Document
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NATIONAL ADVISORY COMMITTEE
FOR AERONAUTICS
REPORT No. 563
~-tF J
CALCULATED AND MEASURED PRESSURE
DISTRIBUTIONS OVER THE MIDSPAN SECTION
OF THE N. A. C. A. 4412 AIRFOIL
By ROBERT M. PINKERTON For sale by the Superintendent of Documents, Washington, D. C. - - - _ _ - - - - - - - - Price 10 cents S ub scription price, $3.00 per year / 1 ., AERONAUTIC SYMBOLS 1. FUNDAMENTAL AND DERI VED UN ITS Metric English Symbol Abbrevia- Abbrevia- Unit Unit tion tion meier __________________ Length _______ foot (or mile) _________ m ft. (or mi.)
l Time _________ second _________________ (or hour) ___ ____ t s second sec. (or hr.)
Force _________ weight of 1 kilogram _____ weight of 1 pound _____ F kg lb.
Power ________ horsepower ___________ horsepower (metric) ______ hp.
P ---------- miles per hOuL _______ {kilometers per hOUL _____ k.p.h. m.p.h.
Speed _________ V meters per second _______ feet per second ________ m.p.s. Lp.s.
2. GENERAL SYMBOLS v, Kinematic viscosity
w, Weight=mg
p, Density (mass per unit volume) g, Standard acceleration of gravity = 9.80665 4 2 2 2 Standard density of dry air, 0.12497 kg _ m- _s at m/s or 32.1740 ft./sec.
15° O. and 760 mmi or 0.002378 Ib.-ft.-4-sec.
Mass = W m, Specific weight of "standard" air, 1.2255 kg/m or g 0.07651 lb./cu.ft.
Moment of inertia = mk • (Indicate axis of I, radius of gyration k by proper subscript .)
Ooefficient of viscosity Il, 3. AERODYNAMIC SYMBOLS i .. , Angle of setting of wrngs (relative to thrust S, Area line) StD, .Area of wing Gap Angle of stabilizer setting (relative to thrust a, line) Span b, Q,' Resultant moment c, Ohord
n, Resultant angular velocity
b Aspect ratio
S· Vl
p- , Reynolds Number, where l is a linear dimension V, True air speed J.I.
(e.g., for a model airfoil 3 in. chord, 100 m.p.h. normal pressure at 15° 0., the cor-
q, Dynamic pressure =~p V
responding number is 234,000 i or for a model of 10 em chord, 40 m.p.s. the corresponding
L, Lift, absolute coefficient OL=:S
number is 274,000) D, Drag, absolute coefficient OD = ~ Oenter -of -pr es::lure coefficient (ratio of distance of c.p. from leading edge to chord length)
Profile drag, absolute coefficient OD. = ~S Angle of attack
Do, Angle of downwash
Induced drag, absolute coefficient OD, = ~S Angle of attack, infinite aspect ratio
Angle of attack, induced Parasite drag, absolute coefficient OD = DS1> Angle of attack, absolute (measured from zero- • q lift position) 0, Or ass-wind force, absolute coefficient 0 = q~ Flight - path angle 0 'Y, R, Resultant force
REPORT No. 563
CALCULATED AND MEASURED PRESSURE
DISTRIBUTIONS OVER THE MIDSPAN SECTION
OF THE N. A. C. A. 4412 AIRFOIL
By ROBERT M. PINKERTON Langley Memorial Aeronautical Laboratory I 71878-36-1
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NA TION AL ADVISORY CO MM ITT EE FOR AE RO NAU TI CS H EA DQU AR TE RS, NAV Y B UIL DI NG, WA HI NG TO N, D. C.
LABO RAT O RIE S, LANG L EY FJEL D, VA.
a p pr o\' ec l )l a l' cb 3, 1915. f or t he s up e rvi s i on a nd di r ect ion of th e eien tifie Create d by act of Con g l 'pss ((', ~ . Colie, Ti lle 50, ~e(; . 151 ) . It" me mb e rs hip \\' as i nc r ea ed to 15 by study of (hl' p r oble ms of fligbt Th e melll ber s a re appo i nted by t he Pl' es ic1 <' n l, and se n e a. sueb ", ij 'b out act app roy C'(l )I n l'ch 2. H)2!l.
COUlpen a tion .
<'H, \R LES A. LI ND BE RG If , L L. D. , .J OSEPH S. AM E. , P h. D., Cha i nlla n, N e w Y o rk ' it,\'.
Balt im o r (:', M d .
\\ ' I L Ll A M P . MA CC RA "I{lo;N, Jr. , L L . D ., DA V ID \ V, T A '1 "LO R, D . Eng., V ice (' /tairl) wn, Wa s hin gto n, D . C.
\ Vas bi nglon, D. C.
A UGUST I N~; \V. R O BI NS, Br iga l li e l' Gen e r ll l , U n i te d S tates Army, ('IL\I{L~;S G. A IlIlO' r, Sc . D. , Chi ef 1\ I u tc ri el Div is ion , A ir Cor p , Wri gb t Fi C' l d, Da y ton , Sec l' C'tar~', Smit h son i an I n>; ti l u lio n .
Obi O.
L YM .\ N .I. BRIGG ., Ph . D., EOGE:" I, L . I' U )AL, '. E. , D i rC'cto r , Natio na l B u r au o( Sta l1ll a rd Dir ecto r of A i r Co mm e rce, D e pa r tme n t of Co mm e r ce.
AR TIIl'R B . COOK , R ear A( lll li ra l, ni ted StatC',' N a v y, E DWARD P. W AB NER. l\I . S., C hi ef , Bu r ra u or Ac ron aut i c;;;, N a \'y D epu r t rn E'lll.
I'e \\' Y o rk C il ,\' .
\\' U.LIS R. \ Y GREGG, B. A ., O SOA R ",, 'EST OI'ER, Ma j or Ge ne r il l, ni te d t ate A r lU Y, U ni tE'd Statc. W eat h er Bur ea u.
Cu id of A i l' Co rp, W a r D E' pa l tm en t.
lIAI UlY F . GUGOElN lJ El M. M. A., O nV H .L I ,; \ V mG II T, Se . D ., Po r t \ Vn ;.; b ing ton , LO lJ g I s la nll , X. Y.
DlI~ ' (on , Ohi o.
SYD NEY 1\ 1. KRA US, C ap ta in, U n i te d States N a v,\' , B u re au (If .\ (,),UlIu lIti <:,; , ::\'a v ~' D e[la r lmr n t.
GEORGE W . L E WI S. ]) ;1'eC I 01' of A.eronat IIi N I/ R (',~e a 1'c h J OHN F . V ICTORY, , ('cre t an) II ENIlY .T . K H ElD, l ~nflineel' in C har ge, L allfl / ey Jl emoria l ..1 ,er01 10 1 1li ca/ L(ll m r% r l/ , T ,on fll e ll F iel,d, V a .
.T OH N .T . I D E. ' l' e(' /l lIi('a / A "8'; . ~ tallt i n N llr ope, f 'a r i .\'. i" 1'fI 1I(,(' TE CH NI CAL COMMITTE ES A IR C RAFT A CC ID E TS AE RODY T A ID CS 1 VEN TIO NS A N D D ES I GNS POW ER PLA N TS FO R A IR CR AFT A IR C RAFT STRU CTU RE S AN D MAT E RI AL Coo l' iZin ali llll Of N e8e(l1'c l! Neeils of Jl i /it a l' lI and C iril Avi at io n ['rep((ration Of R esem'ch Pl'of/r aIl1S A ll oca ti on of Pr obl ems Pl'e' ren t ion of D ttp l ica t ion ('onsil/e r at io ll of I m ' enf'i 0118 O FF I CE OF AERO A U TI CAL I TELLTGE T CE LA NGLEY ME MO RI AL AERO NAU TI CA L LA BORATORY W AS HINGTO N , D. C.
LANG L EY FI EL D, VA.
CO ll ec ti on cl ass ifica t ion , Co m1 ) i1 ation , U nifi C'(l <:oJ1(l u ct, fo r a ll age n Cies, of a nd <Ii 'sem in a ti on o( sc ie ntifi c and tec h - scientific resea r ch on t b e f Ulllla m en ta l n ica l in fo J'm a ti on on a e rona uti cs .
lU'o hl elll of fli gbt.
