REPORT No. 465

DETERMINATION OF THE THEORETICAL PRESSURE DISTRIBUTION FORTWENTY AIRFOILS

By I. E. GAREICK

SUMMARY

Thh report gives the theoratic.aldixtributiun ojpremma4 lijl coq%i.9rdsof O, 0.6, 1.0, and 1.6 for 20 airfoils,ca.hlated on the bami of a rigorous potential theoqt ofarbitrary airf0i18. It d80 ~oviah tdkx fTO~ whichthe characteristic of the airfoile for any angL9of @adin fidimewicndpoteniiu.1*W are reudily calculable.

, The theoreticalvalm of the ang.?aof zero lift, the lifi andmonwni COejikieqtSlad the ideal a@x of aitack areltited and .som.ecompariwnw with ezperinund are indi-caied. Some of the well-known characi%%tia andproperties of airfoih are amoun$edfor in terme of thetheoretical presswredM-ibu#ion curves. (3wrMaiioe de-dwctti are nuub conixrning the causee of breakdownof potentti$ow and the e- of the airfoil in wisc0u3

jlow. The rerm?tspresented may be of value in pre-dicting 8truAmd bad? and&o in a correi?dion of the5-

retid premure gradti &h profi ranktance.

INTRODUG1’ION

Until recently the theoretical distribution of pressurearound airfoils could be determined only for the so-cdled “theoretical” airfoils. Indeed, only in the par-ticular case of the Joukows@ airfoils is the calculationnot unduly laborious. (See refereneea 1 and 2.)The theoretical airfoils, which are defined by speoialmathematical transformations, have, however, seldombeen employed in practice. Their use in a precisestudy of pressure distribution has in fact been due moreto necessity than to desire. The distribution of pres-sure for mathematically “thin” airfoils (i.e., the airfoilis represented by the mean-camber line) can be ob-tained, at least approximately, by the processes givenby Munk, CHauert,and Theodomen (references, 3, 4,and 5). In another report (reference 6) Thecdorsendeveloped a theory readily applicable to arbitraryairfoils. This theory wag extended by Theodorsenand Garrick in reference 7, in a report which gives aunified treatment of the 2-dimensional potential flowmound airfoils of any shape. The treatments givenin references 6 and 7 avoid approximations in the anal-

ysis, and are referred tc for details of the underly@theory of the results of the present paper.

The diilerences exhibited by airfoils in potentialflow, as well as the differences between the actual andideal oases for a particular airfoil, can, of course, becritically studied only if the ideal case is known.Furthermore, it is only on this basis that the assump-tions of the theory itself can be critically analyzedand modified. It is therefore believed that an existinggap in aerodynamieal literature will be, to someextent, bridged by the publishing in the present paperof ecmvenient tables and curves of the theoreticalresults for a number of commonly used and relatedairfoils.

A knowledge of the theoretical distribution of pres-sure for an airfoil is, undoubtedly, a major factor inmaking it ultimately possible to predict accurately thebehavior and efficiency of the airfoil under actual con-ditions, for the theoretical changes along the snrfaeefrom pressure to velocity and from velocity to pressureare very significant in the determination of the dragcharacteristic. Knowledge of the theoretical resultsis of considerable value, too, for guiding experim-ental work whenever the measurements are rathercritical, and such information also direets attention tothe significance and interpretation of diffmencesbetween theory and experiment.

Unfortunately, because of lack of sufficient accurateexperimental data, comparison cannot be made directlywith wind-tunnel results exeept in a few cases. Inreference 7 an interestingcomparison was given betweentheory and experiment of the pressure distributionaround the N .A.C.A.-M6 airfoil at 12 diilerent mglesof attack. Reference 8 may be referred to for quali-tative experimental results for five additional airfoils.A more accurate experimental study of pressure dis-tributions is in progress at the present time at theN.A.C.A. laboratories.

A part of the following work was undertaken at therequest of the Bureau of Aeronautics, Navy Depart-ment, for use in work on structural loads.

433

. .

/-. ... . ..=— .. .. . ..<. -..—-----s-. —-.-+ . - - —

ADVISORY COMMITTEE FOR AERONAUTICS434 REPORT NATIONAL

In makirw the calculations the author was ablyassisted by & Alyce V. Rudeen, of the Committee;sSt@.

SUMMARY OF FORMUL4S USED

The formulas used to obtain the results presented inthe tables and curves are developed in referencw 6 and7. A sample calculation for the N.A.C.A.-M6 airfoilwith a comparison with experimental results, as wellas explanatory figures and diagrams illustrating theuse of the formulas, is given in reference 7. Thefollowing list presents the symbols employed and theirdefinitions:

SYMBOL DEFINITION

(1) (z, y) See discuwion of the choice Of axes inn

(4) E

(5) e’

(6) V

(8) h

(9) a(lo) p

(11) k

(12) +

following paragraph.2 ti%=p+d~”

where p = 1—(s) -c)

2sinh~#=– p+~~ Sincey isgen-erally small for airfoils, the followingequation may be preferable:

sinh+=T& Near the leading (or

m) edge # is given approxi-

Jmately by #= ~ where u is the radius

of curvature at the leading (or trail-ing’) edge.

See appendi& of reference 7 for methodof evaluation.

Obtained graphically from the e, 8 curve.

