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NATIONAL

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TECH171CAL NOTES 1“ “’,!;

~+.. . >.ADVISORY COMMITTEE FOR AERONAUTICS ‘igq ‘*””

T

.—.

No ● 708--------

A SIMPLIFIED METtiOD FOR THE CALCULATION OF

AIRFOIL PRESSURE DISTRIBUTION

By H. Julian AllenLangley Memorial Aeronautical Laboratory

-—-- -—-

wa~hi~gtonMay 1939

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https://ntrs.nasa.gov/search.jsp?R=19930081453 2020-04-11T04:54:15+00:00Z

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NATIONAL ADVISORY COMti.ITTEZl.

---------

TECHNICAL NOTE NO,

---— ----

A SIMPLIFIED METHOD FOR THE

----

FOR AERONAUTICS*

’708

CALCULATION OF

AIRFOIL PRESSURE DISTRIBUTION

By H. Julian Allen

SUMMARY

o

A method is presented for the rapid calculation ofthe pressure’distribution over an airfoil section whenthe normal-force distribution and the pressure distribu-tion over the ‘lbase profile” (i.e., the profile of thesame airfoil were the camber line straight and the re-sulting airfoil at zero angle of attack) are known. Thisnote is intended as a supplement to N,A.C.A. Reports NOS.631 and 634 wherein methods ~re presented for the calcu-lation ’of the normal-force distribution over plain andflapped airfoils, respectively, but not of the pressureson the individual surfaces.

Base-profile pressure-coefficient distributions forthe usual N.A.C.A. family of airfoils, which are alsosuitable for several other commonly employed airfoils, areincluded in ta%ular form. With these tabulated base-profile pressures and the Computed normal-force distribu-.tions, pressure d,istri%utions adequate for most engineer-ing purposes can %e o%tained.

INTRODUCTION

A method is given in reference 1 for computing thechprdwise pressure distribution over both the upper andthe lower surfaces of an airfoil. In this method, a per-=feet nonviscous fluid was assumed and, consequently, theagreement %etween the integrated forces and moments andthose obtained from experimental observations was in ,many

F cases inadequate. In the work reported in reference 2,*, the effect of viscosity was accounted for by an adjustment\ of the mean camber line of the airfoil and the agreement

d“ was improved.for practicalereii adequato

Both methods, however, are tool aborioususe even though the results might he consid-for design purposes.

.

‘i

.

2 N.A.C.A. Technic%l Note No. 70!3

A subsequent analysis of theory and experimental data(references % and 4) resulted in a series of charts where-by the normal-force distri,lmtions, %ut not the pressures Ion the individual surfaces, could be obtained for ordinaryairfoils and airfoils with flaps. In this method, themagnitudes of the various component distributions that areadded to o’bt~in the normal-force distribution are deter-mined from experimental force and moment coefficients.When the calculated normal-force distribution is inte-grated, it then yields normal-force and moment coeffi-cients that must agree in magnitude with the correspondingexperimental coefficients.

The present paper, which is.a supplement to refer-ences 3 and 4, gives a method of utilizing the calculatednorm~l-force distribution to obtain the pressures on theindividual surfaces of an airfoil. Again, the resultingpressure distributions when integrated must yield normal-force and moment coefficients that agree with those ob-tained by experiment.

This method requires, in addition to a knowledge ofthe chordwise normal-force distribution over the airfoilsection (as may he obtained from references 3 and 4), thepressure distribution over the ‘*base profile’! of the air-foil section (i.e., the profile of the same airfoil werethe camber line straight and the resulting airfoil at zeroangle of attack). The method is applicable, to date, tonormal airfoils and to airfoils with plain trailing-edgeflaps . :The base-profile pressure distributions for theN.A.C.A. fami~y of airfoils (as well as for the Clark YAnd the G~ttingen 398) calculated by the method of refer-ence 1 are given in this report.

