Document
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NATIONAL ADVISORY ‘COMMI’rrEEj_ ‘ .:’
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FORAERONAUTICS
.—: .- r. - - ,.
.+ ..- ---- TECHNICAL NOTE . — .
No. 1448 ------ ------- .. .
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SUPERSONIC WAVE DRAG OF SWEPTBACK TAPERED .- WINGS AT ZERO LIFT -r.
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.._ ByXennethMargolis :: ~ ‘- _ -:: — . . .. ......_— ._ .
I, -Langley MemorialAeronauticfiLaboratory LangleyField, Va.
...—— .-. ,—.
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Washington October 1947 . ---- .-:. -- ... . . ... . . .
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:%?+[ ;LEY M WWL4L AEE?h<icA 1.
LAmumicY .- .“ L+iw W Va.
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.- L-, . . . . . . . ..-_ ~~— , .4., l NATIONAL ADVISQRY C0MMD3XE 3URAERONA~ICS ,, .
~AL iwm No.” Ww””.: “. -.: ‘“ “““ : ‘
,.
,.> ,,, -.— ‘.- -, .>. . . . . .
,. ,- — .— > EXiE!ERSONIC WSZEIIR.N2 OF QS4F2TBACK TAPEKED . .
,, . . ..
llm~ ~. ~o L~T” “ “ ,. “- . . ‘..:. . .._.__-.
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BY ~Qmethmgo~is ~ ,, ~~ . .. . - .,, .
>-’ -.
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. SUMMARY ..
.- / . .
-,-.
.,. .
i —- On the’ basis of a rec&&” ctevelojed +lmory forswip”tbaci tin~s / at supersonic velocities, equations are&erived i?or thewavedragof sweptba ck tqered. wings tit$thinqfmmtrical double-we~e sections at zeroMft. Calculati.ous of section wave-drag Mstributicms and.
wingwave&ragarepyesented forfamilieq of tapered. planforms.
.
Distributions of section wavedragalong thesyanof tapered.
win~ are,in general, ve~ 6XIS5.L3V in shape to those of unta~ered planforms.For a Given tap~r ratio Gmd,aspect ratio, an apprecia%h3 reduction in winGwave-drag coefficient. withincreased sweepback is noted fortheentire rangeofMachmmiber oonsi&ered.
For a given -.
.- —._ —.
sweep andtaper ratio, hi~e~ asyect ratios reduce &e wingwave- tiagcoefi’icient at substantially m.ibcritical supet’sc@c Machnmbers~ i At Machnwnbers approaching thecritical value, thatis,a valueequal \ tothe secant of thesweepback angle, thepbn forms of lowaspect .
-.
ratio havelowwr dragcoefficients. .. ->, -.
— .. -.
,, “b i ..--E -i._-._— Calculations farwin.mof equal rootbendinti stress (and h&me different aepect”ratio) i&i.cate-that tqmx?ing tiewingreduces the %fi&wave-drag codficient at-llsch numbers considerably lessthan the critical value but increases thedra~coefficient at Machnumbers / near thecritical values-Comparisons on the~asisof constant aspect ratio, hovmer,indicate an increase of the”wing wave-drag coefficient . .
vithtayer atMachnumbers considembly lessthantheoritical value anda &ecrease of’ thedrafl coefficient YcLth taper at Ma@ mmbors “ .
nearthecriticaz value. .
. .
,,0.
,- -=r~ . ! r .’- : ., .- -—. ,- ..- = ,\ —-.-.— -.. + d iiWRODT~IOIi , .,, .
Recent developments i.n airfoil theory forsupemoni.c speeds (mf~ences1 and$2)’ in~catepronounce~ favorable effects of sweep- backon thewavedrag. In reference 1, a-method M developed for calculating pressmedragat supersonic speeds forswa~tbaek airfoils /- .4 ----- .<. .
;. ..: .. ., * -.
. .
. ...— .— . .
-— .- -r ?. :- having thinsections at zerolift.Reference 3 appliee thismethod to calculate thesupersonic wavedragfora eerles of untapered win~ withsymletwical %iconvex airfoil sections.
. .
..=.
Thepresent paper apylies themethod of”re~enco1 to derive the.generall.zed equatio~ fortheeecti.on wave&ag andtir@%eve &ragof meptback tapered }~n~ withthinsymmetrical double-wedge Hections at zerolift. Section wave-drag distributions andwing wave-drag calculations areTresented forspecific faminesof--tapered planfcmas.Theairfoil sections andwingtipsarechosen parallel to thedirection of fli~t. The angle of sweptack is refmrodto thatof’ tholineofmaximum thickness, andtherange ofMachnwiber considered is between 1 and“the critical. value corresponding to tho condition where theMachl.ine~ areparallel to themax%mm-thiclmess line; t;ha,t--is, to a Machnumb~r,eq~l to me secant of we swe@@ ...=_., ..=.+ L..
,.
angle. —.
-3 ..
sYl@oLs .
.f%axtesian coordinates *7: ~ .
— . .
..
v “velocity in” fl.ight directioq —— ..: . ...—-- ..&.: .-, .-.-— z derwity oPair . ~. . . . .
P -- . .
_____— . . . . .. . . . ..— —— .. ~+:---- .. *.:-. f=-.,-r-z ~-e~s~~--~~”%ent” ‘ ““ “-’- ~ .
., -.
.
dynamic presm.we ;PV2 ‘“”- Q —. — () .
..
M&xnAmnce-veloci+y potential ‘T _.. - -.
.- .
M Machnumber <@l .
. . .
,-”. .- ,.- ~ ..—— —.
:....——— . .—-— -—- fi=JZ~; ‘- ‘ “ ‘“’”-.:””-_ .= -— ...—= .- . .
az/clx slope of airfoil surface” a rootsemiohord, meas~ed@ flight direction a .
c chord lengbh at spanwise e~ationy, measured in fli&Lt -.:- .
. . . .
direction .’ :* . . .
-.”.
t maximum thj. c?mess of section at spamisestationy _.
NACATN No l .~kk8 A an@e of sweep of tine’line ofmaximum Wnickness, degrees .
sloye of line of maxirmm thickness (cot A) ,,.
., .,”. . ..
— j~fijn-~’ “edge - slope of wing ..
rnlIIQ Elloye of wing tam iling edge —-
%2
., 2Rl~-~
(.)
b spanof wing . .
.d ,.” % % ,.:.
‘.
., --- , .< .
., , . “.’ . . .
tingarea. ‘, . ,.
s’ . .
. . . . . ... .