(Denotd” $ in reference 7.)

Obtained graphically from”the #,0 curve.

@enoted }% in reference T.)

Angle of attack with respect to the z axis:The angle of zero lift, given by the value

of Eforo=lr.

eti(1+/)‘= J(sinh’*+sin%) (1+ *’7

Note: k is independent “of the angle ofattack.

The ratio of the local velocity at theairfoil surface to the uniform shamvelocity:

(13) :

(14) c

(15) CL

(16) F

(17) M,

(18) &Q

(19) a,

~=k[sin(a+@) +sin(a+@].

The ratio of the local supemtream pres-suie to the dynamic pressure (the term“supemtream pressure” is used todesignate the difference of the locrdpressure and the static pressure in theundisturbed uniform stream):

2=~.–()

+ ‘and g=:pv?!/The segment of the z axis intercepted by

the airfoil boundary.The lift coe5cient

C.=l L Sr e$o.—sin(a+fl?)

c2P c V2

A point designated the ‘{focus” of the air-foiL We may fit define the complexconstants c1 and Q aa

cl=m~=lll+illl

*O ~‘:{ #(@) (cog #+i sin+) d~

c%= f&+ ’i&

=:? ii~)(~s 2#+i sin Z+) d+

Then writing

b2#7=l+;2+Q

we have

(b’= 1+A’2;~’2

)-I-A2 ‘+ (AJ3, +Z&)z

Then the complex coordinate of F is~2

zF= (Z+iy)p= mea +—ef(2~~Je$’o

The moment at F is constant for allangks of attack:MFE2m P b’ ~sin2(7-/3)

The moment coefficient referred to thepoint F:

Chfp=M.~=4T$ sin 2(7–’f3)

The “ideal” angle of attack:EN+ET

a~- ——2

where ~Mand CT denote, respectively,the values of e at the nose and tail;i.e., for 0= O and 0= ~, respectively.

The ideal angle of attack for thin airfoils has beendefined by Theodonxm (reference 6) as that angle forwhich the front stagnation point is at the leading edge.

THEORDTrCAL PRESSURE DISTRIBUTION FOR TWENTY AIRFOILS 435

At this angle of attack large velocity gradients at theleading edge are avoided and the profile drag is at, orvery near, its minimum value. The definition can benaturally extended to actual airfoils to designate thatangle of attack for which the front stagnation point isat the foremost edge of the mean-camber line. How-ever, aapointed out by Theodorwm, the effective mean-camber @e of a thick airfoil actually alters withchange of angle of attack, and the ideal amgleof attackfor a thick airfoil represents an average of a range ofangles for which the profile drag is very near itsminimum.

PROCEDURE AND ACCURACY OF THE CALCULATIONS

In order to avoid possible confusion it may be wellto state beforehand that the term “chord” is used inthis paper as synonymous with the segment-of the zaxis intercepted by the airfoil. The “standard chord”in terms of which the airfoil is usually empiricallydefined does not, in general, coincide with this above-deflned chord. The angle between the z axis as chosenand the standard chord is designated A and is listedin table I. (See also fig. 1.)

In the procedure of the calculations, the axes ofcoordinates are fit chosen in a definite convenientway, since the ease and rapidity of convergence offurther evaluations depend considerably on the choiceof axes (references 6 and 7). This choice may be madeas follows: If the distance between the leading edge ofthe airfoil and the center of curvature of the leadingedge is bisected at E (the coordinaks of E are (2, O)),and the same is done for the tiding edge at E’ (thecoordinates of E’ are (– 2, O)), then the z b shouldpass through EE’ and the origin bisects the distanceEE’. However, smaU vsxiations from tbk partictiarchoice of axis and origin do not noticeably i.nlluencethe ease of calculation? The quantities given in theheadings of table II are directly calculated in terms ofx and y by means of the formulas previously listed.The angle of attack corresporkling to a given valueof the lift coefficient may be obtained from (15), inwhich c is XN—XT, where XN ~d XT denote the abscis-sas of the leading edge, and tmilhqg edge, respectively.The moment coefficient C&. may be obtained from(18), in which the constants b’ and ~ are obtained from(16) by graphical integration of the # sin ~, # cos ~,#sin 24, and # cos 2$ curves.

The ordinates of the airfoil are given empiricallyto hundredths of a percent of the standard chord for16 stations of the upper and lower surfaces respec-tively. The quantities z, y, #,, and o are defined tothe same degree of accuracy. The ~, o curve is thus a

faired curve through 32 points and e (0) is e&mated tobe of the same order of accuracy as #(0). The deriva-

JNotlcs, however,thrttweham chcan them.9.., th.nxhJthacamotfOIll),.e lwtacon%?R”Yti%&mg%&3::thesrds~o dbechcom fntheunf nemmmtmindfcatedabove(thishasbeendonefor theakfoflstreatedhero, for,fianotherM fs &.lntmt.PmPerIYrelm4tJwmTra .fL.nsiriH%i&e%*w.wm Mftwfthrfamt to thenewoxb.

tives E’ and ~’, beirqgdetermined graphically, whit ofa possible smill error which, however, causes an errorin k of probably less than 2 percent. The angle of zerolift, or the value of Efor 8= ~, may perhaps be in erroras much as 15’, but the influence of this possible erroron tlie theoretical pressuredistribution curves forfixed values of CLis negligible.