THEORY OF THE METHOD

Let it be assumed that the chordwise distribution ofthe filaments of the bound vortices within an airfoil sec-tion are located along the mean camber line and that thecurvature of the mean camber line Ys slight so that theinduced velocities on the upper and the lower surfaces atany given distance, x, behind the leading edge of theairfoil are equal in magnitude but opposite in sign.

The velocities on the upper and the lower surfaces ofthe airfoil at any point, x, behind the leading edge are,respectively,

/

*.

i.

..

N ●A ●“C.A .,N9,$e,,, ”

No.?, 3,...

,, “..,,

where Au is the inbe the velocity thatthe point x %ehindconditions.

,,.

If pf be desi

on, the base profile,

Uu =:“!.,:” ;

UL =.,

,,

,“duce dthethe

...,

gna’te

then

,u~ + Aq,,, :. .,. .:1.,.

~t - Au

~..’, ‘

velocitybass profleading e

d as the

9 by Bern

‘1

\

k,.

Ji?:dge

t

sur

Oul

.,(“” (1

is assumed texperience athe same str

)

otearn

.’

:(I Urwouldunder

fhce p

lils e

\

%.

.

ressure at

quation,

x

Pf:,

pua---2

: P(u’)a.-.— -2“r,,

:’

/.

:

d+.;.,.:-

L. .

..

L

fi’trea

.-

or

Hence()Ut ~

pf=q’-q’” ~-’.,

—----u? = uJi-- Pf

stream, velocity, and

/q, where q is th

quations (1) and (2).

.:..,!

--

$2):.

essure

c pres

u’-=-u

where ‘U ‘is

coefficient,

r!s>

the

Pf

ng e

‘1?

esJ..

‘%

sur”e.

co,mbini gi ves

Uu wx-:-~

—-—.‘u/-i - Pf

-t- ‘+(z)TJ

- Au

!l?hepressureq on th’e upper and, the lowerairfoil at x are then, respectively,

sur fa c of

puu 2----2,

P“z

AU}2&-Pf-----

(u +qJ

( )

P-,-

2

--------(ufi.Pf -

r,

q.,

pu~ 2-—--z,

,“

AU)2‘L.

t

4. “

Nowy ifat x,

,,’

in,

sathe normal force,nr+~ward forti8‘b

,,p=- pu +

,,‘ex’periein~ “co“e’,

pL

enc~ii bynsidered

.,-

the airfoilpositive,

4 N. A. C. A.. Techni.cfil,Note. No. ’708

Then substituting equations (4), canceling, and rearrang-ing gives

Hence

. . .Substituting equation (5) in (4) and rearranging gives

(6)

Thus , if the normal-force distribution over an airfoil Pand the pressure distribution over’ the base profile Pf

are known, then pressure coefficients for the upper andthe lower surfaces may be calculated.

ACCURACY AND LIMITATIONS 03? THE METHOD

At the outset, i.t is necessary to justify the assump-tions made in the development’ of the present method, Tothis end, the theoretical pressure distributions over theN.A.C.A. 4412.airfotil section at two angles of attack werecalculated by the method of reference 1 and are plotted infigures 1 and 2. The theoretically exact normal-forcedistributions were determined from the pressure distribu-tions for each case. The press”ure distribution over thebase profile (N.A,C.A. 0012 at zero angle of attack) wasalso calculated by the method of reference 1. By meansof the normal-force and the base-profile pressure distri-butions, the pre”ssure distributions were then calculatedby equations (6) and are plotted for comparison in figures1 and 2. Similar calculations were made for the N.A.C.A.6512 and 27018 at Ct = O and Ct = 1.0; the resulting

pressure dietrihutions are shown in figures 3 to 6.’ The

*

m.

N.A. C,A, Technicml Note I’?oti-.708, 5.

N.A.C.A. 6512 airfoi,l is representative of the more highlycam%ered airfoils;. tihe N~&.C.A. 2301’8, of fairly thick air-foils; and the N.A.C.A. 4412, of the more commonly em-ployed airfoils. The agreement Q.etween the. theoreticaldistributions and those c~lculate”d by equations (6) forthe se~e~al airfoils ’investigated is, good, as shown by thedistributions in figures 1 to 6. ,

● ,“

Theri’this method is applied to the calculation ofairfq~l p“ressure distributions, the accuracy of the calcu-lation tiill depend principally on:

,(1) The accuracy of the calc~lation of the chord-,wise normal-fo”rce distribution over the air-

.,‘,foil.

