Lab. y. Scheepso:un
REPORT No.
835Technische I-Iogeschool
Dellt
PROPERTIES OF LOW-ASPECT-RATIO POINTED WINGS AT SPEEDS
BELOW AND ABOVE THE SPEED OF SOUND
B 1iii iii T. .JONESSUMMARY
Low-aspect-ratio wings li acing J)O ijited plaii forms are treat((l on the assumption tiwl the flow potentials in planes at rig/it angles to the long axis oft/icaT/foilsare s'iinilar to the correspond-ing two-dimensional potentials. For the limiting case of small (ingles of attack and low aspect ratios the theory brings mit t/ii following significant propei'tu's:
The lift of a slender pointed aiifoil moving in the direc-tion of its lovq axis depends on 11i increase in width of I/ic sec-tions in a downstream direction. Sections behind tire section of in axiinum width (le lelo» vo i ift.
The spanunse loading of such an airfoil is iìirlepenilent of the plan foim and ü/)»rûac/res the distribut'ion gieinq a minimum induced drag.
The lift distribution of a pointed aiifoil tai'thng point-foremost is relatively wualfeefed b?! the compressibility of the a ir
below or above the speed of son ud.
A test of a triangulai aiijoil at a Afach nuinbe, of I .7.5 rerified the theoretical lu/nec of lift and center of pressuie.
INTRODUCTION
The assumption of sirinil disl ii}oiiì's in a t%vo-(Iirn(Isi()!1o1 pOtO1tia1 flov IPO(IS t o I !i \v(11-knovn tlì iii-airfnl iIi otv
Of 1unk (rvfvionce I ) ;iiiil lite I'ia,oItl-(ìlaio'it1II, ( t111i1tues
3 and 3) at speeds less titan Sottie. At speeds above tite speed
)f sound, application o t lie stime assumptions ietIAIS to lie ckeret theory (reference 4) a(cord ¡11g to which the wing 4ections generate pia tie sound waves of small ti ni pl i t tul e. ts is well known, tite Acketet I iteory pied irts a rad ictil (liti Ilse
in the properties of 511(11 wings on transit ion to silpeisortic
velocities and these eiianres have been verified by
exiiiri-rneiits in supersonic Wiit(I luittuis (reference 5).
Both the Ackeret theory a ini t he M tink theory app1- lo Lite case of a wing havi ng a la Ige sparì and a small CI ord. rite present discussion is based on assumptions siinii:ii to
hose used by Ackeict and Munk but covers tite opposite xtreme, namely, tite wing of small span and large chord.
r11 tite latter ease tite flow ¡s expected to be two dimensional
rIten viewed in planes perpendicular to the ciiiectioit of
notion.
A theory foi tite rectangular wing of small aspect ratio
tas l)een given by Boilay (ieleience G). Bollay assumes a eparated, or (lisconl ¡ii toils, pot etitial flow similar to tite veil-known Jircitofi flow aitit
shows that under these
ircurnst:utces tite lift is ProPoll ¡una] to tite squale of lite
ingle of ttttu(k. Boiltv (lOi'S hot consider tite ifiect (if
ontpressibiiity. Tite pioSeli I tieatinettt covers ot.lter plait
orrns and, although lui sed oit (li tfereit t assumptions, is not 7S875-49
ittroitsistenl with Boiiay's theory in tite liinitittg case of small
angles of atta(k.
By limiting tite plan forms to small vertex angles, the
l)ropert.ies of tite wings in conuj)resSibie flov at itigit subsotiir and at supersonic speeds aie also covered. Tsien (reference 7)
lias pointe(i out that Muitic's airship theory (reference 8)
applies to a slender body of revolution at speeds greater tiran
sonic.
The lift and moment of such a body are not
eX-pected to change appreciably with Macli number. The
present paper gives an analysis of tite low-aspect-ratio
air-foil based on similar assumptions and shows that little change of tite lift distribution of an airfoil of pointed plan form lying
itear tite center of tite iachr cone is to be expected.