The numerical data for the Clark Y airfoil are pre-sented in table II. The distribution of velocity andpressure for any argle of attack or at any lift coeffi-cient, as well as other theoretical characteristics, areobtained with a *U of effort from M table.Similar tables for the remainirg airfoils are omittedhere for reasons of economy in printing and also be-

cause it is not known how general the interest in themwill be. They me available on request horn theNational Advisory Committee for Aeronautics.

DISCUSSION

Although the airfoils chosen in this paper are mainlyconventional airfoils (fig. 1) and not extremely radicaltypes, it is nevertheless possible to isolate some of theindividual effects of change of slmpe and comparethese with experimental rcgults. It is believed, ho~-ever, that future experimental work on radical andless conventionti shapes, for which the theoreticalresults are readily available (see, for example, reference7, p. 31), will enable the isolation and analy~is ofeffects which are probably masked and unemphasizedin conventional types.

We may first make some general comments regard-ing the curves of theoretical pressuredistribution givenin the following pages. In each figure the abscissarepresents the location of a point of the airfoil surfacein percent chord and the ordinate gives the quantityP/g, the ~tio of the 10~1 Supemtimm pressureto thedynamic pressure q. It may be noted that negativevalues of p/g are plotted upwards. This is an arbitraryconvention and is made because it is more readilyassociated with the upper surface of the airfoil, whichfor ordinary angles of attack is the surface of suctionor negative pressure. In @urea 2 to 21, inclusive, itmay be noted that the points of the curves above thezero, or normal pressure, line represent suction; thatis, velocities, greater than V. Positive ~a,lues of pfq

denote pressures greater than normal static pressure;i.e.. o< V. The stagnation points at which o= O cor-

respond to ~= 1.

Effect of compressibility.-lh figure 22 there isshown for convenience a curve of the dynamic pressureq in inches of water and in pounds of force per squarefoot against velocity in miles per hour. The valuesgiven correspond to atmospheric conditions at 2,OOOfeet altitude and 0° C. For the ordinary velocityrange of aircraft, say from 45 to 200 miles per hour, qvaries from about 1 to 20 inches of water. For verygreat velocities the effect of compressibility of the air

—-___ .- —:.. —— —— . . . . . Q-

436 REPOliT NATIONAL ADVISORY

l. Clark Y

2.N.A.C.A-M6-— -—-— -

3. U. S.A.2?

4.TJ.S.A.35B -—-— -—--

5.N22x axk

— -—- .— -J

6. G6ttingen387----- .—. -—

7. Gottingen 398 x axis.—

8. N. A. C.A.CYH .—. —. —---

9.C 72

10.Boeing 1~ o F_. —. —-— —-

12.E.A.F.19

13.N.A.C!.AOO1O-—.—.~-~.

14.N.A.C.A.00IZ

15.N.A.C.A.2212 o F x axis. —- —.—- /

16.N.A.C.A.2409 0 F.—. — .— -.

17.N.A.C.A.241Z

18.N.A.C.A.2415

19.N.A.C.A.44120 F x axis.— -- 2

20. N. A.C.A. 6512

~GURE L—TIM20afrfoilndxeen, ~ axedandlomtfonof F.

CO~ FOR AERONAUTICS

becomes sig-niiicant amd the potentkd-theory charac-teristics based on an assumption of incompressibilitymay be considerably altered. However, as is pointedout by Glauert (reference 10), the compressibility ofair has minor influence for velocities under 0.5 thevelocity of sound, or ordinarily about 350 miles perhour. However, it should be noted that cutcertainangles of attack the local velocity may be as much astwo or more times the stream veloci~. Thus, forvery great velocities, the strong suction in the regionof the peak pressures may introduce radical changesin the flow, as the compressibility properties of thefluid become important. This effect is m.sociatodwithMach’s Number (o/c, where c is the velocity of sound inthe medium), and the ordinary Reynolds Numberalone is not a safe criterion for scale effect. Thepotential theory yet remains to be properly modifiedfor the effect of compressibility. Reference 9 givessome experimental results of the distribution of pres-sure over airfoils for very high speeds. The maximumnegative pressure (or suction) obtained in this refer-ence wa9 37 cm of mercury.

Pressure gradients.-l?rom the concept of the idealangle of attack we are led to expect that the thinner theairfoil at the leading edge the greater the velocitiesnear the leading edge for angles dMerent from ar. Thisexpectation is coniirrned by the large negative valueaof p/q attained by the R.A.F. 15 and the N.A.U.A.0010, 0012, and 2409 airfoils. In particular, the pres-sure on the N.A.C.A. 0010 reaches –llg for liftcoeifkient CL= 1.5, whereas the somewhat thickerN.A-C.A. 0012 reaches – 7q at the same lift coefficient.In practice, the value of C~_ for the N.A.C.A. 0012is somewhat greater than that for the N.A.C.A. 0010.Results of force tests of the airfoils treated in thispaper are presented in references 11, 12, and 13.

