1、./v . , -%NATIONAL ADVISORY COMMITTEEFOR AERONAUTICSREPORT No. 572DETERMINATION OF THE CHARACTERISTICS-; OF TAPERED WINGSBy RAYMOND F. ANDERSON.!rREPRINT OF REPORT No. $/2, ORIGINALLY PUBLISHED FEBRUARY 15J_7.f . -i.REPRODUCEDqYNATIONAL TECHNICALINFORMATION SERVICEU. S OEPARFMENFOF Cr_MMERCti1940BtP
2、rovided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-AERONAUTIC SYMBOLSI. FUNDAMENTAL AND DERIVED UNITSLength .Time Fores Power. _ “.Speed .SymboltFP.VMetrioUnit Abbrevia- t tionmeter 2 _ msecond . sweight of I kilogram kghorsepower (metric) _fkilometers p
3、er hour kph1.meters per second_ _._ ropeUnitEnglishfoot (or mile) second (or hour) weight of 1pound iAbbrevia-finnft(ormi)sec(orhr)Ibhorsepower_._ .miles per hour .feet per second hpWgmI#Ss.GbeAVqLDDoD,D,OMass= W . and 760 ram; or 0.00.0378 lb-ft- sec 2Specific weight of “standard“ air,Moment g of i
4、nertia-m_. (Indicate axis of 0.076511b/cuftradiusofgyrationk by propersubscript.) , “ “ . .Coefficientofviscosity2.GENERAL SYMBOLS,Weight=rag v Kinematic viscosityStandard acceleration of gravity=9.80665 m/s _ p Density (mass per unit volume)or 32.1740 ft/sec 2. Standard density of dry air, 0.12497
5、kg-m-4-s 2 at 15 C1.2255 kg/m a or$. AERODYNAMIC SYMBOLSArea of Wing.GapSpanChordbaAspect ratio, _vTrue air speed1Dynamic pressure, _pVLift, absolute coefficient Cz-_Drag_ absolute coefficient Ca- DC DoProfile drag, absolute coefficient vo _-_Induced drag, absolute coefficient Cm=q_Parasite drag, ab
6、solute coefficient C_=q_C= CCross-wind force, absolute coefficient q-_2626 . stQ0, i, _ Angle of setting of wings (relative to thrust line)Angle _of stabilizer setting (relative to thrustline). Resultant momentResultant angular velocityR Reynolds number, p P7 where l is alinear dimen-/asion (e.g., f
7、or an airfoil of 1.0 ft chord, 100 mph,standard pressure at 15 C, the correspon, lingReynolds number is 935,400; or for an airfoilof 1,0 m chord, 100 raps, the correspondingReynolds number is 6,865,000)a Angle of attack. Angle of downwasha, Angle of attack, infinite aspect ratioa, Angle of attack, i
8、nduceda_ Angle of attack, absolute (measured from zero-lift position)Flight-path angleProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-REPORT No. 572DETERMINATION OF THE CHARACTERISTICSOF TAPERED WINGSBy RAYMOND g. ANDERSONLangley Memorial Aeronautica
9、l Laboratory408317 0-41-1Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS_,ADQU._aTlmS, NAVY BUILDING, WASHINGTON, D. C.Created by act of Congress approved March 3, 1915, for the supervision and d/rectton of
10、 the scientific study of the problemsof flight (U. S. Cede, Title 50, See. 151). Its membership was increased to 15 by act approved March 2, 1929. The members areappointed by the President, and serve as such without compensation.VANlvgV_ BUSH, So. D., C“hatrman,Washington, D. C.G_oeGg ;L Mr.av, Sc.