SYMBOLS T7 flight velocity a artgk of at tack S Wit 1g aiea r !/ i) A aspect ratio
(!)
thistaitce along axis of symmetry of l)Oillted aifoii measured downstieain from nose
spanwise (listtlnce, ntetsured from ax s of symmetry vertical distance from dane of wing
tiitie
ll(l(litiflhtitl appatemit flh;iSS (spanwise si tion)
10(111 sintmt (1101(1 density of air Lp
M
C,,, dynamic pressure(pv2)
p (Ji local lift foice (per length dx)
e1 local lift coefficient
(-t) induced drag Vi1 Ir
¡D1 CD1 induced-drag coefltcteitt L tota! lift 6'L
lift coefficient ()
4' surface potential ospanwise-location parameter (-t9)
local pressure differenceMacli itumbei', ratio of flight velocity to speed of
50 tilt (1
distance of center of i)ressuiie trotti nose of airfoil / Pitcliiii montent pitching-moment coefhcieiit I\
qSi
lift itt \ [utcit niiini.,er 71!
L0 lift itt zeto Macit iuitiuiber
maz ntaximrim (used as subscript)
2 1El'O1t'l' NO. 83NA'l'1ONAL APVISOItY COMM ITTEE THE)RY FOR WINGS OF LOW ASPECT RATIO
Tue flow about an airfoil of very low asl)ect ratio maY be
considered twO dinlelIsiolial when viewed in CrOSS sections
perpendicular to the longitudinal axis. With this idealiza-tion, the t icatment of the low-aspect-ratio airfoil becomes exceedingly simple; formulas are obtained that are similar
in some respects to those derived by Munk (reference 8)
and Tsieii (reference 7) for an elongated body of revolution.
Perhaps the simplest case fioin tue analytical point of view is that of the long, fiat, triangular airfoil traveling point-foremost at a small angle of attack. Viewed from a
reference system at rest in the undisturbe(l fluid, the flow pattern in a plane cutting the airfoil at a distance x from the nose is the familiar two-dimensional flow caused by a
flat plate having the normal velocity Va.
(See fig. 1.)Observed i i this plane, the width of the plate and hence the
scale of ib flow pattern continually increase as the airfoil
progresses through the r'Iane. This increase in the scale of the flow pattern requiìs a local lift force i equal to the
downward velocity Va tines the local rate of increae of
the additi( ial apparent mass m', or
dm'
IrVa
dt 2 dm'=Va
dx since dxVdg
By a well-known form tila from t wo-d iniensional-flow
theory,
, b2
m =ir4p dx
where b is the local width of the plate. Hence
(/ifl' b db
and the lift i per length dx will be given by the expression
i
=7raj
P T72b/x
(IXDividing by V2 ami by the area b dx gives the local lift
coefficient
Cjira --
dxdb (1)When this flow is consi(l cred in more detail, it is found from
the two-dimensional theory that the surface potential is
distributed spanwise according to the orduvites of an ellipse, that is,
4 ±
Va/()_y2
r±Vasin O
(2)where cos O and the sign changes in going from the
u pper to the lower surface of the airfoil. (See fig. 2.) An
iiistant later, in the same pliui e, the ordinates are those of li
slightly larger ellipse, corresponding to an increase of . The
local pressure difference is given by the local rate of increase
of , that is,
Lp=2p
°
- -p
=2pt
òb dx (3)where /òb is a function of y. Differentiation of yields
the equation
r
FIIURE 1.Flow pattrn.
FOR AEII()NAU'IICS h
dba
p=2pT72 dxP_
2adb
(J Si1lO dx'l'ue pressure d istrihut ion tlì us shows an infinite peak along
the sloping sides of the airfoil similar to the pressure peak
ii t the leading edge of a C()JIVCJI t tonal airfoil. The
distribu-I oil iiloiig laut ial lines pssitg thllougiL tue vertex of the
t riutigie(lilies of (Otistufit hf2) is uniform (fig. 3), however, utol t lie (enter of pressure (oitI(i(les with the center of area.
