NASA TECHNICAL NOTE

14

PROCEDURES FOR THE DESIGN \

OF LOW-PITCHING-MOMENT AIRFOILS

\

/

,

Raymond L. Burger

Langley Reseurcb Center Humpton, Va. 23665 4.

N A T I O N A L AERONAUTICS A N D SPACE A D M l2 N. & TRATlON

.

i

WASHINGTON, D. C.

t

,AUGUSH975

f

ii

TECH LIBRARY MFB. N M

I

1. Report No.

NASA TN D-7982 4. Title and Subtitle

3. Recipient's Catalog No.

2. Government Accession No.

5. Report Date

Aueust 1975

PROCEDURES FOR THE DESIGN O F LOW-PITCHINGMOMENT AIRFOILS

i

7. Author(s)

P

6. Performing Organization Code

8. Performing Organization Report No.

L- 10114

Raymond L. B a r g e r

10. Work Unit No.

9. Performing Organization Name and Address

505-06-31-02

Y

NASA Langley R e s e a r c h C e n t e r Hampton, Va. 23665 13. Type of Report and Period Covered

12. Sponsoring Agency Name and Address

Technical Note

I 14. Sponsoring Agency Code ~~

National Aeronautics and Space Administration Washington, D.C. 20546

I

15. Supplementary Notes

16. Abstract

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I I

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T h r e e approaches to the design of low-pitching-moment a i r f o i l s a r e t r e a t e d . T h e f i r s t method d e c r e a s e s the pitching moment of a given airfoil by specifying a p p r o p r i a t e modifications to i t s p r e s s u r e distribution. The second procedure designs an airfoil of d e s i r e d pitching moment by p r e s c r i b i n g p a r a m e t e r s in a special f o r m u l a f o r the Theodorsen +function. The t h i r d method involves a p p r o p r i a t e camber-line design with superposition of a thickness distribution and subsequent tailoring. Advantages and disadvantages of the t h r e e methods a r e discussed.

I

x rp

17. Key-Words (Suggested by Authoris) 1

18. Distribution Statement

Airfoils Design Low -pitching-moment airfoil 19. Security Classif. (of this report)

Unclassified

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Unclassified

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Subi e c t Cate gorv 0 1 22. Price'

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PROCEDURES FOR THE DESIGN OF LOW-PITCHING-MOMENT AIRFOILS

Raymond L. Barger Langley Research Center SUMMARY T h r e e approaches t o the design of low-pitching-moment airfoils are treated. The first method d e c r e a s e s the pitching moment of a given airfoil by specifying appropriate modifications to i t s p r e s s u r e distribution. The second procedure designs a n airfoil of desired pitching moment by prescribing p a r a m e t e r s in a special formula for the Theodor s e n €-function. The third method involves appropriate camber-line design with superposition of a thickness distribution and subsequent tailoring. Advantages and disadvantages of the three methods a r e discussed. INTRODUCTION Low-pitching-moment airfoils find application primarily as helicopter rotor blades; but m o r e recently some attention h a s been given to the advantages of low -pitching-moment sections for a "span-loader" vehicle. For such applications a s y m m e t r i c airfoil could conceivably be employed, but cambered a i r f o i l s can offer significant advantages. The usual difficulty that is encountered in the design p r o c e s s s t e m s from the fact that the airfoil shape and performance are sensitive to the p a r a m e t e r s that control the pitching moment. F o r example, an airfoil with z e r o pitching moment but with moderately s m a l l positive lift at z e r o angle of attack deviates significantly from a symmetric section. Similarly i f one attempts to modify the lower surface of a n airfoil to reduce the pitching moment, while retaining the upper- surface shape and p r e s s u r e distribution, he generally finds that substantial modification is required even for s m a l l reductions in pitching moment. In this paper t h r e e approaches to the problem of designing low-pitching-moment airfoils are treated. Generally these methods utilize equations o r procedures that are already in the literature but that have apparently not been heretofore applied in a systematic mann e r , with the required modifications, to the specific problem of designing low-pitchingmoment airfoils.