. .
. .
~2 A aspect” $atio .“ .,, T “() ?$ taper rat to,ratioof tipchorfi to rootchord .
.
..
cd section ‘wave-drag coeff ict.ent at spanwl.se stationy Ix exclusive of tipeffect increment U3. section %t~ve-drag coefficient at spanwise stationy. dueto,tiy.
section wave-wagCoefficient at spanwise Stati.w; y %+c%ip ““ ‘“ ( ) +lng, y&ve-&a&coe?fi.ci9nt” exclusive of ti.y etfect , . . .
,.
.. . .
incremsn% in winewave-drag coefficient dueto tip:” ., . .
,, ., :,. CD” “ wingwave-drag coefficient
(%=+, %tip) ““’-”’:’: ‘“”
. .
;, ,, ,, ., ..” .
. Subsqript ,s refeys to conditiicris at root” ..-.
.. . .
..
,.. .
=* \.
.
.
_. _J , L ...— — .- ___ . .
—— 4 “-: ““ -“”’”” JiAtiA ~ No, 14ti--”““ . . ....
–~ -r.. . -- .— . . .
,.
.= ., .
.— ANAmHs .. .
. .
Theanalysis .i.e, bpsed on mpemimic thiwairfoil theory and on thea&s@ptic5ns of E@LU dieturbandea anda co~tantvelocity of sound throughout thefh.zid. These assumptions leadto theJJnearized equation fcm t% veloc Itypotent ial q ( re:erence 4) —- .. .
.- 1- M2CPa+qJ~+qzz=0 . .
-() -.
where M is theMachnumber Qf tine fkm andthederivat Ivesare taken withrespec;” to thevariables x, y, and z of thei!ertlY3i.an- ooorilnate “system. I&should ho nQted thatthelinearized theory is notexpected to be applicable nearMachnumber unity.On the basi~ of thisUngarized thecuy, a solution fora unifomn EIwept- backllne of souroes in @g yressure field ie derived in referenoe 1.
..—.
Thepressure field associated withthiseolution comesponds to thatovergn.ai~fol~ of wedge section. Thepressure .coefficient .
Ap/q at .a spanwiee station y Eind point.x Uong theWedge IB .
. ,--- where .”,..fil’ is “the slope of theleading $@e”oft.h.e w-!.w~ dzjdx fa theta@entof theha~f-wed$e angle(appmx.equal to haJ.f-we@.
angle-pince theangle h small); ~ =“=~ - theOrwn @ thelinesource. ie taken at (0,0).
Thedis tributlon of pre~suzze oversti%ptback wings of daslmd planfor.u andprof 31eie obtained by superpoeAtion ofwedge-type oolut:.ons. In order to satis~theboundary cof~ditions overthe surfaoe” of a tapered wingof symmetrical doubl~wedge eection, Eemi - Infinite J.ine sources are‘pl.aoed at theleading andtrailing edgeof thewinganda semi-infinite linesinkof twice thestrength is placed along .tfig. lineof.~.iq~ thickness so thatallthree Unee’ intersect at oneyotnt.At thetlpwherethewingis cutoffin thef13ght direotion, a reversed dl~t.rib~tion vf these lines of sinks andsmroes .
areplaced so aff to cagce~. @x&ctly alleffects of theoriginal distribu- tionfai%he-r epanwisg thanthetip. Fi@re 1 ~hm’~ the&L@tributions ..— .. ... . . --, .
.- — — — —.
NAC!A ~ ~0- lkh8 :5 .- of sinks andsources for 3 tapered. @rig,, and~?.asbifieq thesystem of axesandthesyzubo18 associated withthederivation of thedrag - ,,.- eq.titlons.
Thedisturbances caused by theelementary llnesources and sinks areLUni%ed. to theregions enclosed by the3,r Machcones.
Figure 2 shows theMachlineconfigm?ation torthetspered-wing @an formandtndicates theregions of thewirJg affected b~ each linesource sadsink.For purposes of simpl:.f~cation thetapered wings consideredwere res+n?ictea to thoee @.thno tipeffects other thantheeffects eachtipexsrts on it6 o~m halfof thewing. For a Wili~ of taper ratio 0,,notipeffects ~eedbe considered since theMachlines ori@nating at tile tipdo notenclose anypartof the wing.
Thepressure coefficlsnts obtaineii from superimposing solutions of thetiype shown in equation (2)areconverted intodragccmffi- — ..
cients bythe follovi~ relations: .
For sectton &ag at a. spanwise stationy . ..’ Trailing edge Ap &z —— &., : . . .“ (3) Cac= 2 ,..
qdx
r
.,., . .
dLeading edge .“ .’.
-.
...
. .
.,.
.“ . .— where . .
@ -4+ 2’%-2
c=—, %..
$s:.~e chapv~ lwwth a%.y, andtheintegration is ~erforrqe-d along .
t~qchcmiyarallel tQ theflight ~rection. “ * .- . .. . .
------ L.”” .— :S .- — — ..— .= *.
.
6. li@Ati ~0, l.~ .+ The wingwwe-ire~ coefficient i8 obtainecl by integrating the sectim” dragalong the@pananddividing we reeult b~ the*“g .— ,. --- areav - ----—- —: .- ... ,—.
..” . .
..
. . — .— lTip T,?.
‘ ) Tip P 4 4 dz-ti ~ =- —— cdc~. E (4) s qdx Root ,uRoot. L.E.
~ .U.
-. .~ ..— — . . .
. .
—. . . .
. -~ . .- —.
-. - .— .
___ ,.. , . .
..---2 ... .. , w- .
thewing area, an~ theinte~ation withremeetto y dmng the span, .
—— DERIVATION cm cxmmRAL IZED EQUATlXXW3 .
— — . . .. . .= . . .
.- .— By superposition of we~e-t~e solutions (eq.uatd.cm (2) ),the preeaure fieldis obtained fora tapered wingwithloa&Ln6 edge, trailing edge, andlineofmaYJmnn thickness sweptback. Thedrag equations arederived forhalf of thewin~since thedra~is dimtrf - buted symuetrica~ overbothhalves.The induced sffects of the opposite half -wing arerepresented by theccmju~te termain the integrands of the@a~ integrals.
--- .,: ..- - .::.
.: . -— .- :Fora symmetrical double-wedge profile, — - :“ .— ,,= e dz t ?-- =- dx c“
I
-.
.- %,?
~eotion tlqlckqess ratio. The generalf zed equa - where t/c is the .
tlp effects, for thewizIg tivedragiEI obtained tion, ex?jlueive of aB follows : (See f’ig; ~ for itiornation pil%nent to-integration . ...= limits ,). — :+ ,.
