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https://ntrs.nasa.gov/search.jsp?R=19740025318 2018-07-13T23:48:15+00:00Z

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1. Report No. 2. Government Accession No. 3. Recipient's Catalog No.

NASA TM X-3069

4. Title and Subtitle

DEVELOPMENT OF A COMPUTER PROGRAM TO OBTAIN

ORDINATES FOR NACA 6- AND 6A-SERIES AIRFOILS

7. Author(s)

Charles L. Ladson and Cuyler W. Brooks, Jr.

9. Performing Organization Name and Address

NASA Langley Research Center

Hampton, Va. 23665

12. Sponsoring Agency Name and Address

National Aeronautics and Space Administration

Washington, D.C. 20546

5. Report Date

September 1974

6. Performing Organization Code

8. Performing Organization Report No.

L-9558

10. Work Unit No.

501-06-05-07

'11. Contract or Grant No.

13. Type of Report and Period Covered

Technical Memorandum

14. Sponsoring Agency Code

15. Supplementary Notes

16. Abstract

A computer program was developed to produce the ordinates for airfoils of any thick-

ness, thickness distribution, or camber in the NACA 6- and 6A-series. For the 6-series

and for all but the leading edge of the 6A-series, agreement between the ordinates obtained

from the new program and previously published values is generally within 5 x 10 -5 chord.

Near the leading edge of the 6A-series airfoils, differences up to 3.5 × 10 -4 chord are found,

17. Key Words (Suggested by Author(s))

Airfoils

Wings

Rotorcraft

19. Security Classif. (of this report)

Unclassified20. Security Classif. (of this page}

Unclassified

18. Distribution Statement

Unclassified - Unlimited

STAR Category 01

21. No. of Pages

101

22. Price°

$4.5o

*For sale by the National Technical Information Service, Springfield, Virginia 22151

DEVELOPMENT OF A COMPUTER PROGRAM TO OBTAIN ORDINATES

FOR NACA 6- AND 6A-SERIES AIRFOILS

By Charles L. Ladson and Cuyler W. Brooks, Jr.

Langley Research Center

SUMMARY

A computer program was developed to produce the ordinates for airfoils of any

thickness, thickness distribution, or camber in the NACA 6- and 6A-series. For the

6-series and for all but the leading edge of the 6A-series, agreement between the ordi-

nates obtained from the new program and previously published values is generally within

5 x 10 -5 chord. Near the leading edge of the 6A-series airfoils, differences up to

3.5 x 10 -4 chord are found. The program which is given in the appendix will also produce

plots of the nondimensional 'airfoil ordinates and a punch card output of ordinates in the

input format of a readily available program for determining the pressure distributions of

arbitrary airfoils in subsonic potential viscous flow.

INTRODUCTION

The NACA 6-series airfoil sections were developed in the early 1940's, and discus-

sions of the method of the derivations and the resulting ordinates have been published in

references 1 and 2. As aircraft speeds increased, more attention was focused on the

thinner airfoils of this series. However, difficulties were encountered in the structural

design and fabrication of these thinner sections because of the very thin trailing edges.

As a result, the NACA 6A-series airfoil sections were developed, and details of these

have been published in reference 3. Essentially, the modification consisted of a near-

constant slope from about the 80-percent chord station to the trailing edge and an increase

in the trailing-edge thickness from zero to a finite value.

Recently, parametric theoretical studies have been made to investigate the use of

these airfoil sections for both rotorcraft and conventional aircraft. The results of one

investigation are presented in reference 4 and show advantages of an NACA 6-series

cambered airfoil for use as a tail rotor on helicopters. It was tedious and expensive to

make these studies because no method was available for calculating the desired ordinates

rapidly and accurately. Because the 6-series airfoils do not have an analytic expression

for the ordinates, use must be made of the ordinates published in references 1 to 3.

Also, the ordinates are not linear with variations in thickness-chord ratio so that airfoils

obtainedby linearly increasing or decreasingthe ordinates of an originally derived shapewill be approximate, as mentionedin reference 1. The published ordinates havebeencross-plotted as a function of thickness andpublished in reference 5, but the values mustbe read off the graphs and only 26 longitudinal locations from noseto tail were available.

An attempt, using a derivative of the NACA4-digit series, to provide a computerprogram for ordinates of the NACA 6-series airfoils was madein reference 6. However,as stated in the reference, the resulting accuracy of only 3.5 x 10-3 chord is not sufficientfor manyapplications.