SYMBOLS Ao,Al,A2 a

r e a l coefficients

,

b2

modulus of a complex quantity (see eq. (7))

C

chord length

cZ

section lift coefficient

C 270

section lift coefficient at z e r o angle of attack

Cm

pitching- moment coefficient about the q u a r t e r - chord point

Cm, a c

pitching-moment coefficient about the aerodynamic center

cP

airfoil p r e s s u r e coefficient

c17c2

cl/R

and c2/R2

a r e coefficients in the complex Fourier expansion

of *(@) !2

complex quantity (see eq. (6))

M

local Mach number

M,

f r e e - s t r e a m Mach number

pb

basic lift distribution

R

radius of c i r c l e into which an airfoil is mapped by the Theodorsen transformation

t

maximum thickness

X, Y

Cartesian coordinates

yb

mean-line ordinate

a

section angle of attack

P

negative of the angle of z e r o lift

2

I

2Y

amplitude of a complex quantity (see eq. (7)) phase angle

E

function relating angular coordinates of near-circle and exact-circle airfoil transformations

rl

=Y

I

-P

transformation variable (see eq. (10)) angular coordinate f o r points mapped f r o m airfoil surface onto a c i r c l e function relating radial coordinates of near-circle and exact-circle airfoil transformations average value of

+,

L12T0(@) d@ 2n 0

GENERAL CONSIDERATIONS

I

i

k 1

Inasmuch as t h r e e distinct approaches to the low-pitching-moment airfoil design problem a r e discussed in this paper, an initial comparison is perhaps appropriate to provide orientation and avoid possible confusion. The f i r s t method is an application to this specific design problem of the design technique that has been described in reference 1. This technique is applicable both to subcritical and supercritical airfoils. The design is effected by modifying an initial airfoil and providing an analysis of the modification on each iteration. The second and third methods represent the full-thick-airfoil theory and the thinairfoil superposition theory, respectively, applied systematically to the low-pitchingmoment design problems. They a r e both essentially incompressible, and in both c a s e s the design is initiated by specifying a s e t of p a r a m e t e r s that determine certain airfoil characteristics. The p r e s s u r e distributions a r e obtained by an independent analysis program. All three design procedures are inviscid, but in each case an allowance f o r boundarylayer effects can be made. T h i s problem has been discussed for the first procedure in reference 1. For the other two methods, a rough estimate of the displacement thickness

I

i

3 !

effect can be obtained by a judicious u s e of the thickness distribution controls in the design process. However, i f the p r i m a r y boundary-layer considerations are l o s s of lift and i n c r e a s e in pitching moment, it is a simple matter t o overestimate the lift and underestimate the pitching moment in specifying the design parameters. DE SIGN BY PRESCRIBING PRESSURE DISTRIBUTION VARIATION The f i r s t method to be described is applicable to airfoils that have approximately the desired characteristics but require a reduction in the magnitude of the pitching moment. The designer p r e s c r i b e s a change in the known p r e s s u r e distribution of the original airfoil in such a way that the pitching moment will b e changed in the desired manner without destroying the favorable characteristics of the airfoil. The usual procedure is to shift some of the loading f r o m the rear forward by prescribing changes to the p r e s s u r e distribution along the lower surface and possibly to the rear of the upper surface. This method h a s the advantage that by working directly with the p r e s s u r e distribution the designer can avoid those adverse f o r m s of p r e s s u r e distribution that a r e conducive, say, to flow separation o r to shock formation. Furthermore, he can indirectly cont r o l the r e s u l t s of a specified change in the p r e s s u r e distribution; that is, whether it will d e c r e a s e the pitching moment, the lift, etc. However, the effects of a prescribed change on Cm, cl, t/c, etc. are more difficult to control in a p r e c i s e manner; and consequently a number of zttempts may be required to obtain closely specified values f o r these parameters. If the thickness is altered slightly in the design process, it can be adjusted by the incompressible method of reference 2. Figure 1 shows two variations of a basic airfoil that w e r e obtained by this procedure with the use of the design technique of reference 1. The lower-surface p r e s s u r e distribution was altered so as to unload the airfoil near the rear. To compensate f o r the consequent loss of lift the loading w a s increased in the middle p a r t of the lower surface. In the f i r s t variation (fig. l(b)), the pitching-moment coefficient remained constant a t -0.025 while the lift w a s increased by more than 25 percent. In the second variation (fig. l ( c ) ) , the lift remained virtually constant while the pitching moment w a s reduced in magnitude to the more acceptable value of -0.010. In this example, the decrease in thickness ratio r e p r e s e n t s a significant alteration to the geometry of the original airfoil. It is generally true that moderate changes in pitching moment a r e associated with relatively l a r g e changes in airfoil shape especially if the lift coefficient is held constant. F o r this reason, it is often preferable to s t a r t with a design that has near zero, or even positive, pitching moment, and then, if necessary, tailor that design. Such a procedure is discussed in the next section.

