Supersonic Wave Drag of Sweptback Tapered Wings at Zero Lift

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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&LTicA 1.

LAmumicY .- .“ L+iw W Va.

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.- L-, . . . . . . . ..-_ ~~— , .4., l NATIONAL ADVISQRY C0MMD3XE 3URAERONA~ICS ,, .

~AL iwm No.” Ww””.: “. -.: ‘“ “““ : ‘

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,.> ,,, -.— ‘.- -, .>. . . . . .

,. ,- — .— > EXiE!ERSONIC WSZEIIR.N2 OF QS4F2TBACK TAPEKED . .

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llm~ ~. ~o L~T” “ “ ,. “- . . ‘..:. . .._.__-.

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BY ~Qmethmgo~is ~ ,, ~~ . .. . - .,, .

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. SUMMARY ..

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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 ,.” % % ,.:.

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., --- , .< .

., , . “.’ . . .

tingarea. ‘, . ,.

s’ . .

. . . . . ... .

. .

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~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) ““’-”’:’: ‘“”

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;, ,, ,, ., ..” .

. Subsqript ,s refeys to conditiicris at root” ..-.

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=* \.

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—— 4 “-: ““ -“”’”” JiAtiA ~ No, 14ti--”““ . . ....

–~ -r.. . -- .— . . .

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.= ., .

.— ANAmHs .. .

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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 ,.

. .

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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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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°.