II
REPORT No. 563
CA L CU L AT ED AND MEASURED PRESSURE DI STRIBUTIONS OV ER THE MIDSPAN SE CT ION OF THE N. A. C. A. 44 12 AIRFOIL By R BERT M . P IN K ERTON S MMA RY lo cal ve lo c iti e over th. e urface; t h. e pres ures :1 re ca J- cul ated by mean of Be rn oulli's equa tion. Althou gh P ressures were simu ltaneously measured in the varia bl e- this me thod provides an in ex pensive mean or ob tain- density tunnel at 54 orifices di tributed over the midspan in O" th e di tri bu t ion of pre ut e, the re ul t m ay no t be sec t ion oj CL 5- by 3D -inch rectangular model oj the N . . i l.
in sat l facto ry agr ee ment wi th mea ured r es ul ts. Llc h
c. 1. 4412 ai1joil at 17 angle oj att cLc lc rang'ing jrom
di agreemen t, however, is n ot urpr is in O" s in ce the - 20° to 80° at (L R eynolds Num b r oj approximately theory cloes n ot ace LIn t for the e fl" eets of the vi CO ll S 3,000,000. l ccurate data were thus o bt ained jor t udy - bound ary layer .
ing the deviations oj the re ul ts oj potential-flow theory A rea onably acc ur ate met hod of calclu at in g the from mea ured result. T he re 'ult oj t he analysis and pre ure eli t ribu tion over a n airfoil ection is de irable a di cus ion oj t he experimental technique al ·e present ed.
and migh t be ob ta in ed by two pr oce du res. Fi r t, L1 ch It i shown that t heore tical ca lculations ma de eith er at a me th od mig ht be found by t he developmen t of a com- t he effective angle oj ait ac lc 01· at a given ac t ua l l if t do not plete th eory . Surh a th eo ry, however, mu t ta ke in to accumtely desc1·i be the observed pre SU1 'e dist ri but ion over aecollllt all th e fac tors or ph enomena invol ve d a nd an airjoil ection. Th ere is therejore d eve loped a modified mu st give a tisC actory agr eme nt wi th act ual mea U l" e- men t. \. econd pr ocedure, the most fea ible one at t h e01·etica l ca lculation that ag ree rea ona bl y we ll wi th pr e eu t, i the deve lopm en t of a rational met hod of
the mea ured resu lt s oj the tests oj the N. A. c. A. 44 12
co rr e ting the appli cat ion of the potentia l-flow th eo ry ec t ion and that con ist oj making the ca lculations and to minimize the discrepaneie b et ween t he t h eo r e~ i ca 1 eva luating the circulation by means oj the experimentally and mefl.s ur ed r es ults.
o bt ained l if t at the effective angle oj a tt ac k; i . e ., the an gl e It was rea li ze d, however, tba t unll ually relia bl e ex - that t he chord oj the model makes wi th the direct ion o j the perime nt al pr essure -d i st ribu tion d ata fo r co mp ari on with calculation were n ot av aila bl e. Th e ex peri- fl ow in the re gion o j the sec tion under consideration . I n me nt s to oh ta in s ll ch d ata co n i te d of p ressur e the cour e oj the computation the shape pa ramet er e i mea m ement at a large numb er of poin t around on :.'
modified, thu leading to a modified or an effective profile ection of an airfoil. Becau e the i nvest i O"at ion wa hape that d~ffers lightly jrom t he specified shape.
primarily in te nd ed to st ud y devi at ion of the act ual fr om the id eal, or poten tial, fl ow, the te t,s were ma de in I TROD UCTIO the va ria ble-densi ty t unn el over a ra n O"e of va lue 0 f thE Pre sm e-c li st ri bu tion mea uremen t o ve r an airfoil R eynolds umber , r epresen ting va ryin O" efrect of section pr ovide, di rect ly, th e kn owled O"e or the air- fo rce vi co ity. In addition , te t were made in the 24 - in ch el i trib u t io n along tbe ehord that i r eq ui re d for so me high-sp ee d t unnel a t ce r ta in co rr e po ndin g va lu e of purpo e . In addition, uch da ta, when compared wit h th e R eynolds J umb er ob ta in ed by means of high sp ee ds, the re uIt s of pot en tial-flow (non viscous fl.uid ) th eory, th ereby bringing ou t th e e fl" ect of comp re sibili ty.
pr ovide a means of t ud y in g th e effect of vi se- ous forces P ar t of this e xp erim en ta l in ve ti gat ion outs id e the on the fl ow abou t the airfoil section .
scope of thi repor t ar e st ill incomplete.
Th e resu lt of experimen ta l pr e s ur e m ef\, Ul 'em (' nt Th e prese ll t repor t, which pr e en t th e mo t im por- for a few miscellaneou ai rfoil may be fo un d in va rious ta n t oC the ex pe rim en ta l 1' e ult (tho e co rr es ponciin O" to publi cat io n. Th e general appli ation of t hi met hod th e highest va lu e of the R yno ld Jum ber), i divided of ob ta in ing de ign data, however, is li mited becau e of in to t,, -o par t. Th e fir t p art eo m pr i e the de crip- the ex pense of ma lting uch measuremen t .
tion a nd di. seu sion of t he ex pe rim en ta l tec hni que: A m et hod of calculating the pre m e di t l' ibu tion is M ate rial th at are e sen tial to e ta blish the f act th at th e developed in referen e l and 2. This m et hod, based meas ured ee uIts a re sufficien tly ac ur ate a nd reliable to on the " id eal fluid" or pote ntial-flow theory, give th e m ee t the dema nd s of the llb equen t analysi . Th e REPORT A'fIONAL ADVISORY COMMITTEE FOR AERONAUTICS near the test section. The remaining 54 tubes, used second part presents a comparison of theoretically cal- cu l ated result s with measured results and an analysis of to measure the preSSUT e at the orifices on the airfoil, were connected to the tubes leading to the airfoil model.
the differences and probable causes. A method is A lig htti g bt box mounted on the £lat side of the developed to modify the app li cat ion of potential-now semicircle contained drums for holding photostat paper theory in order to minimize discrepancie from the and the neces ary operating mechanism. Th e ma- measured pressure distributions.
nometer was arranged so that it cou ld be operated EXPERIME TAL PRESS RE DISTRIBUTION from outside the tank that hou es the tunnel.
Th e manometer characteristics determined by trial APPARATUS AND TESTS included the time req lured for the mcni cuses to be- The experimental investigation described herein wa come steady and the proper exposme of the photo tat made in the variable-density wind tunnel (reference 3).
paper.
Themodeill s ed \yns a s tandard cluralumin airfoil ha v in g A record of the hei ght of the manometer fluid in the glass tubes wa taken at each of 17 angle or attack FI G oR E I. - D is tribu tio n of pr ess ur e o rifi ces abo ut tb e N. A. C. A . 4412 a irf oil.
the K. A. C. A. 4412 section and a rectano'ular plan form with a span of 30 inche and a chord of 5 in ch es.
I twas modiflCd by replacing a midspan section ] inch in length with a bra ss ection in which the pre sure orifices were located. The 54 orifices, each 0.00 inch in diameter, were clrilled perpendicularly into the air- foil surface and placed in 2 rows about the airfoil. Th e method and accuracy of construction of the model arc described in reference 3. In order to eva luate the pressure force parallel to the chord, a relntively lnrge number of orifice were located at the nose of the airfoil ( fig. 1); well-defined distributions of pressure along a normal to the chord were thus assured. The lo cat ion of the pre sure oriftce are included in table I.
tu bes were connected to the orifices and carried in grooves in the lower surface of the airfoil to the planes of the supporting struts where they were brought out of the model. After the model was assembled, the grooves were covered with a plate carefully f:1ired into the surface. The tubing extended through the tunnel wall into the dead-air space and the part expo ed to the air stream together with the support truts Wfi faired into a single unit (fig. 2). The tubes were connected by rubber tubing to a photorecording multiple-tu be manom- eter mounted in the dead-air space.
FigUJ'e 3 shows the 50-tube manometer, composed of FIG U RE 2 .- Pre ss ur e ·di s tribution model mounte d in th e tunnel.
30-in ch glass tubes arranged in a semicircle and con- nected at the lower ends to a common reservoir. The from - 20 ° to 30 ° at a Re yno ld s umber of approxi- total-head pressure of the air stream was chosen as the mately 3,000,000.
reference preSSUl'e and was measured by a pitot head, In order to keep the results as accurate as possible, mounted as shown in figure 2, to which foUl' equally it was necessary to obtain lar ge deflection of the ma- spaced manometer tubes were connected. The dynamic nometer liquids, which was accomplished by using two pre ssure of the air stream was determined by two liquids of widely diff erent specific gravities.
tubes connected to the calibrated static-pressUl'e orifices Liquid: Spec ific OTa .i t~ Mere ur y _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __ _ _ _ _ 13. 6 used in the normal operation of the tunnel. One tube Tetr abromoethane__ _ __ __ _ __ __ _ __ ___ __ _ __ __ __ _ _ _ 3. 0 was connected to a set of foUl' orifices spaced around The proper choice of the angle-of-attack groups and of the inner wall of the retUTll passage and the other tube the liquid enabled the use of large and comparable to a set of foUl' orifices spaced around the entrance cone - - ~- - -- -- ---- ---, ~------~--- -- -~- PRESSURE DISTRIBUTIO IS OVER THE MIDSPA SECT IO OF THE N. A. C. A. 4412 AIRFOJL :3 deflections througbou t the angle -o f- attack ran ge. R e- and the pitching-moment coeffici ent, which are defined peat tests, using the same and different manometer by the fo llowing expres ion liquid s, provided data on the precision of the tests.
