1、REPORT 1071THEORETICAL SYNUW2TIUC SPAN LoG E To ELAP ELEcTo FOR s OFARIWIRARY PLAN I?ORNI AT SUBSONIC SPEEDS 1BY JOEX DEYOENGSUhlhIARY.A imPled ijiing-wrface them-y is appiied to the problem ofemrlwting span loading due to flap deject ionjor arbitrar.r.rwingplan forms. U“ith the resulting procedure,
2、 the e$eck of j?apde.fl.ection on the span loading and associated aerodynamicchuracten”stics can be ean”ly computed for any wing which isgYmmt”calabo Ut the root chord and which has a straight quar-ter-chord line owr tht wing semispan. The C$PCLSof cornpresR-biity and spanwiw criation of section lif
3、t-curw sopc aretak(m into account by the procedure.For the case f straight-tajwred u-, a sweep parameterdefinvi as .i8=tan-1 (tan .i),lp ancI a w-chord distributionptirarneter H, defined by(qJ .+s discussed fn reference I, a, is not limited h small values but cm be ss Iarge sn em.zleas dwired, provi
4、ded separation does not omur.c The efieets of mmpres$l%fliw errd seecion lift-eume elope ere eqDfmIent to a change inwing plan hrme and em be mken “intoaceouni by a proper adjustment of the a,. wdueswhereK, ratio of experimental section lift-curve slopeat span station v to the theoretical -ralue of2
5、T, both at the same JIach numberPc, *U chord at span station vd, scccIefactor which has the following values:0.061 for v=l.234 for v=?.381 for P=3.320 for v=4Equation (2) can be w-ritten in an alternateH, in terms of U-LP geometry parameterssignificant; thusUH,=d, _ 1, ( A and theequivalent, angle-o
6、f-attack distribution for unit flap deflectionaJtiL for each case is Jisted in tabIe .44. The -raIues of aJ81Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-24S REPOR 1071-NATIONALADvIs0R3” COMMITTEE FOR AERONAUTICSgiven in table .44 are for (cl/c)=
7、1. The results can be easilymodified for the case of (cf/c) 2. This limit to whichda/d8 can be used will be considered further in the sectionDiscussion when comparisons with experimental results arcmade, ._3. Arbitary span.wise distribution of jlap cford cJc) =zmrialdcl: The flap can be divided into
8、 several parts eachhaving constant da/d and the load distribution due to eachpart determined. The total distribution is then t.lus sum_ ofthese individual load distributions.Lift coefficient,-The lift coefficients due to flap deflectionfor the same flap configurations previously discussed cau befoun
9、d as follows:1. Full. wing-chord jfaps (c,/c) = 1: The spanwise loadingdue to flap deflection is, in general, so complicated thatProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-THEORETICAL SYMMEIC SP.0” LOADING DUE TO FLAP DEFLECTION FOR ,WLWGS ATntl
10、nwrical integration based on four spanwise values cannot.br perfomed aceuratelj- with conventional integrationformulas. However, in appemlix A a spwial integrationformula (which needs but four spanwise dues) is developedwhich applies to the dfierent flap spn. Equation (.A6)can be written aswhere, fo
11、r each of the cases of equations (4) and (5), the hmvalues are gi-ren byhl 0.293 0.299 0.300 0.352 0.301h, a 301 0.301.544 .544 .555 .561 .3.56 .536h-35s.725 .725 .734 .Z?2 . .726 .ixh, .392 :% .392 .105 .395 .393 .393,.2. Constant fracfion of un.ng-chord yaps (cf/c) =consfant:The flap effectiveness
12、 is given by(7),. Arbitrary span wise Wribufion oj jZap chord (c.t/c)=wria ble: Flaps for which cf/c varies spamvise on the wingcan he considered as equivalent to a wing-twist distribution.The effective s-ymmetric twist of the wing is gien by(8)whtw daldfi is now a function of spa.nwke position. Ift
13、,quation (8) is a continuous distribution such that it can beplotted by specifykm its value a fo semispan points, thenthv wing can be considered to be twisted and solutions founddirmtIy as for basic loading.?Ilen equation (8) is discontinuous, the ae of attackcan be divided into spanwise steps of eo
14、mtant. angle ofattack and the total lift can be found by the summation ofthe lift due to each spanwise step. The lift of a spamvisePC.,step is obtained from a curve of Lift. coefficient as arequired to obtain spanwise center of pressure and induceddrag are not. dewloped here. Eowewr, the integration
15、formulas given in refererice 1 to obtain these characteristics _can be used -with acceptable accuracy for the case of loadingdue to flap deflection.No rigorous procedure for determining pitching momentscan he developed from this method. As noted in reference1, ho-wever, a good approximation for a wi
16、ng without flapscan be obtained if the wing sweep is Iarge since moments dueto shifts of spanwise load overshadoxr those due to shifts ofchordwise load. Vilere the effect of flaps is to be considered,as in the present case, it ppears unlikely that the shifts inchordwise loads can be sa.feIy neglecte