Equations (1) and (4) show t li:i t tue development of lift by the long slender airfoil depends on an expansion of the sert iOns 111 a (lownstreu UI (li reel ion; hence a part of the
surface havi ng parallel sul (S WOO 1(1 develop no lift. Further-niore, a decreasing width would, according to equation (4). require negative lift with iriíinuite negative pressure peaks along the edges of the narrower sections. In tile actual flow.
however, hie edge behind the maximum cross sectin will
lie in the viscous or turbulent wake formed over the surface
uihiead; and for this reason it will he assumed that the infinite
pressure difference in(licatc(l by equation (3) cannot be de-veloped across these edges.
lt is
this assumption,corre-sponding to the Kutta coridit ion, which gives the plate the
properties of an airfoil as distinct from another type of
body, such as a body of revolution.
C ¶
V
FIGURE 2.l'otentil.
(4)
PROlERTIES OF LOV-ASpE('I'-1tA'I'lo POINTED WINCS Al' Sl'EEI)S BELOW ANI) ABOVE 111ESl'JED (JI' SOUNI)
With the titi of tire Eritta condition, it niny easily IO'
own that. su t ions of tut' airfoil beiritid t he sectioti ofgreatest
lt li tICVP1O ) no lift. A potent irr! flow satisfying bot li t he
u udo ry cor (IitiOfl and t Ile Kiit ta eoial it jolimay he 01) f.iiiiwd
t ho introd i ict ion of a free su ifrire of d iscont iii ti i ty bel i nti
e widest, s etion. This stirfare of discontiiiuil.y (fig. 4)
nild he co irposed of parallel vortices ('Xteli(l ¡ng
(lOWli-enm from 110 videst section of the airfoil as prolongations
t he Vort ¡u s represent ing the d iscontiliuity of potent ¡iii er t lie for :Lrd part of tire airfoil. Tu is sheet, a lt Irougli
ssihly wid r t Iran the downstream sections of the airfoil,
il satisfies tiroir l)Oufldary coiìd itioti, Silice the Ial eral
a !igenion t if the Volt ¡ers iS
such as to give
ii ii iforni\Vflwrìrd Ve )City eq rial to Va Over the etitire W ititir of t lie L'et ineludir g tire rearwar(I port ion of t ire rì irfoil Since
pressure I ifference across flic airfoil is proportional t t)
¡òx and si ice this gradient disappears
as soon as
t liert ices hecoi te pa ra I lei to I ire st ream, no lift is tievelopir i
the rearwa t i sect iolis.
[ntegratioii of t he pressures in a chiordwise (I ¡rection from
leading or ge downstreaun to the wiriest sect ion will give
span load (listriblition arid I lie induced drag. The span
d distrihirl on is
ÒL
r
òyrr ¿/) f/f
ÒL
L second integration of
òy dy across the widest section
s the total lift, which is
IGVRE 3Preure distribution.
pViaF,,ar2 (C)
Tire lilt of tire slender iiirfuih I bruciore (leptnds wily ou tire
wid (hi ii ud not oir lire irren.. 1f t ire lift. is divided by I2AS
nuid i f t lie aspect ratio ¿I is considered to 1)0 t iitri
ir
(7) niid tire md uced-(l rag/coel!icieri t is
rl - I'-
1)--
-, (t(8)
Fuouir equation (S) it iiI)J)l'iirs t liii t I ho risiiltiinit force lies halfway between tire uioiiriitl to LIa surface nial tue riorririrl to (lit nu slreauri.