The large gradient of pressure behind the negativepeak ptiure is very significant for the breakdown ofpotential flow. The deceleration of fluid becomes sorapid that fluid is piled up at the trailing edge and theflow no longer separates precisely at the truiling edgebut breaks off along the upper surface. The flow alongthe lower surface undergoes but little change at highangles of attack except that, after the breakdown ofpotential flow has occurred, the pressure at the trailingedge may be somewhat negative instead of positive.

The change hwm the front stagnation point to maxi-mum velocity occurs within a very small space interval,and all indications are that frictional losses are practi-cally negligible while the fluid is accelerating, as com-pared with losses when deceleration occurs. Thefluid follows the surface boundary more easily in thechange from pressure energy to kinetic energy thanvice versa. This fact is also abundantly confirmed bymperiments with nozzles and venturi tubes. For

THEORETICAL PRESSIJEtl?l DISTRIXXJTION FOR TWENTY AIRFOILS 437

-3.0- Uppersurface Lower surface-— ——. —

-2.0-1 CL-O\

c’ a 0.50(u= -3-35’) (a= 0“33’)

G “ /.00

-IvoJ,N,

?’>>... ——— —— ____

Percent chord Perced chard Ir1.0

Perceni chord20 40 60 80 IIJO 20 40 60 80 100 20 40 60 80 ILTO

,... ——— ~

20 40 ‘60 80’ jmhNJBI! Z—TtitlmI p~ dlstributtonforthe ClarkY afrfofl.

-5.0

t-4.0

-3.0

-2.0

I

Upper surface Lower surface.—. ___

L

c.‘ 1.00(e“ 7“4/7

-.-----//“

per cent chord’20 40 60 80 10 &c.= /.50

[U= 11°42~

~-”---——~‘P;r cent chord20 40 60 80 10

C’no(u= -0”36’)

c.=0.50(ri= 3°32]

-1.0

1--1\$0 +-.-—- --~

Per cent chord1.0 20 40 60 80 /0

Percent chordJO20 40 60 80 I

FIGWEEL—ThmretIraJprmme distributionfor theN.A.O.A.-M6 till.

-3.0

1

-2.0;II cL=o

(e=-4”31’)-1.0-~

\

q

Per cenf chord1.0 20 40 60 80 /00

Upper surface

c.=0.50(u“-0“24)

~\

Lower surfoce.—— ___

c““ /.00

-------------Per cent chord

r

---- —--/-//

204/760801FIGURE 4.-T~ prmnre @butIon fortbeU.S.A. 27afrfoff.

-3.0

-2.0

I-/,0\\

[

Upper surface

[

Lower surfoce.—— ———

——. —----~0 Per cenf cfwrd

20 40 60 80 100

C’=o(u “ -4”14’)

-\

~o9

=%

Percent chord/.0 20 40 60 80 100

FICUJRE&—ThwretIcalpresmredistributionfortheU.&A. MB afrfoff.

.1.. . . –. — - .—.~- - “-~- .--— - .- — -

438 REPORT NATIONAL ADVISO.EY COMMITLWE FOR ADEONA~CS

-4.0

-.30

I

k-.?.0 ,

\ C.=o(a= -4” 7’)

-/.0‘\\

PO --

9Per cent chord

Lo 20 40 60 80 10

LI-2.0

I

[

Upper surface

t

Lower surface——— ———

KOBOFIQmE 0.Jrkmetfcd~ dfshibntfonfor the N 2!2akfoll.

\Upper surface

L

Lower surfioce——— ———

EOEOFIGORE7.—Tkmtfcal PrESSmOdhmhti fm ti a6m w fdifoff.

qoper surface

c.“0.50(Ci=-0°227

h\ ___Percenf chord

20 40 60 80 10

Lower surfoce.— ———.

/ ———— ——--——

Per cent chord20 40 60 80 J

Ram 8.-Tt@Hetfd Presmn dMibnuOnforthe Qatlngenm afrfofl.

:~<oE&;oI?IGUEE9.—TbKmtfcd Pmsmm distributionfor the NA.C.A.-CYH afrfoil.

Uppff surface Lower surface_—— — ——

c.“0.50 c’= 1.00(U=-0”4J

—.—— -————— ——IR

Pwcen+ chord20 40 60 80 J 20 40 60 80 1

__. — —.

~R-;e; cent chord20 40 60 80 100

-/4---~.-

~ per ceni chord20 40 60 80 I

THEORETICAL PRESSURE DISTRIBUTION FOR TWENTY AmFom

-40 -

-.30- Upper surface Lower surfoce -—— ____

-2.0-C.=o c’=0.50 c.- /.00(a=-2”48’) ‘ (Ci= 1’247

-1.0\\

:0~.--—— ——

Percenf chord/

1.0 204060801 20406Q 601

-8.0-

-Zo -

-6.0-

-5.0-

-40 -

-2.0C“=o(a=-2°49)

-1.0\\.

;0

1.0

Fmum?11.—Thematfml~ dhtrfbutionforti BOEIIWlUS-Aahfoil.

-Zo -

-6.0-’

-4.0

-3.0

I

-2.0-\

I C.=o(o!=-8”56~

-1.0 -~

Per cenf chord1.0 20 40 60 80 fOO

[

\

Upper surfoce Lower surface———— ——

Lc.=0.50(a= /“3/7

—--

Percenf chord20 40 60 80 10 k

cc= /.00(e- 5°497

/--——/.— -

Per cent chord20 40 60 80 I

FI131JEEEL-TlwimticzLI ~ dfshibntfonfor tJMR.A.F. U fdrfofl.

