1、REPORT 1050commssuMMmY -INTRODUCTION-_-, _IMETHOD OF THE SUPERPOSITION OF CONICALFLOWS_-_-_-_-_-_-M lines from the ledns- md trefM-edgew-= md m tiePS-.and (d). The case (fig. 1 (a) in vrhich,the Mach numberand aspect ratio are so low that interaction takes place betmeen the tip flow fields will not
2、be treated. An approxi- mate solution to this probkn may be found in referenoe 3. -Whaa a wing with a subsonic leading edge is to be studied,considerable simplification of the probkm may be achieved “-by making” use of the solutions, aaiIable in refermw 41147Provided by IHSNot for ResaleNo reproduct
3、ion or networking permitted without license from IHS-,-,-.1148 REPORT 1050-NAiMONAL ADVISORYCOMMITTEEFOR .4EROItAW1CSand other sources, for the infinite triangular wing.1 Fromthese solutions the aerodynamic characteristics of a varietyof swept-back plan forms can be calculated by the use ofthe super
4、position principle of linearized theory to cancelany lift,beyond the.specified wing boundaries. Two methodsof cmcellat ion have been developed:”on-e,presented in refer-ence 5, uses supersonic doublets and is general enough toapply to curved boundaries; the other, originally due toBusemann (reference
5、 6), canceIs by means of the super-position of conical flow fields. In the present report theccnical-ffow method is used, since it appeaxs to offer someadvantages for the straight-sided plan forms underconsicleration,particularly in determining the integrated lift.The material prwented in this repor
6、t is largely drawnfrom references 7, 8, and 9, with some simplifications sug-gested by practical experience. In particular, the formulasfor the total lift have been reworked tQ substitute, withno increase in computational labor, a combined iprimary”and “secondary” correction for each .of the tprimar
7、y” cor-rections in reference 7. Also, the formulas containing ellipticintegrals have beeu rewritten to tie full advantage ofavailable tables. As in the preceding papers, the final for-mulas W be derived for unyawed wings with tips parallelto the stream, but the application of the general methodand t
8、he basic SOIUtione to other plan forms and problemswill be appnrent. Some numerical examples will be includedin order to show the magnitude of the effects discussed andto summarize the. method. .A table summarizing theformulas is also included.IMETHOD OF THE SUPERPOSITION OF CONICALFLows _A couical
9、flow field is oue in which the velocity componentsu, ,and win heStreamjcro-gt.ramand vertic directions,respect.ively, are constant in magnitude along any ray fromthe foremost point, or apex, of the field. Such flows arefound as solutions of the linearized potential equation forsup) s and any point x
10、,y has the conical coordinate(4)x-m/9sin the field with apex at 8.Other symbols referring to”angular locations -ml -1 0 f m“a m9kwldM the second side is-the extension of a ray from apmO of the wing. Between the leading edge (a=m) and theleading-edge filach line (a= 1), no division of the field isnec
11、e=ary since the lift density is constant in that region,If a sector with apex at A and angle tan- is used to cancelthis unifomn lift, then the remaining superposed fields mustbe used where aa, u=O(5) When t=a 1, u=u=w=O.The solution of the e.upelsonicflow equation satisfying theabove boundary condit
12、ions has been derived in refrrencc 4.3In”the W lane, tlestreamwise component of the velocity isu= kr,p. : cos- , a+t+2af -.t=a (10The sigi refer to the upper and lower surfaces, respectivcly.In fimi.re 5, the eswmtial features of the solution arcshown. - At the top is a detail view of the wing sido
13、cdgo andshows the boundary conditions. In tho center is a typicalplot of the argument of the inywee cosine in equation (10),against ta. mere this quantity is less thtin 1 (i. e.,Otz and t.O)aLeoding-edge con-ecfion - YvOblique troiling-edge correc?im -Symme fric fraifing-edge comecffon - J(a)20 40 6
14、0 80 100Disfance from leading edge, percenf chod(a%tion A-A f3u m-O.4; ?ni.0,6,.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-FORBJITAS FOR THE f31WERS0.NICLOADIXGOF FLAT SWEPT-BACKTVES-G-sWITH LEADIXG EDGES BEHISD MACHLINES 1:63 .=B# Findfoadinq -
15、fLeading-edge correction - Y%ymmefric fraiing-edge correc%-r -1 I 1 I 120 40 “80 80 100Disfunce fmm Ieudinq edie. percenf chord(a) Seetfm A-A BIIjcFO.60i5432B%Ic-I-2- Corecfion forKuf fa condition (estimo+ed)fA - Triongulor-winq loading/Finulfoodng -“Leading-edge correchim -, OMique fraihnq-edqe cwr
16、eciion-2 .Symmefric fraiiing-edge correcfhn -”r r I 1 20 40 80 m 1-00 .Disf ante from leading edge, percenf chord .(b)*dIon B-B Pra-OJWFICmEE IS.Lond dfstribntfons c.ahdated by the aInk8LEowK zm?thd for two streemwfse sm!thns of m mrknpered Z m-O.L .11, fsdhg aIong a mowed inverse cosine curve from