,,

(’2) The accuracy of the base: profile pressure dis-tribution.

.,

(3) !I%e thickness of the boundary layer over theairfoil.

The close agreement het~ee~ the theoretical pressuredistributions of figures 1 to 6 shows that, “for airfoilsof normal profile, thickness, and ca”mber, equations (6) ‘are sufficiently accur,ate. For an airfoil with a plain Iflap deflected through a larg~, angle, the basic assumptionthat the ourvature of the mean Camber line is small is dis-regarded so that the accurac,y o~f equations (6) is accord-ingly less. The same statement is, of course, tru~ forhighly cambered airfoils. Again, the assumption that thebound vorticity may be consider~d as distributed along themean camber line, although it has been shown to be per-missible for airfoils of normal thickness, will probablybe less accurate for the thicker profiles. ,

,,‘ For’ plain air~oils, the accuracy of the method for’

the calgulati.on of the normal.force distribution given inreference 3 is, as a rule, well ~it~in 5 percent; for. air-foi~s tii:h flaps, the accuracy of the method given inreference 4 is, as a ru].e$ within 1“0 percent of the truenormal~force distribution.

The base-profile pressure d.istrilxition calculated by‘the method of reference 1 will probably be sufficientlyaccurate for design purposes for all base profiles in com-mnn use. The base-prof~lq pressure distributions for.theN.A.C.A. family of airfoils, as well as for the Clark Y “

,,, ,,

,.

.

6 N. A. C.A. TechnicalNote No. 708

and the !l~ttingen 398, were calculated by the method ofreference 1 and are given in table I.

That the thickness of the %oundary layer is a factorconcerning the accuracy of this method is apparent whenit is se~n that not only is the base profile effectivelychanged. when the boundary layer is thickened hut also thatthe effective shape of the mean ce,mb~r line and thereforethe effective position of the bound vortices are alteredwhen the boundary layer is thicker on one surface of theairfoil than the other (as for example when the angle ofmaximum lift is approached). This inaccuracy, which af-fects the distribution of pressure between the upper andthe lower surfaces of the.airfoil, exists in spite of thefact that the effe~ of the %oundary layer on the normal-force distribution is partly taken care of by the methodsof references 3 and 4.

A consideration of the several factors would lead oneto expect that the method would be very accurate for thinairfoils with small amounts of camber and would be lessaccurate for thick highly cambered airfoils and particu-larly for airfoils with ordinary trailing-edge flaps. Thatthese expectations are fulfilled is shown in figures 7 to14, where the computed. and the experimental pressure dis-tributions for several airfoils (one with a flap) aregiven. In every case, the normal-force distributions weredetermined by the methods of references 3 and 4 and thebase-profile pressure d.istribution~ given in table I wereused. The experimental pressure distributions over theN.A.C.A. 4412 airfoil (figs. 7 to 10) were obtained fromreference 2 and those over the N.A.C,A. 23012 (figs. 11 to14) were obtained from reference 5.

One limitation should be noted although it is of neg-ligible importance. Equations (6) cannot be used to pre-dict the pressure at the most forward point of the airfoilbecause at this point the slope of the airfoil profile isinfinite and, even though the normal force approaches zero,the solution must, of course, be indeterminate at thispoiht. The nose point should therefore be omitted frompractical calculations.

CONCLUSIONS

When the base-profile pressure-coefficient distribu-tions given in table I and the normal-force distributionscomputed by the methods of references 3 and 4 are used,

. .

,.

●

✎

b

N.A.C.A. Technical Note No. 708 7

the pressure distributions calculated by the method of thisnote are adequate for most engineering purposes. The com-puted pressure distributions not onlY agree in form withexperiment but the integrated normal forces and momentsagree in magnitude with experiment. “

Langley Memorial Aeronautical Laboratory.National Advisory Committee for Ae~onautics,

Langley Field, Va. ,.April 3, 19X9,

.