7.
Y-sof)= —= 8(t/c)2 .
.
.— .
.
{5) -.
.
,, 8 “X%DA m n’o. lJ:V!
where A, B, and. C refqr to thoprcmsure~ resulting &cm the].eatMng lineuources, linesinks~ andtrailing Lineamn’c.es j re%X+cfi~.~elY l .
x-is - 3.n#2Y ‘“l + Cosh - anq / !qy- rnlx ,.- ------ ., .- :.
: ..
----
Pp -
-1
+ Cotm
PIY-rQp-Eq ..-. —. -,- ..<. — . .-- .. -.-— --—= .
am. - TheIimitin&” case — .- andthewingof constant chord (taper ratio 1.0)is obtained by” “ equating m = Theinte~a%ions in equation (5)am yor- forzned and h%=?’ e result ng formvlas forthesection wavetiagandthe wing wave dragfor the;omplete ran~e of conwntlonal tay= (O ~ taper ratto5 1 .0)arepresented in appendix A.
.
-.
._. — It was sta’tdi prev$mmly that tie Iiqiwed wi&13 mtiidared have , no tip~ewbs other thanthose eachtipexerts on itsom halfof thewing. Th3E! implies thattheMachlines fr~ onetiP~.o”n~b enclose any partoftheopposite haU?-wing. Thiscondition Is ,.— eqressed mathm.w.tlcally as follows: . *-— -.
— W?llQ &nll .
Aspect ratto =- ~-(1”+ i)(l+“@@l) — .
.— ,.
.
“EACfi! INNO. j..&Q3 “ . I
-“9 ,.
. .
. .
.
Tip chard where k is thetaper’ ratio , It cenbe seenfrom ) . ( Rootchcrd equations (6) that thi~ “e~mplificatioi dossnotmaterta”U.y limit the rangeofMachmm~er thatmaybe considered, For small tayer ratios thislimiting effect isnegligible andfortaper ratio O there 5.s no lfiti.tation whatsoever qince equations (6)reduce to expressions that arealways valiL.
Tile wave-drag contribution of thetip5-s (Geefig.3) . .
*Y_ -.
‘%
Ddxdy ~(Wq)-ml(a+lW)
I
1.
; [% , -?-
p%’w%)yw%
1, ml(l+p~) ., -1 ~ @2(l+@ul)-aml~ v-.. . .. . ..
(l —- . I
ml(l+~~) + .’ I --— .
(7)
.
— . — :. — .— :_:..-_ .
---- .—.
10 - ;.— .-= . .
where D and .E refer to thepiesmres resulting fromtheleading ‘line d.nk andlinesource,’ respec~ively.
..- -lml(x+ a)-%- -D =.COS1l .
.-’@xi a!
l ~ Fm~lY .“.
. . —— . .-; ----- ““ -“ ::” .=- -. -.+ ~~ .- -— .—-- .“ .- L= .= . .. . . -’...-. . ...
. ..
:-~- p?~(y” - q)) .l” x E = ccmh “ P/Y-~xl” - .... ..— TheMachconeFromthetrailin~ linestnkat the~ip’ doesnotenclose W Partof the wing and, hence, hasno effect on thewavedrag.
Equatl.an (7)is solved fm section wavedragandwingwve drag forthecomplete range of taper andtheresult~ areprasented in appendix B, Thetotal wave-drag coel?ficibnts arethenobtained by thefollowing relatlona: .-.
..
. . ..
. .... . .. . -,
c~..= ‘i”+ cdtlz
------- :.
(8) —.
.-~’= cDm+~;;p ‘ “
...-
J ~ .-
,- — It is found that ~ is identi~lly equal to zeroforall tip cases satisfy ingtheaspect%atio limitatlcms expressed in equa - ‘ticms (6)and,hence,CD = ~m forthe-tapered wings considered.
. . .
“- “-The conditi”ti imposed in ~q~ti<ns(6), althcsugh notmterial~-- limiting me .q&n@”of ?$ach nymiwr for’ tapered winGs, do limit to a certain extent therangeofMachnunibti” forLow-aspect -z’atlo wings of constant chord,Equation (6a) forthiscasereduces to -— Aspect ratio ~“~ .
-1 - ‘. ... .
--- ,;- ,.
since y..~.~.
.-.
= -.
-, .,: - -., - -. — MICA For untaperedwings of aspect ratio 2,.1, ana0.5,the ,lowmt Mach nrmibers thatcanbe consimred witlnout ta~~ng intoaccount additional tipeffects are1.JX3, ,1.41k, and2.236, respectively.
It is desirable, therefore, %o,take intoconsidbratlon forunta~ered..
planforms theinduced effects of theopposite tipwhentheMach lines froaonetipenclose partsof theopposite half’-w5.ng. Figure k shows theMachlineconfigurations fortkese induced effects, andthe dragequations ~e derived in aypendix C. The wtngtivs-drag coef- ficientis thenobtained fromequktton (6)where ~ forthese tip ..
cases incluties theeffects tiucedby theopposite tip.
,.
RESWffSANi DISCUSSION{ Calculations weremadeforfatilies of tapered. planforms, each family characterized by a constant sweepback of themaximum-thiclmess line. Theplanforms wereobtainea by considering themoment of the .
areaabout therootchord divtde~ by thecubeof therootchord to be constant forany@ven family.Theaspect ratio varies with taper ratio because of thisarea-moment parameter.
For a constant thickness ratio theparameter, areamcment divided by theproduct of therootchord andthesquare of theroot thiclmess, is alsoconstant, ThM condition is i.ntenried to imply thatto a first approximation -the”root bending stress is thesame forallmembers of anyfamily having thesamethic~ess ratio.A representative family of tapered planfarms andaspect-ratio varia- tionwtthtaper ratio is shown in figure 5.
“.
Section wavebag.- Section wave-drag distmibuticms forwings !
of ta=atio O, C.5,and 1.0arepresented in figwres 6 to 10 f6ra Machnumber of 1.414 andsimepback of 60°.
The distributions of sectimwavedragof tapered wiigs a.rej in general, verysimilar to those of untapered planforms.As a yoint of in’te~est, theinduced effects of theopposite half-wing andthetip-effect ~stribution areshown i,n fi~e 10 as separate curves.Thetotal.
sectfon wave- dragdistribution is then obtained by adding thetipdragcurve to thesolid-line curve.Thetipeffect is placed correctly as shovn fora wingof aspect ratio1.0;fora wingof aspect ratio 2, this tipdragdistribution shoulilbe shifted 1 semichora to the ri~t.