li is Seoir hint in tue cuse of ii rcctniuigruhnrr plan form tiri'
siiiiphiliid fo,iiiuilni, (equal iou (4)) gives nui infiuuite
roncen-Iruitinii of lift at lite h(urdiuig erige mid no lift elsewhere,
Whieltui5 ii liinie acriuralt t hirtity W'OiiIti siuit' soiuui'
distiihu-t ori (if the lift rearward. If the iii e oï inucreutse tif tire width
innoniuts tori great, t lit flow ruruttiot litt expected lo relliainl (WI) di liielisioiiii I. lt cita lo slir, ii by exnriiiinuntt io,, of I lue
kuiowur t lirer-diiuurtisjrriiuui (tiotiliftinig) Pottlil uil flow ntiouiuid
nui rilipt ir disk ( refere nett ¶t), hiowrvrr, t liai, Lire
(wo-liliiriittiotiurI I hirory gives ru goon ripßooxiiiiiitiuui in tue rase rit ari (lupi irai ieiirhiuig tIlgt wluirlu inidiruirs litai, liii l.lirorv
.1,
o
A puc/ / j/Ñ
ii;n,i: .--(,riijiarjsi,,i ot flrt e1eulaI,'i 1v rt'ii. Il''ry ('r 'ilitt,ic,ai wi,i's if low Lslwct ratio wii,h ,isoiis iii kitiiet (rtrr,ij, iii).
is applica hie over a large rangt of liase shapes. In uiguire S is
Sliownu it r()Iiuparisoll (Jf thurt lift (tiulctihnttrtd by tue presutat
i lueorv for elliptical wings of low aspect ratio with flue
irsuiPs tif the more accurate I iunre-diiuuensional potential-flow
raluiliil inris of Kiienues (rrfirenu(rt IO). 'I'hue results are iii
good ugreernrnit up to aspect rutias ru pproruiiinug h.
Appii-ca I ion of eq uation (4) gi ves nu ccitt 'r f Pressure on (lue
lhipticul pharu fornir at one-sixt lu of (hurt chord. i"igure G also
shows t huis value compared with virI as gi veri by Krienes'
theory. Irr this respect it ttppctirs thiitt the agreement is
not so good as for the lift.
EFFECT OF COMPRESSIIJILITY
lii (untier to show tue offert of coiuipressihility, use will be
unruhe of i lue theory of polen ti i i flow willi small tlistiiibancn_s.
(flatirnt (rrfereiuce 2) unir! l'rirnirl tI (rrfe,rt,icc 3) have
thenìon-stnui I tri t huat, nit subsonic sJ)(Cds, IL r iistrihu (ion of Potential
satisfying Laplace's equation will satisfy tire linearized
compressible-flow equation if the distribution (i, y, 2) is
:
from equa ion (3),
D j2 2 (5)
,,,(I .0
= 2pVçb Oil equal ion (2),
= Va sin O
ncc ÒL/Òy is elliptical and independent of the plan form.
Lb tire ('Il i pl irai span load t hie i 11(1 iJ(ed drag is a ni in ini rim
:\ htdiI (tOile (fig. 7). t1 rund il oit of a sinai! vertex angle i II Ito lìeCeSStLry in O!(iel that tite potential (i istrihution of
i ita t ion (2) may apply lit tite case of a wing with a bI tint-lead tug-edge plan form, al Ill I t citfinges in tite flow fInSe 011 transition to supersonic velocities, aiìd potential flow of the subsonic type no longer exists.
