NASA Technical Memorandum 4741

/ a / ""

Computer Program To Obtain Ordinates for

NACA Airfoils

Charles L. Ladson, Cuyler W. Brooks, Jr., Acquilla S. Hill, and Darrell W. Sproles

December 1996
NASA Technical Memorandum 4741

Computer Program To Obtain Ordinates for

NACA Airfoils

Charles L. Ladson, Cuyler W. Brooks, Jr., and Acquilla S. Hill

Langley Research Center • Hampton, Virginia

Darrell W. Sproles

Computer Sciences Corporation • Hampton, Virginia

National Aeronautics and Space Administration

Langley Research Center • Hampton, Virginia 23681-0001

December 1996
The use of trademarks or names of manufacturers in this report is for

accurate reporting and does not constitute an official endorsement,

either expressed or implied, of such products or manufacturers by the

National Aeronautics and Space Administration.

Available electronically at the following URL address: http://techreports.larc.nasa.gov/ltrs/ltrs.html

Computer program is available electronically from the Langley Software Server

Printed copies available from the following:

NASA Center for AeroSpace Information

800 Elkridge Landing Road

Linthicum Heights, MD 21090-2934

(301) 621-0390

National Technical Information Service (NTIS)

5285 Port Royal Road

Springfield, VA 22161-2171

(703) 487-4650
Abstract

Computer programs to produce the ordinates for airfoils of any thickness, thickness distribution, or camber in the NACA airfoil series were developed in the early

1970's and are published as NASA TM X-3069 and TM X-3284. For analytic airfoils,

the ordinates are exact. For the 6-series and all but the leading edge of the 6A-series

airfoils, agreement between the ordinates obtained from the program and previously

published ordinates is generally within 5 × 10 -5 chord. Since the publication of these

programs, the use of personal computers and individual workstations has proliferated. This report describes a computer program that combines the capabilities of the

previously published versions. This program is written in ANSI FORTRAN 77 and can

be compiled to run on DOS, UNIX, and VMS based personal computers and workstations as well as mainframes. An effort was made to make all inputs to the program

as simple as possible to use and to lead the user through the process by means of a

menu.

Introduction bi

Although modern high-speed aircraft generally make

use of advanced NASA supercritical airfoil sections, c (C, CHD)

there is still a demand for information on the NACA Cl,i (CLI)

series of airfoil sections, which were developed over di

50 years ago. Computer programs were developed in the dx

early 1970's to produce the ordinates for airfoils of any

thickness, thickness distribution, or camber in the NACA

airfoil series. These programs are published in refer- d(x/c), dy, d_

ences 1 and 2. These programs, however, were written in I

the Langley Research Center version of FORTRAN IV

and are not easily portable to other computers. The pur- kl, k2

pose of this paper is to describe an updated version of m

these programs. The goal was to combine both programs

into a single program that could be executed on a wide

variety of personal computers and workstations as well P

as mainframes. The analytical design equations for both R

symmetrical and cambered airfoils in the NACA 4-digit- Rle (RLE)

series, 4-digit-modified-series, 5-digit-series, 5-digitmodified-series, 16-series, 6-series, and 6A-series airfoil r

families have been implemented. The camber-line designations available are the 2-digit, 3-digit, 3-digit-reflex,

6-series, and 6A-series. The program achieves portability t

by limiting machine-specific code. An effort was made x (X)

to make all inputs to the program as simple as possible to

use and to lead the user through the process by means of Y (Y)

a menu.

Symbols

The symbols in parentheses are the ones used in the

computer program and in the computer-generated listings

(.rpt file).

A

ai

camber-line designation, fraction of

chord from leading edge over which

design load is uniform

constants in airfoil equation, i = 0, 1..... 4

Z

Zt

0

constants in camber-line equation, i = 0,

1,2

airfoil chord

design section lift coefficient

constants in airfoil equation, i = 0, 1, 2, 3

derivative of x; also basic selectable

interval in profile generation

derivatives of x/c, y, and

leading-edge radius index number

constants

chordwise location for maximum ordinate of airfoil or camber line

maximum ordinate of 2-digit camber line

radius of curvature

leading-edge radius

chordwise location for zero value of second derivative of 3-digit or 3-digit-reflex

camber-line equation

thickness

distance along chord

airfoil ordinate normal to chord, positive

above chord

complex variable in circle plane

complex variable in near-circle plane

local inclination of camber line

airfoil parameter, @- 0

complex variable in airfoil plane

angular coordinate of z'

angular coordinate of z

airfoil parameter determining radial coordinate of z'
Ilt0 average value of _t, _ _/de

Subscripts:

cam cambered

1 (L) lower surface

N forward portion of camber line

T aft portion of camber line

t thickness

u (U) upper surface

x derivative with respect to x

Computer Listing Symbols

For reasons having to do with code portability, the

computer-generated listing (.rpt file) will always have

the alphabetic characters in upper case. The following

list is intended to eliminate any confusion.

A camber-line designation, fraction of chord

from leading edge over which design load

is uniform

A0 ..... A4 constants in airfoil equation

CHD, C airfoil chord

CLI design section lift coefficient

CMB maximum camber in chord length,

y(m)

pc

CMBNMR number of camber lines to be combined in

6- and 6A-series multiple camber-line

option

CMY m, location of maximum camber

CRAT cumulative scaling of EPS, PSI,

17 RAT(I), I = 1 --+ IT

DO..... D3 constants in airfoil equation

DX basic selectable interval in profile

generation

dy

DY/DX first derivative of y with respect to x, d---x

D2Y/DX2 second derivative of y with respect to x,

d2y

dx 2

EPS airfoil parameter, a -- _ - 0

IT number of iterations to converge 6-series

profile

K1, K2 3-digit-reflex camber parameters, k 1 and k2

PHI _, angular coordinate of z

PSI _, airfoil parameter determining radial

coordinate of z'

RAT(I)

RC

RK2OK1

RLE

RNP

SF

TOC

X

XMT

XT(12),

YT(12),

YTP(12)

XU, XL

XTP

XYM

Y

YM

YMAX

YU, YL

ith iterative scaling of a,

radius of curvature at maximum thickness

for 4-digit modified profile

k2

3-digit reflex camber parameter ratio, kl

leading-edge radius

radius of curvature at origin

ratio of input t/c to converged t/c after

scaling

thickness-chord ratio

distance along chord

m, location of maximum thickness for

4-digit modified profile

location and slope of ellipse nose fairing

for 6- and 6A-series thickness

profiles

upper and lower surface locations of x

x/c location of slope sign change for

6- and 6A-series thickness profile

m, location of maximum thickness for

6- and 6A-series profiles or chordwise

location for maximum ordinate of airfoil

or camber line

airfoil ordinate normal to chord, positive

above chord

y/c location of slope sign change for 6- and

6A-series thickness profile

y(m), maximum ordinate of thickness

distribution

upper and lower surface y ordinate

Analysis

Thickness Distribution Equations for Analytic

Airfoils

The design equations for the analytic NACA airfoils

and camber lines have been presented in references 3

to 7. They are repeated herein to provide a better understanding of the computer program and indicate the use of

different design variables. A summary of some of the

design equations and ordinates for many airfoils from

these families is also presented in references 8 to 10.

The traditional NACA airfoil designations are shorthand codes representing the essential elements (such as

thickness-chord ratio, camber, design lift coefficient)

controlling the shape of a profile generated within a

given airfoil type. Thus, for example the NACA 4-digitseries airfoil is specified by a 4-digit code of the form

pmxx, where p and m represent positions reserved for
specification of the camber and xx allows for specification of the thickness-chord ratio as a percentage, that is,

"pml2" designates a 12-percent-thick (t/c = 0.12) 4-digit

airf6il.

NACA 4-digit-series airfoils. Symmetric airfoils in

the 4-digit-series family are designated by a 4-digit number of the form NACA 00xx. The first two digits indicate

a symmetric airfoil; the second two, the thickness-chord

ratio. Ordinates for the NACA 4-digit airfoil family

(ref. 2) are described by an equation of the form:

Yc-a- O_cJ (x'_l/2 + a l_cJ (X_+a 2kcJ (x-_2 + a3 Ixlc 3 + a4 (x14

The constants in the equation (for t/c= 0.20) were

determined from the following boundary conditions:

Maximum ordinate:

_x = 0.30 -Y = 0.10 dy = 0

c c dx

Ordinate at trailing edge:

x y_ - = 1.0 - - 0.002

C C

Magnitude of trailing-edge angle:

x

- = 1.0 = 0.234

C

Nose shape:

x y_ - = 0.10 - - 0.078

C C

The following coefficients were determined to meet

these constraints very closely:

a 0 = 0.2969 a I = -0.1260

a 2 = -0.3516 a 3 = 0.2843

a 4 = -0.1015

To obtain ordinates for airfoils in the family with a

thickness other than 20 percent, the ordinates for the

model with a thickness-chord ratio of 0.20 are multiplied

by the ratio (t/c)0.20. The leading-edge radius of

this family is defined as the radius of curvature of the

basic equation evaluated at x/c = 0. Because of the term

ao(X/C ) 1/2 in the equation, the radius of curvature is finite

at x/c = 0 and can be shown to be (see appendix)

2

a0(,/c?

R le = -2 _,-O---,_J

by taking the limit as x approaches zero of the standard

expression for radius of curvature:

R = [ 1 + (dy/dx)2] 3/2

d2y /d x 2

To define an airfoil in this family, the only input necessary to the computer program is the desired thicknesschord ratio.

One might expect that this leading-edge radius R(0),

found in the limit as x _ 0 to depend only on the a0

term of the defining equation, would also be the minimum radius on the profile curve. This is not true in general; for the NACA 0020 airfoil, for example, a slightly

smaller radius (R = 0.0435 as compared to R(0) =

0.044075) is found in the vicinity ofx = 0.00025.