It is seenby reference to fi~e 3 thatfigures 6, 8, and10 (fig, 10,A= 1) aresection wave-drag astributi~ns forone-family of wings andthatfi~es 7, 9, and10 (fig. 10,A = 2) arefor another family of wings whose as~ect ratios-are twice as large, .— respectively, ---- ..— L .
.. . . -— -.
,,. _- -==, .: , . .
.-, -..
“ “mci”m”fo.” UA6-
. . . .
It ig interesting to note at this yoint that for a gfvenM@clI number the sectionwave-drag coefficient at the roof 18 a function of thes%mep of themaximum-thickness lineonly; theterms involving leadlng-eke sweep add.dng up,to zero, (Seeseokion dzza~ equation inappendixA fory = 0.)
—., - Typical variations of” wingwve-drag. coefl?i.-
‘-F”’” cient wit Muchnw%er f~.ti.ngs of taper ratio O andtaper ratio 1.0
of the#aimfamily-are shown in I’igurbs lIJ12,and13 for~“, 60°, and70° sweephackj respectively, At sonM Machnumber between 1.0 andthecritical value (~itiul = secA), thedragcurve forthe tapered winghasa d.iscontinucn.m slope,ThIadiscontinuity occurs at thatMachnumber cor~esyonding to me condition where therear —.
MachliDeorosses-the tra~”ing w!@ of MO wing, thatis,w&.ero - ..
.: . .- ,>~ ,..
. . . . ,“ - ---- .
,j=~=~.-% ,.
%%
%2
,.-. ‘.-: i.,.- “
..” . ~r” r.
“.+” -.””’ , =: ..- .. . . .
‘= . ... .:. .!
In thisregion andnearthecritical Mach number (~=:).”ie , ,, theory h note~octedto be applicable because theassumption’ of, small di,eturbances i~ violated, buttheresults arepre~en~ed j.n order tcl give a morecomjlete picture.of thelinearized. thecry.
.-* ,.
It is seeil”fron figures H. to 13 that taper reduces thewing wave-drag coefficient atMachnuuifiers substantially below the .
critical value butincreases thedragcoefficient atiMach ntibero approaching thecritical value.Thistrend is similar to the-one shown by theeffeotof hi~ a“og,ect rat~oon thewavg-drag coeffl- clwrt of wtin.gs j?or a-@.vtin taper ratio, itmustbe remembered thatforthe. families of tapered vzhI@ cons~dered in these calculat- ions, however, thewin~ withgreater taper havehi@er aspect ratios and,hence, ’the offsets of aspect, ratio as wellas tayer A ., areincll.tied in thistrend.
.- -- ,.
Var:lations of wi~gwave-drag coofficl.ent wi’;h taper ratio for different ”swaopback a@l_es at‘~Machmmber 02 1.2areshown in .
fig!me.1.~le Theuntapered win3forthi5family hasm a~pect ratio of 1.0andthevariation of aspect ratio withtaper ratio are prgsemte<L in tabular formin thefigure.I?gr a given meep an@e, thewin~of taper” ratio O hasthelowest dragcoeffi~imt amlthe Un.taper;t. wins theMachn~or epproaches”the thehi@est. As critical value trend would reverse itself andthe 13=~, this ( ) .- ..-
i’ik.wmtiu, Wi3
,, . . . .
.
.
untapemed wingwo~k iiqvi. -k~e ~o%reek’ dra ckfic~entas- canb$
seenby’ ref~encbt~ “t is alsoevldiwk @m - figlmsIl. to 13.’f . .
.
figure.lk thatfor& ~$bti tmw& re%io andamect re%io, an apjreciahle reiuctlW–@tifig-wave-drag ooeftitcient is Gccolqpliahed ,., withincreased. sweepbkick.
Figure 15 yresent~ variat~ons of’wing wavb-dra~ coeFf’icj.ent- withtaper ratio fdrthree families df %dngs basedon untaperbd. p,l.an forms of aspect ratio OJ~,1, and2, resjsctit~l.y, Thertmilts are .
presented.fti 60°~weepback anda M&.ch tiumber of 1.4141~ertinont details of thewings arepre~ented in tabular. formin thefi@r~ to facilitate interpretation of theplotte’d c~es. Theaforementioned.
trend. of reduction in wing~ve-tia~coefficient associated wi”th ‘highasyeot ratios at Machnumbers substantially bel.ow thecritica~ ~Mach nu!aber ~,or a given taper rstioISclearly seenin thisfi~e~.
By choosing points along-these curves bo&F6e@@.nq %6 tir~~ of ‘the. s “ sameaspect ratio, it is”seen thatfor.aconstant aspect ratio ‘- ‘ :‘“’,’ tapering theWag inc~eaees thewinswave-dreg coefficient. By a similar procedure it canbe shown thatforwin~a of constant aspect ‘ rati~ taper retwoes the @.ng%~ve-drag co6ffici.orrb at Machnumbers “ ‘ nearthecritical value.The increase @aspect ratiov5.th taper “.
ratio defined by the area-nmnent’pa rme@r thus hastheef<ect of offsetting tile adverse effeots of tep”er at theJ.owr Wch mmhcre, CONCLXiIOES 1.Distributions of section wavedragalcng thespanof tapered wings are,in general, verysimilar in shape to those of untapered @an forms.
2. The section wave-dxag coefficient at therootis a function of theMachnumber andthesweep of themaxlnmm-thickness lineand N independent of taper.” ., ., ‘.
3. The increment h wingwave-drag coefficient causgdby the tipis identically eq~l to”zero foralJ. tapered andU@apered tin@’ ‘ forwhich-the Machlines frcun onetipdo notencloee enypartof theoppcmite half-wi~.
.- 4, For l.fin~j of equal rootbendi~~ stilww~, tap%r reduces the * f wingwave-drag coefficient at Machnm?ibers considerably lessthen thecri.tioal value- thatis,a value equal to We secant of the smepback angie- butincreases thehag coefficient et Machnumbers ‘ .
near thecritical value.
.
,. ..
. ..” ,.
‘“: .“6. Fora given taper’ratio a~ a~bc%.””r~ti~,,~~ a~precieble reduction in WIQEwave-drag coefficient withjncrea~ed meepbaclf 1s noted~orthe. entire rangeofMach yqbeu com:derecll .