RES LT S A copy of a sample photostat record is shown in
c c= ~ J Pdy
figure 4. Tb e pre sure in inches of manometer fluid were measured to 0.01 in ch. All measurements were made from a reference line obtained by drawing a line
C m c I 4 = ~ [J p(~ . - x )d x+ J PydyJ
connecting tbe meni cu e o[ the four reference tu b es.
The quantitie tbu obtained from the photostat records were: 6p = H -p where H is the tota l-head pr es ur e of the stream and p, tbe pressure at the airfoil orificej a nd q= factor X 6 p.
where q is the dynamic pre sure and 6p s is the differen ce in pre sure between the static -pr ess ur e orifice in the entrance cone and those in the r etu rn passage. Th e factor was previously determined by com parin g va lu es of 6p s with sim ult an eo u va lu es of the dyn amic pr e - sure obtained with a calibr ate d pitot - stat ic tub e mounted in the air st ream in the ab ence of a model.
Finally, the pressure on th e airfoil were computed a rat ios to the dynamic pressure, thereby making the results independent of manometer liquid.
Bernoulli's equation for the undi st urb ed st ream becomes where P oo is the pressure a nd V the ve lo c ity . Th e pr essure of the fluid at the wing orifice is given by p = H - 6p ubstitute for H from the pr eVIO US equation and remember th at ~ p V2 = q, the dynamic pres ur e, then onsider P oo as the datum pre ure. The pr essure coeffici ent then becomes 6 p
p= P-P oo= 1 _
q q where 6p and q are quantities obtained from the photo tat record as previously de cribed. Values of P at eacb orifice on the airfoil ani for all angles of at(;ack are tabul ated in table r.
Figure 5 C a, b, c) present plots of P against orifice FJG ORE 3. - Ph otore co rdin g mul t iple-tub e m anom et er .
position along the chord and again st position perpendic- ular to the chord for each angle of attack . Large-scale where c is the c hord , x is the orifice stat ion along the plots imilar to tho e presented h ere were me c hani cally chord, a nd y is the orifice ordinate measured from the integra(;ed to obta in the normal-force, the chord-for ce , chord. The lower-ca e ymbol Cn, Co, c "'cl4 de ign ate 4 REPORT NA'rIONAL ADVISORY CO MMITTEE FOR AERONAU'rICS section chanwLc ri Lies and rcrer re pectively to the where w ithe indu ced nor ma l \'clocity produced hy the normal-for cc , chord-force, and pitcbin g -moment co- vortex system of the airfoil, including the tunne l-wall efFicients for tho mids pan soetion or the airfoil. interference, and l ' is the v elocity of the undisturbed Plots of these coefficient (see table 11 ) against geo- now. In ordor to ea.lculnte the induc ed veloc ity w, metric angle of attack are given in figufe G. The geo- the di st ribution of thr lift (or cireuln.tion) along the me t ri c angle 01 attack a is mea 'm ed :from the mean pan of the airfoil mu t be determined. \.. theoretical direction of the Dow in the tunnel. This direction is method of obtaining this distribution is gi en in refer- defined as the zero-lift direction of a symmet rical airfoil enc e 4 and , wh en applied to tbis problem, gi ves for in the tunnel and wa s found to be equival ent to 20' of tbe induced angle of attack of the mid pan section upDow. In order to have true section characteri tics a; = 1.5 4 CI (2-climensiona1) for compari on with theoretical cal- n ilation s, a determination mu t be made of the e fl' ec - where CI i the lift coefficient for the midspan sec tion.
tin angle of attack, i. e., the angle that the chord of Thi s lift coeffic ien t is obtained from the pre ~ 1Il'e FIGURE 4. -Co py of sample record. 1. lead in g -ed ge orifice t ub e; S. static-pre ss ure tubes; 'r, trailing-edge orifi ce tube; and Z, reference-pr ess uro tubes.
the mode l makes with the direct ion of flow in the region mea urem e nt by means of the equation of the midspan ect ion of the model.
Th e effectiye angle of attack , corre ponding to the Values of Cl, a i, and a o are giyen in tnble II.
angle for 2-dimen ional flo,,', is given by PRECISION Th e reliability of the 1' e lilt S of the pre me mea llrc- \\ 'here a, is the angle thnt the flow in the region 01 the ments reported he)'ein mn,y be determined by conside ra- aidoil section make with the diroction of the undi s- tion of thc technique of ohtaininO' and measuring the Lurhed now . The nn1O'1Int of thi. clf'Ylntion is mnll pres m'e record, of the deviation o[ t he prc sure nncl ea n he enlculntccl from diagram s ohtained from several te ts at the ame angle w of attack, and of the method of calcul ating the effe bve a, = V angle of attack.
PRESSURE Dr TRIBUTION OVER 'I'HE MID PAN SEC'!,ro J OF THE . A. C. A. 4412 AJRFOIL to become teady and by delaying the taking 0[' the Th e method of obta inin O' the pre me records is a record at each angle of attack un til sufficient time had d.ir ect, simul ta n eo u , photogr aphic record i ng of the clap cd . As a fu r thcr heck, a zero record wa take n hej O'ht of the liqu id in the manometer tubes. ince at th end of eltch tcst r un 'u nder the srUlle conditions.
the pressur e coe ffi cien ts used in the ana ly i are r lttio I ( Chord force Normal force x Chord force Normal force - ::I x Experiment ---- Usuol theory - - -- M odified " a =-8° a ~-16· p p -7 -6 -5 - 4 50 10 0 50 100 o 100 o (a) P er c e nt chor d lfIGt;RE 5a.- Experimental and theoretical pressllre-distriblltion diagrams for the K. A. C. A. 4412 airfoil at so\'er.1 angles of a tt ack.
of ql l ant i tie taken from the same record, the pr imary In addition, the tube we rc checked for leak before s Oll J' ce of ['r o J' therefore ]jes in thc llnequal dampinO' in fl nd after each run. In order to minimize any po sih]e L he t ll be co nn ectin O' the airfo il orifi ce to th manom- e rr ol' in reading the photo tatic record (fig. 4) mea ure- eter . Thi s om ce of erro l' wa m i. nimi ze d by deter- 01 nts of the recorded pre ur e, were made indcpcnd- minin g the t im e required for the liquid in all the tub es e ntl y by two persons. T he l' cltdings were then com- 6 REPORT NATIO AL ADVISORY COMMITTEE FOR AERONAUTICS Normal farce Cha rd force Normal force Chord force -9
Expenmen,.l x
-3 ----- Usual theory - - - Modif ie d " - 8 -2 -7 ct =oo - I -6 -5 I
I
ct =16°
I
I I -/ -
I
I
O~--- ---T--- - -12 P -1/ -/0 -9 -8 -7 -6 -5 -3 -2 -/ O~----------r-------~~r-~~~~---- --~ \ \ I 1 0 100 o 10 o 50 50 100 o /0 Percent chord (h) FIGURE 5b.-Experimental and theoretical pressure-distribution diagrams for the N. A. C. A . 4412 airfoil at se"eral angles of attack.
----- - -~- -- PRESSuRE DIS'l'RIB TIONS OVER THE MID PAN SECTIO I OF THE r . A . C. A. 4412 AIRFOIL 7 pared and a compromiso wa made wh ere difference from everal test at th o arno angle of attack. Figure 7 ccurrcd. Th e difJ'el'cncc between any t wo such present such diagrams at t wo angle of atta ck, _ 4° ind ependent readings raro ly exceeded 0.01 in ch except and 0 T etra bromoethane, because of the larger in tho case of obviou elTo r . Po ible errors due to dene ct ion , O'ave more acelU-ate results, which agreed
- r
Norma/ force Chord force Normal force
~ Ch ~ d r ~"
-/ +
x Exp e rim e nt - 16 - --- --- Usual theo ry --- -- Modified · - /5 - - 6 -/ 4 - - 5 -/ 3 -4 -12 - 3 ~ 1/ - 2 « - 24" - 10 - / • x X -9 O~ -- --------r-------~-- r- ~~-- -------- ~ P -8 p -7 - 6 - (5 I I - 5 I
I
- 4 -4 I I -,J I « =30" I - 2 - - 2 I I I x -/ -/ I XX XXXXX"X X )(X X l( )( X )( X X X )( X X X X )()fl(
r
O r- ____ ______ -r ________ ~LX_X~--~---- ------ --~ J( )( x x . ;t' )( x )( .( .