11、D., 1r/co _hatrmau,Washington, D. C.CH_s G. A_e_,r, Sc. D.,Secretary, Smttbsonlan Institution.HZN_Y H. AR_rOLV,Major General, United States Army,_RoB_a_ E. Domm_r, M. S.,Pittsburgh, Pa.ltovm,z H. HINCxLt-r, A. B_,Assistant Secretary of Commerce.J_oM_ C. HUNS_LK_, SC. D.,Cambridge, Mass.SrvNZT M. Kaa
12、_s, Captain, United States Navy,Deputy Chief of Staff, Chief of the Air Corps, WarDepartment.GEomg IL B_rr, Major General, United States ._rmy,Acting Chief of the Air Corps, War DepartmentL_ 3. B_Qos, Ph. D.,Director, National Bureau of Standards.Do_xu_ H. CO_SO_LT, B. S.,Administrator of Civil Aero
13、nautics.Bureau of Aeronautics, Navy Department.Faa_cxs W. R_cn_vmr_, Se. D.,Chief, United States Weather BureavLJOHN H. Towns, Rear Admiral, United States l_avy,Chief, Bureau of Aeronautics, Navy Department.EDWA_ W_, Sc. D.,Washin_gton, D. C.0zv_ W_nr, So. D.,Dayton, 0lflo.Gzo_z W. I_wm, D_rector of
14、 Aerow_utlcal Researob S. PAUL Jon_s_olv, _oord_tor of Research_on_ F. VIcfo_r, georetarv_aY J. E. R_n, _er-_arge, Lang|ey Memorial Aero_ttea_ Laboratory, Langley FteZd, Va.S_r_ _. DmFgaNcL Hngtneer-in.Chc/rge, Ames Aeronautical Laborator1/, Moffett Field, _J/If.AERODYnAMiCSPOWER PLANTS FOR AIRCRAFI
15、AIRCRAFt MATERIALSTECHNICAL COMMITTEESAIRCRAFT STRUCTURESAIRCRAFT ACCIDENTSINVENTIONS AND DESIGNSCoordination of Researoh _eed_ of Military _d _i| Avb_tto_Preparation of Researc?t ProoYa_Allocation of ProblerasPrevention of Duplication_oaside,_tto,t o! Inve_tio_LANGLEY MEMORIAL AERONAUTICAL LABORATO
16、RY AMES AERONAUTICAL LABOI_ATORYLANGLEY FIELD, VA. MOFFETT FIELD. CALIF.Conduct, under unified control, for all agencies, of scientific research on the fundamental problems of flight.OFFICE OF AERONAUTICAL INTELLIGENCEWSHINGTON, D. C.Collection, classification, compilation, and dissemination ofscien
17、tific and technical information on aeronauticsProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-REPORT No. 572DETERMINATION OF THE CHARACTERISTICS OF TAPERED WINGSBy RAYMOND F. ANDERSONSUMMARYTables and charts for use in determining the character-istic
18、s of tapered wings are presented. Theoreticxd Jactorsare given from which the following eharacteristics oftapered wings may be jound: The span lift distribution,the induced-angle-of-attack distribution, the llft-curveslope, the angle of zero lift, the induced drag, the aero-dynamic-center position,
19、and the pitching moment aboutthe aerodynamic center.The wings considered cover the complete range of taperratios and a range of aspect ratias from _ to 20. Thefactors given include the effects of sweepback and tudstand apply to Wings having a straight taper plan form withrounded tips and an elliptic
20、al plan form. The generalformulas of the usual wing theory are also given fromwhich the characteristics of a wing of any form may beex_ulx_ed when the section characteristics are knownfrom experiment.In addition tp the tables and charts, test results aregiven for nine tapered wings, including wings
21、with sweep-back and twist. The test results verify the values com-puted by the methods presented in the first part of thereport. A final section is given outlining a method forestimating the lift coe_isnt at which a tapered wingbegins to stall. This method, which should be useful forestimating the m
22、aximum lift coel_cient of tapered wings,is applied to one of the wings tested.INTRODUCTIONA large amount of work has been done on the deter-mination of tapered-wing characteristics from airfoiltheory. Glauert has given some of the characteristicsof wings with straight taper for a limited range ofasp
23、ect ratios (references 1 and 2). Hueber has givenother characteristics of wings with straight taper for alarge range of aspect ratios (reference 3). Severalother papers have given various characteristics oftapered wings. The data of all the papers, however,kave been limited by one or more of the fol