REFERENCES

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,

*.

Theodorsen, Theodore: Theory of Wing Sections of Arbi-trary Shape. T.R, No. 411, N.A,C.A., 1931.

Pinkerton, Robert M. : Calculated and Measured Pres-sure Distributions over the Midspan Section ofthe N.A.C,A.,4412 Airfoil. ToR. No. 563, N.A.C.A. ,1936.

Jacobs, Eastman N., and Rhode, R. V.: Airfoil Sec-tion Characteristics as Applied to the Predictionof.Air Forces and Their Distribution on Wings,T.R. ~0, 631, N.A,(l.A. , 1938..

4. Allen, H. Julian: Calculation of the Chordwise LoadDi-stri%ution over Airfoil Sections with Plain, Split,or Serially Hinged Trailing-Edge Flaps. T.R. ~T,o.634, N.A,C.A. , 193s.

5. Wenzinger, Carl Je, and Delano, James B.: PressureDistribution over an N.A.C.A. 23012 Airfoil witha Slotted and a Plain Flap. T.R. No. 633, N.A,C.A. ,1938.

4-

“,

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.

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4’.

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*.

a

1

*“

● -

N, A. C.A. Technical Note No. 708

.....-, ~A~LE I

>

VALUES OF (1 - p~) FOR N,A,C.A. YAMILY OY AIRFOIL SECTIONS---— -—--

Station(pe;~ent

--------,

1,252.557.5

101520253035494’550556065 ‘707580859095

---------

— ----

6.-—--.—

1,1181.1901.2191.2221 .2%1.2091.1921,1801.1651.1501.1371.1221.1091.0941.0811.0701,0561.0411.0211.002.981.949

.--—-,

----—.

9

1.0751.2321.3131.3J01.3301,3761.2981.2771.2561,2331.2101.1891.1671.1461.1251.1041.0821.0591.0321.002.970.922

-—---,

Thi<————- .,

a12---—.

1.0081.2571.3951.4251.432_...1.42.81.4071.3801,3521.3211.2901.2561.2251.1951,1671.1381.1071.0741.0411.001.955.895

—— ---

aClark Y; G~ttingen 398.

kness-——---

15------

0.9321.2501.4601.5131.5341.~3_&1.5181.4901.4531.4131,3721.3281.2881,2481.2081.1711.1321.0911.0481.000.941.865

-—-—-.

(nercent c)

o.s~y1.2161.5051.6011.6351.647....-..1.6341.6041,5601.5101.4571.4011.3511.3001.2501.2021.1571.1051.053.998.929.83’4

--——-

.—-----———-.

21 25

0.775 0.6991.170 1.1051.529 1.54X1.675 1.7601.737 11.8611.76.0 1.914_.. -,_.-=-1.749 1.9051.715 1.870”1.667 1.8131.603 1 ● 7401.540 1,6561.473 1.5701.410 1.4921.350 1.4171.293 1.3431.239 1.2791.182 1.2111.122 1.1411.059 1.069.995 ,990.911 .890,807 .‘?64

-——-—4 --——-

30------

0.6201.0611.5501.8282.0002.106.--.....-2.1062.0702.0011.9051.8051.7021.6051,5091.4111.3261.2401.1601.074.982.860,.712

-—--

35-----

0,573.982

1.5531.8992.1152,3002,3092.2712.1932.0751,958,1.8351.7111.5971.484‘1.3781.2711.180“1,085..972,829

~ ,663

.11.A.O.A. Toohniocl Xote MO. 708 ?ig. 1

4.

-12

-11

-lo

-9

-8

-7

-6

;-5#

-4

-3

-2

-1

0

1

ao .1 .a .8 .4 ..5 .8 .7 .s .9 1.0

Xfo

~lgure1.- TheO~et~e~lpremure diotrlb=tlon. %.A.O.A. Ul~ tirfoil; a , 13° 5?1.

