NASA NACA-TM-948-1940 A simple approximation method for obtaining the spanwise lift distribution《获得顺翼展方向翼展向升力分布的简单近似方法》.pdf

1、U.S, DEPARTMENT OF COMMERCENational Technical InformationServiceN62-57948A SIMPLE APPROXIMATION METHOD FOR OBTAININGTHE SPANWISE LIFT DISTRIBUTIONNational Advisory Committee for AeronauticsWashington, D.C.Aug 40Provided by IHSNot for ResaleNo reproduction or networking permitted without license from

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4、GAL MEMORANDUM NO. 948A SIMPLE APPROXIMATION NETHOD FOR OBTAININGTHE SPANWISE LIFT DISTRIBUTION“By 0. SchrenkPRELIMINARY P_EMARES_eIn this paper a simple approximation method is pre-sented for rapidly computin_ the lift distributions ofarbitrary airfoils. The numerical results are comparedwith those

5、 obtained by an exact method and for many pur-poses show a satlefac_ory degree of accuracy. The latter,for all practically occurring cases, can be estimated atthe start of the computation work with the aid of the com-parison examples _iven.The method described below enables the approximatedeterminat

6、ion of the lift diztributlons in a few minuteswith an accuracy sufficient for many purposes. It is alsocharacterized by a certain simplicity which is useful inthe clarification of many questions and is in accord withthe englneerls point of view. Finally, the method is ap-plicable to cases for which

7、all other methods are entirelyunsuitable (for exampie, wings with end plates).Simil_r methods, as the author subsequently found,have already appeared elsewhere. The surprising acc-aracyof such simple methods is, however, generally unrecognized,so thmt a presentation of comparlson computatLons whichp

8、rovide a measure of the degree of accuracy obtainable,should lead to an extended application of the method.The author wishes to express his appreciation to hisco-worker at G_ttlngen, Mr. N. Hiorth, for carrying outthe laborious computations required for the comparison.-, I!*“Ein einfaches l_aherung_

9、verfahren sur Ermittlung yon Auf-triebsverteilungen lanes der Tragfl_elspannwefte.“Luftwissen, Bd. 7, Nr. 4, April 1940. pp. 118-120.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-2 NACA Technical Memorandum No. 948YUNDA_ENTAL IDEA 0 THZ METHODThe p

10、lausible assumption is made that the real liftdistribution lies between an_ideal distribution independ-ent of the wing shape and a distribution determined in asimple manner by the wing shape. The ideal distributionis that with minimum induced drag and constant induceddownwash velocity - that is, for

11、 the usual monoplane, theelliptic distribution; while the distribution dependenton the the shape is proportional to _ t at each positionof the wing.COMPUTATION PROCEDUREIn the case of the untwisted wing the angle of attackis not absolutely necessary. There is drawn instead (forthe monoplane) the sem

12、iellipse of equal area with thechord-dlstribution curve, and the llft distribution is ob-tained by forming the arithmetical mean between the twocurve s.In the case of twisted wings (and similarly for wln_swith aileron or flap deflection) there is first determinedthe zero llft direction of the entire

13、 wing. A sufficientapproximation for this is the direction of the mean aero-dynamic twist8(x) tCx)S=tWhere the bars denote mean values, 8(x) and _ arethe twist angles between the reference direction of theentire airplane and the local and mean zero lift direc-tions, respectively. For all further com

14、putations, theangles of attack and twist are reckoned from the zero llftdirection _Iven by 8._ith angles reckoned as indicated above, the llft isdecomposed into a component without twist and a twist com-ponent without lift. The second component is determinedon the basis of a mean value which has the

15、 zero line in-stead of the ellipse as the ideal distribution, and forwhich the twist angle must always be considered.The general case with tWAst is most conveniently com-AProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-fe%NACA Technical _emorandum No

16、. 948 3puted with the aid of the following formula, which requiresno explanation:dA i d_c_ 4 “_q _ tCx)+-._ 1- -_dm W1 dca8(x)t(x)+_ q _-idc_In the above relation - andd_be taken as constant along the span.rate formula is given below.The trial computation of _cadm_Ca can generallyd_coA somewhat more

17、 accu-dC aand d.-_ may be some-what refined by the introduction of a correction factor(1 + _) setting:de ade_ 1 + dca F (l + _)The constant _ for various taper ratios was determinedby trial in s_ch a way that the lift determined with ourapproximation method is _iven as correctly as possible(fi_. 1).