17、thedue of the error at the trailing edge to ZWO, w-M zerosIope, at the boundary of the region afEected. Vilth tbieboundary (the llach line from the point x2,y2), it is possibleto draw a. satisfactory estimate (dotted curve) of the correc-tion needixl to brirg the pressure once more to zero at thetra
18、iling edge.The untapered wing -with the same sweep (m=O.4) rehit.iveto the Xheh lines is show in iigure 18, with the load dis-tributions cslcukted at the same stations.Four section lift distributions are pre;ented (fig. 19 for?n=o.% Ak =0.15 only the rear 60 percent is influencedcoby the subsonic tr
19、ailing edge. The reflection of this infhenceat- the leadiug edge alters the pressure over the rear 40 per-cent of the sectiom At section B-B, the leading- andtrailing-edge interaction afEecta the entire section. A furtherreflection of this effect at the trailing edge must be estimated.At section C!-
20、C the iufhmnce of cancellation of the leading-edge correction at the trailg edge extends over the -ivholeof the chord and any estimate of its magnitude Todd beneceasariIy arbitrary. AIao, B cond pair of reflectionsmust be taken into account. The final pressure dktributionhas therefore been dravrn as
21、 a band within which the truecurve may be shown to lie. Its height is the error introducedat the trailing edge by the firstt leading-edge correction,except very near the leading edge,.where an infinite negative “-correction is ?mowu to be introduced by the second leaSymme +ricfroiling-edge, .a-corre
22、cfion -”) 1 1 I 1 !20 ongr- wing 1000ingF“Leading-edge corm iion -ObJique +iling-edge correction-.b) 1 I 1 I t20 40 60 80 /00Llkfance from” leading edge, percen f chord/ -Trianguior - wing 100ding!.Symmetric froifing-eoge,d) corfj-k”/ 1 II 120 I. .40 60 80 /00=.tonce from Ieocfingedge, percen t“”cho
23、d(b) Eecton B-B j7uk3-025(c) %tfon C-O pJk4=0.S5(d) Seetion D-D pr/cO.46FICNJRE19.L6addistributions calmfatod by the oonicsl-fiows method for four s expressed as afraction of the chord. and B is a constant. If the section ofthe swept-back wi is taken perpendicuhw to the stream(r constant), the chord
24、 leh is; Jnz-?nt(x-c,) (49)Illldmx-fly=mr-7nt(r-cJ.Substit utiot.1 for q in equation (48) gives(51).Then the strength of the leading-edge sin=gularityin u isIIe letding-edge singukwity in the loadirg on the a-iiept-back wing is initially gion I, fig. 16) that in the triangnlar-vring loading. -Introd
25、uction of the leade .conical. .$t station B-B, howeer, the agreement is +erygoocl. At sections Cc and DD, rhere the exact theoretical.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-REPORT 105E-NATIONAL ADVISORYCOMMITTEEFOR AERONAUTICSldzzdfE-”-I(8)I
26、 1 1 1 )u m 40 60 80 100OiStunce from ldi m, percent chord“r- Correcfed fm-!-Conicol-flowstrefkdSimplesweep fheory-“b) , 1 ,100Disto%e from finq.edp”ent %od -(a) Seetion A-A(b) Section B-BFIfiuRE W-bad distributions on the tap?rd wing aa mlculetwl by theconical-tires method, eomed with the twc-dhens
27、ionfd apprcmfnmtion.loading had not beeu determined, the two-dimensional-typoloading lies within the band prescribed by the ccmicrd-flowcalculations. Since the discrepancy between the correctedtwo-dimensional loading and the exact theoretical distribu-tion is already, at section B-B (fig. 22 (b), le
28、ss than the widthof the bands in figures 22 (c) and (d) and must diminish tozero at infinity, it may be supposed that the correctcd two-dimensional curve is at least as satisfactory an approximationto the correct curves at sections C-C and outbotird asat section B-B. lt is probably more satisfactory
29、 thtin canbe obtained by a limited application of tho conirxd-flowmethod.The load distributions derived by simple sweep theoryare included in the last part of each figure to show the mrqyli-tude of the plan-form effect and also, in the case of the un-tapered wings, the curves that the load distribut
30、ions musL5r 4 -1: C2mkaf-fbwsmethal ,I,3 - A, /, Bg-p,2 - BI - Corrected fwo- .-” “%ciih7ensiwltheory-.*(a) .I I I 10. 20 40 60 80 fm “Distonce fnnn koding edge, percen(_chord/!g,.;-.%mpk sweep fheory.-Slender-wing Mew-y (reif3r-Conicot-flO* m acurve for m=O.9 was therefore inserted in the charts fo
31、rvalues of mm equal to or greater than 0.5. T1hen mm.tisIese than 0.5, m=O.9 represents, if the Ieacling edge extendsbeyond z,w, such extreme taper that the successive reflectionof the 31ach lines (at z%,X6,. . .) take place vrithin a er.smaUfraction of a chord Iength and no useful curve can bedram.
