred E., 225-227

Wenzinger, Carl J., 194, 202, 215

Wing sections, combinations of NACA

sections, 123

concept, 8

experimental characteristics, 124, 449

families, 111

N ACA four-digit, 113

NACA five-digit, 115

N ACA modified four- and five-digit,

116

NACA T-series, 118

NACA 6-scri~,s, 119

NACA 7-series, 122

ordinates, 406

unconservativc sections, 175

Wings, 1

aerodynamic center, 5, 19

angle of attack, 8, 11

angle of zero lift, 16

applicability of section data, 28

693

INDEX Wings, aspect ratio, 6

drag, 3, 16, 27

induced, 8, 16, 27

forces, 2

lift, maximum, 19

lift-curve slope, 11

'lift distribution, 9, 25

pitching moment, 16, 27

Wragg, C. A., 215

y Yarwood, J., 258

Young, A. D., 101, 109

z Zalovick, John A., 168

\. '-v

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DOVER BOOKS ON PHYSICS

TIE 0EYEL0PMENT OF X-RAY ANALYSB, Sir Lawrence Bragg. (67316-2) $8.95 SURvEY OF PHYsIcAL THEoRY, Max Planck. (67867-9) $7.95 INTRooucnoN TO HAMn.ToNIAN Oenes, HA Buchdahl. (67597-1) $10.95 Mmtoos OF QuANnJM FIELD THEORY IN STAmTICAL PHYsIcs, A.A. Abrlkosov et a1. (63228-8) $9.95 El.F..cTRoDYrw. AND CLASSICAL THEORY OF FlEws AND PAR11CLF5, A.O. Barut. (64038-8) $7.95 QuAN'llJM TIIEoRY,.David Bohm. (6596g.{» $14.95 ATOMIC PHYsIcs (8TH EDmON), Max Born. (65984-4) $12.95 EINsTEIN's THEoRY OF RE1.Anvrrv, Max Born. (60769-0) $9.95 MA11t£MATICS OF CLASSICAL AND QUAN11JM PHYSICS, Frederick W. Byron, Jr. and Robert W. Fuller. (67164-X) $18.95 . MEawucs, J.P. Den Hartog. (60754-2) $11.95 INvEs11GATIoNs ON mE THEORY OF mE BROWNIAN MOVEMENT, Albert Einstein. (60304-0) $5.95 THE PRINcIPLE OF RE1.Anvrrv, Albert Einstein, et ale (60081-5) $6.95 THE PHYsIcs OF WAVf!), William C. Elmore and Mark A. Heald. (64926-1) $12.95 THERMODYNAMICS, Enrico Fermi. (60361-X) INTRODUCTION TO MODERN Omes, Grant R Fowles, (65957-7) $11.95 Dw..oouEs CONCERNING Two NEW SCIENCES, Galileo Gali1el. (60099-8) $8.95 GRouP THEORY AND ITS APPucAnON TO PHYsIcAL PROBLEMS, Morton Hamermesh. (66181-4) $12.95 EI..EcrRoNIc STRUC1lJRE AND THE PROPER11ES OF Souos: THE PHYSICS OF THE CHEMICAL BoND, Walter A. Harrison. (66021-4) $16.95 Souo STATE THEoRY, Walter A. Harrison. (63948-7) $14.95 PHYsIcAL PRINcDv.s OF mE QuAN'ruM THEORY, Werner Heisenberg. (60113-7) $6.95 AToMIc SPEcTRA AND ATOMIC STRUC1\JRE, Gerhard Herzberg. (60115-3) $7.95 AN INTRooucnoN TO STA11S11CAL THERMODYNAMICS, Terrell L. HIli. (65242-4) $12.95 OPTICS AND OPTIcAL INSTRUMENTS: AN INTRODUCTION, B.K. Johnson. (60642-2) $6.95 THEORETICAL Souo STATE PHYsICS, VOLS. I & II,WilliamJones and Norman H. March. (65015-4, 65016-2) $33.90

THEORETICAL PHYsIcs, Georg Joos, with Ira M. Freeman. (65227"() $21.95

MOLECULAR ROTATION SPECTRA, H.W. Kroto. (67259-X) $10.95

THEVARIATIONAL PRINCiPLEsOF MECHANICS, Cornelius Lanczos. (65067-7)

$12.95

A GuIDE 1'0 fEvNMAN DIAGRAMS IN 11iE MANv-80DY PROBLEM, Richard D.

Mattuck. (67047-3) $11.95

MAlTER AND MOTION, James Clerk Maxwell. (66895-9) $6.95

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