—- ,

.

,0

.

., I.A.0.A. TOObBiOQ xoto uo. 708 Pig. 8

Q~_y Oal”@date ip eetkd.o r

~% .0: w-s

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As

1 0# ,/

/)‘

o/’,

/ -+ .

/

J

1

0 .1 .a .s .4 .s .6 .? .a .9 1.0do

Mgure 3.- Theomtiecl pzeseure &e$riWtiem. B.A.O.&. SS18 airfoil; a , -f’4’; el ,0.

., Y.A.O.A. TOChlliCd )lOte~00 708

“,

<’

-1.0

-.5

0

;hi

.5

1.0

0 .1 .2 .8 .4 .5 .6 .7 .8 .0 1.0Xlo

MWS 4.- Th.OmOt;Od pSOWUO distznu*ios. I. A.O.A. asza ASfoil; a , 1° 101; o% ‘ 1“00

)

. S.A.O.A. TOOhlliO~lMote ~0. 708 rig. 5.

-1.0

-/ 5

.6

1.0

I 1 1 1

I r

0 .1 .2 .3 .4 .8 .? .8 .9 1.0

:0

Yigur. So- Ttioz@tioal pre~mre dhtributioa. 11.A.C.A. 23018 airfoil; d , -1° St ; 01 , 0.

.

~od Ouptbdp mothnl of refa !ono@ 1x---

-3equationa:6)

/ !,/

-a Cl

,

-1

:*P

0 — -—. ,--

1

L0 .1 .a .s .4 .5 .6 .? .8 .9 1.

do

,

.

0

r:gu?a6.- ?harotioslPSO#,uro 4i,tzibution. Y .A. O .A. 830M airfoil;U , 60Sa$; c~, 1.00

.

> \. .

A *’

-4

-s

0

“o .2 .4 .6 .8 1.0Xla

Figure 7.- Prmrwm distribution. E.A.O.A. 4423 sirfniU,a,-tto.

-1

10 .a .4 .6 .8 1.0

x/oPigure 8.- p~ ~~tfibu~ion. X.A.O.A. 441a airfoil;

.

X/OTiglm $.- Prca4y &imtribu*iom.I.A.O.A. UIS aixfoil;

u,

-.0 .9 .4 . .6 .8 1.6

XIcFigure 3,0.-Prom& d@tribmtion. X.A.O.A. 4412 8irfoil;

d,

. .

M ,

. ●

-1

<J-.---~u

d?

*Do-–--Exp● lmtafromrefos●1095— OalImlated by ●Wa tions (6)

10 .a .4 . .6 .6 1.0

Mguro 11.- Pr~mre distrll%on. ~.A.O.A.a~12 =$rfoilwithXbporocnt-ohord pldn flap; a , 0 ;6,0°.

~,-. , , 1 1

- Ro--”-- xxperimend tit B fro m

tref rene85

.“—0 Bloulbtedby eqL18tionc(6)

-2 1

\

A-

0 .2 .4 .6 .8 1.0Xlc

FigOre 12.- Preseuro distribution. M.A.O.A. S2012 airfoilwith20-peroont-ohordplainflap;a , 80; 6, 0°.

-2

1

o---- dm b fro ● r. monoBsIletioIU(6f

-—

/-1+ Pu

12I1--- ---- .--4, b 1—.— Q~b.

\-@.r--‘--- ----.-,.--

‘L ~+=-.—

‘o .2 .4 .6 .8 1.0x/o

rignrola.- Premaure dimtribntion.I.A.O.A.22012aArfoil with60-peroent-ohordplainflap; d , @; 6 , 15°.

-2

-1

Iaa

1

o--~– eri nenta L datb froD r“eformo B5aatiome(6)

<...

I -&z,.\

t> 1)- O-%

A.~_- --- --- -- --~

‘L -.k 2 ~

d

0 .2 .4 .6 .8 1.0

~ignre14.- Promure diatribution?~.A.O.A.2S012 tirfoil tith20-percent-ohordP3.eAnflaw d , 0°; 5 , ~“.

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