32、 l-o curves are drawn for values of m, smalIer than0.2! because of the tip-interference limitation mentioned inthe introduction.Calculation of tip effect.-The foregoing assumption oftwo-dimensional flow can be extended to give fakly simple.6.8.thut is, is rm extension of the -reIocity distribution c
33、alculatedfor the tip section aIong Jines parallel to the leading edge-For this purpose the apprcmimate Ioad distribution “gi-renby equation (60) is used, still further aimplifled by assumingr to remain constant at-its mdue at the Ieading edge of thetip section. (Where the wing is tapering to a point
34、 and uis changing very rapidly, the tip region is so small t-hattheentire cahxlation of tip effects could probably be omitted.)The assumption of constant c results in a failure to cancelexactly the lift along the tip. The assumption of cylindr caflom, while reasonable for the untapered wing (compare
35、 ifig21 (a) tith and 7n=0.4, 7n:=0.6, fls=0.86c0. .The tip effect has been cahdatcd for each wing at 13y=0.8c0,where the loading wu. prewioudy calculated (s. (17b) and(18b) assuming the wing to extend indefinitely. The tiplocations were seIected so that in each case only one reflection “. of the pri
36、mary tip effeck ected the sion at 13y= 0.8e0.Fureh5 shows the rssults of the calculations. The heavydid curve in each case was calculated entirely by thecorrected two-dimensional t.hecrg-th is, by equations(60), (73), and (31). As a oheck on the accuracy of the _cylindrical-flow approximation for th
37、e flow outboard of thetip. Iocation, the accurate theoretical loac was calculatedl?umm ZS.-Loacl clktrilmtfomowr sttenmwke section near tip as dcukted by lwe+llmen-sfenel hrmuks. com with mm ecmmte theomtfml ralues. -. .-:.-.-. .Provided by IHSNot for ResaleNo reproduction or networking permitted wi
38、thout license from IHS-,-,-1172 REPORT 105fiXAT10NAL ADVISORYCOMMITTEEFOR AERONAUTICS -.3P-.CoFrec12dlmensk4WW-H-F! 1 , . ,., 1 iI“1-tth-l-1-2-o 20 40 60 8 /00(h) Tarwed wln m=0.4: m,-0.6. Section at 68-O%, or 9Spero?nt s?misparrFIWCRE25-CantInuedfor one point within the region”of influence of the t
39、ip in eachcase. he procedure employed for the exact caldation wasas foIlows:The accurate Ioadings with no side-edge effects had alreadybeen caIculated, as has been noted, by the conical-flowmethod. A primary tip correction was cakulat ed for eachcase by equation (15), This correction is the cffect o
40、f ctincel-ing the unmodified triangular-wing loading off the tip sta-tion. The remaining pressnre dscwences to be canceledconsisted of those int.roclucedby the leading-edge and trail-ing-edge corrections. These pressures Were computed tJymeans of equations (26) and (46) of the present report. andcan
41、celed b.y the method of reference 5.The ruhs arc designated by the circled points on rmhfigure. At the point at, which the seclion cntms the tip larhcone in each casej a second circled point. indica tea the acrllra( t!theoretical loading. The vtdue cliflers from Lhut cdcuhlt Mlby the appiosinate for
42、mulas only as the two lottdin willwu tip effect%flcr. “ “It may%e pointed otit in concluding this ion 011localculatic that, while t.hp formulas have bum dovelopcI(l for”plan foti with streamwise tips, tho procedure rnrty IXJadated -”obvioti” nieans to raked tips as WC1l. lliiWm;Irin- eveiy Fikk the
43、deviqtio in the tip regions-of thti physiwlflow frofiie assumed potential flow mus bc I.wrnc in mind,.-. . III-LIFT E.-Gfi*ERAL PROCEDUREFOR CONICAL FJ,OWS.The total lift for any Iting is, of course, the inlqywl of Lh(loarg oerthe wiw area Iu gc%crl,l!oywr, it i+lirul(to ohtaiyan analytic expression
44、 for thr M b ti dirwt” “”integrati;3:j-maiz Va (85a)Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-1174 REPORT 105 va K, (m,)1 . (98)qaThe effect of canceling the pressure field induced by thespmetrimI wake correction at the wing tips k obtainedwith
45、 the aid of the previously derived formula (equation(78) for the lift associated -with a single tip eIement. Thepsrameter defining the boundary of the canceIiug tip ele-rnent is now tOinstead of a, and the veIocity on the canceIingsector is .-d:t:)o.dtO=K(m,) &$ (h-m (99)The distance from the apex of the mncehg sector to thetrailing edge may be expre=ed as)11.8(m, to (1 00)Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-