NACA 4-digit-modified-series airfoils. The 4-digitmodified-series airfoils are designated by a 4-digit number followed by a dash and a 2-digit number (such as

NACA 0012-63). The first two digits are zero for a symmetrical airfoil and the second two digits indicate the

thickness-chord ratio. The first digit after the dash is a

leading-edge-radius index number, and the second is the

location of maximum thickness in tenths of chord aft of

the leading edge.

The design equation for the 4-digit-series airfoil

family was modified (ref. 4) so that the same basic shape

was retained but variations in leading-edge radius and

chordwise location of maximum thickness could be

made. Ordinates for these airfoils are determined from

the following equations:

yc- __- a0_,cJ (x_ll2+al(x]+ a2(c)x 2 +a3(X) 3

from leading edge to maximum thickness, and

y = do+dl( l_x]+d2(l_x]2+d3(l_x) 3

from maximum thickness to trailing edge.

The constants in these equations (for t/c= 0.20) can

be determined from the following boundary conditions:

Maximum ordinate:

x y_ dy_

- = m - - 0.10 -- - 0

c c dx

Leading-edge radius:

2

x ao -=0 R=--

c 2

3
Radiusof curvature at maximum thickness:

x (1 -m) 2

-=m R=

c 2d 1(1 - m) - 0.588

Ordinate at trailing edge:

x y_

- = 1.0 - - d o = 0.002

c c

Magnitude of trailing-edge angle:

_x

= 1.0 -" = d 1 = f(m)

c dx

Thus, the maximum ordinate, slope, and radius

of curvature of the two portions of the surface match at

x/c = m. The values of d 1 were chosen, as stated in reference 4, to avoid reversals of curvature and are given in

the following table:

m d1

0.2 0.200

0.3 0.234

0.4 0.315

0.5 0.465

0.6 0.700

By use of these constraints, equations were written

for each of the constants (except a 0 and dl) in the equation for the airfoil family and are included in the computer program. As in the 4-digit-series airfoil family,

ordinates vary linearly with variations in thickness-chord

ratio and any desired thickness shape can be obtained by

scaling the design ordinates by the ratio of the desired

thickness-chord ratio to the design thickness-chord ratio.

The leading-edge index is an arbitrary number

assigned to the leading-edge radius in reference 4 and is

proportional to ao. The relationship between leadingedge radius Rle and index number I is as follows:

t/c I 2

Rle = 0.5(0.29690-_ _)

Thus, an index of 0 indicates a sharp leading edge

(radius of zero) and an index of 6 corresponds to

a 0 = 0.2969, the normal design value for the 20-percentthick 4-digit airfoil. A value of leading-edge index of 9

for a three times normal leading-edge radius was arbitrarily assigned in reference 4, but I = 9 cannot be used in

this equation. In reference 4, the index I is not used

in computation. Instead, the index I = 9 was assigned to

an airfoil where a 0 = 0.29694r3 = 0.514246, which 2

(because Rle = a0/2) thus has a three times normal

leading-edge radius. The computer program is written so

that the desired value of leading-edge radius or the index

I is the input parameter. The value of a0 is then computed

in the program.

NACA 16-series airfoils. The NACA 16-series airfoil family is described in references 6 and 7. From the

equation for the ordinates in reference 7, this series is a

special case of the 4-digit-modified family although this

is not directly stated in the references. The 16-series airfoils are thus defined as having a leading-edge index of 4

and a location of maximum thickness at 0.50 chord. The

designation NACA 16-012 airfoil is equivalent to an

NACA 0012-45.

Thickness Distribution Equations for Derived

Airfoils

NACA 6-series airfoils. As described in references 9

and 10, the basic symmetrical NACA 6-series airfoils

were developed by means of conformal transformations.

The use of these transformations 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 11. The basic airfoil parameters

and e are derived as a function of _, where 0 - _ is

defined as -e. Figure 1, taken from reference 10, 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 e which

give the desired velocity distribution were obtained

through an iterative process. Then the airfoil ordinates

can be calculated from these parameters by use of the

equations presented in references 10 and 11. Thus, for

each prescribed velocity distribution, a set of basic airfoil

parameters is obtained. However, as stated in reference 10, it is possible to define a set of basic parameters

and e which can 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 airfoil family (i.e., 63-, 64-, or 65-series), there is one basic set of

values for _ and a.

NACA 6A-series airfoils. The NACA 6-series airfoil

sections were developed in the early 1940's. 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 12. Essentially, the modification consisted of
z-plane

z'-plane

Z r _ = e(_-_o)+i (O-¢p)

Z

_ = Z' q- --

Z = ae _O+i _

Z I

a2

Z'

= ae_+iO

y

;=x+iy

Figure 1. Illustration of transformations used to derive airfoils and

calculate pressure distribution. From reference 1; a is basic

length, usually considered unity.

a near-constant slope from about the 80-percent-chord

station to the trailing edge and an increase in the trailingedge thickness from zero to a finite value.

Calculation procedure. For each NACA 6-series

airfoil family, a unique curve of _ and e exists as a function of q_.This curve can be scaled by a constant factor to

provide airfoils of different thickness within this family.

A computer program could therefore be developed to calculate the airfoil ordinates for given values of _ and a.

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 11 from

published airfoil ordinates. To provide more values of

and e for storage in a computer subroutine, a fit to the

original values was made with a parametric-linked cubicspline-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 arrays for

each airfoil family (one for e as a function of _) and one

for _ as a function of _). For the new program, the

enrichment was completely redone and the q_ array is

stored in the program only once for all 6- and 6A-series

profiles.

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

determination is calculated. Then, _g and _ and are multiplied by this ratio to arrive at a new airfoil thicknesschord 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 computed thicknesschord 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 "DERV" in the program. Although these ordinates and slopes are calculated

at more than 200 internally controlled chord stations, a

subroutine is used to interpolate between these points (by

use of a piecewise cubic curve fit labeled "NTRP") so

that the output will be in specified increments of chord

stations.

Calculation of leading-edge radius. The values of

leading-edge radius of the derived airfoils, published in

references 9 to 12, 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. (See fig. 2.) This tilted ellipse is

described by the basic ellipse function plus an additive

term, linear in x, which vanishes at the origin. The tilted

ellipse expression thus has three arbitrary constants,

which are determined in the procedure. 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

(even smaller in the current version than in the original

computer program (ref. 1)). The ellipse is defined so that

it has the same ordinate and slope as the airfoil surface at

the 12th (of 201) tabulated value of _ in the airfoil

parameter subroutine. (The l lth stored point, which is

actually the second point of the original tabulated values,

was used in ref. 1.) This tangency point is usually located

at about 0.005 chord but varies with airfoil thickness and
.02 .10

y(x) .01

I

I

I

I

I

I

I

I

I

I

Curves match at

12th point _,,

,,

I --°"

li "/'"'

_I......... 6 series

.,°--r

,°,-°°° i

i

Tilted ellipse ,

I

I

.010

x/c

Ellipse, y(x)

[2]-----------_Profile, y(x)

............ Radius, R(x)

/-°

°.°

°.-"

o/

°°,.°°

o/'

°,°'°

profile

(a) 12-percent-thick example, NACA 64012 airfoil.

.05

0

.020

R(x)

y(x)

.03

.02

.01

0

I

I

Curves match at I

12th point

//°'"

I ,,/"

I /"

. ....o"('°

.....-'"'"'° I

Tilted ellipse I

[]

/

°/

o/"

/

o/

_rofile

I

.010 .010

X/C

Ellipse, y(x)

D-----g3 Profile, y(x)

............ Radius, R(x)

(b) 24-percent-thick example, NACA 64024 airfoil.

Figure 2. Leading-edge fairing for 6-series and 6A-series profiles.

.09

0

.020

.06

.03

R(x)

6
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.

Camber-Line Equations

2-digit. The 2-digit camber line is designated by a

2-digit number and, when used with a 4-digit airfoil,

would have the form NACA pmxx where p is the maximum camber in percent chord, m is the chordwise location of maximum camber in tenths of chord, and xx is the

airfoil thickness in percent chord. The NACA 2-digit

camber line is described in reference 3. This camber line

is formed by two parabolic segments which have a general equation of the form:

-Y=c bO+ bl(X/+b2(X) 2

The constants for the two equations are determined

from the following boundary conditions:

Camber-line extremities:

x y_ x y_

- = 0 - - 0 - = 1.0 -- 0

C C C C

Maximum ordinate:

x y_ dy_

-=m --p ---0

c c dx

From these conditions, the camber-line equations

then become

-Y: P(2mX__ (x/2

c m2_, c)

forward of the maximum ordinate and

Y m

P ( c

(l-m) 2[ 1-2m)+2mx-/x)21

aft of the maximum ordinate. Both the ordinate and slope

of the two parabolic segments match at x/c = m. Tables

of ordinates for some of these camber lines are tabulated

in references 9 and 10. The ordinates are linear with

amount of camber p and these can be scaled up or down

as desired.

3-digit. To provide a camber line with a very far forward location of the maximum camber, the 3-digit camber line was developed and presented in reference 5. The

first digit of the 3-digit camber-line designation is

defined as two thirds of the design lift coefficient (in

tenths, i.e., 2 denotes Cl,i = 0.3); the second digit, as twice

the longitudinal location of maximum camber in tenths

of chord; and the third digit of zero indicates a nonreflexed trailing edge.