(:x
)(X X X 50 0 1 0 0 50 100 0 10 Percent chord (c) F IG UR E 5c.- Experimental and th eo retical pre ssure-dis tributi on diagrams for tb e r . A . C. A. 4412 airfoil at several angl es of at tack.
hl'inkage of the record were avoide 1 l;y the u e of the vory close ly with the mean value obtain d from ratio of two pre mes obtained from the arne record; I' peated mercury tests, or which the greatest devia- namely, the ra t io of the pre ure at a wing orifice to mean va lu e was approximately ± 3 per- tion from the the dynamic pre ure. c ent of the dynamic pre me. Thi s deviation is not a Th e pr eei ion of the mea ured 1'e ult is indicat ed random seatte1' ino' of point from any given te t but is by the variation of tho pro uro diagrams obtained a con istent dift'oronce between repeat test and may 7J 78- 36 -- 2 REPORT T ATJONAL ADVISORY COMMITTEE FOR AERO AUTICS because of the fact that the tips o[ a recta ngular wing he partly accounted for by a po ible mall difference in angle of attack, Fi gure 7 ( 1) ) also includes the rf' ul ts ca rry a larO"er proportion of the load than is indica ted by the theoretical ca l cu l ation on which the method is of test made before anel afte r ca refully poli hin g the ba ed, To make an ac urate experiJllental dete rmina- mid span section of the model. The c han ge in s urfa ce tion of the lift eli tribution on which to base tbe induced- s moothne ss and a s li gh t change in [aime s had no dis- angle calcul at ions would require pr e u re mea me- cerni hI e eO'ect on the eli trih\ltion ; the eli O 'e renccs were 48 ment at eve ral sections along th e pan , es pe cially 2.4 ne ar the tips, . \.n est ima te can be' mad e, howeye 1', of the po sible e rror in the in lu ce d an gle of attack given
I
20 40 herein by compa ri so n of the ded ucod slopes of th e lift c UrYe for infinite aspect r at io obtained [rom these te ts I I anel from the be t force-te t dat a ava ilabl e, uch a ~ \ comparison indi ca te that the indu ce d angle of attack
j
I
m ay be approx imately two-third of th e cal culated
If f
1\
va lu es gi ven h ere in , whic h would mean a pos ible error
V 1/
\~ of approxima te ly W for a lift coefficie nt of l.
I
It i e vident , therefore, that the erl"ective angle of - attack arc u bject to a considerable error of un ce rtain
V II
en
\
> magnitude, I\ pproximate possible elTor hav e been
\
I \
I
08 -2 C
/r c
\
L - - .I
•
i
\ I . .
/ I Mer cu ry (4 tests ) -~
II
l- l o x Tetrabromoe thane
~ I " / I I I I
,/
~ /1 _ x =+_+_+ -t _ + _ +-+ ..... +
- I
V C mC / 4 ............. + - +
•
0' =-4·
/
V ~ x •
~
t -,8
•
o
t_ -16 -8 0 8 16 24
•
Angle of ottock, « ,degrees F"Gl'RE 6. -:'onllal- and chord -f orce coefficients, and pitching-moment coefficient> P • ahout the quaner-chord point. The numerital , -a lu e of C, should be prefixed hy b minus sign.
(a) I I le ss than tho e obtained by rep eat te ts of the same + Mercury (b efore pol is hing) -1.8 ..
0 (aft er ; 2 tests) surface, ~ " x T~ trobromoe thon~ Th e determination of the effectiye ang le of a tta ck t , ~!! ~ + of the mid pan sec tion e ntail s certa in a umptions that e are sub je ct to co nside rabl e unce rtainty, First, the - I O' ~8° angle or attack of this section m ay he in e rr or beeausc of th e as umption that the deyiation of t he air-stream axis from t he tu nnel axis is uniform along t he pan of t he mo l eI ; i. e. , th at the geo metri c angle of attack a o is the snmc for all sec tions alon g tJw span . .\ ct ually ~ there is so me yaria tion of the air- tream direction it tleros the t llnn e l. Be ca use of th e interference of tb e • s uppor t s trut, the cleflection of the st ream in this ~ (b) region migh t reaso nabl y be expecte cl to exceed the ~ .
50 100 deflection at the mids pan ection ; hence, the de fle ct ion Percent chord at the mid span section is probably l es th an the effective P'Gt:RE 7. - Pr ess ur e -di st ribution diagrams f rol11 several test at two a ngle of attack .
mean ya ille . Furthermor e, a zero deflection of the stream at the mid 'pan ection would brin g the angle e timat ed and ummariz e cl as fo llows: Th e va lue of of ze ro lift obtained from the pr ess ur e tests in to agree- the angle as g iy en may be too large by a con tant ment ,, - ith force -t e t r es ult, error of approximat ely W be ca u e of a p os ible e rror .A second ,1 nd rathcr large ource of error lies in the in the a s um ed direction of the stream , On the other de te rminati on of thc induccd ang le of attack, The ha nel , the an gles may be too sma ll by approximate ly method used pr obably pr duc es erroneous res ulL s cdZo, o \\ -ing Lo the error in Lhc inc! llce l-angle cal ulation .
PRESSURE DISTRIB UTIONS OVER THE MIDSPA SECTIO OF THE . A. C. A. 4412 AIRFOIL 9 THEO RETICAL P R ESSU RE DISTRIB UTIO mined by means of the sa m e transformation' . R efer- POTENTIA L -F L OW TH EO R Y ences 1 and 2 pre ent d eta iled discu sions of the under- ly in g theory and th e derivation of the necessary equa- A theoretical d ete rmina tion of th e distribu tion 01' tions for the calcula tion of th e char acte ri tics f the pr e s ur e a bout an airfoil Eection ha been develop ed pot e ntia l fie ld aboll t the airfoil.
fo )' pote nti al flow and a sumes an ideal flu id that is ct ~ _4 ° a ;-8° _\ ct ',= - 7.4" \ o I do ; -4.0° . [ potential flow, some angle Theoreflcol pressure _____ potential flow, some l if t contours - ___ - - - modified (low C( = 8° , do ~ 6.4° ExperImental pressure/ vectors
- -
-
-
/ / / / / f I \ \ \ \ \
"
"
"
'- a, ~/J.5· FIG U RE .- Pressure-vector diagram for the . A. C. A. 4112 airfoil at several angles of attack.
nonvi co us and in co mpr e ibl e. Briefly, the method The general e qu at ion for the local el oc it y about an con i t of the co nformal tnmsformation of the airfoil airfoil section in a p ote nti al flow flS given in reference 1 section into a c il' I e. T hen, ina much as th e flow IS ab out the circle can readily be calc ul ate d, the fl w (1) characteri st i cs abou t th airfoil sec tion can be det,er- REPORT NATIO AL ADVISORY COMMITTEE FOR AERO A TICS where where p i the pre sm e at the airfoil urface and p 8 the pre ure of the free tream.
(2) COM PARJ ON OF THEORY AND EXPElHME T The theoretical distributions of pre ure have been calcul ated for the 2-dimen ional angle of attack corre- V i the velocity of the undisturbed ponding to the mea ured distribution on th J. A.
tream.
C. A. 4412 a irf oil. ompari on of the ca lculated and a, the angl of attack (2 -dimen ional).
mea ured di tribution are pre ented in figure .5 (e x- r , the circula tion.
ILlding the diagram after tbe airfoil ha tall d ) and 0, >/I, f, parameters that are functions of the in figure . Figure:5 pre ent the 1I ual normal- a nd airf oil coordinates.
hord- omponent Ire ure diagrams and provide s a >/1 o, the mean value of 1{1.
means for a general tudy of the difference be tw en R = ae"'o , the radius of tbe conformal circle the theory and experiment a a fun ct ion of angle of about which the flow is calculable.
attack. Figure provide s a more detailed tudy at 11 In order to calcul ate the veloci ty field from equation few ang les of attack an 1 pre ent vector diagram for (1) th e circu lati on mu t be eva lu ated. Thi evalua - the angle of - 0, _4°, 2°, 0, and 16 °. The se dia- 2.4 o-rams we re obta in ed by plotting the pr e me coefficien t
/
normal to the airfoil profile ; tbe perpendicular cl i sta nce Usual theory from tbe profile line repre ent the m.agnitu Ie of the I-- - 2.0
lL -
Mo dified theory _ ____ / coeffi.cient. Th e experimental pressure ar repre e nte 1 I-- by the drawn vector and the theoretical pre s Lire by Experiment 0 + tc ,,,- or<> tbe solid contour line. The other contour line repre- 1 / I ent ertain modified calcul ation to be discu edlater.
/ It is i mmediately evident that the theoretical re ults
/ I
1.2 do not ati factorily agree with the actual measure- I I: ments except for angles of attack near - 0, corre pond- f io
/ / ing approximately to the ang le at which the experi-
men tal and theore tical lifts are the same (fig . g). Th e
/0'
I
comparisons in figure 5 how , moreover , that with in- / I crea ino- angle of attack the difference between theory and experiment be come larger as predicted b y the !: higher lope of th theoretical lift curve. A detailed 1 ) 0 + + o study of the vector diagrams (fig. 8) shows how the e
j -+ -+
-+- difference vary around the profile of the airfoil. The +-+ -+ '+ .- + ......
c mc / + 4 l argest differences occur in the regions of low pres sures,
'j
-.4 or the high-veloc ity areas, and as previou ly tated they increa e with in crea ino- angle of attack. Furthermore,
of the per centage diff rence in pre ss ur e is larger near the
-:8 / trailing edge than in the reg ion of the n ose, indica t ing a progre iye influence on the flow a it move over the -/6 -8 0 8 /6 24 EffectIVe angle of attack, a. ,degrees aiI-foil urface.
