1、REPORT 1307LIFT AND CENTER OF PRESSURE OF WING-BODY-TAIL COMBINATIONS AT SUBSONIC, TRAN-SONIC, AND SUPERSONIC SPEEDSBy _AILI,IAX (. 1)ITTS, JACK N. Nll_:L.qLX, “tad (h,zOR(_F E. _.AATTARSUMMARYA _ethod i._ prf plessme position, in.lift force, lblift on tail set,lion due to wing wwlires, II)lift on b
2、ody section t)etween wing and tail due towing vorti(es, 11)(.otangent of lea(ling-e(lge sweep anglepit(_hing moment, Ill-in.fr(,e-stream Math numt)erstatic l)rcssure differen(,e between top and t)ottolnof wing, ll)/sq in.free-strealn dynamic pressur(, Ill/s( t ibody radius, in.1)ody radius at shouhl
3、er of nose, ibody radius at wing, in.body ra(lius at tail, in.Reynolds nlnnber t)ased oil ? of larger liftingSuIfflcCSRNr_WV:t:_ !I, 7JhtdIrmALEA TEP+B6(I-NFN2VWAS_(T)B(W)STT(B)lnaxhnun scmislm, of wing (b) IIfI17LBir) 3fIIc)I Wing vortex -_ I/-r(vl- I _/ Ib. “, II(at Parts of a wing-body-tail combi
4、nation.(b) Lifts without wing-tail int(:rfcrence.(c) Lift, s duc to wing vortices.Fuwrtr; 1 ,-Parts and lift con)po)ents of a wing-body-tail combination.The terminology is in(li(-ate(l in tigme 1 (at. The nose isthat part of the body in front of the wing. However, whenthe wing is mounted on an expan
5、ding section of the body,the nose is taken to be the entire expanding part of thebody. For the purpose of analysis, rite lift of the wing-body-tail combination is taken to be the sum of the seven569l)rin(il)d (.()ml)On(,ms imli(nie(l in paris (I) an(l () of figure I“These (olllpOll(qll_ _|i(:l, Lift
6、 on nose in(h.lil)g forel)ody, Lv2. IAf( on wing in l)r(,s(,n(, ()f l)ody, Lwc.)3. lfif( oit l)o(ly due to win_, Lm.-4. l.if( ()n (aft in l)r(,senee of l)o(ly, Lr5. IAft on l)ody due to (all, L,(r)(i. l,ift on tail due 1o wing vortices, Lr,Kw(,), and K.v whMt represent the ratios of (.tie 1)ody lift
7、,wing lift, and nose lift. of the eomt)ination to (tiat of thewing ah)n(,.I(+K,. a o (4)I(“(“)= Lw- (_).,= (Kq, L, - ( a=0Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-57O REPOIVi 1307 NATIONAL AI)VIS()H,Y (7OMMITTEE F(H AEIONAUTICSThe factors/x_()
8、v) and Iw,) are defined for the case in whichthe angle of attack of lhe (.ombim/tion is varying but thewing- (or tail) incidence angle is zero. For ihe ease inwhich the ineid(,)we angle is varying l)ut (h(, angle of attackof (he body is zero, two amflogous factors art, (h,fi)eil.n, (k,),(,). _=:0 (7
9、)+b_=L, = (._o),.(YL,_,_( _), =r4 1l ,“ r“ 1 1“2 .“ L- - ,r ,_+ _j x-L r ,_/ ,_.jj7r 1 r _ (,.(The assumption is made that no negative lift. is developedbehind the maximum wing span. Jones (ref. 16) has pointedout that for wings, at least, the negalive lift, predicted onthese sections l)v slen(h,r-l
10、)o(ly theory is l)r(,ve)He(l l)ysel)aralion.) This function is I)lotled in (harl i. 1) (hi,limiti)lg (,as(, of r/._-0 the coral)ira, lion is all wing and thewdu( of Kwm-1. As r/._“at)lroaches unity, theft, is a v(,rysmall exl)oS(,(l wing. For this small wing, the I)()(ty iNellectively a ve)lical rel