This camber line is also made up of two equations so

that the second derivative decreases to zero at a point r

aft of the maximum ordinate and remains zero from this

point to the trailing edge. The equations for these conditions are as follows:

d2y- kl( c-x r)

dx 2

from x/c = 0 to x/c = r, and

d2y - 0

dx 2

from x/c = r to x/c = 1.0. The boundary conditions are as

follows:

Camber-line extremities:

x y_ x y_ - : 0 -- 0 - : 1.0 -- 0

C C C C

At the junction point:

x_ : r (Y)N = (Y) (d-_xxlN = (d-_xx)

c T T

The equation for the camber line then becomes

yC - kl6I(X/3 _ 3r(X/2 + r2(3- r)X 1

from x/c = 0 to x/c = r, and

from x/c = r to x/c = 1.0. These equations were then

solved for values of r which would give longitudinal

locations of the maximum ordinate of 5, 10, 15, 20, and

25 percent chord. The value of k 1 was adjusted so that a

theoretical design lift coefficient of 0.3 was obtained at

the ideal angle of attack. The value of k 1 can be linearly

scaled to give any desired design lift coefficient. Values

7
of k 1 and r and the camber-line designation were taken

from reference 5 and are presented in the following table:

Camber-line

designation m

210 0.05

220 0.10

230 0.15

240 0.20

250 0.25

r k 1

0.0580 361.400

0.1260 51.640

0.2025 15.957

0.2900 6.643

0.3910 3.230

3-digitreflex. The camber-line designation for the

3-digit-reflex camber line is the same as that for the

3-digit camber line except that the last digit is changed

from 0 to 1 to indicate the reflex characteristic, which is

the normally negative camber-line curvature, becomes

positive in the aft segment.

For some applications, for example, rotorcraft main

rotors, to produce an airfoil with a quarter-chord

pitching-moment coefficient of zero may be desirable.

The 3-digit-reflexed camber line was thus designed to

have a theoretical zero pitching moment as described in

reference 5. The forward part of the camber line is identical to the 3-digit camber line but the aft portion was

changed from a zero curvature segment to a segment

with positive curvature. The equation for the aft portion

of the camber line is expressed by

d2y-- = k21 c -x r)>0

dx 2

(xr)

By using the same boundary conditions as were used for

the 3-digit camber line, the equations for the ordinates

are

k2 _r)3_x_ 3x 1

klI(x - 3_ +r 3

-Yc= 6L\c-r _1(1 c r -c

from x/c = 0 to x/c = r and

yc = -ffL \c- r/3- _1 (1 -r)3X-c r c3X* r 3]

for x/c = r to x/c = 1.0. The ratio k2/k 1 is expressed as

_ 3 k2 3 (r- m) 2- r

k 1 1 -r

Values of k 1, k2/k 1, and m for several camber-line

designations from reference 5 are presented in the following table:

Camber-line

designation m

221 0.10

231 0.15

241 0.20

251 0.25

r k1 k2/k 1

0.1300 51.990 0.000764

0.2170 15.793 0.00677

0.3180 6.520 0.0303

0.4410 3.191 0.1355

g-series. The equations for the 6-series camber lines

are presented in reference 9. The camber lines are a function of the design lift coefficient Cl,i and the chordwise

extent of uniform loading A. The equation for these camber lines is as follows:

y_ Cl, i i__._1 I_1 (A _ _Xc/2log2(A _x )

c 2n(A + 1)

_x/2loge(1

-_(A-X)21 -x- l°geX+g-hX}c c

where

g= 1-A1 IA2(_ logeA __)+_1

h- 1 1AI_(1-A) - 2 loge(1-A)-_(1-A)21+g

As was true in reference 1, the program is capable of

combining (by cumulative addition of y/c) up to 10 camber lines of this series to provide many types of loading.

16-series. The 16-series cambered airfoils, as

described in reference 6, are derived by using the 6-series

camber-line equation with the mean-line loading A set

equal to 1.0, which is

Y- Cl, i(l _X)21Oge(l _X I

c 4_\

This equation is the one for the standard mean line

for this series.

6A-series. The 6A-series cambered airfoils, as

described in reference 12, are derived by using a special

form of the 6-series camber-line equation. This special

form is designated as "the A = 0.8 modified mean line."

The modification basically consists of holding the slope

of the mean line constant from about the 85-percentchord station to the trailing edge. As the reference indicates, this mean-line loading should always be used for

the 6A-series airfoils. This camber-line equation is given
asoneof theoptionsin sdectionof meanlinesin the

program.

Calculation of Cambered Airfoils

To calculate ordinates for a cambered airfoil (including the limiting cases of zero camber (Ycam(X) = O) and

zero thickness (yt(x) = 0)), the desired mean (camber)

line is first computed 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 8 are all

functions of the original independent variable x/c. The

upper surface ordinates on the cambered airfoil (x/C)u

and (y/C)u are given by

= - sin 8

(Y)u = (cY ]cam + (c)t y cos 8

where 8 is the local inclination of the camber line and

(y/c)t is assumed to be negative to obtain the lower surface ordinates (x/c)l and (y/c)l. This procedure is also

described in reference 10.

The local slopes of the cambered airfoil are

and

(dY)u

tan8 sec 8 + - k.cJtL d \cJJ

y d8 x

see( )tan

tan 8 sec 8 -

(dY)t dx +(Y-] \c Jt Fd6(Xltan Ld \c JJ 8

(dY)l = y d8 x

sec 8 + (d_/ttan 8 + (c )t I-_- (c )]

by parametric differentiation of (x/C)u,l and (y/C)u,l with

respect to the original x/c and use of the relationship

(dY)u d(y/C)u/d(x/c) d-x = d(x/C)u/d(x/c )

Although specific camber lines are generally used

with specific thickness distributions, this program has

been written in a general format. As a result, any camber

line can be used with either type thickness distribution so

that any shape desired can be generated.

The user should understand that these equations for

combining a thickness distribution with a mean line,

while mathematically stable, can produce profiles that

are geometrically discontinuous and aerodynamically

useless. An NACA 2224 airfoil, for example, will have a

small region on the lower surface where (y/c)l is triple

valued.

The density of the final output of the procedure (as in

tables I and II) is determined by the user-supplied value

of the basic x/c interval dx. As the leading edge is

approached, the increments become smaller to provide

good definition of the leading edge of a blunt profile.

Ordinates are printed at increments of dr/40 chord from

the leading edge to x/c = 0.01250, at increments of dx/4

chord from x/c = 0.01250 to 0.1000, and at increments of

dr chord from x/c = 0.1000 to the trailing edge. The user

is offered the option of setting dr and the smaller increments for x/c less than 0.1 will be reset in the same ratio.

This procedure is devised to provide rational x values for

dr = 0.01, 0.05, and 0.10.

Results and Discussion

Program Capabilities

AIRFOLS is the computer program which was

developed to provide the airfoil shapes described in the

section "Analysis." AIRFOLS, the NACA Airfoil Ordinate Generator, is the result of merging two previous

efforts reported in references 1 and 2. This program calculates and then reports the ordinates and surface slope

for airfoils of any thickness, thickness distribution, or

camber in the designated NACA series. AIRFOLS is a

portable, ANSI FORTRAN 77 code with limited platform dependencies. Parameters describing the desired

airfoil are entered into the software with menu and

prompt driven input. Output consists of a report file and a

data file.

Provisions have been made in the AIRFOLS program to combine basic airfoil shapes and camber lines

from different series so that nonstandard as well as standard airfoils can be generated. The analytical design

equations for both symmetrical and cambered airfoils in

the NACA 4-digit-series, 4-digit-series modified, 5-digitseries, 5-digit-series modified, 16-series, 6-series, and

6A-series airfoil families have been implemented. The

camber-line designations available are the 2-digit,

3-digit, 3-digit-reflex, 6-series, and 6A-series. The program achieves portability by limiting machine-specific

code. The two exceptions are a compiler-specific format

edit descriptor and a terminal-specific (display device)

set of screen control codes. The user's targets, selected

from the available platform configuration and terminal

device choices, are communicated to the software

enabling the machine-specific setup to be performed

internally with coded logic. The program attempts to be

completely self-guiding and to reduce required input to

9
theminimumnecessaryfor therequestedairfoilsection

andmeanline.Theconventionusedfor thetwooutput

file namesconsistof combiningauser-suppliedfile prefix respectivelysuffixedwith".rpt" and".dta."Theoutputreport(.rpt)isanASCIIfile containingthetabulated

nondimensional anddimensional airfoilordinatesaswell

asotherinputandoutputparameters.Theoutputdata

(.dta)is an ASCIIfile of X-Y pairsrepresentingthe

dimensional airfoilordinates.

To executeAIRFOLS,entertheappropriatecommandthenfollow the programprompts.A message

printedby theprogramat thebeginningof execution

shouldreiteratethe platform,operatingsystem,and

FORTRANcompilerto whichthepresentexecutableis

targeted.Examplesof thesupportedplatformconfigurationoptionsare

CONVEXC2usingCONVEXFORTRANcompiler

IBM compatiblePC using IBM FORTRAN/2

compiler

Sun-3usingSunFORTRANcompiler

VAX780usingVAXFORTRANcompiler

IRIS-4D using Silicon Graphics FORTRAN

compiler

Sun-4usingSunFORTRANcompiler

Gateway2000PCusingLaheyFORTRANcompiler

Generic,useonlyif otherdevicesunavailable

Theprogram'sinitialrequestis foridentificationof

thedisplaydevicebeingused.Examplesofthesupported

displaydeviceoptionsare

ANSIterminaloremulator

IBMpersonalcomputerorcompatible

TEKTRONIX4107

SunMicrosystemsworkstationconsole

DECVT terminal50

DECVT terminal100

DECVT terminal200

DECVT terminal240withREGIS

IBMpersonalcomputerVTERMpackage

TherearetwoformsofinputforcontrollingtheAIRFOLSprogram.Theprogrampromptstheuserto enter

textualinformationor selecta choicefromamenu.The

primarytechnicalinputrequeststhatauserseesare

Choiceof standardornonstandardairfoil

Selectionfromairfoilsectionoptions

10

Choiceof airfoilwithcamberornocamber

Selectionfrommean-lineoptions

Promptforairfoiltitle

Promptforoutputfileprefix

Prompts for specific airfoil and mean-line

parameters

Theprogramalsomakesavailablea nontechnical

tutorialin theformof a genericmenuexampleto show

theuseof menucontrol.To viewthetutorialreply"F"

(false)to themenuquery:"USERIS EXPERIENCED

WITH AIRFOLS."Replyingto anymenuquerywith

"/EX" (respondwithexample)will alsoexecutethetutorial.Thetutorialinterruptstheuser'sinteractivecontrol

to executetheexamplemenutypequery.Thedesired

repliesareenteredby thetutorial,alleviatingrequirementsfor userinteraction.Eachreplyproducesa program responseand a tutorial explanationof that

response.Programcontrolis returnedto theuserwhen

theexampleterminates.Thetutorialexamplewill displaytheprogram'sresponseto anumberof correctand

incorrectentries.Theresponsesto theprogramcontrol

charactersprintedwitheachmenuarecoveredaswell.