4

J

DESIGN BY SPECIFYING €-FUNCTION PARAMETERS

A second procedure f o r the design of low-pitching-moment airfoils is based on a formula used in reference 3 for the E-function of a c l a s s of airfoils. Basically, this formula E ( @ ) = A1 s i n ($I - 61)

+ A2

s i n (2$I - 62)

(1)

represents a simplified €-function with only two Fourier components specified in t e r m s of the amplitudes and phase angles. F o r this E-function, the angle of z e r o lift -p is approximately determined f r o m the relation

p

=

E(T)

= A1 sin 6 1 - A2 s i n 62

The conjugate function to different) is

E(@)

(see ref. 3, eqs. 11 and 12, where the notation is slightly

where +o is the average value of complex numbers a r e needed:

RJ:7r 7r c1=-

+.

In o r d e r to compute the pitching moment, two

+(@) ei@d@ = AIR c o s 6 1 + i A l R sin 6 1

(3)

and

"," I:'+(@) e2i@d @ A2R2 c o s 62 + iA2R2 sin 62

c2 = -

=

(4)

where R is the radius of the circle into which the airfoil is mapped by the Theodorsen transformation. Now the real number a is related to R by

i

L

R = ae*o

and the complex quantity g defined by n

c2

5

is represented in polar f o r m as

in accordance with the procedure of reference 3. Then the pitching moment about the aerodynamic center is given by (see ref. 3, eq. 51)

P) It is c l e a r that the value of Cm,ac depends on the angle q = y - P ; specifically, cm,ac = 0 when q = 0. Now one can e x p r e s s q = y - /3 f r o m equations (2) and (7) i n t e r m s of AI, A2, 61, 62, and +o by means of equations (3) to (6). Thus a unique airfoil can be determined by specifying the five p a r a m e t e r s AI, A2, 61, +o, and q in the equation y - P - q = O

and solving it for

(9)

62. This highly nonlinear equation is solved by interval halving.

Varying each p a r a m e t e r produces a c l a s s o r family, of airfoils. The value of q chosen controls the pitching moment according to equation (8). The selection of the other p a r a m e t e r s r e q u i r e s some c a r e . Although varying any one of these p a r a m e t e r s influences to some extent all the airfoil characteristics, each individual p a r a m e t e r has a dominant influence on a particular property of the airfoil. The value of A1 provides the basic thickness distribution, which is then modified by the choice of A2. The effect of varying A1 can be seen in the example shown in figu r e 2(a). ( F o r all the airfoils shown in fig. 2 the pitching moment about the aerodynamic center is essentially zero.) Very small values of A1 yield a shape very much like an ellipse, whereas l a r g e values produce negative thickness near the trailing edge. When A2 is varied, the distribution of thickness is modified, as shown in figu r e 2(b). The magnitude of A2 also influences the extent to which the second t e r m in equation (1) affects the airfoil performance. Since this t e r m is the one that involves 62, equation (9) may not be solvable f o r 62 if A2 is too small. On the other hand, l a r g e values of A2 (relative to A i ) tend to produce impractical distorted airfoil shapes. T h i s effect is seen in figure 2(b) where f o r A1 = 0.1 and A2 = 0.06, the airfoil becomes too thin in the 75-percent chord region. The p a r a m e t e r +o affords a control over the maximum thickness (ref. 2), as is seen in figure 2(c). The p a r a m e t e r 6 1 primarily controls the lift, as indicated by figure 2(d), where varying 61 f r o m 0.1 to 0.9 radian h a s the effect of changing p f r o m 0.0027 to 6