11、lecti,)n plane “m(l the angle (if at l,ckis 2a (lue to Ul)wash (as is discussed later). This mnl_(,sIw (,) : 2.t is clear that the values of Kw(,) should lie salisfacloryfor shin; s|ender-1)ody lh(,ory fi)r slender triaugular wing _u_d1)ody combiuatious, aml an (,xn(t linear theory solution forrecta
12、ngular wiug nnd body (ombinalians. The slender-body result based on lhe load dislributiou given in Al)l)endixA gives the following expression for kwu_ in terms of r, thesemispan-radius ra!io, ,_/r:i, _ r: (,+l):+_-!_+l: +-I -“eT+l)+:,_,_ sin-, r: t-1- r(r-l)T(r-I)(r-l) ilg 2r JThe wdue of kwtion to
13、an equivalent pbmar case is desirable betore calcula-tions are lwrformed. The body is imagined now l,_ be co-lapsed to a pl,ne and the Mach helices of figure 4(a) becometim Maeh lines of tigure 4(b). The lifting area of the bodyis the shaded area of figure 4(b) which is at zero _mgle ofattack. This
14、area is equal to the horizontnl projection of thelifting tm,a of the actual body surface (fig. 4(a). lte lift (mProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-LIFT AND (IqNIEH. (iF Ill, ESSU.IE OP VING-BOI)Y-TAIIJ COMIINATIONS 573d=4. ,i,rcl 0 r _ -
15、 7iic:-._iiic,i._, ,_ ,r? tly tim he cnluulait,d simply by init,graiing I)rtssurestl.e It) lht, half-win_ ov_,r ltu, shnthtl mt,_ nn,I doullli.bmar ltlen, holh suhsonie and SUlWrsonit_ h,ading edges nr,.,roilsideretl. Tip efft,els are not _.onsithwed, mid tht analysisis confintd to the t.ase in wlli
16、uh lht, XI,dl line emanatingfrom lht, h,mliilg _,dge of lhe wing lip f.lls Iwhind t ht, regionof litl_ earry-ov,.,r onlo lhe body. This eontliiitm imposest,he reslrirlion,;l(l f X)(_lm+l)4 (22)on the wings for whMl the mtqhod is to al)l)ly.The value of lift iransniitted to llie llody 1)y a half-wing
17、with a SUl)trst,fit leading edge is given (usrift the solulionof ref. 20) asL_(w) 4l:qr Jm f t +_t_r,f-m_:i: d_ _+7,r#= d,7/ cos -_ , _ . d, (2:1),f,iin lerm._ of lht, roordinalt svslt,ni of figure 4 (b). This resultis douhh,tl to at.count for the lift of lwo hMf-wings _n.t dividedby tw lift of the
18、wing alone to ot_lMn K_-i. For lfll s,ipt,r-sonic Math mlmlwrs tm) is d ,l-tm-(_m-11)/_/ (7m-I1)L Ii/ “e,- JVl_:“ m_7m- (Td_ecshc,! (1 -t-;,t) 1 -r-_m.Bmcos-. ,(l_7,n/“_, ) ()4,wht,l_ m_-l. Nimilarlv for sul,sonie h, luling edges lhere isoblMned using the Iqfl)rol)riale coni_M lifting solution front
19、rt,f/,lt,ll( , 2 , givingL _8%a.,(jm):L, id=tldv.It. is it be iloled that /tsJilr) in equlilitiliS (:2-t) itiid (:2li)depends _tli il lilllllHel tlf iiilliililtqtrs, ill which ftllll life/_ ,nly ._j ,rod _J. This quantity i_tpresent.ed is l_ funcliml or 2;7r,c, for eon._tllnt. Yahies of m/7in chart