7. For a“give~ sweep aiti-taper:ret%o, hi&er asyectratios reduce ”,t’he wing wave-dragcoelTfioj.e@ ,qt. i3ubf3tqntially subcrltica”l Maoh n.xnbotis. ~At Machnumbers apprca@i.n& thecr~tical. value, the p@q f!?~. of lowaspect ratio haveIower. dragcoeff:cientso ,- The. generalizec$ equatione prmantod ~n tile ap~andixml lm”y be used. to tilctiate-the ”sub.critical cmpersonic wa%mitrag. at %er~. lift ftianji conventionally tapered or unta~ered wingwithsymmatr3.cal double -m”dge. airfoil sections andwithloqdin~ ud~e,’~.lin~ ed,se, andline’- @ guax@mthicknese mep.t~a ok:, . . ;.
. . . . .
.,.
,- ; Lan@BF_ Membrial A.monaut icalLaboratwy . . ,., ., Ne.tiond Ad.viscm~ Caamittee forAeroWu*ice ~ ..
Langley Field, Va,,April 7, 19)*7 ..
.
..:.
.
., .
., .- .. a----- .
,, ..
.
.
. . . . ..
.— - ,.
. .
.— ,.
., .“, .,.’ . .
,-.
,.. . . .
..
., :.. ..- .. .
/=’
““ ~
.
NACA TN No. 1~8 ’15 ,.,-,. ,.
‘%%+
Al?PENDZX A EVALUATK)N OF EQUA3?ION (.5) WA~ IM.AG EXCIJJSl?LE Section wag for .
l$CdmC cA’ for 4(t/c)2 =A+B .
=JL+B <-C ., ,,.
. -.
, =A+B+C!+D tiere .
.
. .
.
... .”# :-?” :*= .— .. .. ”-— .— __— .-~ . .
I(j ... ... . ... :-_ : ..= .......
,.
-“y L - ml~~‘)+ 2“%2 ~ (, J --- , .,.
..
., .
. .. .
.
.
—- —.
-~ 1 + lJp2 ,—.
“2y .
cosh 2Pm’ — -~m .- .— — — i? “, t ,.,
w% ‘%) + WY%? 2’
+ ——— ‘2 {.
b “ p’%’
_..
.-.— .— .- . . ,. ,- ., .
l— .. . — . —— .:
-i 1+ %2(32
y(~-~]+aw ~o~h-lY 1 -*P 2)+9 ~ ( - 4Y coah —+ ‘m(J
%2
qy(~~ -%) - =%% [ -
..._. .-:
1 ——.—- —
;:’- . .
— 7 ..- —.
. .. . .. ~--- -—-,-—- . ...+.. -,. .
. .
_ -— .. —- —— .
-, ,* L . ..— -, -.
.. .. . . .-. — .. - - .
. -. ..-.
.. .
l..
.
,.
.
B= yl -
Wl~2) - aml ‘[
(
>
!qYpi~ - %) + ~rnr%l 1
“d /-
J c=
... .
13% +%)- a*”l
P/Y(q) +%)- awl
m&- i3%#
L
., . ... , .
.
-m@#)-EqJ
and .
.
cd-l-l Y(’- ~Y%@2) - !2Ciml
— .- -1 .
., .
, ,-,. -.
--- -.
. . .“. -.
.
— . . . ... .
.. .. . . . , -... . . .. ..,.-, -=Z - “-.
4“’.
“; ““’%12EF:.}.’: := ‘~ . -., “ --=g . .
. . .. .:..:.
: SEl!t
,-, .- .-.:-: :... . ..
k -. ‘::-s.- : ==-:
gl -
. — -. —--- ---- i----&--- —— — ,-”” ----- -. =-., “b- .“-” - – ---, , -._.
, .’1 . . . ..- .= ;-.
..-. . —. *—.. -w-- .,. . :--, pfl.
. . .. . . . ., -.=.
0. ..~.-.~. -7,.zp “ :: ~; *?,?? # . :.T. S ~ ,.
. . .- .>- , ‘-> 4 --.
— Q.. ::” ““: _.. .“..:” .... /:::’” ‘.’:+++-:;!2 -“:+:<
.- — .— -....,:- .-...-~~
18----- #ii -“
*..K,. >438 -.: .- . .
. .
. ~“ ~ -Wt~ Dra@forO < I@r Retls < 1 .
. “~y”‘ — . . . . . ..- . .. . . . ..— L ,--<. ~. .. . . .,..+ -=. .- im.....+~ ~..+ .. —- *-.:” :-*- . . . :.. ._ ._.=._ J. --- :. -. ... . . —.- —— .— . . . . . —+-. --<, .% *:W _-wg.-~ --- :. .:: ---g ..= -->- --- -. . . . . .
., ----- . .. ..
._Tl%:+z.+ .-: .,> .= < >.., —+. . . .. .:, *<W--”- .“ .’-..
Elnil —=’A - *’- .; ‘ - .
6~q-”c___ for . .
,, .. ..
8(t~c)2 ““ i.: ,-, ~ J _ .
1 - 13?,n.
., -, ----- = . .---- .-:: .L ,.
-** -“ -- --Q #-f ~ . . ..A.
,. :-:. (..
I __=+_., 2;* .+?.5 . - ““ :,-i?-.::ti i.i:i:..,’zG:.7:*f-z.:, T.”@ = -, \ ~.. ---.+’-:J :-- : - ..:.. , -...--,-. : ..- .-., :.. . .*:.
----- ,“”._ :.& ---- ...-; --- “—._ - G.” aq
:;: .=?”-A”” + .-5-: ::>1~
*“
-.
, “ -“-<”q “~- ..,---- -.
. .- ..’
7,.;-’%. - ,. ;-- “. l :.-+ .
1: --J3DIL 1-P%
q:”;,=. ..” ‘. . ““”
G.. ... ., .. ..–-*.- ? - . ..-. -.+- -, “.
x“-” .- .-+ ~ I .}. ; ,.< —.
‘2 ““i :“’
,:’;% &(ii ,+ -
A , --... <., -.
.@I::;:.-l %(1 - ~zt@)+.:~ -.. . .
:- ... ... . . ... .... . .=-. .:,.. —.-= --- .<->
..-—-- --
-
.p~~~(d -,e)- ~~1”. .
‘-: ‘“=2?%’- % )
.—.
-“ -=. , , _. ..= --., -,... --- .
. .*-.
--- . . .. —-— .
-------- 7—s:[’+. -42-”’? .:. s- “-~ -– ::.-:, ,---- -:?–:–:-’-= : — ~ ‘ti2[~ ~~(d +“~j]2T ““”’ -1 %(1 ~ Wp;) + ~“ ‘ - -,,.