j

i

0.03 radian. Notice, however, i n figure 2(b), that the variation of A2 h a s very little effect on the lift. Since five p a r a m e t e r s can be v a r i e d i n this design procedure, it appears that a wide variety of shapes and c h a r a c t e r i s t i c s is attainable. However, the fact that the €-function is represented by only two Fourier components is a significant restriction. Furthermore the availability of numerous p a r a m e t e r s is in one sense a disadvantage in that the designer might spend a considerable time "toying" with the p a r a m e t e r s in a n effort to obtain exactly some desired design characteristic. These difficulties can usually be circumvented in actual practice. For example, the airfoil shown in figure 3(a), which was designed by this method, w a s too thick near the trailing edge. I t s other properties - lift, pitching moment, and maximum thickness w e r e satisfactory. Therefore a smooth analytic fairing was made, starting at the 0.60 chord station and proceeding to the trailing edge, so as to reduce the thickness i n t h i s region while maintaining the same mean line. The resulting airfoil is shown in figure 3(b), together with its p r e s s u r e distribution. (The viscous p r e s s u r e distributions in figs. 3 to 5 w e r e computed by the method of ref. 4.) The lift, pitching moment, and maximum thickn e s s are essentially unchanged, but the trailing-edge angle and consequently the p r e s s u r e distribution near the trailing edge are improved. Of course, not every a r b i t r a r y combination of p a r a m e t e r s yields a solution of equation (9). Furthermore, as h a s been seen, even those combinations that yield a solution do not necessarily correspond to a practical airfoil shape. DE SIGN BY GEOMETRIC SUPERPOSITION P e r h a p s the simplest approach to the design of airfoils is to design the mean line and then superimpose a thickness distribution on it. In reference 5 it is shown that, if the variable e* is defined by the relation

x = -C( I 2

G

COS

e*)

(10)

then the basic lift distribution (that which is dependent only on the mean-line shape and not on the angle of attack) can be represented by a Fourier sine series Pb = 4

2

An s i n (ne*)

(11)

n= 1 Then r e f e r e n c e 5 a l s o shows that the distribution of slope of the mean line dyb(8*)/dx at the ideal angle of attack is the conjugate of Pb(67/4 provided that both functions a r e extended to the interval (7~,2n)with dy/dx symmetric about 7~ and Pb antisymmetric

7

The situation is s i m i l a r to that in thick-airfoil theory where the €-function can - t,bo can then be calculated t o determine the airfoil be prescribed and its conjugate geometry. Here, a basic lift distribution can be prescribed and the corresponding mean line calculated. For a lift distribution expressed as a sine series as in equation (1l), the conjugate of ~ b / 4 is about

T.

+

dyb = dx

An

COS

(ne*)

n= 1

Naturally some experience would normally be required to design a lift distribution that provided the desired lift and pitching moment as well as a reasonable mean-line shape. However, a simpler, more direct approach is available. From reference 6, equations (4.7) and (4.8),it is seen that the lift coefficient a t z e r o angle of attack is simply

where

and the pitching-moment coefficient about the quarter-chord point is

Here A1 and A2 are the f i r s t two coefficients in the Fourier s e r i e s of equation (11). Thus, in the design of a mean line, the lift coefficient can be controlled by specifying the value of A1 and the pitching-moment coefficient is proportional to the difference between A2 and A i . Specifically, A2 = A1 gives a pitching-moment coefficient of zero. Families of mean lines can be derived by specifying various values of A1 and A2 in a simple 2-component lift distribution. However, it should be noted that l a r g e values of A2 yield impractical distorted mean lines; consequently, l a r g e values of lift cannot be specified if the pitching moment is required to be near z e r o o r positive. F o r each mean line so derived, a family of airfoils can be obtained by specifying v a r ious thickness distributions. It is in this phase of the design that the superposition procedure of this airfoil theory displays i t s limitations. These limitations appear whenever the assumptions of thin-airfoil theory a r e violated; specifically over the entire airfoil if it is sufficiently thick and near the leading edge for any airfoil. The f o r m e r problem is not as troublesome as the latter.