The purpose of this paper is to review the basic designprocedure for the NACA6-series airfoils and to describe a program which will generate sufficiently accurateordinates for airfoils of anythickness, thickness distribution, or camber with an accept-able expenditure of computer time.

SYMBOLS

a basic length, usually considered unity

A mean-line designation, fraction of chord from leading edge over which design

load is uniform

c airfoil chord

C airfoil chord on computer-generated plots

CLI design section liftcoefficient

X distance along chord

X distance along chord on computer-generated plots

airfoil ordinate normal to chord, positive above chord

Y airfoil ordinate normal to chord on computer-generated plots

complex variable in circle plane

Z _

5

complex variable in near-circle plane

local inclination of camber line (or mean line)

2

ii

e airfoil parameter, _b - 0

complex variable in airfoilplane

angular coordinate of z'

_b angular coordinate of z

airfoil parameter determining radial coordinate of z

average value of @, _ _.1r _ dq5z,77

t_ o

Subscripts:

u upper surface

I lower surface

t thickness

cam cambered

ANALYSIS

Basic Airfoil Derivation

As described in references 1 and 2, the basic symmetrical NACA 6-series airfoils

were developed by means of conformal transformations. The use of these transforma-

tions to relate the flow about an arbitrary airfoil to that of a near circle and then to a

circle had been developed earlier and the results are presented in reference 7. The

basic airfoil parameters _ and e are derived as a function of G where 0 - _b is

defined as -E. Figure 1, taken from reference 1, shows the relationship between these

variables in the complex plane. These parameters are used to compute both the airfoil

ordinates and the potential flow velocity distribution around the airfoil. For the NACA

6-series airfoils, the shape of the velocity distribution and the longitudinal location of

maximum velocity (or minimum pressure) were prescribed. The airfoil parameters ¢

and • which give the desired velocity distribution were obtained through an iterative

process. Then the airfoil ordinates could be calculated from these parameters by use of

the equations presented in references 1 and 7. Thus, for each prescribed velocity distri-

bution, a set of basic airfoil parameters is obtained. However, as stated in reference 1,

it is possible to define a set of basic parameters _/ and e which could be multiplied by a

constant factor to obtain airfoils of various thickness-chord ratios while maintaining the

minimum pressure at the same chordwise location. Thus, for each NACA 6-series air-

foil family (i.e., 63-, 64-, or 65-series) there is one basic set of _ and _ values.

Calculation of Symmetrical Airfoils

There is a unique curve of _ and e as a function of _b for each NACA 6-series

airfoil family. This curve can be scaled by a constant factor to provide airfoils of dif-

ferent thickness within this family. A computer program could therefore be developed to

calculate the airfoil ordinates for given values of _ and e. Although the values of

these basic airfoil parameters were not published, tabulated values existed in files or

could be computed by the method of reference 7 from published airfoil ordinates. For the

6-series airfoils, values of _ and c were available for 21 values of _b, and 26 values

were available for the 6A-series airfoils. To provide more values of _p and e for

storage in a computer subroutine, a fit to the original values was made with an existing

parametric linked cubic spline-fit program and nine additional values were obtained

between each of the original values. This process was carried out for each airfoil

series, and the results were stored in the computer program as two subroutines for each

airfoil family.

To calculate the ordinates for an arbitrary airfoil, the program first determines

which airfoil series is desired and calls for the subroutine for this series. The airfoil

represented by the stored values of _ and e is calculated and its maximum thickness-

chord ratio is determined. The ratio of the desired value to that obtained in this deter-

mination is calculated. Then, _ and e are multiplied by this ratio to arrive at a

new airfoil thickness-chord ratio. The iteration is repeated until the computed thickness-

chord ratio is within 0.01 percent of the desired value, or until 10 iterations have been

performed. Usually convergence;occurs within four iterations. After the iterative

process has converged within the limit established, any residual difference between the

,.omputed thickness-chord ratio and that desired is eliminated by linearly scaling the y

ordinate and its first and second derivatives by the appropriate scale factor. The first

and second derivatives of the airfoil ordinates as a function of chord are computed by a

subroutine labeled "DIF"in the program. Although these ordinates and slopes are cal-

culated at more than 200 internally controlled chord stations, a subroutine is used to

interpolate between these points (by use of a vertical axis parabolic curve fit labeled

"FTLUP") so that the output will be in specified increments of chord stations. As the

leading edge is approached, the increments become smaller. As programed, ordinates

are printed at increments of 0.00025c from the leading edge to x/c = 0.01250, at incre-

ments of 0.0025c from x/c = 0.01250 to 0.1000, and at increments of 0.01c from

x/c = 0.1000 to the trailing edge.