. . .. . . ..-.
-~ -~(%”+~) C“$h + a “,”—.”- ,...- -..— _. ..- -------- .. . . --- ----- . .
:. ;-— ----- .- ..”!J. -.
.? -- -e-— -.-- +7*=+”>w~:.f- .-. -.,--;-:-.~~ . .:. , - . = .
___ r., -i:.”. “.. ““. .“” +.”:+.
~~;d” = ~.
% [% + “Ul(a + d)]2 -1 d(l+ ~~lj .--a--- ‘.
cosh .
2(% + %) ~[~ + ml(d+~” +,& - ~2m12 -–- - ---- ““ ‘“ .::. “,., . ..:1” .:-. -_ .._.._~—-~ ““””-{ ,:’. ..- T -z++~=;.=~:-== ._=<~.+:- “J- ~.. - “;”~=z=””” - Y- ,.= . .
..”.-..
..:.~==—-=–– .. -:.—: :-.
— ——— . - -~i ::, -. .. .- ,, .:-A 2&:: 4~:4.3. . .
.+ -.~:+: . . . . ... ,.=. .
—.
I -— .
.
-.
.. ...
.— --l ..+ .- .
. .
-.
.- ) ., . :4 _., .. :=:4 4 . .
—- . .
.
--” “’2 2 -k%3m1 “e ~ ““ “,,<< p ).+ ,& ‘- -’-,,,. ‘ “: ‘ “~:-:--:=j ,,>~i:l ..% :
J
. . . .
. .. . . .. . . .
—+ .
., . .
I .,:, ~ ,.
- ml?
2@m22.
..
%2 \.. . ..- ..- . — — . . . ..=_— . . :=?
.
+ / .._ -—. .. . ..~l”d ,-. .
.s=., ,’. -
‘“-’k”’
,, ~o~h-l -.”=’-- ,., . .-, ...
---
“ “y”. ““:-IX:+ “z.=: ‘: ‘q!?) 1 . : . ... . ..... ... .
-#!
~ ?#m +&. l”
2 -., ,.
.P.mi7-~3m~- %) 1“ b - ----- . .;; ‘: ..4; .,
‘w
—.
(%+ %)lJI=@
. .-[ > . . :..””
,’ .. .. . -- -.-’ . . .
.— --- .. ..
. ..
.“, ..::.. . :, * ,“ .’:~:”.fl ‘ ‘ ‘. ~ ~.-’!-” ~lbq,)-,,““ $2:,..., .:. :-., “,~...:”y:” ~“ : -- ; .,::= ,.. ’.,,.. ... .
~o~h-.~ & .
..
‘%k - .
). .:..-. ..._....’ _l - p%; , ~% +,:!:. a 7- --,_ \ ..
,.
(3% - %)(% : ‘1) 1 t ‘,.. .
J., .
- ~@’, ‘ ,.,; ,:, - , “, . . . , ;-,: “, . ., _ “.::..-. -.-.:.””...,:, -, ._.; . ‘:;~en.~-l -, ~292.,. ~~ ~e~tive“~egl’e~k’ ‘i~~~ .~”ked with ‘asterisk .. -.: .- .-.
-1 1 ~ m>2~2 in values ‘forA andusetherifition- ‘“ ~ -&2~2 cosh- ‘2p~ ) .
3.
“ ( a8 -lx~ -iCosh ‘1x fOr allterms involving ,. ship Cos . .
1- l.U@ ~ d _..
multiplication factor.
-—- :::, ?-:_y f .- ..
-.
,—
J“’
,“ /“ r- .- ,-’ -1 cosh .
apm12 .— -.
._ .
. .
.- ..*S pMO /d(ml -
%) “- a% I
%(% - %)
.- .
J
.
.
-.
.
2(IIQ -_ IQ) ‘- \ .-— .- .-. .
-, and D= .- .--: --- -.
,., :. .
.-.
--–s . .-
drq)(l +- .lnlm&2) - auq
. !l?iil + ;“)-. ,&p#J’ -’q
. cosh
.
Pyn@q + q) -“ 2aJqmp[
%!(%.+- ~). “- , ---
....... ...—.-
“.=% “*-= W7WY’?
. “J, -...
- “}
:.. =4T!. - . .,-.. ;...--.,- J.* .r, - !. .i~. - ,p .-. . ..+:. **,.*.
:. , ..,1 * ,..,., :: :–.i_i- -_ . ..”.
. .
., ;: l +-. :;” :--G. +++-. , - .2.- ---- . . . . .
“.
.-. = .~:.,.
,- -. -.’ . . .
- ..
. . . . . . . . ,.. :.:<;-:~ .. ... ..M. ..} .:; .
< “.. -”.,.- ..-.
-— ..— --- _ -. —-.
..T. .
. . ,,:$. -... . . .
.. . ...+ , .-:-’ : ““~ =.= .-+ -.+ , ...=—. .
,:” ,, ,- ,L.
. .
., .“! -..
,. +.
. ..
.
.
,.
. . .
.
# . -.
.* “ . .
,. .?.
‘J;!+ ;
Wnguragfu rl!aperli atloo ~’
. ... .. .
8(t/c)2 ~,; ,(:,,”, jf ‘ s. ’l.- .,. - J,]:*” ., l iim .1 N P f; i ,, ’,;, ,, 1,.,,!, ., 1,;1 ,1. , . --y .,” - -, :-, - .
.: .-.
, .- .“:. .- .= --- ““ UhCAm r?o~ - ,.- For ~<p<—, use A+B+C where ;“ %2 2a&2ml
.11- m@lE$2
[
A= $=x%% - %)(%2 “ %2) ‘(m’ -‘] ‘O”’ ‘(% -“)
..- -a.:~. .-:. .= -- .-.
—. .— .----- 1- qyq+-.
.
~
1 - 2m.”2 oosh-~
>-_ — -.
. . .
J*X.7 -t e .— . . . . .
—.
—- -- — r“ .— - . . . ..-. .— -. -—.. . . ...— . . . .-
+’ (aq - mo)(~ - I!q) C08-: ~
P% — — ..- - . .
,- ml 1 - ‘m~qj32 +ml-~ .
( .)
2@1(~ - q) ,..
For 13=+, use A+B % .- — — .
..
I
D. a..%~~’ ~11- ~ +
? ,.,2(g%-%)]
(% - %)(%- w
.- -- NAM ~ ~0. ~ -“ ,ForP = ~, use B+~,, &usD ..: ,, ..% ‘,. “.” . .. . .“ “.”: :. - — ,- -.