1 I

F o r a thick airfoil the lift and pitching moment do not appear to be very sensitive to thickness, even though the velocities due to thickness and camber are not simply additive. Furthermore, a thick airfoil generally h a s a l a r g e leading-edge radius and consequently a relatively smooth p r e s s u r e distribution. Therefore, desired adjustments i n the p r e s s u r e distribution can be made fairly simply with a design m e a o d such as that of reference 1. I

F o r thin airfoils, on the other hand, the superposition of velocities is valid except near the nose. Low-pitching-moment cambered airfoils generally have a mean line with considerable slope at the leading edge. T h i s l a r g e slope, together with a small leadingedge radius, often r e s u l t s in a lower-surface suction spike near the leading edge. T h i s effect is seen in the example of figure 4, f o r which the camber line is determined by A1 = A2 = 0.025 and 4 = 0.0106, which correspond to = 0.15 and cm = 0.0, with the NACA 65A010 thickness distribution (ref. 6, p. 369). The possibility of lower-surface boundary-layer separation a t small negative angles of attack is introduced by this type of lower-surface p r e s s u r e distribution. Furthermore, the modification of an airfoil to eliminate such a suction spike is not a minor modification, inasmuch as the required change in local p r e s s u r e coefficient is large. Of course, a certain amount of modification is possible, as shown by the example of figure 5. At an angle of attack of zero, the lower surface of the original airfoil does not display a high leading-edge suction peak, but it does have a kind of p r e s s u r e distribution that rapidly f o r m s a spike a t negative angles of attack. Thus, by making the p r e s s u r e l e s s negative in this region (fig. 5(b)), the performance at small negative angles of attack is improved. The method of reference 7 was used to make this adjustment. CONCLUDING REMARKS T h r e e approaches to the design of low-pitching-moment airfoils have been discussed. The f i r s t method is applicable to a wide variety of airfoil types in compressible flow; but control of lift and pitching moment is indirect, by means of specifying appropriate changes in the p r e s s u r e distribution and consequently several attempts are sometimes required to obtain the desired values of the p a r a m e t e r s .

\

1

The other two methods which a r e essentially incompressible provide a closer cont r o l over such p a r a m e t e r s as maximum thickness, lift, and pitching moment, but the airfoils generated fall within r e s t r i c t e d families and often r e q u i r e tailoring. This tailoring, either to the geometry directly o r to the p r e s s u r e distribution, can often be accomplished without significantly altering the values of the airfoil parameters.

9

The design methods are essentially inviscid, but it is possible to make a n allowance f o r the boundary layer with each method. Langley Research Center National Aeronautics and Space Administration Hampton, Va. 23665 May 30, 1975

1

REFERENCES

1. Barger, Raymond L.; and Brooks, Cuyler W., Jr.: A Streamline Curvature Method f o r Design of Supercritical and Subcritical Airfoils. NASA TN D-7770, 1974. 2. Barger, Raymond L.: On The Use of Thick-Airfoil Theory To Design Airfoil Families in Which Thickness and Lift Are Varied Independently. NASA TN D-7579, 1974. 3. Theodorsen, T.; and Garrick, I. E.: tions. NACA Rep. 452, 1933.

General Potential Theory of Arbitrary Wing Sec-

4. Stevens, W. A.; Goradia, S. H.; and Braden, J. A.: Mathematical Model f o r TwoDimensional Multi-Component Airfoils in Viscous Flow. NASA CR-1843, 1971.

5. Allen, H. Julian: General Theory of Airfoil Sections Having Arbitrary Shape or Press u r e Distribution. NACA Rep. 833, 1945. 6. Abbott, Ira H.; and Von Doenhoff, Albert E.: Inc., c.1959.

Theory of Wing Sections. Dover Publ.,

7. Barger, Raymond L.: A Modified Theodorsen €-Function Airfoil Design Procedure. NASA TN D-7741, 1974.

i

10

-1.5

-1.o

/"

=

leO

-.5

cP

0

~

.5

1 .o

1.5

1.0

0

x/c

(a) Original airfoil. t/c = 0.095;

cI = 0.038; Cm = -0.025.