FIGURE 9.-Lift and pitching-moment section characteristics ror the . A. C. A.
Th e e ffe ct of these difference in the pre ure di tr i- 4412 airroil.
bution on the pit c hing-moment characteri tics is hown tion i done by the u e of the Kutta condition, which in figure 9. The theoretical pitching moment about require that the velocity at the trailing edge (O=7r) be the quarter-chord point was obtained by integrating zero 0 that equation (1) become theoretical pre ure diagrams. The result show an in- creasing diving moment with increa ing angle of attack, ( la ) wherea the diving moment actually decreases.
where fT is the value of f at O=7r (t railing edo-e).
The co mpari on have thu far been made at the The angle of zero lift is equal to - f T ' ame r e lativ e angle of attack, that i , for the ang le of Th n __ ary eq uation and a step-by- tep de cription attack in 2-dimensional flow. Another cond iti on of of the calculation of the velocity field are given later .
compari on that ha been u ed more or less regularly The pre ure coefficient are computed by mean of in previoLl s tuelie i ugge ted; it allows a comparison Bernoulli ' equation, at th e ame lift and con i t in com paring the theo- retical distribution calcul ated at an ano- Ie of attack (3 ) that give a theoretical lift eq Llal to the experimental PRESSURE DISTRIBUTIO S OVER THE MIDSPAN SECTIO OF THE . A. C. A. 4412 AIRFOIL 11 potential one; probably the pre UTes may also be value. This method has been u ed fOT the di aO'ram considered a being tran mitted undiminished through in figUTe and the di tributions thus ca l cu l ated are the thin boundary layer. The actual flow might there- repre ented thereon by the long- and -shor t- da h con- (ore be replacerl by a potential flow about a shape tour lines. Again the difference are too large to be lightly different fr om that defined by the aiTfoil neglected, especially at angl es of attack where a large coordinates, which would require the determination of lift i obtained. At - 0 the curve coincides \ovith the the boundary -layer thickness to define the effective previou ly de cribed contour, ince the angle and the profile shape. The pressure about the new shape could lift are the same, while at _ 4 the di str i bution cal- then be computed by the potential theory. Boundary- culated on the ba is of the ame lift is approximately layer calculations, however, are at present subject to the same a the da hed contour represent i ng a third unr.ertainties that , would ca t doubt on the validity calculation presented herein. At the higher angl es of of the results and, in addition, the eomputations are attack the calculated distr i bution depart progres- ively in hape from the mea ured di t ri but ion . It difficult and tediou .
may therefore be concluded that, on the basis of the e M ODIFI E D TH EORETICA L CALCUL ATIONS compar isons, the usua l cal cu lations from the potentia l A simpler and more practical method of calculating theory do not give an accurate determination of the the pressure over an airfoil se tion has been developed di tribution of pre lire about an airfoil.
The inaccurate prediction of the forces on an airfoil by the usual potential-flow theory i not urpn rng -3 ince the theory neglect the frictional force of the 01 =8· vi cuous fluid acting on the airfoil. The direct efl'ect Experiment x Usual theory of tbi force, which act tanO'ential to the direction of - -- - -- Reduced circulation
1\ - - -- - Modified theory
tbe local £low, is important only on the drag and -2 contribute what i known a the "skin -friction" drag.
i'--....
Because of the small magnitude and the direction of p
this force, the component in the lirection of the li ft is ~ ~ ~---;:~
~.,.."""" ........ -
-/ probably negligible, the lift being determined en- x ~~~l:---- ...... x tirely by the pre ure force. The indirect effect, how- I ---X...::..
~ ever, of thi friction forc is the deceleration of the air "j(~ in a thin layer near tbe urface of the airfoil and the
o ~~
production of the so-called "boundary -l ayer" phe- x h ~-
,-
\ Af'. ~ nomena , which are importan t in the developmen tor lift by an airfoil. In ttl boundary layer the velocity
If
hanges rapidly from zero at the surfa e of the airfoil to tbe value of the local stream vel ocity at the ou tel' limit of the layer. The loss of energy involved in ove1' - 100 o 50 Percent chord cominO' the friction force re ult in a cumul at ion of F, GU RE JO .- EIT ect of an a rbitrar y reduction of th e ci rculati on OJ] t he cal cu lated slowly moving air a the flow move back along the pr essure di st ribution.
airfoil; hence the boundary-layer thickness increa e toward the trailing eelge. Thi eUffiulative efi'ect is as a re ult of the foregoing analysis. The anaiy is indicated by the progre ive increa e in the dill'erences how that theoretical distributions calculated at the b tween the theoretical and measured preSSUl'e . true angle of attack are imilar in shape to the true From this discu sion it is not to be pre umed that distributions but give too high a lift. Conversely, agreement between the mea s ured and calculaterl results when the theoretical eli tributions are calculated at an should occur at zero lift, except approximately for a anO' le of attack that give the same lift as the e:\."peri- symmetrical airfoil ection. The velocity eli tribut ion mental distribution, the two eli tributions are di similar over the upper and lower lU'Jaces of an a ymmet rical in shape.
ection are not the same, ~ven at zero lift.. The viscou The modified calculation i made at the effective effects on the flow over th e two urfaces at the calcu- anglo of attack but the circulation is determined from the e:A"perimentally mea urecl lift instead of by the lated angle of zero lift arc therefore different and a lift is measlU'cd, wlei ch i nega tive 1'01' mo t ection . Kutta -Joukowsky method. The preliminary calcula- J .. tually, then, the xperimental and theoretical angles tion made on this ba is re ulted in an exce ive velocity of zero lift are not the samc and for normal ections and a con equent high suction pre ure at tho trailing the two lift cmves inter ect at a negative value of tl 1f' edge, as shown in figme 10. Thi unsati factory result lift coefficient. .
( hown by the dot-da h line in fig. 10 ) wa finally Outside the boundary layer the vi co us forces can avoided by means of a further modification subsequently probably be con idereel negligible and the flow a de cl'ibed.
REPOR1' TA 1' IO T AL ADVISORY OMMI'l"l'EE FOR A ERO I AU'l' I CS ince a change in the e O' ectiye pl'on.1e . hape ha s be en tions obta in ed by means of the modified c akuia tions are given by the ela hed line. Th e relative merit of pr ed icted by boundary-layer con ie!erations, an :11'bi- tL'al' Y modification of t he hape parame ,ter € i made so the unalter cl pot e ntial theory a nd the modified method L h at t.he ve lo city becomes zero at {) = 7r. ( eo eq uation for the calcul at ion of the pr es ure distribution about an (1).) The shape i 't hu altered to s ati fy again th e airfoil ection is hown in figures 5, 8, and 9.
Kut ta-Jonkow ky co ndi tion . In order to maintain the Th e following tep-by- tep description of the compu- ccntinuity of th e € c urv e, a s tudy ha been mad e of the tation required to obtain the calculated pre s ure dis- manner in which € should be modined, Th e indicat ed tribution i g iven in uffi ci ent detail to e nable the calcu- cum ulativ e e fl'e ct of the vi co ns forces toward the lations for any airfoil to be mad e. Th e local ve locity tmiling edge how that most of the chan ge in € houle! a bout the airfoil is co mputed by mean of e quation (1) \ / \ 1 \ / \ I
II
k ' I 1\ II \ I /
"-
..........
..........
V / k ' --..!... k ' I- I ~ I -~ I I ~ I o I
I I
1 , --...
~ .20 V I'-.- .16
""
I i"-.
./2 ........
/ (, /'
"'"
I/J
--
/ "- /" / \ .08 1 "'-.., /' "- k;:- E /" V \
"""
/\'
E "
r "'-.., "-..
I/Ii / .04 If; .......... V
~ j
I- I-- r- r- ........
--- I
"'-
-
d E 0
'"
d O j'-" /
II
d E .