17、d. Therefore anestimation of pitching moments due to flaps should not onlyincIude the moment due to spanw-ise load redistribution,which this method -wilIgive accurately from the longiudinalmoment of the center of load, but also should include anestimation of the moment due to chordwise redistributio
18、n.Additional considerations,Several items pertaining to -the usage of the method should be considered.1. Addifice nafure oj loading and spanwise angle of attack:The Jinear relation between angle of attack and loadingdistribution of equation (1) states that all loadings areadditive if the respective
19、angle-of-attack distributions areadded.Thus, the aerod.ynamie coeflkients that restit from theintegration of the Ioading distribution are additie if theydepend linearly on the loading distribution. Hence, lift.coefficient, rollingmoment coefficient., and pitching momentare additive; whereas induced
20、drag, spant+e center ofpressure, and aerodynamic center are not.The additive concept is very useful for the determhationof loading due to flap deflection. The loading due to anarbitrarF span flap having arbitrarF position on the wingsemispan is simply found by adding or subtracting knownIoadings due
21、 to flaps at other Iocations on the wing. Forexamde, if the spa-nwise ends of the flaps ar at the same.,span stations, thing-clord inboard flaps for flap spans measnred from theplane of symmetry outboard. The lift clue to outbomdflaps for flap spans measured from tbe wing tip inboard canbe obtained
22、from figure 5 by use of the relations of equation( 10). For fuII wing-chord flaps located arbitrarily on tlewing semispan, the lift can be obtained from figure 5 as in-dicate in the following example sketch:Throughout the figures. . . . is the comtant spanwise-section lift-curve sloI)edr the aerageo
23、f a small variation. For Iarge spanwise variations ofKthat foowthe function given in cqua.tion (B5)developed m appendix B, tbe pzrmneters 5.4/x_ and x can be reacedby theparameters - and X,respectively.For large spanw-ise wriatfons of K thst do not follow the curveof equation (B5), the sinm.taneouse
24、quations fortbegenend solution cm besolmdfor arbitrary distributions of., vauesof H, can be obtained conveniently from figure zWith the full wing-chord vtdues given above, tllc lift dueto conshant- fraction of wing-chorcI and flaps of arbitraryspanwisc chord distribution can be lound through Ilsc of
25、equations (7) and (9) with the da/d it, is shovn I1o;various forms of symmetric. Ioading can be found quicklythrough use of simplified lifting-surface thcoI y and vtilues ofthe a,m coefficients which are presented in graphical form.It can aIso be sho;vn that if the symmetric. load distributionis kno
26、wn, from any computational method or from experi-ment, then the same a. coefficimts can be used to find cor-responding values of the downwash in t.ho center of the own-wash wake. It is the purpose of this section of the reportto make available the method for completing dcmmwashwithout att wupting to
27、 explore the possibilities or limitationsof the method. In vimv of the importance of being al)k t _predict, downwash, it is considered that inclusion of this matwrial herein is justifiable although not directly related to themain purpose of the report. It is important to rememberthat the method can
28、bc used for any case of symmetric lofid- _ing and is not restricted to that discussed in this report.As noted previously, the a,. coefficients provide a verysimple means of finding the clew-nwash in the center of thedownwash wake of the wing due to any symmetric spa.nviseIoading_ distribution for an
29、y sweep angle. The downv-ashcan be found at spcrific points in the region bctirecn thowing tips from near the quart er-c.herd line t o an in fhlit edistance downstream,Equation (1) was derived (referece 1) to fintl the down-wash at the effecti-c three-quarter-chord line for which thedistance downsre
30、am from the quarter-chord line is K(c/z).The gemeral expression for the distance downstream fromthe quat,er-chord line isxy tan A$where z is the longitudinal distance measured in the hori-zontal plane from the Jving-root quarter-chord point to anarbitrary downwash point. Substituting this general CX
31、-pression .in equation (2) givesd,11($/0) 7Ptan ith theH, values given by equation (13).It should be noticed that, because thc chordwise loadingis concentra-ted at the one-quarter-chord line, equation (13)is independent of aspect ratio and spanwise distribution ofwing chord, w-bile from equation (14
32、) it can be seen that,dowmvash depends on these wing geometric values onlythrough the loading distribution (7.,Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-THEORETIC.4L SYMMETRIC SPAN!ilquatio; (14) fl give the downwasbLOADING DUE TOange le to ari
33、.t rary s-mmetric loading distribution for all values of $ q,tan . for .42, as dcussed earlier. However, from thelinearized compressible-flow theorj- the aspect ratio becomesa-n effectire aspect ratio giren by B.-1.which approaches zeroas M+l .0. Thus the use of two-dimensional da/d thus a deflected