Figure 1.- Example of changing pitching moment and lift by prescribed changes in the inviscid airfoil p r e s s u r e distribution at M, = 0.75 and a = 0.80.

11

II

,. ,,,_._

I1 111 111111 I I

_.._

.. . ....

_.

-1.5

-1.0

-

r

M

= 1.0

-.5

cP

0

Upper surface

7

-

.5

1 .o

1.5

1.o

0 ./c

2 (b) First variation.

c1 = 0.049;

c,

= -0.025;

Figure 1.- Continued.

t/c = 0.091.

-1.5

-1.0

/

- M= 1.0

- .5

C

P

0

.5

1 .o

1.5

c (c) Second variation.

cL = 0.035;

c c,

= -0.010;

t/c = 0.090.

Figure 1.- Concluded.

13

AI

AI

=

=

0.05

0.10

(a) Effect of varying the leading coefficient AI. A2 = 0.0;

61 = 0.0;

qo = 0.1.

Figure 2. - Examples illustrating the influence of various parameters in the E-function formula on the airfoil shape. q = 0.0.

A2

=

0.05

A,

=

0.04

L

(b) Effect of varying A2. A1 = 0.1;

6 1 = 0.5 radian;

q0 = 0.1;

Figure 2.- Continued.

computed values of

P of 0.0173 f 0.0002.

-.---.-

_ c -

#o

=

0.14, t /C

0.143

=

-_

_ I _

$

=

0.10, t / c

=

0.108

#o

=

0.06, t/c

=

0.075

0

((*) Effect of varying

011

__

--------___I.

the thickness ratio. A1 = 0.1; A2 = 0.05; 61 = 0.9. Figiwe 2.- Continued.

__*_--__I---

61

=

0.9, p

=

0.0300

-6,

bl

=

=

0.5, p

0.1, p

=

=

0.0173

0.0027

(d) Effect of varying 61 with computed values of p. Angles in radians. A1 = 0.1; A2 = 0.05; Figure 2. - Concluded.

qo = 0.1.

(a) Unmodified airfoil with corresponding p r e s s u r e distribution. CZ = 0.10; cm = 0.00; t/c = 0.20.

Figure 3.- Example of tailoring airfoil by a n analytic fairing without altering design parameters. P r e s s u r e s calculated by method of reference 4 a t M, = 0.1, cy = 0, and a Reynolds number of 44.0 X 106.

18

I l l 1

o Upper surface

M e r surface

1~~1'1

(b) Airfoil modified by reducing thickness aft of the 0.6 chord station

with corresponding p r e s s u r e distribution. t/c = 0.20.

cz = 0.10; c m

=

0.0;

Figure 3 . - Concluded.

19

lpper surface L

C‘

“-e ”

...

...

..,

-

-

Figure 4.- Example illustrating p r e s s u r e distribution typical of an airfoil designed by superimposing a thin airfoil thickness distribution on a camber line designed f o r z e r o pitching moment. c ~ = 0.15; c , = 0.0; NACA 65A010 , ~ thickness distribution, P r e s s u r e s calculated by method of reference 4 a t M, ’= 0.1, CY = 00, and a Reynolds number of 44.0 X lo6.

f I

Upper surface Lower surface

1

H

i!

I

'I

i

L i

. x/c

I

I

i .Ed

(a) Unmodified airfoil.

C Z , = ~

0.08; Cm = 0.00; t/c = 0.12.

Figure 5. - Example of tailoring airfoil designed by geometric superposition. P r e s s u r e s calculated by method of reference 4 at M, = 0.1, ci = Oo, and a Reynolds number of 44.0 X lo6.

21

o Upper surface Q

.

Lower surface

x/c

.lj

'

x/c 1

(b) Airfoil modified by reducing lower-surface suction near the leading edge. CZ = 0.08; Cm = 0.00; t/C = 0.12.

Figure 5.- Concluded.

22

NASA-Langley, 1975

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