IUpper surfoce

II

\ c.=0.50

-\ (u=-4”4.q

\

Per Cent chord20 40 60 80 100

Lower surfoce—— ____

hCL= 1.00(a=-0°4f7

\

.< /- a~----Per cenf chord

20 40 60 80 JFIQVEE lx—TkdfcII ~ dktrfbntfonforthe RAF. 19afrfcdl.

439

/-- ---—/ Per ceni chord

2047260801

t

t

t

P%c.= f.50(u= 3“29~

‘\

>e7ZZZZZ20 40 60 80 I

~- .——------ - -._.. —-— .

440

-ll.o–

-10.0-

-9.0-

-8.0-

-Zo -

-6.0-

-5.0-

-4.0-

-3.0-

-2.0-

-1.0-

REPORT NATIONAL ADVISORY COMWT’I’EE FOR AERONAUTICS

C.=o((x= O“oy -

1 Per cent chord1.0 J2040608010

UppW surfoce

c.=0.50(e: 4°127

Percent chordJ20 40 60 80 10

l?mwm 14.-Thwretiralm=sme dbtribntionfor theNA.OA. WIOdrfofl.

.

Lower surface___— ——

%

CL-“ 1.00(a= 8“2G)

,/-— —/

Percent dmrd204060891

-7.0-

-6.0-

-5.0-

-4.0-

-.30-

-2.0-C.=o

(&= 0“0’)

‘/.=)O

Upper surfoce

\c.“0.50(CY“ 4- 8’)

Percent chordJ20 40 60 80 10

Lower surfoce—————.

&c.-1.00(&= 8°/87

--— ——.,/

r Per cenf chordc!20 40 60 80 IO

--———-

‘%eHr cent chord/.20 40 60 80 I

FIomrr 15.-ThmretIcd ~ dkibutlon fcatheN.AO.A. IXI12afrfoll.

-4.0

-3.0

-2.0I

THEORETICAL PEESSURE DIS1’EIBUTTON 41

Upper surfoceI

Lower surfme——— ———

A

CKOKOKOFmmm 16.—Theomkal~ dktrflmtionfcutbeNA.O.A. 22VJafrfoil.

-8.0-

-Zo -

-6.0-

-50 -

-4D -

-3.0-

-2.0- t

Upper surfoce.

CL-O(a=-z”z’)

I

c’=0.50(t-in 2“137

-4.0

-3.0

-2.01

Lower surfme——————

c.“ 1.00

%

(e= 6“.29J

_- —- ——— ../

Per cent chordc!20 40 60 80 /0

!#rOWM17.—The0rettral~ dhtrfbntfon hr the N.A.CA. 2#3 afrfdl.

IUpper surface

c’ = 0.50(a= I055’)

l-nCL” O(a= -2”15~

-1.0\

;0 -.

Per cenf chord1.0 .?040608010 b —--

Per cmf chord20 40 60 80 10

1 .Lower surfoce——— —__

c.“ 1.00(a= 6°77

.- ——— ———/

20 40 60 80 I

__— — —//

------,~HP;r cent chord

FIOUEBlR-Themetld pressme dMribntfOn for tke liA.O.A. 2412&foil.

4076S-3*29

--- —.. ~,- . -. . ---- - .——... .—- -—--— --

REPO13T NATIONAL

tiOflS used in plTLCtiCS without auxiliary

ADVISORY

devices, apressure ratio pig of about – 3 or –4 is the bumattained. It may be observed here that since theleading edge of the airfoil and the upper surface nearthe leading edge experience very large pressuregradients, they are critical regions, to be especiallykept

free horn unnecessary protuberances and roughness.It may be noted in the figures that very large

pressure gradients exist in the region near the rear

-40

-3.0

-2.0[

kCL =0(a=-l”M’)

-1.0\

:0--

Per cenf chord1.0 20 40 60 80 10

I

.

upper Surfoce

COMMI!ITED FOR AERONA~CS

Some effects of oamber.-It may be observed that aproperty common to all airfoils is that the negativevaluesof p/g mount rapidly nmr the leading edge on theupper surface after CL= 1.00 is exceeded. However,for the highly cambered airfoils, as the R.A.F. 19 andN.A.C.A. 6512, it may be noted that even for OL= 1.60the p/g negative peak is but slightly above – 2. Theseairfoils are high-lift airfoils, and the lift is well distri-bu~d over the whole chord. Eowever, they have

I Lower surfoce——————

c.=0.50(CY- 2-57-B- 20 40 60 80 10

FIGURE19.—Thmr0tical~ dfstrfbntin for the N.A.OA. 2415afrfofl.

-3.01- Upper surface

LL

1

Lower surfoce——————

-2.0I\

CL=O c.=0.50 c.“ 1.00(a=-4-3’)

-f.o \\\

PO!?

(aF-0°26?h ‘“= 3“4’)

Lo1! Per cent chord20 40 60 80 jL10 Percenf chord ~k

!

204060801Per cenf chord

20 43 60 80Ffomm m—Th&uwfceJ~ distribution fortheN.A.oA. U12afrfofl.