The lift and lift (list rihu t ion for rectangular surfaces at
supersonic speeds have bitt t (alCil lated by Sci t ii(liting (refer-ence II). Figure 7 shows titi variation of lift-curve slope wit lt \Íac1t number as obi iii ted from Schlichting's results fut
teetangitlar wings of two (li li('l('tl t aspect ratios and for lin' lange of speeds in wit lt t lit two \iacit cones froni tite tips
ib not reach hie center of tite wing. Iii I lI(' subsonic range,
values given by tite Pratid t i-Ula tort rutl(' are shiovn. These rElives fl'f toinpared Wit it titi' values m(iirated by the present t lteory for a triangular wing lying neat tite center of tite '\la'ht cone. Figure S shows t lic t ravel of titi' center of l)res-sitio foi tlt('S(' phiri forms. It is to he noted that, with tin' hiiltlt-lea(ling-e(ige plait faillIs, t lie renter of pressure travels fiotti it J)oiitt neat tite (llaIte, (lioni to a point near tite mid-(1101(1 when titt veloci t V is iilcI(tased above the speed of
Soli t It i.
/
,
i) li;c/i cone.,,i.
,1 i » Pee langui (refeenceI/) it;och rumbe,-, AI 3.0FICURE 7.Variation of lift willi \!tcli number for different plut ferino.
/0 2.0
J
E/«ofico
A
=»
4'ec fan gula,- ('-efer-encei/
I.UU
[i
u....
u....
.uuuu
u
a.0
u....
4 I1EPOIrI' NO. S35NA'I'IONAL ADVISORYCOMMIt TEE FOR AEw)NAIj'II(S
u » a
/pd ra!à, A
Fu U RE fiCoi arison of cii tr of issu ri eticiiktted by prusin i thiory for elli i o i igs el low aspect ra it, wilt osti I Is of Krimis (rtlrtnee IO).
Ioresiìortei d along tite (i ¡IN! ¡ùii of mot ion by i lu
transforma ion
a X y'==y
z'=:
nus fact, ii be applied in a (IÌlCUllltiOfl prOr((lnrc by start-ng with a lititioiis liiIfOil lostart-nger iii t lie z-direction t Itatu tite rue One a d calculating I iìe ¡)oteultial (list tihution for titis uirfoil by methods of i Il(Offl piessibie flow. Frite (O tier t
limensjons and correr! (lisirihution of art' liten obtained ,vlieiì tite t ansforma lion is applied.
For tite long slender airfoil, tite poteti t ial d istribti t ion a t
'ach seri oit is similar to that for an iitíi,tiI ely long hotly lierefort' ò/òx and hence tite local pressures Vary in inverse )roportioIl to the length. The foregoing calculation pro-edure gives a null result in titis rase, since tite pressures alculated for the fictitious airfoil at M=O will be redured n the same ratio that the length is increased and the Lorent z ransforniation to restore tile correct length will also restore
lie same pressures as those obtained at íW=O. Since
/òz is unchanged by the transformation, the normal
'elocity component and hence tite angle of attack are
un-hanged also. These results can he obtained by referring
irectly to tite linearized equation for the potential
2 2 2
òx2 2
,2
ee reference 3.)
If the airfoil
is sufficiently slender, 2/òx2 can be neglected in comparison with ò/òx except ear the edge. Since the lift is proportional to ò/òz, the tcrease of the lift with Mach number ran therefore be eglected iii comparison with the lift,It is important to note that the theory of small disturb-flees is no.t limited to subsonic velocities and that, so long m the term (1M2) in equation (9) remains small, tite )llltion in tile region of tite wing will continue to hegiven y tite potential (equation (2)). Evidently tite Macli nium-ir cannot be increased indefinitely, for titen tite eoefFuient
will become so large that the first terni will no longer negligible. Tite required contlition will he satisfied, how-7er, by adopting a pointed platt form willi tite vertex angle t small that the entire surface lies near the center of tito
1.0 2.0
/1'aC/7 ru,nbet- A! 30
FIBRE .'rraveI of renter of pressure with Mach number for different pian forms.
2.