18、 In the case of the rectangular wln_ the valuethu_ determined a_rees with the theoretically determinedvalue of Glauert. For other taper ratios the agreementhas not been checked and in this connection is not required.A further refinement in the value of K, nevertheless,seems unnecessary.The computati

19、on procedure thus consists of the follow-ing steps :a) Coz_putat_on of 8 by forming the mean.b) Trial computation of dca/d_ and dCa/dO_.c) Computation of the lift distribution by theformula for d.A/dx.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-4

20、 NACA Technic_l Memorandum No. 94BACCURACY OF THE METHODThe comparisons _iven in figures 2 to 13, between theaccurate value_ computed by the method of Multhopp and theresults of the approximation method, show in _eneral a sot-isfactory, and to some extent even surprising, agreement.The error arising

21、 through the assumption of a constantvalue of dCa/dC_ along the span in unfavorable cases, canbe eliminated by computing a mean value_c,(x) t(x)dca dc_and then computin_ the lift by the only slightly alteredformuladx 2 q _ dc_. t(x) + d_ T,t 1 -I dCa(X)+ _ q d_= 8(x) tCx)The sharp difference between

22、 lift without twist and twistwithout lift, to be sure, no longer arises. This correc-tion has not been applied in the example here given.Further refinement through additional computations,at the expense of briefness and simplicity, does not ap-pear to be of advantage. The deviations, moreover, occur

23、at such positions where the exact theoretical solutiondoes not agree with actuality.- for example, for unroundedwing tips and at transition positions of ailerons andflaps.The deviations at the transition positions of ailer-ons _nd flaps can be readily balanced by hand with the aidof examples _iven i

24、n the figures.POSSIBILITIES OF APPLICATIONThe method, as mny be seen, is suitable for all prob-JProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA.Technical Memorandum N0948lems where too great accuracy is not required; that is, ingeneral, for inve

25、stigations with regard to the maximumlift coefficient Camax, stalling, and static equilibriumproblems. By the decomposition into an ideal, plan form,and twist distribution, simple and time-savin_ relationsmay be set up for frequently repeated computations of bend-ing moments, transverse forces, and

26、torsional moments alongthe sp_n. For the computation of the induced drag and forthe downwash computation, the method is not directly appli-cable.The method is suit,_ble for the determination of liftdistributions also in cases for which the usual methodsf_il completely. Thus, it is applicable to mono

27、planewings with end plates. Ideal distributions that take theplace of the ellipse c_n be computed on the basis of aninvestigation of the Aerodynamic Experimental institute(reference l) - the distributions there given being forsmallest induced drag and constant downwash. The relationfor the lift dist

28、ribution now becomes:&A 1 dca _ _I_ _i : _ q 2_- _ t(x) + i -b3_. f f(x) d(x)b2I dC a+ _ q -_-=8(x) t(x)_here f(x) is the function denoted by Mangler as the“ideal function“ for the giwBn case with end plates,The method should likewise find application to bi-planes and other arrangements.SUMMARYThe a

29、pproximation method described makes possiblelift-distribution computation_ in a few minutes. Compar-ison with an exact method shows satisfactory agreement.The method is of greater applicability than the exactProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IH

30、S-,-,-#L6 NACA Technical Memorandum No. 848method and includes also the important case of the win_with end plates.Translation by S. Reiss,National Advisory Committeefor Aeronautics.i.REFERENCEMan_lor, W.| The Lift Distribution of Wings with EndPlatos. T.M. No. 856, NACA 1938.I!|Provided by IHSNot fo