-3.0~I Upper surfoce Lower surfoce——————I

-2.0J\ ~L”o c.=0.50 cc= 1.00\ [Lr=-7”4’) (a.-.2”577 @= 1“IO?

-1.0-\L, \

~oq

-.—/ ‘--’

.—_— ---. -_,__ _

Per cenf chord . %rc.snf chord1.0

Per cent chord20 40 60 80 100

FmmE 2L-Themetkaf ~ dktribntfonfortheNA.CA. 6612drfafl.

stagnation point, within the shadow of the 100 percentchord line. The rapid deceleration of fluid as thusshown to mist theoretically at the trailing edge mostprobably does not occur to any such extent in practice.The flow prob~bly recombines at the trailing edge atvelocities not much below normal, and the pressurecurves are rounded off at a small positive pressure asshown in the figures. There is in this fact no essentialviolation of the Kutta condition for fig the circula-tion, the primary purpose of which is only to avoidiniinite velocities at the trailing edge.

--— ———./

.--— — -——

R~’nj.50(o!5 5“21’)

-—— —— —--(H Per cenf chord

20 40 60 80 I

usually unfavorable pitching moments and rather widetravel of the tauter of pressure, as evidenced by thetheoretical moment coefficients at zero lift, which arerespectively –0.210 and –0.185.

Further effects of camber maybe observed in figures15, 18, 20, and 21 for the N.A.C.A. 0012, 2412, 4412,and 6512 airfoils, where in every case the maximumthicknwa is 12 percent of the chord and the maximummean cambers are, respectively, O, 2, 4, and 6 percent~f the chord. The theoretical moment coefficientsC&pare, respectively, O, – 0.055, – 0.110, and – 0.185.

THEORETICAL PRES81JIU3 DISTRIBUTION FOR TWENTY AIRFOIIS 443

The ideal angle, also, increaseawith camber and, hence,the optimum lift coefficient increases in general withcamber. We may observe that the high-cambered air-foils, and more especially thin high-cambered airfoils,me not efficient at low values of the lift coefficient, as isevidenced by the high negative peaks in the pressuredistribution for zero lift. In fact, the flow aroundmany &gh-crunbered airfods (for example, the R.A.F.19) is known to burble on the under surface at lowlift coefficients. Indeed, large gradients of decelera-tion of fluid may eve~here be associated with de-crensed ef%ciency. The bringing of pressure gradientsand profile resistance into a precise correlation is asignificant problem for further investigation. Auniform gradual change from velocity to pressure, as infigure 2 for the Clark Y airfoil at CL= 0.5, gives prob-ably the optimum lift distribution and occurs verynearly at the ideal angle of attack. “

The experimental curves of lift coefficient againstangle of attack for high-cambered airfoils like theR.A.I?. 19 and N.A.C.A. 6512 are well rounded nearmaximum lift, whereas for airfoils like the N.A.C.A.0010 and R.A.E’. 15 they are likely to be sharp andjagged (reference 12). The former airfoils lose theirlift rather gradually after maximum lift is attained,while for the latter airfoils this effect is likely to occursuddenly.

Effects of thiokness.-In fiegures17, 18, and 19 wemay note some effects of the airfoil thickness. TheN.A.C.A. 2409, 2412, and 2415 airfoils have a commonmean-amber line and maximum thicknesses, respec-tively, of 9, 12, and 16 percent of the chord. For theN.A.C.A. 2415 it may be noted that the pressure onthe under surface is generally leas positive than forthe 2412 and 2409. Also, we may observe that at liftcoefficients C.= 1.00 and CL= 1.50 the down gradientof pressure along the first 15 percent of the chord isgreatest for the 2409, while for the rest of the chord itis greatest for the 2415, indicating that an optimumeffect for thickness lies perhaps between the extremeslisted, The theoretical slope of the lift curve in-creases somewhat with thickness, and for the aboveairfoils has values of 6.75, 6.90, and 7.10, respectively.h experimental result that merits closer investigationis the fact that.after a maximum thickness of about 12

percent is attained, ~ decreases somewhat with fur-

ther increase in thiclmess (reference 1!2). A partialeqdanation lies in the fact that, ti general, a thickerboundary layer exists on thick airfoils, decreasing theiraerodynamic eiiiciency. It has, indeed, puzzled someobservers employing various schemes for removing theboundmy layer that the experimental slope of the liftcurve for infinite mpect ratio often excaeds 2zr,whichis the value given by the approximate thin-airfoiltheory. The a~erage theoretical slope of the curvesof lift against angle of attack for the airfoils listed in

this report is approximately 6.90 01 about 1.10X 2ir.This value is about 15 to 20 percent greater than theexperimental value of the slope of the lift curve foridnite aspect ratio, and indicates that the airfoil is ingeneral from about 80 to 85 percent eflicient withregard to lift.

Moment properties.-The theoretical momentM~ is, in most of the cases studied in this paper, fromabout 10 to 20 percent greater than the &perimentalvalue of the moment for zero lift. (See refererws 12and 13.) On the basis of (16), the position of the focusF, at which the theoretical moment is constant for allangles of attack, was calculated and is shown in theiigures of the airfoils. (See @. 1 and also table I.)In every case its abscissa is very nearly at 25 percentof the chord from the leading edge, the maximum for

28[ [ I 1 [ 1 I [ 1 I I 1 f -h

FIGUBEZL-Vekity Vagabt dymmio ~ g.

the airfoils considered being near 27 percent and theminimum near 24 percent of the chord line. How-ever, it is important to note that, in general, the ordi-nate of F does not fall on the chord line but is usuallylocated at a small distance from it.