20、4 (l/) whi_qL is to serve lls it th,._igli diint ill th,ler-nlinin_“ As_,w sulrl lo the it,slrirlion .f t,qul/liolt (22 .Ft/r lltt 1)urliuse trl_iling-edgt, swt,t,). rhe (liSt, t)f x=0 (.or-responds 1o llili, lLffuhi_r VillgS (tig. 5 (_)7, X= 1 1o leelangtlillrProvided by IHSNot for ResaleNo reprodu
21、ction or networking permitted without license from IHS-,-,-574 REPO1VI. 1307 NAPI()NAI_, ADVISORY COMMITTEE F()R AEIC, ONAUTI(?Swings (lig5 (t), _m(I _=|,r:2 10 lral)0zoidal x_ings (fig.5 (c). For triangular wings, lhe (urv(, of K., t)v Ill(!present 1l(,o15 _ for 15.1-0 is sligdly gr(,aler lhan l_mv
22、 ) a._iv(,n 1)y sh.n(h,r-1)ody lh(,ory _md has not I)(,elt in(lude(I inthe figures, sinre for such. small values of B, 1 sh.i.h,r-bodytheory is lhe more vMid. n(idemally, lh(v.slvi(li(m ofequal ion (-)21)is me( by all lriangular wings wit h no tndling-odge swoet). An examination of figuro 5 (b) for
23、reclangularwings shows good _gr(,(,men( b(lween sh,nder-l)odv iheory_md 1ire )r(sem theory a( _.t-_2, lhe lowosl asl)e.i ralio forwti(.lt llw I)Ves(,nt theory is al)l)li(._d)(, 1o r(langular “_,II lgs.In it(, (_:o of lhe trapezoidal winks (fig. 5 (), 1It(, veslriel ionof equation (22) impos(,s the c
24、ondilion lhal _1 4:_. For av_due of _.1 of 4/3 thor(,is no apl)r(,(.ial)le difl(.rencc b(lween5 .4.3/f(o!0 .I .2 .3 .4 .5_ody radius wmg-$emsp_,n ,ct,o, r/(_) Triangular wing-body eonddnaliou,_.Fmt-m_ 5.-Oomparison of/x_ front (h_ut 4.Sine(, r(tangpl_lr a)M iriangulnr wings _ue very (mHn()n,and sinc
25、e (B(_.,)., is known in (h)s(,d form for th(se )h_nforms, Sl)t(Miz(I v(._ults ()in r(,_tdily t)e ot)t_ine(l fr()m.8Z8.7.2_A2Slender-X; I bad ytheory,35/.2 .3 .4 .5_Jody radius, wing-semispon roho, f/s(I)I lb,oi:_ngular _ing-body comlinalions.Fmlrll,: 5. (_ontinu(_d.Provided by IHSNot for ResaleNo re
26、production or networking permitted without license from IHS-,-,-:JoIIXII II IIIGii=I !II I, IG2I IO-: o=, I_ ou.=-_ o-,-oKS(W)or /(B(T)_._ 0_I yroI -LPO 4_C_+- i ! “-“G+ _/_. 2-.+ !_! _iIt _ Ji- 3-li _ -_i_ _ _. ,I-_,_ _ I =I I. ,-,- I_,J.zl_._. I -I I t_l_i_l_ I T _-I-._ _ I , ,.,_ I-I,/T_Cc_ _. .7
27、._ M3 _Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-57( REPORT 1307 NATI()NAL ADVISORY COMMITTEE F()R AER()NAUTICSTim case fro no afterbodv behind the wing (au also be(ahulat(,d for the high-asp(,ct-tati() raNg(, at _-_ it isiv(lear that lhc lift
28、transmitl(d lo the t)ody is th(, :ame as for( :d=-_ so that /f.r_(e;.o),l(x+l,(-i._., )is t)h)t(ed ll_ a funcli(m of m3a.d “23r/c. in (harl 4 (I).A comparison of I(.:_,) as dct(,rminod from chart 4 (atwith lira( from chart 4 (t) gives an irl(li(alio, of the traitor-tan(, of the afterbody for any l)a
29、rticular (outigtnalion.For small values of the ratio 2B(r/C_)w llwre is very littleefftTt of the af(,rl,o(ly on /fn:v) but, for large vah.,s, iheefte( can 1re as larffe as :everal hm.lred 1)erc(t. At sub-sonic siltotis rio distinction is mad(, b(,lv,(,(,ir Ih(. aft(,rt)o(lyand no-af(erl)o(ly cases.