.
2ql~ 0% 2a~%a1 (% -%)$- - (% - %)(% + Ml) .
[ Section DragforTaper Ratio1i~ .- =6A for 4Kr@/c.)2 -.
=A+B - ., .’ .
~.
=A+ B-I-C —<7<= 1-D% ., ,, . .
:.
, where “ ~ -- -.._ ---- ..- - .. . . .
. . . .. .
A=4(2y+”a@ . .
. . . .
..
-1 cosh -+ ha -1 cosh --2a ..— . . # ,.
., ..
— ..- -–f . .
-i--- ..
. .- .“:. ,,*,,;.. .. . ,.:. ,..=;q~:~ .; i .> ’.. .“. .
.-i .- .
-.
.- ..-. .
.. —-
~i & N;. ‘~ “-”‘“
=-.
.
.
Y(l- (32%2) - 2a~
qY”a.mo) C a: 2%” “c~~h-l
2apl# % —— —- ,.
.,. , ,. .....
—— .- Win$DragforTape& ,Ratio 1.0 . —.
— . . .
=A .- c=A+B
<dm#2!%_
1- I@) =A+B+c .
.- -.
. .
. . ...” == -- .,.
— .- . . .
., -.
--. — .— .
.,- - a%F(3 - Rn#) -1-2,(, - ;2%2)j - ;’ p - Eulf)d +,, cosh
------- .-
l-p~s ,“.-,--’ ,
m%
_.
:\\l\ ., . ..-.1, .
:.:”-.” (l+,?qf)d+ ,, a)2 .
@@+a) .- -- . .
~-.= .
._._ . . . ....-—+ ~.. .
t .
.4 .- .
.
.
‘1: . . .
(LJ
2q3
@. ,%+.fp 2amAn@.-;i2 cod “~’;’$y;; a “ -1-
.
—
,.
-— ,-=.
, - ..-= —., ..
and (&) %~(3 - B%*) - ‘M(4 “ &Il#J’J “ cod b “ I+%’)* -2.
,,, . c=- -..
f%@~
1“-$~2 “’”.’”
Tjiij .::., ,/ -:_ ., .- w “-J&%-%ba’+ ~a2 ““ ~ ““- /’ .. . . . .
.- .
.- ,-- {71j n --- ,.,./:: ..” ‘ - ;_ .:
-~-1 [1 -1- IA&p “ 2/3
..— ~(d-a)2 .
— 2fI~(a - a) .
p=a, When use thefollowing eqrfession “%
D=
.— .,, . .
, -..
. .
.— “. .,. , .<.
,...___ ___ . ..
,, ,, .- ... -- -.. ... .. , :..
-- ,.
.
.
‘.. - — . .
.- . .. -
,, NACA m“ I& lwi
.
.
.’ ~mn,B , .-. . ,. + .
. .— . .
. .
.- i.-. “.
. -== ,-, , ..’ - ‘i - . a-.=, .- .>_-,_____ ..
. . . . .,, . ,. -. “. , !
. . . .
. ..r ~TALUATION”- OF EQT.jl@XON (~)F@ TM *MS ~ ~ ~ ‘ “,
[ ‘]
% Section DragJ@rememt for’ O”<Teqjer Ratio < 1 . . . . . . .
.-:. -..— — .- . ..- ,.
.-., ,,. . ..- . ...- ., .- . . -.
.—. .—— .-; . ‘.– - -- Foraspect-ratio lfmitati~, see eQlt?timEl (6) , --- .- “C%tipcg ,,
i(’t/c).2
~.. . _ ,- ,= >.’. . .
+ for .
“(
+ c
for
(
where .“ ,.
.“ .
_ .= -.
.=- .
, .- :. .—.—— ,.. = .-M .
, ..-..
.< .,-.
.. 7. -r— -. -m .— I .
.
and -ly:~(a-d) =- 2(Y - ~) cdl c ~U@?j-Y] “ “,” .:, Sectmrl Drag Iilcramant fac Ta@r Ratio 1.Q; Aa?ect Ratio ~ ~
I
. . ..— ,-- ....— ~. .C ..+— . .
..r.
— .- ..
. 28 where A= ., — .- .- 2ap~2 . .
.— y - ~(d -‘a)
-+ - ~%?(r - dlnJ cdl-’
.
B= $q-@q.J-Y)’ % -.
.
(
- Ua cosh-l .- --- .- ~?here iu” no tip effect tiatsoeve~ for the WLWof tep9r ratio 0, andtheincrement in wingwave ..tia g caused by thetipis identically equal to zeroforallcages satisf’ykng these a~pect-ratio l.imitations.
.- --- .
. . .
.—.. : .-&q ------ . .
.— 4 .>. . .“-.4.- — — ..
.- . . . .
. .. i-.
.+ :- . .
L.
.<.—.
;_,:: ..- —. .-. —.- .-.
.. .. ~ . . . .
.- -.= =- . ., .
.
-_-.~ ..— .- ..- ..: i .. .. -i---”= .- .— . _—. . .- == .]
, .
N?PEmlx c EQuAT.mJs Tal Amcc?mm TIP EETEcIcs FOR WING SUJ?
CONSTANI CHORD &ee fig. 4] I ., ; .. ..
a ? d&’02
= ~(a - d)2 cosh-l’a - ‘*P2 + ~(d + a)2 Coati-l
f?~la- al 9111@+ a) ; l-o u) I ,, ~,. . .-. ..;....~ .>. :; . . . .
.— — “’” - . . . ., : . ..- ,,. -.
.+ . ..- .
.. >- .~ .++= CaseH ..— ---- ..
-..
-. .— . . .“- ,.
-= ,.. . .. .— .— .-: -.— ,-. = ..
—- ..* -- .S .- .
---- .- .
a -.— ..
.-., - .....
. . ...:.
,- — -T.; .: -.-’ -,.
2a - d(l+~2P2)l . “- -— . ..
.- .- . . , .,.
. .
.
>. ., .- - ., —.
.
.:+.. .
. .
, .- -+ .— .— ,, . .“. - .. --:.
,- 7- . .. . .
Case ZII . .
. .
,..
,. .
l . .
J -..
,.
—- --— __ ——--- ., . .
.- -.— . . .
-— .
..— -.
..
., :-a . .
.-— .— j ..
..— .- --e -— .=. ..— .
—-.
.- i“ .1 a -~2P2 —- (a-z!d)’ cash-’ =
.~ (a - d)z cosh
~~la-dl
!