Theprogramcontrolcharactersandthetypicalresponse

toexpectare

cr (orEnter)

/

Clears screen then redisplays menu

Responds with system prompt options

menu

? Provides additional details on prompted

request

E (or e) Exits from submenu to previous menu

level

Q (or q) Terminates program execution

AIRFOLS was written with emphasis on user friendliness, self-direction, and informative error processing

with the goal of producing a valuable software tool for

the researcher.

Accuracy of Results

Analytical airfoils. All the airfoils and camber lines

generated by this program are defined by closed analytical expressions and no approximations have been made

in the program. Thus, all results are exact. Many cases

have been run and compared with previously published

results to check the procedure, and for all cases the

comparisons were exact except for occasional differences beyond the fifth digit caused by rounding

differences.
Derived airfoils. In reference 1, 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 9, 10, and 12. 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

9 x 10-5 chord near the leading edge, but from about

x/c = 0.10 to 0.95 the accuracy is about the same as for

the 6-series. The equations for the 6-series airfoil geometry dictate that the trailing-edge thickness be zero; however, the 6A-series airfoils have a finite trailing-edge

thickness. For the 6A-series airfoils, the profiles obtained

from the program are not useful past x/c = 0.95. For construction or grid generation, the user would probably

modify the last 5 percent of the chord by using the ordinate and slope at x/c = 0.95 and extrapolating to the trailing edge.

Early Release Version

A version of this program (0.A, March 31, 1992)

already in use produces the same results for actual construction of an airfoil from the output coordinates. However, this early version does display small discrepancies

in the second derivative of the thickness distribution

upstream of 0.5 percent chord. In the current version, this

problem has been corrected by allowing the tilted ellipse

fit loop to go one step further and by replacing the interpolation subroutine NTRP with an improved procedure.

Users concerned about very small perturbations, as in

grid generation, may want to be sure they have the version documented in this report (1.01, April 1996).

respectively. Printed at the top of the first page for each

table is the airfoil and camber-line family selected, the

airfoil designation, and a list of the input parameters for

both airfoil shape and camber line. Both nondimensional

and dimensional ordinates are listed. The dimensional

quantities have the same units as the input value of the

chord, which is also listed at the top of the page. First and

second derivatives of the surface ordinates are also presented for symmetrical airfoils, but only first derivatives

are tabulated for the cambered airfoils.

Concluding Remarks

Computer programs to produce the ordinates for airfoils of any thickness, thickness distribution, or camber

in the NACA airfoil series were developed in the early

1970's and are published as NASA TM X-3069 and

TM X-3284. For analytic airfoils, the ordinates are exact.

For the 6-series and all but the leading edge of the 6Aseries airfoils, agreement between the ordinates obtained

from the program and previously published ordinates is

generally within 5 x 10 -5 chord. Since the publication of

these programs, the use of personal computers and

individual workstations has proliferated. This report

describes a computer program that combines the capabilities of the previously published versions. This program

is written in ANSI FORTRAN 77 and can be compiled to

run on DOS, UNIX, and VMS based personal computers

and workstations as well as mainframes. An effort was

made to make all inputs to the program as simple as possible to use and to lead the user through the process by

means of a menu.

Sample Output Tabulations

Sample computed ordinates for both a symmetric

and a cambered airfoil are presented in tables I and II,

NASA Langley Research Center

Hampton, VA 23681-0001

August 29, 1996

11
Appendix

Convergence of Leading-Edge Radius of

NACA 4-Digit and 4-Digit-Modified Series

Airfoils

The demonstration follows that in the limit as

x _ 0, the radius of curvature at the nose of the NACA

4-digit-series airfoil profile (or the 4-digit-modified,

which has the same equation at the leading edge) is finite

and equal to a2/2, where a 0 is the coefficient of the

1/2

x term in the defining equation:

1/2 2 3 4

y : aOx + alx + a2x + a3x + a4x

1 -1/2

Yx = 72a0 x + al + 2a2x + 3a3 x2 + 4a4 x3

1 -3/2 2

Yxx = -_aOx + 2a2 + 6a3x + 12a4x

The leading-edge radius is the radius of curvature at

x = 0 (Rle = R(0)):

2 3/2

(1 + Yx)

R(x) -

Yxx

2 3 x=0

R(X)2 = (1 2yx)

Yxx

This expression for the radius of curvature cannot be

evaluated at x = 0 by direct substitution. Evaluating the

numerator and denominator separately in the limit as

x --_ 0, where all terms in x e>O must vanish and all constant terms become negligible compared with the terms

in x e<O, gives

For the numerator,

2 1 2-1 -1/2

l+Yx = _aOx +alaOX

2 3 (1) 3 6 -3 ,,//1) 2 5 -5/2

(1 + Yx) : _.]l) aOx + J_.4) alaOX

,,('1) 2 4-2 3 3 -3/2

+ J_ala0x +alaOX

For the denominator,

2 fl) 2 2 -3 ,,¢'1) -3/2

Yxx : _,_) aOx - z_._)a2aox

,,(3) -1/2

- l.Ty3a°x

Multiplying both numerator and denominator by X 3

gives:

R(x) 2 : (1/4) 3a6 + 3(1/4)2ala5x 1/2

(1/4)2a 2 ...

where the ellipses represent vanishing terms in X e>O.

Thus,

1 4

R(0) 2 = _a 0

1 2

R(O) = _a 0

This result clearly depends on the a 0 term in the original

equation with its fractional power of the independent

variable. This term prevents the first and second derivatives from vanishing at the origin. Instead, both the first

and second derivatives increase without limit at the origin but in such a way that the radius of curvature is finite.

As x _ 0, the variation of R(x) is such that

dR -1/2

dx ala°x

so that even though R(0) is finite, Rx(O) is not. As a

2 • •

result, R(x) does not appear to converge to ao/2 unnl x is

very near zero, and then the curve for R(x) approaches

a2/2 along the axis x = 0. Furthermore, because a 1 < 0

(for 4-digit or 4-digit-modified airfoil), Rx(O) < 0 and

thus R(x) must have a minimum Rmin(X ) < R(0). A general expression for this minimum is probably not worth

the labor of the derivation. For the NACA 0020 airfoil, it

is approximately Rmi n = R(0.000345) = 0.04304 compared with R(0) = a2/2 = 0.044075 in nondimensional coordinates.

12
References

1. Ladson, Charles L.; and Brooks, Cuyler W., Jr.: Development

of a Computer Program To Obtain Ordinates for NACA 6- and

6A-Series Airfoils. NASA TM X-3069, 1974.

2. Ladson, Charles L.; and Brooks, Cuyler W., Jr.: Development

of a Computer Program To Obtain Ordinates for NACA

4-Digit, 4-Digit Modified, 5-Digit, and 16-Series Airfoils.

NASA TM-3284, 1975.

3. Jacobs, Eastman N.; Ward, Kenneth E.; and Pinkerton,

Robert M.: The Characteristics of 78 Related Airfoil Sections

From Tests in the Variable-Density Wind Tunnel. NACA

Rep. 460, 1933.

4. Stack, John; and Von Doenhoff, Albert E.: Tests of 16 Related

Airfoils at High Speeds. NACA Rep. 492, 1934.

5. 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 Rep. 537,

1935.

6.

7.

8.

9.

10.

11.

12.

Stack, John: Tests of Airfoils Designed To Delay the Compressibility Burble. NACA Rep. 763, 1943.

Lindsey, W. E; Stevenson, D. B.; and Daley, Bernard N.: Aerodynamic Characteristics of 24 NACA 16-Series Airfoils at

Mach Numbers Between 0.3 and 0.8. NACA TN 1546, 1948.

Riegels, Friedrich Wilhelm (D. G. Randall, transl.): Aerofoil

Sections. Butterworth & Co. (Publ.), Ltd., 1961.

Abbott, Ira H.; and Von Doenhoff, Albert E.: Theory of Wing

Sections. Dover Publ., Inc., 1959.

Abbott, Ira H.; Von Doenhoff, Albert E.; and Stivers, Louis S.,

Jr.: Summary of Airfoil Data. NACA Rep. 824, 1945.

Theodorsen, Theodore: Theory of Wing Sections of Arbitrary

Shape. NACA Rep. 411, 1931.

Loftin, Laurence K., Jr.: Theoretical and Experimental Data

for a Number of NACA 6A-Series Airfoil Sections. NACA

Rep. 903, 1948.