The tendency for more constant Centeraf-pressureproperties may be observed in iigure 4 for the U.SA.27, where the lower-surface pressure curve has a smallinflection or bend. For the N.A.C.A.-M6 (fig. 3),which theoretically has practically zero travel of thecenter of pressure, the double bend on the pressurecurves at zero lift may be observed. Alternate re-gions of suction and pressurethus exist on ti surface.The N.A.C.A.-CYK shows the same tendencies to a

lesser degree. The double bend in the pressure-distribution curves is probably common to mostreflexed airfoils.

COMMITTED FOB AERONAUTICSRFIPOET NATIONAL ADVISORY

Experimental results on the angles of zero lift showconsiderable discrepancies and indicate a change withthe Reynolds Number. In general, the consistentexperimental result is obtained that the angle of zerolift increaaas (algebraically) with increase in the Rey-nolds Number. For this reason we may only indicateu comparison with experimental values. The valueslisted in table I of this paper consistently fall withinthe range of values obtained by experiment and seemto agree more closely with experimentalresults obtainiidat moderate Reynolds Numbem (about 2 x 10B) thanwith those taken at very large Reynolds Numbers.This fact should be investigated further.

CONCLUSIONS

The preceding discussion shows that, to a largeextent, the general properties and characteristics ofairfoils, such as effects due to camber, thickneM, orchange of shape, may be accounted for by the theoreti-cal preasuredistribution curves. The theoretical pres-sure-distribution curves at definite lift coefficients maybe safely used for structural-load considerations. Acorrelation between pressure gradients and prcdileresistance, affecting the efficiency of the airfoil, hasbeen qualitatively indicated. Further theoretical andexperimental investigations should be concerned withthe significance of the differences between theory andexperiment.

,,

LANGLEY MEMORIAL AERONAUTICAL LABORATORY,

NATIONAL ADVISORY COILAKMTDE FOR kONAUTICS,

LANGL13Y l?EDLD, VA., JW32, 1933.

REFERENCES

1. Blumenthal, Otto, and Trefftz, E.: Preeeure Distributionon Joukowdd Wmga and @aphio Caatruction of Jou-kowski Wmge. T.M. No. 336, N. A. C.A., 1925.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

Perring, W. G. A.: The Theoretical Preesure Dfatribution

around Joukowski Aerofofls. IL & M. No. 1106, BritishA. R. C., 1927-

Munk, Max M.: General Theory of Thin Wing Seotions.T. R.. No. 142, N.A.C.A., 1922.

CHauert, H.: A Theory of Thin Aerofoila. R. & M. No.910, Britieh LR.C., 1924.

Theodorsen, Theodore: On the Theory of Wing S.cotiona

with Particular Reference to the Lit Distribution. T.R.No. 383, N. A. C. A., 1931.

Theodoraen, Theodore Theory of Wing Seotiona of Arbi-

trary Shape. T.R. No. 411, N. A. C.A., 1931.Theodoraen, T., and Garrick, I. E.: General Potenthd

Theory of Arbitrary Wing SeOtfona. T.R. No. 452,N. A. C. A., 1933.

Jacobs, Eastman N., Stack, John, and Pinkerton, RobertM.: Airfoil Preamure Diatnbution’ Investigation fn theVariable Denai~ Wind Tumel. T.R. No. 353, N. A.C.A,1930.

Briggs, L. J., and Dryden, H. L.: Pressure Dfdributionover Airfoils at High Speeds. T.1% No. 265, N. A, C.A.,1927.

Glauert, H.: The Effeot of Compreaeibility on the Lift ofan Aerofoil. IL & M. No. 1135, British A. R. C., 1928.

Anderson, Raymond F.: The Aerodynamic Charaoterlstimof Airfoils at Negative Angles of Attaok. T.N. No. 412,N. A. C.A., 1932.

Jaoobs, Eaetman N., Ward, Kenneth E., and Pinkerton,Robert M.: The Characterietica of 78 Related AirfoilSections from Teste in the Variable . Density WindTunnel. T.R. No. 460, N. A. C.A., 1933.

Jacobs, Eaetman N., and Anderson, Raymond l?.: Large-Scale Aerodynamic Charactmistice of Airfofls aa TeEtedin the Varfable Density Wind Tunnel. T.R. No. 352,N.A-C.&, 1930. -

TABLE I

THEORETICAL CONSTANTS FOR THE TWENTY AIRFOILS

.4fl-fOm &

- 11”d Cc

cur ar‘ !GGiGw)~r N 5 ,

C’.I m 8 A

— —0, .,

Olmk Y..–.-.--–—--— --------- L 112 4.031 &m ;+ >:S& U& -0.. 03sNAOA—ML-.–—.–-.. ---–– L 117 4.040 o.95j~