‘4 .- ... - -.
.- .1a+ -%2 ‘“a +~292 (d+a)2cosh-l —- (2d+a)2cosh + ..
~(2d+&) -j3~(d+ a) .— . .
:. .:= .- -..,.,- ....
.
.- ,., — O<A< .
. .
The lower limit fcm y is changed to O in thefirst ltie@@-
of caseIIIandtheresultant expression forthedragis . .
., . . .
., a-2dm@3* ;. ‘ -1a+ ~2P2 +(d+a) “cosh — - (a- 2d)2 cosh”~ - ‘ p~(d+ a) . .
._—. = - ...= :.
.,a+ 2dq3%2 -a(l-#%#) ..- -a2GoshN1 * - (2d+ a)2cosh P~(2d+a) .
-.
. .
- :- ,.—. — .-. .— .
.- —.
.
-12a-d -~ Gosh — 2 $~d )} . . .<= .-.
.
IiJioA TIT No.. IN@ ,.
CaseV
% ‘“
and,A<—————— wl<~, .
.’.
limits for y am Q1 four lover ., ., a - Mn2P2
%@+ - &’&
—=~ (a - 6)2 coa-l
{ .[ &l&/c)2
., 2P2) a.- 2*2P2 “-l 2a- d(l +lqJ - Cosh - (a.- 2iF ‘1 -2 Gosh-.-——— 2p?ll@ - al
1[
‘- a(l+ P2DI(-)2) ~ ~a+ # ~o~h.l 8+- %2~2
-1 a
- -coeh .— P%(d+a) j3~la-2dl
. 1
)
(
a.+2%292 p ~mh.l a - a 1-%2~2 _+a - (M-+ .# cosh”~ afi~ @4J2d+a) [ .
.
_—- _—.
—.. ..- --. .-— .—.—.
.. . . .
.. —= .
34 -=
i.ki23””’
-. .--— -
.—
—., ..--—
,. -.. .- . . . . .
-.. —
.- ,“ .-
=.
1, Jones, Robert T.: Thin Oblique Airfoils at Supermnic Speed.
NAtiA TN NO, 1107, 1946.
-.-A 2..l?uckett, Allen E.: Supersonic WaveDragof ThinAirfoils.
Jourc Aaro,Sci c,VOL.13,no.9, Sept.
1946jpp. 475-4&* -.
— 3. Hax?non}9iduey M,,andSvanacm, Margaret D.: Calculations of th:, SGpereQ_ni c WaveDragofIfcml.if%ing WingswithArbitrary Ehwepback andAspect Ratio.WlnGsSwept behind theMach .~O l 1319 , lg~~? * –* Lines.NACATN — -.— ... .
k. Pmndtl, L,: General Consideration on theFlowof Compressible i Fluids c NACATM NO. fb~, 1936, . ..- .—- . . .. —.— .. . —— -.. . -. .-— . ..— -— _. —..— ---- _ -.
.“___ - .. .- -:- . .
-— .. ... =.?= , —.
----- . .
i- .
,- . ...— .
.— .— . :.- _,... -— . ..
=, .. .----- ,.. . - ----- .
_____ .._— . .
.— — -—— -%-E. +’-= _.. —— .- .”- .— .
:— . . . . .— - ---- # I
Yt
l -
d
lb / tia~je
v——
x———
\ \ \ —.
NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS a tapered wing.
of sinksandsourcesfor anddistributions Figure 1.- Symbols .- / /
+’
/ / / / /
/’
/ //
“XL
‘\\, ‘y+’+y ,,,
\\ \\’\ \\ \
\\\
\
\
\
\\
“\
ii i \ \ ,\
NATIONAL ADVISORY WHMITTEE F(MAERONAUTICS Figure 2.- Mach line configuration fora tapered plain form.
I
. .
* ,.
Y-v+be am, l-pm, / 2 am, /*, /
k ‘
/’ /Y !4?
x——
Figure 3.- Information pertinent t~integration limits inequations (5)and(7).
co m //
I
Case I Case III OcA <2L P L #&- Ag2
.?
-P /+ ma
F
-7- A27 /t m.
(km? IV
/77
NATIONAL ADVISORY COMHIUEE m AEROHAWKS g Figure 4.- Additional tip effects for ~ ofconstant chord (taper ratio 1.0).
I i?
., , , .
, !;, .
.U1 j
I Ill
-.
< .
/ g4 .
3.+6 /4, C?CT
s
I’@ co -I Z/xr /4s~ct ndio Wlo 1.00 Al /. o ~tJ ~:g .5 1.63 .4 /.84 .2 $fg 0.
•l
NATIONAL ADVISORY CONMITTEE FOR AERONAUTICS Figure 5.- Family of taperefl planforms used for calctiaiions. Planformsshownhave same area.
w a :1 .—.
. .
Y/$
NATIONM ADVISORY COWllEE FC4 AERONAUTICS Figure 6.- Section wavedrag distribution for wingoftaper ratio O. Mach number, 1.414; aspect ratio, 3.46; sweepback angle, ~“.
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d z Section wave-drag distribution forwkg oftaper ratio O. Mach number,1.414; ~ Figure 7.- P aspect ratio, 6.92; aweepback @e, W“.
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Y/ COMMITTEE FORAERONAUTICS — Figure 8.- Sectionwave-dragdistribution for wingof taper ratio 0.5.
Machnumber,1.414;aspect ratio, 1.63; sweepbackangle, 60°.
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Figure9.- Section wave-drag distribution for taperratio 0.5. Machnumber, 1.414; aspectratio, 3.26; angle,60°.
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I L-A’” v //’ t--H Figure 10.- Section wave-drag dist~ibution forwingofconetant chord (taper ratio 1.0).
Mach numhsr,1.414; sweepbckamgle, 60°.
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Figure 11.-Variation ofwingwave-drag coefficient with Mach number. Sweepback angle, 50°. ~ .- -.
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Figure 12.-Variation ofwingwave-drag coefficient with Mach number. Sweepback angle, 60°.
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Figure 13.-Variaiion ofwingwave-drag coefficient with Mach number.Sweepback angle, 70°.
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CCWITTEE FORAERONAUTICS Figure 14.-Variation ofwingwave drag coefficient with taper ratio fordifferent sweepback angles. ~ Mach number, 1.2.
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A Figure 15.-Variation ofwingwave-drag coefficient with taper ratio for three fwnilies ofwing plan forms. Mach number, 1.414; sweepbsck angle, 63°.