13
Table I. Sample Computer File "64012.rpt" for Symmetric Airfoil

AIRFOLS ORDINATE INFORMATION REPORT

for 64012 airfoil

PROFILE: 6-SERIES CAMBER:

PROFILE PARAMETERS:

0.120000 = THICKNESS / CHORD (TOC)

0.010000 = BASIC X INTERVAL (DX)

6.000000 = MODEL CHORD (CHD)

RAT(I) = 1.000000 0.581272

OF ITERATIONS (IT)

T/C RATIO, INPUT TO COMPUTED

CUMULATIVE SCALING OF EPS,PSI

MAXIMUM Y/C (YMAX)

PEAK LOCATION (XYM)

.060000 = SLOPE SIGN CHANGE LOCATION

: NUMBER

999994 :

571965 =

060000 =

375975 =

374842 0

006717 =

011216 =

780074 =

010034 =

X/C

000000 0

000250 0

000500 0

000750 0

001000 0

001250 0

001500 0

001750 0

002000 0

002250 0

002500 0

002750 0

003000 0

003250 0

003500 0

003750 0

004000 0

004250 0

004500 0

004750 0

005000 0

005250 0

005500 0

005750 0

006000 0

006250 0

006500 0

006750 0

O07000 0

007250 0

007500 0

007750 0

008000 0

008250 0

X/C FIT OF ELLIPSE

Y/C FIT OF ELLIPSE

SLOPE FIT OF ELLIPSE

RADIUS OF ELLIPSE,

Y/C X

000000 0 000000

002217 0 001500

003153 0 003000

003861 0 004500

004446 0 006000

004972 0 007500

005434 0 009000

005867 0 010500

006262 0 012000

006633 0 013500

006986 0 015000

007316 0 016500

007631 0 018000

007935 0 019500

008224 0 021000

008501 0 022500

008770 0 024000

009030 0 025500

009279 0 027000

009521 0 028500

009758 0 030000

009987 0 031500

010209 0 033000

010425 0 034500

010637 0 036000

010843 0 037500

011045 0 039000

011241 0 040500

011433 0 042000

011621 0 043500

011806 0 045000

011987 0 046500

012164 0 048000

012337 0 049500

none -

0.984284 0.999700

(SF)

(CRAT)

(XTP,YM)

(XT(12))

(YT(12))

(YTP(12))

ORIGIN TO

Y

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

XT(12)/C,YT(12)/C

DY/DX

000000

013300

018918

023166

026674

029834

032604

035199

037571

039798

041916

043893

045788

047611

049344

051005

052620

054180

055675

057127

058545

059923

061256

062550

063819

065061

066271

067446

068595

069726

070836

071923

072983

074022

4.582653

3.151760

2.557617

2.218630

1.960144

1.791810

1.646746

1.535877

1.443982

1.361467

1.295341

1.236006

1.181388

1.134461

1.093128

1.053807

1o017726

0.986034

0.956632

0.928518

0 902220

0 878265

0 856161

0 834813

0 814474

0 795407

0 777865

0 761235

0 745170

0 729842

0 715419

0 702071

0 689594

(RNP)

D2Y/DX2

-10133.497070

-3253

-1787

-1187

-813

-637

-497

-408

991943

510742

118042

015991

092957

648315

647339

-346.249725

-291 917206

-255 805206

-225 954163

-199 190826

-178 795593

-162 901077

-147 842957

-134 695694

-124 466530

-115 500816

-106 995934

-99 390312

-93 122375

-87 877068

-82 808548

-78 015030

-73 602821

-69 673355

-65 977486

-62 376503

-58 941341

-55 742962

-52 852318

-50 213284

14
0 008500 0 008750 0 009000 0 009250 0 009500 0 009750 0 010000 0 012500 0 015000

0.017500 0 020000 0 022500 0 025000 0 027500 0 030000 0 032500 0 035000 0 037500 0 040000 0 042500 0 045000 0 047500 0 050000 0 052500 0 055000 0 O5750O 0 060000 0 062500 0 065000 0 067500 0 070000 0 072500 0 075000 0 077500 0 080000 0 082500 0 085000 0 087500 0 090000 0 092500 0 O95OOO 0 097500 0 i00000 0 ii0000 0 120000 0 130000 0 140000 0 150000 0 160000 0 170000 0 180000 0 190000 0 200000

00000000000000000000000000000000000000000000000000000012508

012676

012842

013004

013163

013320

013475

014909

016179

017331

018394

019392

020338

021242

022109

022944

023748

024527

025282

026014

026724

027414

028084

028736

029371

029989

030591

031178

031752

032313

032862

033400

033925

034440

034946

035443

035930

036408

036878

037340

037794

038241

038681

040368

041955

043446

044851

046174

047421

048594

049701

050742

051721

0.051000

0.052500 0 054000 0 055500 0 057000 0 058500 0 060000 0 075000 0 090000 0 105000 0 120000 0 135000 0 150000 0 165000

0.180000

0.195000

0.210000

0.225000

0.240000

0.255000

0.270000

0.285000

0.300000

0.315000 0 330000 0 345000 0 360000 0 375000 0 390000 0 405000 0 420000 0 435000 0 450000 0 465000 0 480000 0 495000 0 510000 0 525000 0 540000 0 555000 0 570000 0 585000 0 600000 0 660000 0 720000 0 780000 0 840000 0 900000 0 960000

1.020000

1.080000

1.140000

1.200000

Table I. Continued 0 075048 0 076058 0 077051 0 078026 0 078981 0 079921 0 080850 0 089454 0 097075 0 103985 0 110366 0 116349 0 122027 0 127449 0 132653 0 137662 0 142490 0 147161 0 151690 0 156082 0 160345 0 164481 0 168504 0 172417 0 176224 0 179933 0 183545 0 187069 0 190512 0 193879 0 197174

0.200398 0 203550 0 206640 0 209678 0 212660 0 215580 0 218446 0 221268 0 224037 0 226761 0 229444 0 232084 0 242209 0 251727 0 260673 0 269105 0 277045 0 284524 0 291562 0 298208 0 304450 0 310329

0 677483 0 665794 0 654589 0 643933 0 633884 0 624186 0 614691 0 536935 0 482244 0 441078 0 410865 0 387887 0 369581 0 353842 0 340221 0 327704 0 316412 0 306573 0 297353 0 288415 0 279875 0 271898 0 264468 0 257319 0 250416 0 243961 0 237823 0 232108 0 226970 0 222084 0 217262 0 212446

0.208084

0.204270

0.200514

0.196759

0.193007

0.189404

0.186288 0 183229 0 180203 0 177304 0 174424 0 163658 0 153665 0 144683 0 136288 0 128548 0 120930 0 113927 0 107368 0 100969 0 094978