::; a 110 4; & ;;

u.sA. n-------------------- L 116 4KQ4 .%% 057.079 200 00

.ES3U.SA 35B–_--____––––..-. L lfQ Lms

-. ma i 1: .@ .117 5s0 Ow414022 i% .m -. lm o 49 .m .127 47 10 130

N.=.---––.–— --------------L lIS 4.024ciati 3s7—

47 0 1s Lf03 .s3’4 –. ma Ow .lx 4s0 lfa—-.-..--.-_–– L 153 4.m4 % 039 LKC .867 -. l!u ob5 :: .173 40 10 20

IR;R#EOti:zl::x:xzz :% Ml

7.m 037 L031 .s70 -.102 035 .61e.95

,140 49W 23

—------------------- pG& 402!3044 LIXII .s27 -.017 10 .34

a.91 ;Boll.m4 40 0 0 62

.mBodn#lm ~

-.107 025 .113 47 30 20—--. ——— —---- 5.s4 :%’ .W

RA . 15------–.-–-.---.--–-–– L071 $% an148 0 13 -.076 1 17 :fl .m 49 m 1 11

.950iti~f M .944

-.062 lls .47RAF. 19.--.----..-.-––--––– L 116 4m &97

.m F3 10 00-.210 20 1.3JI

N.A.O.A. mail___________ LW2 4.023.m 6S30 O 24

.910 .Om 00N.A.O..&. mid_____–_______ Ll12 4033 1% ::00 xl .m

.Wl 00 00.033 00 :: .ml 00 00

NAO.A. !22=.--. -—----------- L 119 4.034 &’mlfao30 .s92N.&O.L -—------------------- L@3 4.019 5%7e 22:4 i% .922

-.039 I u. .-33 .057 00 10 00

NA.O.A. 24U_--_––--___– L KC 4.m4-.052 022 ,2s .ms 45 50 00

2 16N.A.O&. 241L–__________–_ L 142 4.m ::

.fw –. us Om AJ .m 2s34 00:E

N.A.O.A. MM--.--...––-––––– L 114 4.m4 rAQ4.675 –. Ofa 032 .116 25m 00

ifi12! .s95 .&m -.110 033 .02NAO.A MU---------------------- L 119 4m4 0.2s 7400 .S93

. KS 49 M 00.s%7 –. las o 19 .s9 .148 IM2J3 00

445THEORETICALPREssmm DISTRL131JTION FOR TWENTY AIEFoILa

TABLE II

CLARK Y AIRFOIL

UPPER SURFAOE

4’ k

+4+ IRrcmt, z o v

am.m.110

:E.021

–: E-. cm–. cm–. a30–. W-.103-.114–. 114–. 10I–. 044

10,radians

n 2030LQ33

!%2L 733L&.11L 430L236..W

:%3-.378-. m-L mL 505L 7W.2 Mll

o 0.am .04M.1240 .1037. w .184.Zza .Z31.m .281.310 .B13.345 .I?s5.372 .833.374 .955;% .’W3

.262 :E

.I&l .011

.OK.S .3J8

.a’m . lM6o 0

:a3&

.W

.0433

.0457

:8%. a407. OHS.0256.0311. W!51.0L57.0125.@

:E

o.222

:E.557.044.7E3

i%L35SLE30L 7.54

;%2.4s325?s3.142

0.173.182.195.237.212.215.216.214.Xr2.190.176. 16s.I.3n.142.0793

:%%

-a CL34e–. Cm5–. CH40–. Oml–. m–. 022s–. am

. 0K!3

.0430

.W

.0772

.0218

.101

.109

. Im

. Uwl

.03m

am

:M’.133. lm.145.123. 1?3.111.a?a

:M.040.013

.–. om–. w–. 077

:%2!X0

HiL’ZRL IXIIL626L 318L 210L 172L l&lL233L 374L 709Z340

-aC&5. 14%5.2.53.403.521.622.793

i%?

;%L&91Z070ZmZ694z 781&m4

iw

M’h 61&46

wQ’i39.28a 747.72.2%4.47Z40LBo

I

LOWER SURFAOE

ao3Lt& &m0.074

. oli3 h w

.0131 : y4

.01045ss3

%!Jk%

.m h 133

.cm’d

. a313 2%

. oz33 4.516

.M05 4.301

.W Loal3.7.33

:8 3.fa3.0310 a 142

a Kc!. &w

o–. &317–. m–. Ws9-. 1a5–. llo–. 114–. 112–. 0%9–. aw–. 0718–. W70–. cM39–. 03Gz-.0161–. mo

0.042e.Ow.178.m5.347. 4S%

:%’

i%.Qm.842

:%!. KU3

o——

a 173.148.131.114.102.0332.a315

:E

:%!.am. U224.0173.Olu.0K9J. a310

-ye&

–. 110–. 103–. m?!–. lm–. 0276–. lwls–. m-. m–. 053s–. 0425–. 0232–: ~

.0265

.Mm

am.171.149.m. mo.032.Ct3u

:E.044.040.ml.a?s.014.017

–: E

ao?o

–: R!–.024–. 031–. 023–. M-.044–. 057–. cm-. 0s7–. U37–. m–. M–. W.a5–. 100–. 077

451–18 8–23 36–31 10–37 1-41 Q3–69 !23–57 48-70 27–81 65–s3 4-m 41-114 59-423 31-142 30–152 40-176 24

hsn6%?35.6375.6s)64025.276&m4&13i.sm4.47342764022Km3.ma3.mi