V '"
1· \ V -.04 dO /' V
I, "'"
.-/
""
f-- ~ -: 08
-
---
........ V
--
~
-
r-- -: 12 I-- o ./ .2 .3 .4 .5 .6 .7 .8 .9 1 .0 e .1I 1.2 /.3 1.4 1.5 1.6 I. 7 /.8 /.9 2.0 7r FIGl1HE ll.-Theoretica l paramete rs req uir ed to compufe the theoretical pressu res on the N. A. C. A. 4412 ai rf oil.
p robabJy be made in that region. In asmu h as the mod ifi ed a indic at ed by the preceding disc ussion. T he e ffe ct of changing € i not c riti cal for different eli s- detailed form of the modifications are introd uced a tributions of th e chan ge, proyirled that mo t of th. e they a ppear in the cour e of rout in e computation .
change i made nea l' the trailing edge of the airfoil, a In order that the tran forma tion from the airfoil to pur e ly arbitrary di tribution is eho en that pe rmit it conformal circle may be of a co nv enie nt form , the r eady application, nam ly, ainu oidal yariation with e. coo rdin ate axe are elected 0 th at the profile i a Th e € curve an 1 ub eque ntly the other pa ram eter nearly as po ible ymmetrical about them. ( e re fe1' - mu st be modified for each anO'le of attack. Thi modi- ence l. ) Th e x a:\is is chosen a the line joinin the fi cation ha s been mad e and the corre ponding pr e ur e ce nt er of th e l ea clin g- and trailing-edge radii. The di tribut ions determined for everal angle of attack. origin i located mid way b et ween a point bi ec ting (See figs. 5 flnd .) At - 0 th e eli tr ibu tion i the th e di tance from the leading eelO'e to the e nter foJ' ame a that hown by the olid line repre ent in g the the lead in g-edge radiu s and the corre ponding point unaltered theory. In the other diagram the di tribu- at th e tra ilin e elO' e; the c oordinat es of th e e point s are P RE S U RE DI S'1' RIB U '1'IO S OVER THE MID SPA T E C '1'l0 . OF THE T. A. C. A . 44 12 A IRFOIL 13 re pocti ve ly (2a, 0) a nd (- 2a, 0). In th e following
Th e v alu e of E at e = ~ ~ i given by th e [o lJ o win g
n di s C'u ion th e c oordin a t e c al e h as been cllo en 0 th at e quati on.
a i unit y. ( For pr ac tic al purpo e it i prob a bl y s uffi- cion t to hoo se th e c hord joining th e e xtr e miti es of th e
E ,,= - ! [ ~( ~ ~) n+ 1.09 J ( .J; n+L- tf; n- l)
mea n line as th e x axi .)
+ 0.4 94 ( .J; ,,+2- .J; n-2) Th e following eqllUtion ex pr ess th e r ela tion hip + 0. 313 b e tw een th e a irf oil cOO l'lin a te pr ev iously de c rib ed + 0.217 a nd th e par am ete r e and.J;.
( ) + 0.15
x = 2 cosh .J; co e
+ 0.11 5 (4)
y = 2 inh.J; s in e
+ 0.0 04 + 0.05 11
In o rd er to co mpu te va lu e of e co rr e p ndin g to any
+ 0.025 1 ( .J; n+9- .J; , /-9) ] gi ve n po in t on th e a irf oil pr of il e, equ a tion (4) ar e oh ed for s in e.
wher E' t be ' ub c ript s de ig nate th e parti c ular e at which
(5) the named quanti ty is tak en. A pl ot of Ea a run c tion
0 1" e I" o r th e N .. \ . C . . \ .4 4 12 airfo il j give n in ri gure
II -h ere 1l. Thll far tbe calculations arc iden t ic al wi th t bo c
I - 1 (. r)t (V )2
I ~- - - - - m ade I" or t he po ent ial t heo ry.
2 2 .\ s s tater l in th e eli cu sion o f th e modifi ed L h eo r eL ica l caieul at ions, t be circula t ion i eya Ju ute d by th e E'x peri- A. imil ar so luti on Jor sinh .J; can h E' ob ta in ed bu L mentnJl y known lift of th e airfo il ect io l1. Th e wcll - expe ri ence h a h o wn t ha t a more u a ble so luti on i kn own eq ua tion rein tin g th e lift a nd th e c ir c ul at ion is oi ven by th e equ at ion b elo, AI 0 by cl e fini t ion (6) A l)lot of .J; a a fl mct ion of e f l' th e N. \.. C. A . 441 2 E x pr es sin g th e c ir C llla tion in te rm of tb e lift coe ffi cien t, a irf oil is gi ve n in fi gure 11. Th e fun ct ion.J;o i gi ve n by
1 J' z". a nd f i na ll y
.J; o= 2 7r 0 .J; de r c (9) l 47rR V =, 7ri l a nd c an b e d ete r m in ed g ra p11i cn Jly fr om th e .J; c ur v or by a U1lm e ri cal eva lu at ion . Th e valu e of .J; o for th e ub s ti t utin o- th e num c ri ca l va lu e Jo r th e N . j t. C. \ .
J . C_ A. 4412 a irf oil i 441 2, (9 <1 ) .J; o= 0.1044 Th e pr edi ct ion o f unr ca. ona blc ve locitics ar ound til e
Th e pa ram ete r E as n. fun ct ion 0 1" e is giv n b)T th c
t railin g edge i av o ided by a lt erin g th e E fun c ti on so d efinite in tegral, t ha t th e Yel oc i ty i z er o at e= r. . Th o a lt ercd fun ct ion (7) i de ign ate d E a and i a rbi traril y a s um ed to be given
1 J 2~ e- e "
€ n= - ~ .J; cot - ?- de by ~7r 0 ~ D.E r ( )
E a= E+ - I - cos e (J 0)
wh ere th e lib c ript " r efer to th e pa rti cular valu e of
e for which th e co rr e p o ndin g va lu e of E is to be det er-
wh er e D.E 1' is th e in cr em e nt o( E re quir ed to gi ve zero min ed . A 20-po int num erical eva lu a tion of thi int e- veloc it a t e= 7r a nd is a fun c tion of th e an gle of at ta ck .
g ral i d e ri ve d in r fer nee 1 a nd i includ ed h ere for Th e qu a n t it y D.E r i o- i ve n by con ve m en ce. Th e in teg r al i eva l uate d a t 20 equ al int p- l" va l va lu e of e, n am ely, wh ere E ar i d ete rmin ed by equ ating equ at ion (1) to zero a nd ub tituting from equ a tion ( 9)_ in ( 7r + a + E a , I' )+ ~ R c, = O o So lvin · for E ar g ives, 14 REPORT NATIONAL ADVISORY COMMITTEE FOR AERO AUTICS Th e parameter e and if; are conjugate fun ct ion of 0, Differ e ntiatin g equat ion ( 10 ) and if; is given by
de a _ r!::.+D.f7" {J
do - do 2 sm v
1 J 2.- 0- 0"
if; n= 27r 0 e cot- - dO + if; o Plot of ~; and k' as fun ct ions of 0 for t he . A. O. A.
where the definite integral can be evaluated in the 'd' t f th 4412 airfoil are given in figure 11. Equation (1) for Th same manner as equat ion (7). e COOl rna es 0 e hI' . h' f'l f'l" t e ve OClty at any pomt on t e arrOl pro 1 e IS now profile corresponding to the modified e function can be wTitten obtained from the new if; function by equations (4)· (lb ) Figure 12 gives the modified hape obtained by thi method fOT the . O. A. 4412 airfoil at a = 0 and 16 .
Th e pTofiles given in figure 12 are, of co ur e, only Th e generality of the preceding method of cal- effective profiles corresponding to the ca l cu lations. culating the pres ure di tribution a bout an airfoil Th e actual profile about which a potential Dow might ection i upported by the following evidence. Fir t, be considered as being established would be blunt at no restri ting assumption h ave been made in the the trailing edge and would have the thi c kne of the development of the method. Second, the ci.rculation is determined by a known quantity , th experimentally waJ.ce at that point. Th e thickness of the boundary la yer on the upper urface, however, i greater than mea UTe 1 lift . Third , the c hang e in th e cfl"ective air- fo il hape is in the dir ect ion indi cated by boundary- that on the low er urfne ; therefore, if the trailing edge layer con ideration. Fin ally, the computed a nd mea - were taken a the midpoint of the wake and the after UTed pre s ur es agree at i fa ctor il y.
portion of the proftle were fau'ed to that point, the a' ~ --1 6.
~-(Jo , LANGLEY : MEMORIAL AERO AUTI C AL LABORATORY, j N.A.C.A. 4 412 NATIONAL ADVI ORY OOMMITTEE FOR AERONAUTIC F,G U RE 12. -Change in profile shape associ ated with tbe modified theoretical cH lcu- LA NG LEY FIELD, VA., March 25, 1936.
lation of pressure.
R EFE R ENCES re ulting h ape would be imilar to the e{feeti e profiles in fio-ure 12.
1. Th eodor en, T. , and Garrick, 1. E.: General P otent ial Th eor y The influence of the c han ges in if; on the value of lc of Arbitrary Wing ections. T. R. o. 452, 1 . A. C. A.
1933.
are fO lmd to be negligible 0 that ka may be writt en 2. Theodor sen, Th eodore: Theor y of Wi ng Sections of Arbi- trary Shap e. T . R. No. 411, . A. C. A ., 1931.
3. Ja cob, Ea s tman ., and Abbott , Ira H.: The I . A. C. A.
Variable-Den s it y Wind Tunn el. T. R. 1 o. 416, . A.
where C. A., 1932.
4. MiUikan, Clark E. : On the Lift Di tribution for a Wing of Arbitrar y Plan F orm in a Circular Wind Tunn el. Pu b- li cat ion No. 22, C. T. T., 1932.