-47.681225

-45.311497

-43.162434

-41.292366

-39.757317

-38.392265

-37.067284

-25.528528

-18.988861

-14.018546

-10.439953

-8.075612

-6 734070

-5 837592

-5 192567

-4 773360

-4.212404

-3 777347

-3 617945

-3 505087

-3 305694

-3 064729

-2 922085

-2 802585

-2 674942

-2 528763

-2 358885

-2 180O8O

-2 010549

-i 917779

-i 948437

-i 813631

-i 664815

-i 503816

-i 480895

-i 525902

-i 444046

-i 350303

-i 246098

-i 189765

-i 180359

-I 181516

-I 176315

-0 964989

-0 953014

-0 846358

-0 839168

-0 769506

-0 740389

-0 645912

-0 689126

-0 582494

-0 631849

15
Table I. Continued

0

0

0

0

210000

220000

230000

240000

0 250000

0 260000

0.270000

0.280000

0.290000

0.300000

0.310000

0.320000

0.330000

0.340000

0.350000

0.360000

0.370000

0.380000

0.390000

0.400000

0.410000

0.420000

0.430000

0.440000

0.450000

0.460000

0.470000

0.480000

0.490000

0.50OOOO

O.510000

0.520000

0.530000

0.540000

0.550000

0.560000

0 570000

0 580000

0 590000

0 600000

0 610000

0 620000

0 630000

0 640000

0 650000

0 660000

0 670000

0 680000

0 690000

0 700000

0 710000

0 720000

0 730000

0 052639

0 053499

0 054302

0 055048

0 055739

0 056377

0 056960

0 057493

0 057972

0 058400

0 058781

0 059112

0 059395

0 059628

0 059809

0 059931

0 059993

0 059991

0 059916

0 059772

0 059548

0 059260

0 058897

0 058475

0 057991

0 057450

0 056858

0 056218

0 055534

0 054806

0 054032

0 053222

0 052372

0 051484

0 050561

0 049607

0 048621

0 047601

0 046553

0 045476

0 044374

0 043249

0 042098

0 040923

0 039733

0 038520

0 037287

0 036040

0 034776

0 033499

0 032206

0 030905

0 029592

1 260000

1 320000

1 380000

1 440000

1 500000

1 560000

1 620000

1 680000

1 740000

1 800000

1 860000

1 920000

1 980000

2 040000

2 i00000

2 160000

2.220000

2.279999

2.339999

2.399999

2.459999

2 519999

2 579999

2 639999

2 699999

2 759999

2 819999

2 879999

2.939999

3.000000

3.060000

3.120000

3.180000

3.240000

3.300000

3.360000

3.420000

3.480000

3.539999

3.599999

3 659999

3 719999

3 779999

3 839999

3 899999

3 959999

4 019999

4 079999

4 139999

4 199999

4 259999

4 319999

4 379999

0 315837

0 320993

0 325812

0 330291

0 334436

0 338264

0 341760

0 344957

0 347833

0 350401

0 352689

0 354675

0 356368

0 357769

0 358853

0 359588

0 359957

0 359948

0 359499

0 358630

0 357290

0 355559

0 353381

0 350849

0 347946

0 344698

0 341145

0 337309

0 333204

0 328834

0 324192

0 319330

0 314233

0 308906

0 303368

0 297639

0 291726

0 285605

0 279317

0 272859

0 266245

0.259491

0.252586

0.245540

0.238400

0.231121

0.223724

0°216239

0.208654

0.200992

0.193236

0.185431

0.177554

0

0

0

0

0

0

0

0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

0 088733

0 083079

0 077596

0 071720

0 066492

0 060994

0 055913

0 050420

0 045442

040442

035488

030786

025781

020854

015181

009271

003000

003694

010928

018412

025435

032712

039309

045457

051196

056922

061447

066273

070572

075264

079229

082976

086970

090484

094048

096797

100354

103489

106118

109077

111324

113851

116378

118140

120230

122256

124178

125457

127102

128568

129713

130663

131876

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

.591514

.560491

.569729

.567787

.525667

.569880

.508531

.557001

.506901

.474289

.500163

.465019

.488108

.534137

.583586

.618653

.617353

.728268

.686004

.779822

.657702

.775743

.598958

.617973

.567208

.501878

.466768

.467965

.444957

.459214

.360220

.400605

.369812

.352421

.310705

.298187

.364253

269771

299605

263943

229142

262963

237618

161841

234471

197908

142382

160029

142993

157516

076584

132782

048956

16
Table I. Concluded

740000

750000

760000

770000

780000

790000

800000

810000

820000

830000

840000

850000

860000

870000

880000

890000

900000

910000

920000

930000

940000

950000

960000

970000

980000

990000

000000

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

028272

026946

025617

024285

022952

021622

020294

018974

017665

016366

015084

013819

012574

011354

010160

008996

007864

006772

005721

004720

003771

002884

002073

001351

000739

000264

000000

4

4

4

4

4

4

4

4

4

4

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

.439999

.499999

559999

619998

679998

739998

799998

859998

919998

979998

039998

099998

159998

219998

279998

339998

399998

459998

519998

579998

639997

699997

759997

819997

879997

939997

999997

0.169634

0.161675

0.153701

0.145708

0 137712

0 129730

0 121766

0 113846

0 105989

0 098198

0 090504

0 082913

0 075443

0 068122

0 060962

0 053979

0 047186

0 040631

0 034324

0 028319

0 022626

0 017303

0 012437

0 008106

0 004435

0 001581

0 000000

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

132190

133017

132997

133368

133111

133023

132388

131488

130493

129051

127493

125490

123375

120714

118055

114893

111270

107152

102876

097767

092050

085061

076628

066925

055016

039331

001854

-0.071199

-0.014410

-0.020850

0.003955

0.023193

0.039957

0.072903

0.116263

0.107131

0.177069

0.174549

0.201894

0.252569

0.275818

0.318184

0.301135

0.392689

0.412245

0.584972

0.498944

0 584660

0 725263

0 904850

1 038957

1 359224

1 985244

21 079264

17
_ Table II. Sample Computer File "64412.rpt" for Cambered Airfoil

6-SERIES

0.984284 0.999700

AIRFOLS ORDINATE INFORMATION REPORT

for 64412 airfoil

PROFILE: 6-SERIES CAMBER:

PROFILE PARAMETERS:

0.120000 : THICKNESS / CHORD (TOC)

0.010000 = BASIC X INTERVAL (DX)

6.000000 : MODEL CHORD (CHD)

RAT(I) = 1.000000 0.581272

4 = NUMBER OF ITERATIONS (IT)

0.999994 = T/C RATIO, INPUT TO COMPUTED (SF)

0.571965 = CUMULATIVE SCALING OF EPS,PSI (CRAT)

0.060000 = MAXIMUM Y/C (YMAX)

0.375975 = PEAK LOCATION (XYM)

0.374842 0.060000 = SLOPE SIGN CHANGE LOCATION (XTP,YM)

0.006717 = X/C FIT OF ELLIPSE (XT(12))

0.011216 : Y/C FIT OF ELLIPSE (YT(12))

0.780074 = SLOPE FIT OF ELLIPSE (YTP(12))

0.010034 = RADIUS OF ELLIPSE, ORIGIN TO XT(12)/C,YT(12)/C

CAMBER PARAMETERS:

1 : NUMBER OF CAMBER LINES TO SUM (CMBNMR )

CLI(I) = 0.400000

A(I) = 1.000000

UNCAMBERED

X/C XU/C YU/C XU

0.000000 0.000000 0.000000 0.000000

0.000250 -0.000119 0.002260 -0.000712

0.000500 -0.000024 0.003246 -0.000146

0.000750 0.000108 0.004003 0.000648

0.001000 0.000261 0.004635 0.001565

0.001250 0.000423 0.005209 0.002540

0.001500 0.000597 0.005717 0.003579

0.001750 0.000775 0.006194 0.004648

0.002000 0.000959 0.006634 0.005754

0.002250 0.001147 0.007049 0.006884

0.002500 0.001339 0.007445 0.008032

0.002750 0.001534 0.007817 0.009204

0.003000 0.001732 0.008175 0.010390

0.003250 0.001931 0.008521 0.011587

0.003500 0.002133 0.008851 0.012799

0.003750 0.002337 0.009169 0.014024

0.004000 0.002543 0.009478 0.015256

0.004250 0.002749 0.009778 0.016497

0.004500 0.002958 0.010067 0.017749

0.004750 0.003168 0.010349 0.019008

0.005000 0.003379 0.010624 0.020273

0.005250 0.003606 0.010895 0.021634

UPPER SURFACE VALUES

YU

0 000000

0 013558

0 019476

0 024017

0 027813

0 031253

0 034300

0 037165

0 039804

0 042293

0.044671

0.046904

0.049052

0.051125

0.053105

0.055011

0 056869

0 058669

0 060403

0 062091

0 063743

0 065369

(RNP)