24、lowingfactors: Range of aspect ratio and taper ratio, numberof characteristics given, and omission of data on wingswith sweepback and twist. In order to provide morecomplete information, data are given in this report fora large range of aspect ratios, for the complete rangeof taper ratios, and for w
25、ings with sweepback and twist.As airplane wings are usually rounded at the tips, thedata are given for wings with rounded tips.In addition to the theoretical characteristics, theresults of tests of nine tapered wings, including wingswith sweepback and twist, and a comparison of someof the test resul
26、ts With theoretical values are presented.The characteristics are given for wings having astraight taper and rounded tips and for wings havingan elliptical plan form, with an aspect-ratio range from2 to 20. For these wings, formulas are given usingfactors that are presented in tables and charts. From
27、the formulas and factors the following characteristicsof tapered wings may be determined: Span lift distri-bution, induced-angle-of-attack distribution , lift-curve “slope, angle of zero lift, induced drag, aerodynamic-center position, and pitching moment about the aero-dynamic center.METHOD OF OBTA
28、INING DATABASIC CONCEPTSWhen obtaining the data used to determine the char-acteristics of wings, a tapered wing is considered to con-sist of a series of airfoil sections that may vary in shape,chord length, and in angle of attack from root to tip.Each airfoil section is assumed to have an aerodynami
29、ccenter through which the lift and drag act and aboutwhich the pitching moment is constant.With the section characteristics as a basis, character-istics of the entire wing are obtained by integrationacross the span. Formulas for the integrations wilIfirst be given for a wing of any shape and zero di
30、hedral;that is, the aerodynamic centers of all the sections alongthe span lie in a plane which passes through the rootchord and which is perpendicular to the plane of sym-metry. Wings of particular shape will be consideredlater and a method for including the effect of dihedralwill be given.For any t
31、apered wing the span lift distribution maybe considered to consist of two parts. One part, whichwill be called the “basic distribution,“ is the distribu-tion that dends principally on the twist of the wingand occurs when the total lift of the wing is zero; itdoes not change with the angle of attack
32、of the wing.1Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-2 REPORTNATIONALADVISORYThe secondpart of the span lift distribution, whichwill be called the “additional distribution,“ is the liftdue to change of the wing angle of attack; it is inde-pen
33、dent of the wing twist and maintains the same formthroughout the reasonably straight part of the lift curve.In the designation of the characteristics of a wing,lower-case letters will be used for section characteris-tics and upper-case letters for the characteristics of theentire wing. The basic and
34、 additional section lift coef-ficients are then c_ and c_o. A complete list of sym-bols follows. It is convenient to find the additionallift coefficient for a wing CLof 1 and it isthen designatedc_o_. The two coefficients are related by C_=CLc_Q_.The total lift coefficient at any section is found fr
35、omthe basic and additional coefficients fromC_o=eZ_-t-CLC_azwhere c_o is the lift coefficient perpendicular to the localrelative wind at any section as distinguished from cvwhich is perpendicular to the relative wind at a dis-tance. For convenience, however, c_ will be used andmay be considered equa
36、l to cz0.8YMI_OL8A, aspect ratio, b_/S.b, span.c, chord at any section along the span.c, tip chord (for rounded tips, ct is the fictitiouschord obtained by extending the leading andtrailing edges to the extreme tip).c, chord at root of wing or plane of symmetry.S, wing area._, angle of sweepback, me