. A. C. A. 4412 AIRFOIL 15
PRESSURE DISTRIBUTIONS OVER THE MIDSPA SECTIO OF THE T ABLE L -E X PE RIM E TAL DATA- . A. C. A. 4412 AIRFOIL [Average pressure (standard atmospheres): 21; average Reynolds Numhe r : 3,100,000] Values 01 pressure coefficient, p=P-:"" lor different angles 01 attack Orifices Sta- tion Ordi- 'v (per- nate Des- cent c (per- 160 1 200 1 _J201 1 _6 0 2 _40 2 _2 ° 2 2 80 2 12° 1 1 8° 1 24° 1 30° 1 _20° I 1 _8° 0° 2° I 4° I igna- -16° Irom rent c tion L. E. above 01 chord) chord) - -- - - --- --- -- - --- --- --- --- - -- --- --- --- --- -- -- - - - -- --- -- 0.134 0.101 0. 010 -0.062 -0.173 -0.466 -0 . 513 -0.42t -0 . 199 0.114 0.198 0. 217 0.204 0.207 0.200 0.181 0. 158 28 100.00 0 .224 .1 1 .178 .183 .164 .157 .167 . 140 .121 .094 . 0 49 -.291 -.304 -.16 -.454 -.251 .159 1 97.92 _179 _127 .1 5 .151 .166 .154 .156 .1 80 .166 .166 - .16 0 -.167
29 : f~~ 94. -.16 - .466 -.291 .107 1.52
.1 53 .122 .128 . 140 . 152 .160 .203 .199 .231 .237 .224 - .030 -.036 2 - .22 -.505 -.330 .074 89.90 .270 .107 .072 .082 .098 .118 .211 .2 12 .257 .283 . 049 .042
: ~! ~ 84. 94 -.28 -.382 .035
_ 348 :t~ -.043 . 055 .049 .068 .095 .126 .136 .231 .261 .322 . 374 . 179 .179 31 =:~~~ 74.92 -.52 -.4 54 .453 .002 .000 . 028 .062 .104 .120 .154 .244 . 2 3 .374 .407 .270 _289 64.94 -.84 -.564 -.539 -.101 .452 .492 -.082 -.063 -.024 .021 .072 . 1 00 .157 .250 . 309 . 414 .348 .36 8 54.48 -1.24 - .671 -.64 3 - . 199 .472 - .252 -.115 -.099 -.053 - . 005 .09 1 .134 .252 .3 16 .426 .531 . 38 1 .407 -1."14 -.571 - . 695 33 49.98 -.160 -.128 -.075 -.0 17 .0 .140 .268 .342 .459 .505 . 670 .413 .4 46 4 :~~ -1.64 -.571 -.721 - . 304 -.169 -.105 -.04 1 .03 1 .071 .136 .265 .362 .4 5 .544 .609 .466 .498 -1 .86 -.558 -.754 -.368 ~:~~ -.217 -.146 -.073 .010 . 066 .133 .290 .387 .516 .676 . 642 .504 .544 =:~~~ -2.10 -. 55 1 -.773 -.447 5 34.90 - . 330 -.2 74 -.190 - .105 -.011 .048 .116 .293 .414 .551 .609 .687 .557 .596 - 2.30 -.54 5 -.76 35 29.96 .4 33 .589 .661 .726 .609 6 -.427 -.367 - .266 -.165 -.054 .025 .115 .313 .64 6 =:~ - 2.54 - . 545 - -.244 -.111 .321 .62 7 .68 7 .752 .642 .700 -.551 -: 819 -.591 -.490 -.365 -.Oll .0 93 36 i~:~~ -2.76 - . . ~~
:m -.799 - .663 -.502 -.348 -.053 . 076 .345 .713 .785 . 57 .733 .778
-2.90 - .558 -.825 -1.17 -.lSO
7 14.94 -1.143 - .946 -.71 6 - .501 -.279 -.111 . 059 .402 .616 .818 3 24 .876 -2.86 - .551 -.832 -1. 660 9.96 l : ~ig .961 3~ -1.407 -1.153 67 -.596 -.333 -.131 . 071 .462 96 .902 .941 -2.72 -.577 -.916 -2.070 - 7.38 1.013 1. 046 .948 -2. -1.861 -1.490 -1.106 -. 777 - .428 -.150 .109 .90 .9 0
:m 4.94 -2 .46 -. 5 71 -.897 07
3~ . 948 :~ra -1.931 -1. 30 - . 932 - . 467 -.098 .231 .948 .993 .909 3 .941 2.92 -2.06 -.702 -1. 242 -3.745 .791 .696 .433 .602 -4.940 -2.478 -L 709 -L059 -.436 . 028 .409 .916 .974 .713 =~:1~ 1. 66 -1.60 -1.053 -L947 .264 -.173 -.518 .003 -6 . 177 -3.770 -2.765 -1.812 - . 995 -.266 .254 .643 1. 013 .831 .244 .92 -1.20 -2 . 082 -3.212 -1.379 -2.285 - 3.012 -1.671 -7.337 -4.052 -2.732 -1.559 -.631 .156 .639 .924 .905 .094 -1.059 40 .36 -.70 -3.204 -4.300 -1.555 - 3.648 -5.060 -6.073 -3.695 -3.433 -6.480 -2.397 -L 232 -.296 .356 34 . 99 .952 . 157 -2.382 11 0 0 -2.623 -3.250 -6.230 -7 . 775 - .941 -2.625 -.538 .184 . 681 1.010 .854 .473 -1.000 -3.730 41 . 68 -1.178 -1. 549 -3.738 - 5.961 -7.125 -7.954 = ~: ~~~ -.043 .765 .955 .994 .720 .336 -.202 -1. 740 -2.552
12 : ~;~ .44 1. 56 .322 .231
-3 . 399 -6.110 - 6.61 - 3.81 .596 .974 1.009 .939 .770 . ' 168 .055 -.456 -1. 793 -2.006 42 .94 2.16 . 739 .720 =t;~~ -3.05 3 -5.190 -5 . 620 -3.010 .8 3 1.000 .939 .782 .569 .246 -.148 -.611 -1. 743 -1. 249 1. 70 2.78 .928 .935 -2.637 -3.765 -4.285 - 4.562 -2.200 .974 . 896 .761 .559 .332 .0 18 -.336 -.728 -1. 647 -.76 2.94 3.64 . 987 1. 000 -2.343 -3.190 -3.570 -3.731 - 1.529 96 . 713 .542 .333 .110 - . 179 -.4 5 - 13 - 1. 547 -.695 4.90 4.68 . 922 .935 -2.057 -2.709 -2 .981 -3.060 -1.235 : 752 .498 .344 .139 -.066 - . 312 -.568 - 31 -1.432 - .644 7.50 6. 74 .804 .798 -1 . 912 -2.440 -2 . 662 - 2.681 -1. 059 .374 . 208 .0 17 -.168 -.388 -.623 -:872 -1.391 -. 630 9.96 6.56 . 687 . 687 -1. 02 -2.240 -2.415 -2.32 -1.007 :~~~ .263 .089 -. 091 -.271 -.468 -.676 - 99 -.611 45 12.58 7.34 .576
=Ugg -1. 769 -2.149 -2 .285 - 2.10 -.95 5
:~g~ .4 5 .407 .178 .014 -.152 -.309 -.500 -.700 -:912 -.604 16 14.92 7.88 -1. 272 -1.620 -1. 952 -2.062 -1.94 -.910 -.604 .407 .329 .100 - . 052 -.210 -.360 -.721 -.910 46 17.44 8.40 .414 -1. 239 -1.548 -1.841 -1.927 - 1.815 70 =:~~ .335 .257 .036 - . 111 -.262 -.402 -.740 -.914 - -.698 17 19.96 8.80 .335 -1. 758 -1. 22 -1.65 - 51 .257 . 172 - .024 - . 176 -.322 -.452 -.609 -.769 -.930 -1. 224 -.591 47 22.44 9.16 . 263 -1. 640 -1.692 -1.592 - 25
=i:m .211 .140 -.063 -.196 -.332 -.454 -.599 -.746 - 95 -1.163 -.591
18 24.92 9.62 .212 -1. 347 -1.573 -1. 391 - 12 _165 .100 -.096 - .228 -.355 -.471 -.606 -.H2 - 1 -1.122 -. 591 48 27.44 9.62 .166 =U ~~ -1.280 - 1.463 -1. 254 - .76 .133 .0 68 -.114 - .241 - . 364 - .469 - . 594 -.722 51 -L071 -. 591 19 29.88 9.76 .114 - -.473 -.693 -.92 -1.144 -I. 269 -1.255 -.760 -.591 .036 .055 .009 - . 154 -.275 -.3 81 - .596 - 04 49 34.98 9.90 -~:~~ - 1.007 -1. 099 - 1.059 -.727 - .030 -.173 - .272 - . 370 -.447 -.542 -.635 - . 732 - 0 - .584 39.90 9.84 -.017 .009 -.902 -.961 -.910 - .655 -.720 -.591 -.095 - .044 -.069 -.194 - .291 -.371 -.439 - . 519 - . 6J9 -.691 -:S09 50 44.80 9.64 - .759 -.76 -.734 - .538 - .715 -.121 -.0 56 -.075 - .173 -.256 -.329 -.39 -.455 - . 525 -.595 -. 690 -.591 21 49.92 9.22 -.64 9 -.649 -.584 - .473 - .700 - .147 -.069 -.075 -.161 - . 238 -.303 -.3 51 -.406 -.471 -.527 -.601 -.591 51 54.92 8.76 - .576 -.55 1 -. 460 -.414 -.199 - . 095 -.161 -.244 -.298 -.3 ' 12 -.391 -.487 -.541 -.591 22 59.94 8.16 -.460 -.414 -.343 -.369 -.51
=: }~~ =:~~~ -.225 -.082 -.128 -.214 -.264 -.296 - . 334 -.421 -.456
52 = :~~~ 64.90 7.54 -.375 -.316 -.264 -.337 -.62 - .252 -.02 -.115 - . 11 -.225 -.250 - 2 -.319 -.351 -.371 -.54 23 69.86 - .264 -.212 -.212 -.310 -.65 5 -.277 -.056 -. 082 -.14 8 -.183 -.200 -.222 -.252 -.279 -.285 - 4 24 ~:~ =:i~ 74.90 -.180 - . 147 -.173 - .291 -.642 -.297 -.147 -.069 -.076 - .11 5 -.144 -. 155 -. 1 69 -. 191 - .210 -.199 -.57 53 79.92 -.082 -.02 - .14 0 -.271 - .604 -.330 - . 154 -.024 -.024 -.068 -.091 -.09 4 -.101 -.116 -.113 -.106 -.565 25 ~:~ 4.88 -.009 - . 004 - . 043 -.114 - .246 -.565 -.552 2.74 -.161 .022 . 028 - . 006 -.019 -.016 -.017 -.026 - .032 54 9.88 .079 . 062 -.016 -.095 -. 226 -. 519 -.519 -.174 .075 .100 .073 .069 .078 .0 2 .076 .0 70 26 =:~~ 94.90 1.48 .147 .1 27 .1'20 .088 -.004 -.075 -.200 - .479 -.506 .68 -.4 34 -.200 .127 .165 .141 .13V . 150 .143 27 9.00 28 100.00 1 ' rest, variahle-density tunne l 109 ; manometer Hquid, mercury. 2 Test, variable-density tunnel 1099-4; manometer liquid, tetrabromoethane.