30、“Phi. diff(.r(,ncc bctw(.ir the two,whi(h is usually siuall in terms of total lift at :.l)(,sonicSl)(Is, is furl lrer redu(d at subso.i(, Sl)e(Is l)(aus(, of the(,ss(,r (,r.h,n(y of lift to b( (arvi(l (hw)r_(r.,am.Wing=incid enee angle.- From (,qua i io)i (7 )( i:,.,- _,:,_( 1.o),a, (13-)The ()fly g
31、o)wral m(,tho(l for d(t(rmining /:.w is slender-l)ody theory-. It ha_ been shown in iof(r()w( 22 by nse of are(il)ro(al th(,()r(,m that fo/“ (:oml)inatio)is with cylin(hi(all)()(li(,s tl., f,)lh)_ in, cqmdity is vali(l .r.h,r (h(, ,s., a.lwm) (“811 Ilr Ilia(It. If it i._ ,qSSllllltd ltltll the wiirg
32、 1.rarrSlllil,;a certain fraction of its lift lo (he body irr(,spev(iv(: i)fwhether the lift is developed by anglt, of altavk or wirl_-ir.idcn(, .ugh, .)r approximai(- value for k., isI(.-w,:s_,Th( vnlu(,s ()f/_,:w) and U_.) as (h,ir,rmi)(,(l fr()m (,(lqali()ns(33) and (34J d() not (lifter by rot)re
33、 than ().01, ,i quantit Nthat is i)ra(li(alv i.(listinguishab(, iN (dm)t I. This smil(lifl(,rl,rw(, is .lu( to thi, (liir(,r(,.c(, i. the forms of _h(, load(listribu(ion (). ihe _ving f()r lifts du(, io angle of at t.rk q)(lwin_-in(i(h, tw(, a ngh.LFT ()N TAIL SECTION l)lJE TO WIN(; VORTICE, SWing-t
34、ail in(,rf,r(, results fr()m d()wnwash iN ih(,)(ffi()n ()f the tail caused l)v th(, wing vorli(es. The l)rol)h,mof adof each of the conligurations investigated is insensitive h) lhedetails of the vortex llow although the downwash behindthe wing and the spanwise distribution of tail load are not.This
35、 conjecture is sutstantiated in part by the theoreticalwork of Morikawa, reference 9, who has calculated the t,aillifts of slender wing-t)ody-tail comlfinations using onefully rolled-up vortex per wing t)anel and using a flat vortexsheet. Only for fully lolh, d-up vortices in the immediatevicinity o
36、f the tail tip does any appreciable difference betweentile two cases o(!ur. The results of I,omax and Byrd, refer-eric(; 10, for a family of swept wing-body-tail (omt)i,uttionsare in accord with the findings of Morikawa. It was on thebasis of this evidence and t)ecause of its great simpli(.itythat t
37、ile use of one wing vortex pet“ panel was adol)ted. Theadequacy of this assumption amt its range of apl)lication issubsequently determined t)3- comparison between experiment.and theory.Th( circulation distril)ution at lhe wing trailing edgedetermines lhe strength lm and the Sl)anwise position Jwof t
38、he vortex at the trailing (tgc. The actual circulationdistribution is replaced I)y an equivalent horseshoe vortexcorresponding to the l,elmerlz model for uniform loading.Figure 7 illustrates this mod(l. Note that iigurc 7 containsthe tacit assumption thai the maximum value of the (.ircu-Provided by
39、IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-%IEPORT lB07 NATI()NAL AI)VISORY ()MMITTEE l,()l AEI()NAUTI(S%.201.5g_o _l.O0ggo ._, , , ! I lTheoretical lateral position from chart 6-Tip vortex-/-:Theoreficol asymptoticlateral posMEonto)_Jiw7f dO C t.Inboard vo
40、rtexij / f-.-._“,-4 - Body, vortexTheory:Vortex path in free-stream directionI IVortex path corrected for cross-flow-and induced effectsTip vortex -FxInboard vortex - J5“/ _- Body vorte_0 4 8 12 16 20Angle of ottack, cz, deg(a) Lateral losilion of voriex.(h) Yertical position of vorl0x.Fi_llrliE 0.