DYU/DXU

-5 929416

19 701138

6 884831

4 681640

3 733716

3 117455

2 756694

2 468120

2 260172

2 095460

1 953106

1 842616

1 746078

1 659309

1 586308

1 523163

1 464067

1 410659

i 364378

1 321960

1 208261

1 172252

LOWER SURFACE

XL/C YL/C XL

0.000000 0.000000 0.000000

0.000619 -0.002112 0.003712

0.001024 -0.002972 0.006146

0.001392 -0.003612 0.008352

0.001739 -0.004132 0.010435

0.002077 -0.004597 0.012460

0.002403 -0.005000 0.014421

0.002725 -0.005376 0.016352

0.003041 -0.005716 0.018246

0.003353 -0.006032 0.020116

0.003661 -0.006333 0.021968

0.003966 -0.006610 0.023796

0.004268 -0.006875 0.025610

0.004569 -0.007129 0.027413

0.004867 -0.007368 0.029201

0.005163 -0.007597 0.030976

0.005457 -0.007818 0.032744

0.005751 -0.008031 0.034503

0.006042 -0.008233 0.036251

0.006332 -0.008429 0.037992

0.006621 -0.008620 0.039727

0.006894 -0.008807 0.041366

VALUES

YL

0 000000

-0 012671

-0 017833

-0 021670

-0 024792

-0 027584

-0.030002

-0.032254

-0.034293

-0.036195

-0.037995

-0.039661

-0.041251

-0.042773

-0.044211

-0.045580

-0.046908

-0.048184

-0.049400

-0.050575

-0.051719

-0.052842

DYL/DXL

-5 929416

-2 468801

-i 928516

-i 651175

-i 475005

-I 330605

-i 231427

-i 142463

-i 072166

-i 012317

-0.957306

-0 912326

-0 871268

-0 832875

-0 799420

-0 769589

-0 740886

-0 714266

-0 690659

-0 668566

-0 691887

-0 672141
Table II. Continued

0.005500

0.005750

0.006000

0.006250

0.006500

0.006750

0.007000

0.007250

0.007500

0.007750

0.008000

0.008250

0.008500

0.008750

0.009000

0 009250

0 009500

0 009750

0 010000

0 012500

0 015000

0 017500

0 020000

0 022500

0 025000

0 027500

0 030000

0 032500

0 035000

0 037500

0 040000

0 042500

0 045000

0 047500

0 050000

0 052500

0 055000

0 057500

0 060000

0 062500

0 065000

0 067500

0 070000

0 072500

0 075000

0 077500

0.003834

0.004063

0.004292

0.004523

0.004754

0.004986

0.005219

0.005452

0.005686

0.005920

0.006155

0.006391

0.006627

0.006863

0.007099

0.007336

0.007574

0.007812

0.008050

0.010446

0.012864

0 015296

0 017739

0 020189

0 022644

0 025104

0 027569

0 030036

0 032507

0 034980

0 037455

0 039933

0 042413

0 044895

0 047379

0 049865

0 052352

0 054841

0 057331

0 059822

0 062315

0 064809

0 067303

0 069799

0 072296

0 074793

0.011158

0.011414

0.011666

0.011913

0.012154

0.012390

0.012621

0.012848

0.013072

0.013291

0.013506

0.013718

0.013926

0.014132

0.014335

0.014535

0.014731

0.014924

0 015116

0 016906

0 018517

0 019996

0 021375

0 022679

0 023922

0 025115

0 026264

0 027373

0 028446

0 029487

0 030499

0 031483

0 032440

0 033372

0 034280

0 035166

0 036031

0 036875

0 037699

0 038505

0 039294

0 040067

0 040825

0 041568

0 042296

0 043011

0.023002

0.024376

0.025754

0.027137

0.028524

0.029917

0.031313

0.032713

0.034115

0.035522

0.036931

0.038344

0.039759

0.041177

0 042596

0 044018

0 045443

0 046870

0 048299

0 062677

0 077183

0 091776

0 106431

0 121132

0 135866

0 150627

0 165411

0 180216

0 195040

0 209879

0 224733

0 239600

0 254480

0 269373

0 284276

0 299189

0 314113

0 329045

0 343986

0 358934

0 373890

0 388852

0 403820

0 418794

0 433774

0 448759

0.066947

0.068485

0.069996

0.)71478

0.072927

0.074338

0.075724

0.077088

0.078430

0.079748

0.081038

0.082305

0.083557

0.084793

0 086010

0 087208

0 088385

0 089546

0 090695

0 101435

0 iiii00

0 119975

0 128253

0 136073

0 143533

0 150689

0 157582

0 164239

0 170678

0 176924

0 182994

0 188898

0 194642

0 200232

0 205683

0 210998

0 216183

0 221247

0 226193

0 231028

0 235763

0 240403

0 244951

0 249409

0 253778

0 258068

1.139723

1.109937

1.081424

1.054469

1.029364

1 006378

0 )84707

0 963901

0 944149

0 925636

0 908545

0 892608

0 877205

0 862395

0 848242

0 834814

0 822171

0 810003

0 798130

0 701787

0 634606

0 584201

0 546903

0 518194

0 495051

0 475074

0 457689

0 441755

0 427353

0 414682

0 402833

0 391434

0 380586

0 370446

0 360983

0 351909

0 343177

0 334988

0 327200

0 319913

0 313279

0 306958

0 300754

0 294604

0 288962

0 283920

0.007166

0.007437

0.007708

0.007977

0.008246

0.008514

0.008781

0.009048

0.009314

0.009580

0.009845

0.010109

0.010373

0.010637

0.010901

0.011164

0.011426

0.011688

0.011950

0.014554

0.017136

0.019704

0.022261

0.024811

0.027356

0.029896

0.032431

0.034964

0.037493

0.040020

0.042545

0.045067

0.047587

0.050105

0.052621

0.055135

0.057648

0.060159

0.062669

0.065178

0.067685

0.070191

0.072697

0.075201

0.077704

0.080207

-0.008987

-0.009161

-0.009331

-0.009497

-0.009658

-0.009814

-0.009965

-0.010114

-0.010260

-0.010402

-0.010540

-0.010675

-0.010807

-0.010938

-0.011066

-0.011191

-0.011313

-0.011432

-0.011551

-0.012628

-0.013558

-0.014384

-0.015134

-0.015828

-0.016480

-0.017097

-0.017686

-0.018249

-0.018788

-0.019307

-0.019807

-0.020291

-0.020757

-0.021207

-0.021643

-0.022064

-0.022472

-0.022867

-0.023250

-0.023621

-0.023983

-0.024335

-0.024678

-0.025012

-0.025338

-0.025656

0.042998

0.044624

0.046246

0.047863

0.049476

0.051083

0.052687

0.054287

0.055885

0.057478

0.059069

0.060656

0.062241

0.063823

0.065404

0.066982

0.068557

0.070130

0.071701

0.087323

0.102817

0.118224

0.133568

0.148868

0.164134

0.179373

0.194589

0.209784

0.224960

0.240121

0.255267

0.270400

0.285520

0.300627

0.315724

0.330811

0.345887

0.360955

0.376014

0.391066

0.406110

0.421148

0.436180

0.451206

0.466226

0.481241

-0.053921

-0.054965

-0.055987

-0.056982

-0.057949

-0.058882

-0.059792

-0.060685

-0.061559

-0.062412

-0.063240

-0.064049

-0.064845

-0.065627

-0.066394

-0.067144

-0.067876

-0.068595

-0.069304

-0.075768

-0.081351

-0.086306

-0.090805

-0.094967

-0.098878

-0.102583

-0.106114

-0.109492

-0.112727

-0.115840

-0.118844

-0.121743

-0.124542

-0.127242

-0.129855

-0.132384

-0.134830

-0.137201

-0.139498

-0.141727

-0.143895

-0.146008

-0.148068

-0.150074

-0.152026

-0.153935

-0.654041

-0.637241

-0.620886

-0.605199

-0.590415

-0.576767

-0.563769

-0.551145

-0.539048

-0.527631

-0.517052

-0.507151

-0.497503

-0.488160

-0.479181

-0.470628

-0.462557

-0.454751

-0.447082

-0.383874

-0.339260

-0.305722

-0.281486

-0.263426

-0.249329

-0.237316

-0.227034

-0.217565

-0.209067

-0.201791

-0.194959

-0.188266

-0.181836

-0.175846

-0.170291

-0.164922

-0.159714

-0.154872

-0.150274

-0.146030

-0.142297

-0.138761

-0.135243

-0.131686

-0.128535

-0.125888
ba Table II. Continued o

0.080000

0.082500

0.085000

0.087500

0.090000

0.092500

0.095000

0.097500

0.i00000

0.ii0000

0.120000

0.130000

0.140000

0.150000

0.160000

0.170000

0.180000

0.190000

0.200000

0.210000

0.220000

0.230000

0.240000

0.250000

0.260000

0.270000

0.280000

0.290000

0.300000

0.310000

0.320000

0.330000

0.340000

0.350000

0.360000

0.370000

0.380000

0.390000

0.400000

0.410000

0.420000

0.430000

0.440000

0.450000

0.460000

0.470000

0.077291

0.079790

0.082290

0.084790

0.087291

0.089793

0.092295

0.094798

0.097301

0.107319

0.117345

0.127376

0.137413

0.147454

0.157500

0.167550

0.177604

0.187661

0.197720

0.207782

0.217846

0.227913

0.237982

0.248052

0.258124

0.268198

0.278272

0.288348

0.298425

0.308503

0.318582

0.328661

0.338741

0.348822

0.358903

0.368984

0.379065

0.389147

0.399229

0.409310

0.419391

0.429471

0.439551

0.449629

0.459707

0.469782

0.043715

0.044406

0.045085

0.045751

0.046408

0.047054

0.047690

0.048317

0.048934

0.051309

0.053550

0.055665

0.057666

0.059559

0.061350

0.063043

0.064648

0.066165

0.067599

0.068952

0.070227

0.071428

0.072553

0.073605

0.074587

0.075497

0.076341

0.077116

0.077823

0.078469

0.079049

0.079566

0.080020

0.080406

0.080720

0.080959

0.081122

0.081197

0.081189

0.081089

0.080911

0.080645

0.080307

0.079894

0.079411

0.078863

0.463748

0.478742

0.493740

0.508743

0.523749

0.538758

0.553772

0.568788

0.583808

0.643916

0.704067

0.764255

0.824476

0.884726

0.945003

1.005303

1.065623

1.125963

1.186319

1.246692

1.307079

1.367478

1.427890

1.488312

1.548744

1.609185

1.669634

1.730090

1.790553

1.851020

1.911492

1.971968

2.032448

2.092930

2.153415

2.213902

2.274391

2.334881

2.395371

2.455860

2.516346

2.576829

2.637306

2.697777

2.758240

2.818694

0 262288

0 266437

0 270507

0 274509

0 278450

0 282326

0 286141

0.289901

0.293604

0.307854

0.321300

0.333992

0.345999

0.357356

0.368100

0.378260

0.387891

0.396988

0.405597

0.413715

0.421365

0.428566

0.435317

0.441630

0.447522

0.452983

0.458047

0.462694

0.466940

0.470814

0.474296

0.477397

0.480118

0.482437

0.484322

0.485756

0.486731

0.487184

0.487136

0.486537

0.485466

0.483870

0.481841

0.479363

0.476463

0.473180

0

0

0

0

0

0

0

0

0

0

0

0

0

0

O.

O.

O.

O.

O.

O.

O.

O.

O.

O.

O.

O.

O.

O.

O.

O.

O.

O.

O.

O.

O.

O.

O.

O.

--0.