4

!

Calculation of the Leading-Edge Radius

The values of leading-edge radius of these airfoils, published in references 1 to 3,

were initially determined by plotting the ordinates to a large scale and fairing in the best

circle fit by hand. Values of the tangency point between the circle and airfoil surface

obtained in this manner were not published. To provide smooth analytic ordinates around

the leading edge for the computer program, a tilted ellipse has been used. This tilted

ellipse is described by the basic ellipse function plus an additive term, linear in x,

which vanished at the origin, and thus has three arbitrary constants. The resulting fit

to the airfoil ordinates is exact for the ordinate itself and the first derivative, and quite

close for the second derivative, though examination of the second derivative in the region

of tangency generally reveals a small discrepancy. The ellipse is defined so that it has

the same ordinate and slope as the airfoil surface at the eleventh tabulated value of _b

in the airfoil parameter subroutine. (The eleventh stored point is actually the second

point of the original tabulated values.) This tangency point is usually located at about the

0.005 chord station but varies with airfoil thickness and series. By use of this method a

smooth transition between airfoil and ellipse is produced, the tangency point is known,

and there is a continuous variation of leading-edge shape with thickness-chord ratio. The

nondimensional radius of curvature of the ellipse at the airfoil origin is also calculated

in the program and its value is in close agreement with the published values of the

leading-edge radius for known airfoils.

Calculation of Cambered Airfoils

To calculate ordinates for a cambered airfoil, the desired mean line is first com-

puted and then the ordinates of the symmetrical airfoil are measured normal to the mean

line at the same chord station. This procedure leads to a set of parametric equations,

where (y/c)t, (y/C)cam , and 5 are all functions of the original independent variable

x/c. The ordinates on the cambered airfoil, (x/C)ca m and (y/C)cam , are given by

(x/C)ca m = (x/c) - (y/c) t sin 5

(y/C)cam = (y/C)ca m + (y/c) t cos 5

where 5 is the local inclination of the camber line and (y/c) t

to obtain the lower surface ordinates. This procedure is also described in reference 1.

The local slopes of the cambered airfoil can be shown to be

tan5secS+ (_)-t (y)t(_)tan5

/, \u 5- dl--I tan 5- y d6see

\ax/

is assumed to be negative

and

._/ = t _c/t\dx]

1 sec 6 + (d_) tan 6 +(Y)t d(_ )t c

by parametric differentiation of (x/C)ca m and (y/C)ea m with respect to the original

x/c and use of the relationship

(==_) cam : d 'Y/C)cam'_ /: d (x/c) cam'_=k 7/l /

The mean line for all cambered airfoils of the NACA 6-series is the single analytic

expression presented in reference 1 and is a function of the design lift coefficient and

type of loading desired. The calculation of these camber lines has been included in the

program so that any desired combination of airfoil family, thickness-chord ratio, design

lift coefficient, and type of loading may be obtained. The design lift coefficient and type

of loading desired are input variables. The A = 0.8 modified mean line which is used

with the NACA 6A-series airfoils (see ref. 3) has also been incorporated. As the refer-

ence indicates, this mean line loading should always be used with the 6A-series.

The standard mean line loadings for the 6-series airfoils consist of loading uniform

over the entire chord (A = 1.0), or a uniform loading to a given chord station followed by

a value decreasing linearly to zero at the trailing edge. By combining two or more types

of loading, many different types of mean lines can be obtained. For example, refer-

ence 8 presents data for airfoils which combine two mean lines to give zero loading to the

60=percent chord station followed by a linearly increasing load to the trailing edge. This

procedure produces the so=called S-type mean line, having negative camber forward and

positive camber aft. Other references have combined up to four mean lines to produce

desired types of loadings. The program presented herein can combine up to 10 different

mean line combinations if desired.

RESULTS AND DISCUSSION

Program Capabilities

The program which has been developed from the analysis described is presented

in the appendix. The output of the program consists of tabulated ordinates, computer-

generated plots of nondimensional ordinates, and punched card listings of the ordinates.