31、r ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA Technical Memorandum No. 948 Figs.l,2,3,4,5,60.20 0.4i02 /“0.6ta, tl0.8 1.0Figure 1.- The values of Kthat occur inthe formulas for dA/dx. Fornon-trc_ezold shaped wingsthe n_rest value of thetaper ratio il used for thed

32、etermination of K.%ApproximationErect by Mu_Ithopps methodo._ _.I0 C2 0.6 1.0_-b,.0.1“ _ Trapezoidal _| t a. i:i=0.25 _-_0 0.2 0.6 1.0 0 0.2 0.6 1.01.41.00.60.20Fi_:es 2 to 5.- Comp_.rlson curves with b2/=6.67 for variousttp_ ratios. The comparison foz ta.ti=0.5 isgiven on Fig i.6. 1% !t_ist“ ti/_.Z

33、:.r._ -._ _cal e “/ “-_._-. _ en/ar_ed - Approximation-. “-. 3 =0.I 1,o L =0._ J o_ _. ,./ b2Trapezoidal ta:ti=0.5 _-b/ZI t I I I I I0.2 0.4 0.6 0.8 1.0Exact byMul thopp smethodfigure 6.- Comparison curves for ta_ti=0.5 for varioud valuesof F/b 2. No twist.Provided by IHSNot for ResaleNo reproductio

34、n or networking permitted without license from IHS-,-,-Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA Technical Memorandum No. 948 Figs.7,8,9,10Jlo0.160.080.0.16-o.o8 _-:I I , t I, , -._0.2 0.4 0,6 08 1,0o -k. _ t I , !0.2 0.4 06 08 1.0Figure 7

35、.- Comparison curvesfor a wing withouttwist with cut-out in center.All three curves encloseequal areaKey for Fig. ?.Chord distributionApproximationExact by Mul_hoppsmethod.Figure 8.- Comparison curvesfor a rectangularwing with semicircularrounding, without twist. Thedeviation of the approximationmet

36、hod at the wing tip for thenon-rounded rectangular winghere almost completelydisappears.Key for Figs. 8,9,10.ApproximationExact by Multhopps methodJo o.2 o.4o,_.%,_ T2“_oFigure 9- Comparison curvesfor a wing wlthta:ti=O.5 and bB/F=6.67 forvarious values of Ca with alinear twist which is 0 atthe cent

37、er and S downwardsat the wing tip.1vU- O.016 Ca=0_ _ _ X-_0.2_-_._ O.6 b72 1.0_ _.-0.016Figure I0.- Larger scalerepresentationof the curves for Ca=0 frmFig, 9.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-)Provided by IHSNot for ResaleNo reproducti

38、on or networking permitted without license from IHS-,-,-NACA Technical Memorandum No. 948 _tge.ll,12,13M4_O.024 _a=O0.016-1.0, ,/Figure ii.- Results for Ca-Oas in Fig. I0 witha twist which similarly amountsto S at the wing tip butincreases parabalically.- Key for Figs. Ii,12,13-ApproximationExact by

39、 Multhopps method9tkl=O. 2 “t ._ .c8kl=14.3 o _J/),.-0.8 _ -0.160.08-_6“ I I I0.2 0.6 1.0.0.08Figure 12.- Comparison curves for a rectangular wlng with b2/F=5 foran aileron deflection B=0.25_14 with and without lift.The corners of the approximation curve can practically be well roundedoff by hand fo

40、llowing the example given and thus considerably betteragreement 18 obtained The rolling momentm of the non-rounded offapproxlmatlon curve very well agree with the exactly computed values.0.320.24O.160.080 8kl 60 tkl O.0.2 0.4 0.6 0.8 1.0.Figure IS.- Comparisoncurves for atapered wt.ng with ta:tt=0.5 and b2/F=6.67 with60 deflected split flapalong center half ofspan.YProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-