-0

-0

-0

-0

-0

-0

-0

278975

274066

269193

264503

260338

256258

252237

248368

244544

230216

216959

204963

193763

183395

173302

163956

155171

146648

138624

130426

122893

115594

107964

101032

093881

087189

080128

073616

067116

060692

054547

048126

041807

034767

027513

019917

011919

003398

005358

013647

022178

030027

037426

044414

051386

057165

0.082709

0.085210

0.087710

0.090210

0.092709

0.095207

0.097705

0.100202

0.102699

0.112681

0.122655

0.132624

0.142587

0.152546

0.162500

0.172449

0.182396

0.192339

0.202280

0.212218

0.222154

0.232087

0 242018

0 251948

0 261876

0 271802

0 281728

0 291652

0 301574

0 311497

0 321418

0 331338

0 341259

0 351178

0 361097

0.371016

0.380935

0.390853

0.400771

0.410690

0.420609

0.430528

0.440449

0.450370

0.460293

0.470217

-0.025968

-0.026273

-0.026571

-0.026862

-0.027148

-0.027428

-0.027703

-0.027973

-0.028239

-0.029249

-0.030191

-0.031067

-0.031886

-0.032649

-0.033360

-0.034021

-0.034639

-0.035211

-0.035743

-0.036233

-0.036683

-0.037096

-0.037470

-0.037806

-0.038105

-0.038366

-0.038593

-0.038782

-0.038934

-0.039056

-0.039142

-0.039193

-0.039210

-0.039188

-0.039122

-0.039009

-0.038846

-0.038623

-0.038344

-0.037999

-0.037602

-0.037144

-0.036639

-0.036086

-0.035487

-0.034851

0.496252

0.511258

0.526260

0.541257

0.556251

0.571241

0.586228

0.601212

0.616192

0.676083

0.735933

0.795744

0.855523

0.915273

0.974997

1.034697

1.094377

1.154037

1.213681

1.273308

1.332922

1.392522

1.452111

1.511688

1.571256

1.630815

1.690366

1.749909

1.809447

1.868979

1.928507

1.988031

2.047552

2.107069

2.166584

2.226097

2.285608

2.345118

2.404628

2.464138

2.523652

2.583169

2.642692

2.702221

2.761758

2.821304

-0.155806

-0.157638

-0.159424

-0.161172

-0.162890

-0.164570

-0.166219

-0.167840

-0.169432

-0.175495

-0.181145

-0.186403

-0.191314

-0.195893

-0.200158

-0.204124

-0.207832

-0.211264

-0.214457

-0.217398

-0.220101

-0.222577

-0.224820

-0.226834

-0.228630

-0.230195

-0.231555

-0.232689

-0.233607

-0.234335

-0.234849

-0.235158

-0.235261

-0.235130

-0.234734

-0.234054

-0.233078

-0.231741

-0.230065

-0.227996

-0.225614

-0.222863

-0.219835

-0.216514

-0.212924

-0.209105

-0 123263

-0 120608

-0 117923

-0 115358

-0 113247

-0 111167

-0.109096

-0.107128

-0.105158

-0.097827

-0.090992

-0.084936

-0.079273

-0.074097

-0.068901

-0.064193

-0.059821

-0.055511

-0.051525

-0.047204

-0.043408

-0.039720

-0.035581

-0.032040

-0.028181

-0.024699

-0.020763

-0.017310

-0.013802

-0.010311

-0.007047

-0.003453

0.000086

0.004397

0.008966

0.013916

0.019307

0.025257

0.031471

0.037231

0 043255

0 048599

0 053497

0 057986

0 062465

0 065736
Table II. Continued

0.480000

0.490000

0.5OO0OO

0.510000

0.520000

0.530000

0.540000

0.550000

0.560000

O.570OO0

0.580000

0.590000

0.600000

0.610000

0.620000

0.630000

0.640000

0.650000

0.660000

0.670000

0.680000

0.690000

0.700000

0.710000

0.720000

0.730000

0.740000

0.750000

0.760000

0.770000

O.78OO00

0.790000

0.800000

0.810000

0.820000

0.830000

0.840000

0.850000

0.860000

0.870000

0.880000

0.890000

0.900000

0.910000

0.920000

0.930000

0.479857

0.489929

0.500000

0.510069

0.520136

0.530200

0.540263

0.550323

0.560381

0.570436

0.580489

0.590539

0.600587

0.610632

0.620674

0.630713

0.640749

0.650783

0.660813

0.670840

0.680864

0.690885

0.700903

0.710917

0.720928

0.730936

0.740941

0.750941

0.760939

0.770933

0.780924

0.790911

0.800894

0.810875

0.820851

0.830825

0.840795

0.850761

0.860725

0.870685

0.880643

0.890597

0.900548

0.910497

0.920443

0.930387

0.078256

0.077591

0.076869

0.076089

0.075260

0.074378

0.073445

0.072464

0.071439

0.070370

0.069253

0.068095

0.066895

0.065657

0.064381

0.063067

0.061715

0.060335

0.058916

0.057464

0.055983

0.054471

0.052931

0.051360

0.049765

0.048143

0.046498

0.044829

0.043141

0.041432

0.039705

0.037962

0.036203

0.034431

0.032649

0.030857

0.029058

0.027253

0.025443

0.023632

0.021820

0.020007

0.018193

0.016384

0.014577

0.012777

2.879139

2.939574

3.000000

3.060413

3.120813

3.181201

3.241576

3.301937

3.362285

3.422617

3.482934

3.543235

3.603521

3.663790

3.724042

3.784277

3.844495

3.904696

3.964878

4.025041

4.085186

4.145311

4.205418

4.265504

4.325571

4.385617

4.445643

4.505649

4.565634

4.625598

4.685542

4.745465

4.805367

4.865247

4.925108

4.984948

5.044768

5.104569

5.164350

5.224113

5.283856

5.343582

5.403290

5.462982

5.522658

5.582321

0.469537

0.465547

0.461215

0.456535

0.451558

0.446268

0.440672

0.434786

0.428634

0.422218

0.415516

0.408569

0.401372

0.393940

0.386287

0.378401

0.370292

0.362007

0.353498

0.344786

0.335900

0.326826

0.317585

0.308160

0.298593

0.288860

0.278986

0.268975

0.258846

0.248595

0.238233

0.227773

0.217218

0.206586

0.195896

0.185141

0.174349

0 163518

0 152660

0 141793

0 130917

0 120039

0 109157

0 098302

0 087462

0 076665

-0.063242

-0.068796

0.000000

-0.079969

-0.084982

-0.090246

-0.095037

-0.099886

-0.103931

-0.108789

-0.113238

-0.117193

-0.121490

-0.125091

-0.128987

-0.132898

-0.136064

-0.139576

-0.143045

-0 146434

-0 149204

-0 152366

-0 155379

-0 158101

-0 160661

-0 163523

-0 165522

-0 168079

-0 169832

-0 172031

-0 173654

-0 175511

-0 176884

-0 178067

-0 179239

-0 180054

-0 180858

-0 181335

-0 181835

-0 181943

-0 182239

-0 182244

-0 182042

-0 181656

-0 181501

-0 181001

0 480143

0 490071

0 500000

0 509931

0 519864

0 529800

0 539737

0 549677

0 559619

0 569564

0 579511

0 589461

0 599413

0 609368

0 619326

0 629287

0 639251

0 649217

0 659187

0.669160

0.679135

0.689114

0.699097

0.709082

0.719071

0.729063

0.739059

0.749058

0.759061

0.769066

0.779076

0.789089

0.799105

0.809125

0.819148

0.829175

0.839205

0.849238

0.859274

0.869314

0.879357

0.889402

0.899451

0.909502

0.919556

0.929612

-0 034180

-0 033477

-0 032742

-0 031975

-0 031183

-0 030366

-0 029522

-0 028656

-0 027771

-0 026869

-0 025944

-0 025005

-0 024050

-0 023083

-0 022105

-0 021116

-0 020117

-0 019117

-0 018107

-0 017091

-0 016076

-0.015058

-0.014042

-0.013026

-0.012017

-0.011012

-0.010016

-0.009030

-0.008058

-0.007101

-0.006161

-0.005243

-0.004346

-0.003477

-0.002639

-0.001834

-0.001068

-0.000342

0.000338

0.000966

0.001540

0.002053

0.002503

0.002877

0.003170

0.003370

880858

940423

000000

059587

119186

178798

238423

298062

357715

417382

477065

536764

596478

656209

715956

775721

835503

3 895302

3.955120

4.014957

4.074812

4.134686

4.194580

4.254494

4.314426

4.374380

4.434354

4.494349

4.554363

4.614398

4.674455

4.734532

4.794630

4.854749

4.914888

4.975048

5.035228

5.095427

5.155646

5.215883

5.276139

5.336413

5.396706

5.457014

5.517337

5.577674

-0 205080

-0 200861

-0 196453

-0 191848

-0 187101

-0 182193

-0 177132

-0 171937

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-0.161211

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-0.002055

0.002025

0.005797

0.009238

0.012320

0.015015

0.017260

0.019021

0.020218

0.069311

0.072354

0.000000

0.078496

0.080977

0.083703

0.085940

0.088221

0.089674

0.091933

0.093757

0.095062

0.096686

0.097583

0.098747

0.099895

0.100259

0.100933

0.101523

0.101987

0.101784

0.101921

0.101851

0.101431

0.100783

0.100362

0.099005

0.098118

0.096340

0.094902

0.092784

0.090773

0.088150

0.085190

0.082053

0.078379

0.074485

0.070031

0.065329

0.059930

0.054349

0.048055

0.041047

0.033235

0.024877

0.015202
__000000"0

60L90T O- IET600"O

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18E8LT O- I6LEEO 0

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9L96LI O- CgTSSO 0

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69_I9L S

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Z£9S00 0

809L00 0

061600 0

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0 O00000"I

0 8E0066"0

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80Z096"0

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O00000"I

000066"0

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O000L6"O

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000096"0

p_pnlouoo "II _lqe,L
Form Approved

REPORT DOCUMENTATION PAGE OMB NO. 0704-0188

Publicreportingburdenforthis collectionof informationis estimatedto average 1 hourper response,includingthe timefor reviewinginstructions,searchingexistingdata sources,

gatheringand maintainingthe data needed, and completingand reviewingthe collection of information.Send commentsregardingthis burdenestimateor anyother aspect ofthis

collectionof information,includingsuggestionsfor reducingthis burden,to WashingtonHeadquartersServices,Directoratefor InformationOperationsand Reports,1215 Jefferson

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1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED

December 1996 Technical Memorandum

4. TITLE AND SUBTITLE 5. FUNDING NUMBERS

Computer Program To Obtain Ordinates for NACA Airfoils

6. AUTHOR(S)

Charles L. Ladson, Cuyler W. Brooks, Jr., Acquilla S. Hill, and Darrell W.

Sproles

7. PERFORMINGORGANIZATIONNAME(S)ANDADDRESS(ES)

NASA Langley Research Center

Hampton, VA 23681-0001

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Washington, DC 20546-0001

WU 505-59-10-31

8. PERFORMING ORGANIZATION

REPORT NUMBER

L-17509

10. SPONSORING/MONITORING

AGENCY REPORT NUMBER

NASA TM-4741

11. SUPPLEMENTARY NOTES

Ladson, Brooks, and Hill: Langley Research Center, Hampton, VA; Sproles: Computer Sciences Corp., Hampton,

VA.

Computer program is available electronically from the Langley Software Server

12a. DISTRIBUTION/AVAILABILITY STATEMENT

Unclassified-Unlimited

Subject Category 02

Availability: NASA CASI (301) 621-0390

i12b. DISTRIBUTION CODE

13. ABSTRACT (Maximum 200 words)

Computer programs to produce the ordinates for airfoils of any thickness, thickness distribution, or camber in the

NACA airfoil series were developed in the early 1970's and are published as NASA TM X-3069 and TM X-3284.

For analytic airfoils, the ordinates are exact. For the 6-series and all but the leading edge of the 6A-series airfoils,

agreement between the ordinates obtained from the program and previously published ordinates is generally within

5 x 10 -5 chord. Since the publication of these programs, the use of personal computers and individual workstations

has proliferated. This report describes a computer program that combines the capabilities of the previously pubfished versions. This program is written in ANSI FORTRAN 77 and can be compiled to run on DOS, UNIX, and

VMS based personal computers and workstations as well as mainframes. An effort was made to make all inputs to

the program as simple as possible to use and to lead the user through the process by means of a menu.

14. SUBJECT TERMS

NACA airfoils; 4-digit; 6-series; Platform-independent; FORTRAN

17. SECURITY CLASSIFICATION

OF REPORT

Unclassified

18. SECURITY CLASSIFICATION

OF THIS PAGE

Unclassified

NSN 7540-01-280-5500

19. SECURITY CLASSIFICATION

OF ABSTRACT

Unclassified

15. NUMBER OF PAGES

23

16. PRICE CODE

A03

20. LIMITATION

OF ABSTRACT

;tandard Form 298 (Rev. 2-89)

Prescribedby ANSI Std. Z39-18

298-102