The punched cards are in the format of the input of the program described in reference 9

so that pressure distributions over the generated shape may be readily obtained. To

show graphically the capabilities of the program, sample computer plots are presented

6

in figures 2 to 9. The subscript designationsof the lift-coefficient range of minimumdrag for these airfoils, as described in references 1to 3, havebeendeleted in the com-puter plots andtables. Figure 2 illustrates the possibility of changingthe thickness-chord ratio for a fixed series. Figures 3 and4 showthe series variations within theNACA 6 and 6A families of airfoils, respectively. The variations in design lift coeffi-cient with a constant mean line loading and the variations of mean line loading for a con-

stant design lift coefficient are shown in figures 5 and 6, respectively. By combining

more than one mean line for a given airfoil, the variations illustrated in figure 7 may be

obtained. If a thickness-chord ratio of 0.0 is specified, the shape of the mean line or

combination of mean lines is calculated. The results of this procedure are shown in

figures 8 and 9. Note that the mean lines of figure 9 are those for the airfoils of figure 7.

Sample Output Tabulations

Sample computed ordinates for both a symmetric and a cambered airfoil are pre-

sented in tables I and II, respectively. Printed at the top of the first page for each table

is the airfoil designation and a listing of the input variables. There follows a summary

of parameters such as the longitudinal location of maximum thickness (the point when

the slope changes sign), the values of the location of the nose ellipse fit and its radius-

chord ratio at the origin, and the number of iterations and scaling factor used to deter-

mine the airfoil from its basic parameters. Both nondimensional and dimensional ordi-

nates are listed. The dimensional quantities have the same units as the input value of

the chord. First and second derivatives of the surface slope are also presented for the

symmetric airfoils, but only first derivatives are tabulated for the cambered airfoils.

Accuracy of Results

About 25 cases, including several from each airfoil family, were computed for

thickness-chord ratios from 0.06 to 0.15 and the results were compared with the values

published in references 1 to 3. For the NACA 6-series airfoils the agreement was

generally within 5 x 10 -5 chord. The NACA 6A-series airfoils show differences of as

much as 3.5 x 10 -4 chord near the leading edge, but from about x/c = 0.10 to

x/c = 0.95 the accuracy is about the same as for the 6-series. A plot showing a com-

parison of the present method with published ordinates for the first 0.05 chord of an

NACA 64A-015 airfoil is shown in figure 10. This is the case of poorest agreement found

in the comparisons made. The equations for the airfoil geometry dictate that the trailing-

edge thickness be zero; however, the 6A-series have a finite trailing-edge thickness.

The best result for these airfoils can be obtained by using the ordinate and slope at

x/c = 0.95 and extrapolating to the trailing edge.

Card Input Format

The input to the program is in a card format as follows:

CARD 1 - Tabulated data title card. Any designation may be used in columns 2 to 80.

CARD 2 - Airfoil and camber line series designations are as follows:

NACA airfoil family Card designation* Columns

9, 1063 -series

64 -series

65-series

66 -series

67-series

63A-series

64A-series

65A-series

63

64

65

66

67

63A

64A

65A

8, 9, 10

Camber line Card designation* Columns

NACA 6-series

NACA 6A-series

63

64

65

66

63A

64A

65A

19, 20

18, 19, 20

*These are Hollerith cards; designations must be in exact columns.

8

CARD3 - Airfoil parameter card. (Notethat cards 3 to 6 are in floating point mode.Numbers are entered with a decimal point.)

Description Variable Columns

TOC 1-10Thickness-chord ratio of airfoil(i.e., 0.120)

Publishedleading-edge radius maybe entered if desired (not usedinprogram)

Model chord usedfor listing ordi-nates in dimensional units

Design lift coefficient (i.e., 0.20);set to 0.0 for a symmetrical airfoil

Meanline chordwise loading (use0.8 for 6A-series airfoils)

Number of meanlines to be summed(if only one, leave blank or insert1.0)

LER

CHD

CLI

A

CMBNMR

11-20

21-30

31-40

41-50

51-60

CARDS 4, 5, and 6 - Up to nine additional mean lines may be summed on these cards.

These cards are not necessary for only one mean line.