41、(!uInptiri_on between theory and experinic, nt for hiteraialid vertical positions of wing vorlex 1.8 c. behind wing of aepeeiratio 2 lriangllhir wili_ lind body conll)illatJoll; .l/ao: 2.0, 1“i1,_=0.33.lalion is at the win_-body junclule. Since ttie lift of theI)ound Yortox is # 1“o1“, per unit span
42、, the value of .,(,1111 lie oslilllalOd frolli the followin_ _elies _lf eqllllliolis:Ltv(.fw_r,v)-2p_ Ihqn of deteimining lhe lateral i)ositions of thewing vortices niu._l lie solved before the foregoing equationtllli be used lo Ovilhllilo F The assuniplion is illitd0 thati,he vorlices of liio wing
43、in conillintilion ltio di_cliluged atlho center of wnliciiy of the panels of lhe wing alone ils;7 I i /“I !r rj C+,- /,:o :_ i/“F“ I :;,a) ,u,_:x(: m, _, 2 C ? (Angle c)f attack,_l,degl(a; ,alt.ral peril,Jan of vorlex.ilu Vertical posithJn of vortex.?ic, UllE 0.- (Joniplirisl)li between theor.v and
44、eXllOriinonl f, lr t/tlerlilaild verlieal posilioiis of wing voriex 1.8 c, lwhind Wili_ of aspuclratio 4 lri_iigilhir wing and body cointfilialion; Joo 2.0, r/._:-l).70,dolerlnined I,v lifting-line lheorv or linear theory. Thislissunll/liOli i_ lilut_lii heciiil._e lhe uiicuhtlion _lim iibiil ionis
45、not genern 3“ known for the wing-l.ldy _!olnhhiiltioli.Tile valhliiy of ttiis liSSllliipliOll Clili lie exunhied for slen_lorwhig-body !onilliliilthlii,_ for wiiiuh llie Slnili lou_ling i_ kilolvnillld flOlll whhh llw llileial posilioli of the erie (lilt 1,t,detorntined, lit fael, lile hilerlil w)il
46、ex lloSilion eli tile I)tisis(if sender-I)ody lhiOlV iS :., ,. I- (,),7, -(:),:.:-t_) -,:/ -t- . sin- ,._“ “ ,/,-(:t-/ +(.c.),J(-:)=., _,This tqullliou gives the lateral position of the vortex ll._ tifriiclioli of lhu ._eniispliii of lhe exposed wing panel nnd llS ilfunction of the radhis-seniispan
47、ratio. Tile, niaxiniunidevialion belween the values given by this equation and filLwing-alone viilliO of 0.7N0 (or lr/4) s lit)Olll : poleOlil. Thisresult is ilidellendenl _lt“ tile phui fOllll of the wJlig or body hifront, of lhe iiilixiintiui ._plili posithm silica in sleli_hq-bodvtheory lhe latent ial liiid, leliCe, lhe cirmiltilion del)eiid_ onlyon liie cro._sthlw plune illi(h,r considertilion.Fat nonsh,ildel wing-body (olnlillltlions the lateral posi-tion (all easily he deterlnined if ill0 lift