34、 flapresuIts in a discontinuity in the spanwise distribution ofangle of athldi. Spanwise loading for a discontinuous dis-tribution of angle of attack can he found through the simul-taneous solution method of equation (1). The number ofequations, however, must be greater than four to obtainaccurate r
35、esults and then the computation efforts becomeexcessiw. (Reference 1 gives the method for an arbitrarynumber of equations.) llowever, an alternate procedure,originally developed in reference 2, provides sufficient accu-racy with a reasonable number of equations and this proce-dure is the basis of th
36、e following development.The zero-aspect-ratio theory of rcferce 3 shows thtit fora given sweep tmgIe, as aspect ratio approaches zero, span-wise loading becomes independent of plan form. Thus, thealtered boundary conditions (see Development of lklet,hod)required for the presen theory to give spanlvi
37、se loading forgiven flap spans can be determined from he loading givenby reference 3, which applies to all plan fortns. Thesealtered boundary conditions are obtained from equation (1)using the a,. values for zero aspect ratio and the loading dis-tribution for zero-aspect-ratio Wings given by referen
38、ce 3for gicn flap spans.The zero-aspect-ratio values of a., are given ly figure Ifor 11,=0 or are given by the 13,. coefficients of refwwnee 1.These a, coefficients are t.abulatd as follows:TABLE Al% (FOR nt=7 AhTD A.=0),+A 41-10.4624 2.0720 -0.22422 3.8284 5.656s3. 0 -: 62S42.3ss3 4.329EL.- -:.154s
39、-.2928 0 -1.7072 4.0000With these ar coefficients and equation (1), symmetricspanwise Ioading can bc found for zero-aspect-ratio wings.As a comment on the accuracy of the simplified lifting-surface theory for determining additional loading Widlm=7, the solution of equation (1) with .4=0, a values ga
40、ve,to the accuracy of three decimal places, the c,lliptic loadingdistribution characteristic of zero-aspect-ratio wings.The flap end is arbitrarily chosen for the presen t.hcoryas the mean value of the spanw-ise trigonometric coordinatesof the points at which the boundary condition is appliccl findw
41、hich bracket the discontinuity or fiap spanwise end. Thusfor m=7, three flap spans can be defined for both inboardand outboard flaps. Let q, be the flap span, and 6 the spnn-wise point at the end of t.h ihpj thenqr=cos o for inboard flapsv,= 1cos 0 for outhoad flapsThese flap spans are given in the
42、following table:TABLE A2DEFINED FLAP SPANS.Inboard OutbofirdI COW -11111 111, IV Y W“-Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-THEORETICAL SYMMETRIC SPAN LOADING DUE TO FLAP DEFLECTION FOR WINGS AT SUBSONIC SPEEDS 253For the flap spans listed
43、in table .42, the exact span load-ing distribution for zero.pect,ratiocan be found fmreference 3. With the a,s values listed in table Al and theexact values of Q1, (72, G3, and G4determined from reference3, equation (1) gives the boundary condition or twist re-quired for the present. method to produ
44、ce the loading dis-tribution for each case listed in table .42, that is,;=10.4524 $3.8284 O.2928 =_.0720 +5.6568 ! ,.,1. 111 w v“ “ jw+- _h, 0.2991 0.2994 0-2929 0.30;0 o.3)14 0.3009-,;hj .5541 .5544 .5549 .5605 .5563 .5556h .7248 .7250 .7252 .7339 .7275 .7259-Qh .3922 .3921 .3922 .4050 .3950 .3930
45、INTERPOLATION FUNCTION FOR ,SPANWISE LOADING DISTRIBUTIONThe method of this report gives spanwise loading due toflap deflection at four span stations. Since the completeloading distribution is not sufficiently defined by these values,the values of loading at otJcr span stations cannot be inter-polat
46、ed accurately by direct use of the interpolation equationand table of reference 1,The direct use of the interpolation table of reference I canbe obtained by converting the present loading due to flapdeflection to a loading approxtimaing additional Ioading dis-tribution. The conversion factors are ob
47、tained from valuesof the zero-aspect-ratio theory and given by the ratio ofequation (Az) to sin pn for each of the four given spanstations. Define ,=(G/ 0:0229_ _._. -=- . .,=.-L. -. :?. - .-The interpolated loading at span station k is derminedfrom the summation_ . ()=n+-enk - (A9), ,. $:-where equ
48、ations (A2) and- (A8) provide values of Rk vhichare tabulated as follows:.-. . TABIJ3,+8 .,. -.- .-4_46 .8201- .,/7 .135i.3324 .7882 .9556 . M42 .2136 .? = _With the Rz values of table i18 and the loacIing ratios determined from equation (9), the loading G, can be foundat span stations k=;, $ and (q= O.981, 0,831, 0.556and 0,195, respectively).APPENDIX BAIDS TO THE METHODLINEAR ASYMPTOTE OF u,For large vahms of H, the a,n functions become linearlyproportional to /+, (see,e.g,fig.1). Since this linear charac-Provided by IHSNot for ResaleNo reproduction or