L
pnop: :ln'IES OF' LO\V-AspEC'J'-JL'I'IO I'OIN'ID WINGS STS OF A TItIANGULAR AIRFOIL AT SUPERSONIC SPEED
tS a test of the foregoing analysis. a small triangular
air-I in tut' fornì of a steel plat e wit Ii rounded lead ing edges
s const ril(t cd and tested in t he La ngh'y model sii person je
mel. Time tests were mimado a t a Macli n umber of 1 .75.
mre 9 shows the (letails of time model and figure 10
sum-l'iZOS time results of the test. At zero angle of attack a
all lift and a small pitching moment occur, which are pre-nimbly time result of the camber given the airfoil by
round-off the leading edges in time manner shown by section
A in figure 9. In gemmerai, the results are in good
agree-mit vitlm the theory if an allowance is ¡nade for this camber,
showii ¡ im figmi ie 10.
CONCLUSIONS
The lift f a slender, pointed airfoil moving in time
di-t ion of idi-ts ing axis depeii Is on the increase in wid (h of
sections i t a downst rea in direction. Sections behind
section of i maxim um vid i Im il evelop no lift.
Tiie_ Spiim \ViSC loading of such miti a imfoil is indepemideim t
:Iue plan f um and approaches the distribution giving a tilnum indircd drag.
Time lift ( strihu tion of mi poi lit (( i a ¡ ifoil t ravel ing )ol ¡it-most is rel it ively ulm fleet t'ti by t lit' (Oliipl(sSi bili y of t lic below oral tove time speeti of Sotilid.
;GLEY MEMOItIAL AEltON,& UTICA L LAIIO1t..\TORy,
ATIONAL LDVISORV CoMil...Eli FOIS AEItONAUTICS,
LANGLEY FiELD, VA., May 11, 1.945.
R EFER ENC ES
Mtink, Max M.: Eleitmetits of the Wing Sect iou, Theory aru.I of lie Wing Theory. NACA Rep. No. 191, 1924.
lauert, 11.: The Effect of Conipressibiliry oit the Lift of an
Itero-foil. R. & M. No. 1135, British A. R. C., 1927.
raîidtl, L.: General Considerations oli the Flow of Compressible
Fluids. NACA TM No. 805, 1936.
ckeret, J.: Air Forces on Airfoils Moving Faster Than Sound.
NACA TM No. 317, 1925.
ravior, G. I.: Applications to Aeronautics of Ackeret's Theory of
Aerofoils Moving at Speeds. Greater Than That of Sound.
R. & M .No. 1467, British A. R. C., 1932.
lollay, Villiain: A Theory for Rectangular Wings of Small Aspect Ratio. Jour. Aero. Sci., vol. 4, no. 7, May 1937, Pp. 294-296. 'sien, Hsue-Shen: Supersonic Flow over ali Inclined Body of
Revolution. Jour. Aero. Sci., vol. 5, no. 12, Oct. 1938,
PP 4S0-483.
AI' SlEEDS BELOW ANI) ABOVE 111E SPEED OF' SOUND
S. 1\1ii,i1, i\'Lax M.: The Aerodvitaitije Forces oli Airship Hulls.
NACA Rep. No. 184, 1921.
t). L:inih, Horace: Hydrolyitaiiiics. Sixth cd., Cambridge Univ.
Press, 1932, pp. 14G-153.
IO. kranes, Klaus: l'ue Elliptic Wing Basedou tite Potential 'I'iicory.
- NACA TM No. 971, 1941.
11. Schlichting, H.: Airfoil Theory at Supersonic Speed. NACA TM
No. 897, 1939.
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Mounting attachment)
Sec tian A-A
ti'u_TRE 9..iir(otl est,,l iii i iigl,v ziiol'l supersonic lunilel.
U. L GOVLRNMLNT PitITING OFFICE: II4
02 . o (J 0-. E o E .02 u 5 ,lf--/j cc,I,..
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A
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P' Ea-per-irnontoip;lching momo,,t
o Exprirnr,f ai lift
Angie of otock, a: dog-/ o / 3 FIGURE 10.lestortriangular airfoil in Langley model smupersoulic tunnel. Mach number.