TABLE n. - INTEGRATED AND D E RIV E D CHARAC- TERI TIC S- N. A. C. A. 4412 AIRFOIL lX C. c, CI lXi CmcJ' "" ------ --- --- -- - --- --- Degrees Degrees D egrees -20 - 0.592 0.0318 0.030 -0.54 5 -0.9 -19. 1 -1 4 .
-16 -.767 -.0170 . 0 35 -.74.2 -1.2 -10.8 - 12 -.722 -.1264 -.092 -.732 - 1. 2 -7.4 -8 - . 372 -.0445 -.096 -.374 - . 6 -.210 -5 .7 -6 - . 0151 - . 096 -. 211 -.3 - 4 -.0256 -4.0 .0043 -.095 - .0255 -.0 -2 . 146 .0107 -.092 .146 .2 -2. 2 0 .338 .0098 -.091 - .5 .338 .6 2 .501 -.0034 - . 087 . 501 1.2 4 . 677 - .0258 -.07 .677 1.1 2.9 1. 020 -.1003 - .084 1.024 1.6 6.4 12 1. 275 - .2043 -.074 1. 289 2.0 10.0 16 1. 548 -.3357 -.068 1. 579 2. 5 13.5 1 1. 626 -.4040 - . 063 1.671 2.6 15.4 20 1.640 - .43 74 -.080 1.690 2.7 17 .3 24 1. 212 -.1838 -.141 1.182 1.9 22. 1 30 1.009 -.0776 -.146 .913 1 .4 28.6 16 REPORT A TIO AL ADVISORY C OMMITTEE FOR A ERO ! AU TICS T ABLE IlL - TH E ORETICAL PARAMETERS - . A. C. A. 4412 AIR F OIL
Station I Ordinate I 0 Ordinate I
E. I y
0+, x 0+, x (percent (percent - V- (percent V V- 11" 11" e) e) e)
I
I I I I I I
I Upper surface Lower surface deg. min.
I deg. min.
2.032 0 0 O. J78 -4 JO 2.032 0 0 0.17 -4 JO -- ---- ------ ----- - 2.031 . 0121 -.012 0 0.62 .007 - 2 4 0 2.031 -.0129 -6 20 -- -- --- -- - .25 1.25 2.021 .0375 -.034 .034 2 I -.00 2. 021 -.037l -JO 49 ---- - ------ .5 1.64 2.011 .05.32 .046 2. Oil -.0484 -.043 -12 52 4 35 - ------ ------ 1.25 2.44 1.981 .0855 -.069 -1 .073 .1 9 48 -1.43 1.980 -.0706 .162 1 2.5 3.39 I. 931 .1239 .103 -.09 -23 48 .194 15 29 -1.95 I. 930 -.0916 .151 5 4 73 I. 830 .1784 .146 .201 -2.49 1. -. 1130 -. 140 .133 -31 47 23 4 29 7.5 5.76 1. 730 .2203 . J79 . 205 -2.74 1. 729 -.1227 -.172 . 119 -37 39 30 19 10 6.59 1.629 .2542 .207 -2.86 1.628 -. 127l -42 .20 35 -.200 .10 36 15 7. 9 I. 427 .3075 .254 . 2J3 25 -2 . 1.426 -. 1271 -.249 -51 11 45 .090
I
20 0 1. 225 . 3446 .296 .213 -2.74 1. 224 -.12 10 -.292 -58 32 53 56 .076 25 9.41 I. 024 .3696 .333 .211 61 26 -2.50 1. 023 -. IIJO - .330 .064 -65 2 30 9.76 24 .3 45 .369 .208 32 -2.26 20 -. 1 005 - . 366 .055 -71 6 40 9.80 .<ll .3873 .435 .1 97 1 42 -1.80 .41 7 - .0807 -.435 .04 1 -2 47 50 9. 19 .0149 .3640 .500 .181 94 32 - 1. 40 .013 7 -.0633 -.500 .030 -93 45 . 14 -.389 60 .3229 .560 .1 63 106 11 -LOO - .390 -.0· 1 60 -.560 .023 -103 50 -.792 .2655 70 6.69 .628 . 143 119 0 -.65 -.793 - .0307 -.629 . 0 17 - 11 5 12 4.89 - 1. 196 . 19 ·W 0 .702 .12 1 132 4 -.39 - I. 197 -.0190 -.70· 1 . 012 -127 32 85 3. -1.398 .151 .745 .106 140 32 -.30 - L39 - .0149 -.746 .010 - 1 34 28 90 2.71 -I. 600 .10H .793 .089 149 23 -.22 - I. 600 - . 0 1 09 -.795 .009 -142 25 -1.802 95 1. ' 17 .0577 .855 .065 1 59 56 -.16 - L802 -.00 J - . 856 .009 - 152 31 98 .68 - 1.924 .0262 -.912 .910 .047 J69 J -.14 - 1.924 - . 0069 .0 12 -16 1 34 J OO . 13 -2 .003 0 1.000 .025 1 4 3 -.13 -2.003 - . 0013 -1.000 .02 5 - 1 75 57 TABLE IV.- TH E OR E TICAL PARAM ET E RS - N. A. C. A.
4412 AIRFOIL (/ , a", E.
V- k' dO d8 0 0 6.201 -0.0727 0.0600 .1 . 0611 3.04 1 -. 05 4 .0755 .2 .0395 1.7 77 -.02 · 11 35 . 3 -.0 11 6 1.326 .0120 · J 220 .4 -.0527 1.139 . 0492 · j()95 .5 -.0866 1. .0797 . 0800 .6 -.0942 1. 147 . 1 002 .0515 . 7 -.1028 1. 350 . 10 7 .0230 -.1166 1.856 . 1109 -.0130 .9 - .1 016 3.528 .0975 -.0720 1.0 -.0590 .0706 -.0960 1.1 -.0249 3.589 .0403 -.0970 1.2 .0020 l. 7 .0115 -.0925 1. 3 .0169 I. 372 -.0153 -.060 .02 4 1.4 1. 1 67 -.03 -.0785 .0434 -.06J2 1. 5 1. 1 09 -.0720 1. 6 .0661 1.163 -.0837 -. Q6. 1 0 .0976 -. 1029 1.7 1. 36 1 - .0505 1. . 12 11 -. 11 26 1. 45 -.02 10 1.9 . 1 361 -. 1070 3.205 .0640 2. 0 0 6.201 - . 0727 . 0600 H . S GOVERNMENT PRINTING OFFICE: 19 36