Des c ription Variable Columnsm

Design lift for second mean line

Loading for second mean line

Design lift for third mean line

Loading for third mean line

Design lift for fourth mean line

Loading for fourth mean line

Design lift for fifth mean line

Loading for fifth mean line

CLI

A

CLI

A

CLI

A

CLI

A

i-I0

II-20

21-30

31-40

41-50

51-60

61-70

71-80

CARD 7 - Title card for plot of airfoil ordinate.

columns 1 to 80.

Any designation may be used in

CONCLUDINGREMARKS

A computer program has been developed to calculate rapidly the ordinates for air-

foils of any thickness, thickness distribution, or camber in the NACA 6- and 6A-series.

The program is included as an appendix to this report. Comparisons of the computer-

generated ordinates with previously published ordinates for the same airfoil show that

the agreement is generally within 5 × 10 -5 chord. Exceptions were noted for the leading-

edge region of the 6A-series airfoils, where differences of as much as 3.5 × 10 -4 chord

occurred. The program will also produce plots of the airfoil nondimensional ordinates

and a punch card output of ordinates in the input format of a readily available program

for determining the pressure distributions of arbitrary airfoils in subsonic potential

viscous flow.

Langley Research Center,

National Aeronautics and Space Administration,

Hampton, Va., June 25, 1974.

-- 10

11

APPENDIX

COMPUTER PROGRAM FOR ORDINATES OF NACA 6- AND

6A-SERIES AIRFOILS

The program presented herein is written in the Langley Research Center version of

FORTRAN IV and has been used on the Control Data series 6000 computer systems. The

computational program, the basic airfoil parameter subroutine, and the plotting routine

are presented. In the airfoil program, two subroutines (FTLUP and DIF) are used. The

first subroutine is used to interpolate between a series of consecutive points using a

parabolic curve fit, and the second subroutine is used to define the slope at a given point

in a consecutive series of points. Any standard subroutines which have these capabilities

can be substituted for those used herein. Also, several unlisted subroutines are used in

the plotting routine, which is presented as a guide for users. The program requires

about 730008 storage locations and takes about 20 seconds to compile. Each case takes

approximately 12 seconds to execute on the Control Data 6400 computer system.

11

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REFERENCES

1. Abbott, Ira H.; Von Doenhoff, Albert E.; and Stivers, Louis S., Jr.: Summary of Air-

foil Data. NACA Rep. 824, 1945. (Supersedes NACA WR L-560.)

2. Abbott, Ira H.; and Von Doenhoff, Albert E.: Theory of Wing Sections. Dover Publ.,

Inc., c.1959.

3. Loftin. Laurence K., Jr. : Theoretical and Experimental Data for a Number of NACA

6A-Series Airfoil Sections. NACA Rep. 903, 1948. (Supersedes NACA TN 1368.)

4. Robinson, Frank: Increasing Tail Rotor Thrust and Comments on Other Yaw Control

Devices. J. Amer. Helicopter Soc., vol. 15, no. 4, Oct. 1970, pp. 46-52.

5. Patterson, Elizabeth W.; and Braslow, Albert L.: Ordinates and Theoretical

Pressure-Distribution Data for NACA 6- and 6A-Series Airfoil Sections With

Thicknesses From 2 to 21 and From 2 to 15 Percent Chord, Respectively. NASA

TR R-84, 1961. (Supersedes NACA TN 4322.)

6. Kinsey, Don W.; and Bowers, Douglas L.: A Computerized Procedure To Obtain

the Coordinates and Section Characteristics of NACA Designated Airfoils.

AFFDL-TR-71-87, U.S. Air Force, Nov. 1971.

7. Theodorsen, Theodore: Theory of Wing Sections of Arbitrary Shape. NACA Rep. 411,

1931.

8. Von Doenhoff, Albert E.; Stivers, Louis S., Jr.; and O'Conner, James M.: Low-

Speed Tests of Five NACA 66-Series Airfoils Having Mean Lines Designed To Give

High Critical Mach Numbers. NACA TN 1276, 1947.

9. Stevens, W. A.; Goradia, S. H.; and Braden, J.A.: Mathematical Model for Two-

Dimensional Multi-Component Airfoils in Viscous Flow. NASA CR-1843, 1971.

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II

z -plane

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Figure 1.- Illustration of transformations used to derive airfoils and calculate

pressure distribution. (From ref. 1.)

91

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NASA-Langley, 1974 L-9558 101