37、asured between thelateral axis and a line through the aerody-namic centers of the wing sections. (Seefig.i.)(,aerodynamictwistin degreesfrom rootto tip,measured between the zero-liftdirectionsofthe center and tip sections,positiveforwashin.z, longitudinal coordinate, parallel to the rootchord.y, lat
38、eral coordinate, perpendicular to plane ofsymmetry.z, vertical coordinate in the plane of symmetry,perpendicular to the root chord.x_._., x coordinate of wing aerodynamic center.a, wing lift-curve slope, per degree.ao, wing section lift-curve slope, per degree.m, wing llft-curve slope, per radian.m0
39、, wing section ;ft-curve slope, per radian.a, angle of attack at any section along the span.a, wing angle of attack measured from the chordof the root section.ao, absolute wing angle of attack measured fromthe zero-lift direction of the root section.ab, angle of zero lift of the root section.a_0_ an
40、gle of zero lift of the tip section.COMMITTEEFOR AERONAUTICSa_cL=0),wing angle of attack for zero lift.a, section induced angle of attack.cz, section lift coefficient perpendicular to thedistant relative wind.Subscripts for c_:0, refers to section lift coefi_cientperpendicular to the local rela-tive
41、 wind.b,referstobasiclift(G_=O).a,referstoadditionallift(anyC_.).al,refersto additionallift(CL-I).cd_,sectioninduced-dragcoefficient.c_0,sectionprofile-dragcoefficient.c_,.,sectionpitching-moment coefficientabout sec-tionaerodynamiccenter.l,sectionlift.mz,sectionpitchingmoment due to additionalliftf
42、orces.Alr_o,wing pitchingmoment due to additionalliftforces.C,z, wing pitching-moment coeffi_cient due to addi-tional lift forces.C,_, wing pitching-moment coefficient due to basiclift forcesC,I wing pitching-moment coefficient due to thepitching moments of the wing sections.C,o.,., wing pitching-mo
43、ment coefficient about itsaerodynamic center.Cz., wing lift coefficient.Cv_, wing induced-drag coefficient.GENn_ rORMt_ASFormulas in terms of the section characteristics.-The induced angle of attack at any section is obtainedfrom c_ byc_m0The section induced-drag coefficient is obtained froma, and c
44、_ fromand the induced-drag coefficient for the entire wingmay be obtained by integration across the semispanfrom the section values:C_ 2 fb/_=S J0 a,c_cdy (1)In order to obtain the aerodynamic center and thepitching moment of the wings, a system of referenceaxes was used; the origin w_ at the aerody
45、namic centerof the root section and the axes were as shown in figure1. The x axis (fig. 1 (a) is parallel to the root chord,and the y axis (fig. 1 (b).is perpendicular to the planeof symmetry with positive directions following thevectors. The wing axis is the locus-of the aerody-namic centers of the
46、 sections and lies in the z-y plane.The lift I and the coefficient c_ of any section along thespan are represented in figure 1.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-DETERMINATION OF THE CHARACTERISTICS OF TAPERED WINGS 3A typical section wi
47、th the aerodynamic centerlocated at a distance x from the y axis has a momentarm ofZ COS _sand a pitching moment about the lateral axis (fig. 1)due to the additional lift force ofml._ -X COS ets/abut the lift increment of any section isl,=c_.qcand the pitching moment for the entire wing is obtainedf
48、rom!b/2Mt=-2q cos a, jo c, cxdyW/ng oerodynam/c center, Aerodynom/c cenCer, of any sechonRoof-s.ecfion . _ CL between, roof ond /poe/oaynomic Cemter , _ _. IRoof- x, ectJon chord“ , ,_r,p- sect/otv chord .j(a) Aerodynamic cenfer of cor_frucf/on t/p secion(a) Detecmtnaflon of twist.Root-seer on aerod
49、_nqmic centerLoI_(b) , ct, _5_/_ , _-_; wing oeroG),r_Tm/e OemTer c,_3nsrrucfion Hp secT/on(b) Stralght-taper wing with rounded tips.i - -(e) Distorted elliptical wing.Fzovam l.-Form of wing38.845 8M 808 877 887“894.9011.2871.2_31.2261.204I.187I.1761.16_1.152I.143I.1301.1271. 1181.1)91.2601. 2211. 198I, 131I. 1691. 257L 148L 13
