FOR AERONAUTICS _. REPORT No. 865 :- : I METHODFORCA~$ULiiTING~$VI~~~CHAiACT~~iSTi&I I 1. " BY LIFTiNG-LINE +IiEilI$k tihNG'~&tiEAR, I 'SECTIONLIFT.DATA,- / By JAMES C. SIVELLS and ROBERT ,H. NEELY , 1947 ~_ i~ao~~umc sm5oLs -, 1. FUNDAMENTAL AND Dl!W@D UNITS c ‘. Meti Y-. ,jbgliak - r - symbo! :.. _ unit / ._’ - YEiT -. foot (or mile)-,: ______ .& (or mi) ----- ------;----- _ eeoond (or. hour)-,-y,, sea (or hr) --Fore ___-____ weight of 1 kilogram-,,-- kg weight of l-pound----- lb ,’ - ~ - -. sayer _____ ,, P -- homepower (met&)-- ___ __________ horsi$oyer __________‘ - hp kilometer6 per hour------ . kph ,mileeperhour ________ mph, _- w -------’ __ .. -v .- {meterkper~ndl_,___ T- .mgks, f-eetperyond _____& fpa .- : - 2. G@ElZA% ~SYMEOLS . ,, ),- -I.~ / W w&&t=& ..- _ *’ ’ Kinematic vi&osity 0 Standard acceleration of gravity~~9.36665.m~s~ -p’ 1 Density (mass per unit volume) . ,or 32.1740~ft&%x .-. .--: Standard density of dry air, OJ2497 k&m43 at 1s” C ‘L ,M&&&E~ ., ,-- -and .76O.mm; or 0.002378 lb-ftA sd na .- - Specific weight of “stqlard” air, I,2255 @/ma -or_ ; , ‘- Momen! of .mertia--mk’. @Indicate axis oft 9.07651 Ib/cu ft I ‘/ radius of gyration-k by proper’subtiript.)’ -. G Coefhcient of -viscosity , : _ - ; .,,-I- , ., .~ .a; ‘~&D&,& &&&j’ ,:- / - m: -_ ‘/ - .i, -A.&&of setting of wings (relat&e to’thrust line) irea -= . &~es of 6. SC -Angle of’ stab@& setting (relative ‘to thrust ‘:. Gapi .i liI@) -- -. ._ ;>-- _’ Sp&;n -r ,,. --. , & Resultant moment .‘t. _- .’ ’ 52 _ Resultant angular y,elocity- -: . _ ,, -. - _ Reynolds-number, _‘vI where Zis alinear dinion- .- < he&ipe(?d .. ,:. ‘. sion (e& for an a$?oil of 1.0 ft chord, 100mph< -c: ‘: -standard pressureatt 150‘ C, the dorresponding !z .. Dynamic pressure, &VI ’ .- --. .- / Reynolds number is 935,460; or for an airfoil ’ of i.0 m chord; 100 mps, the xorresponding L. Lift, absolute coefEcient CL=$ ,. Reynolds number is 6;865,060) \ , D .- - -1. a! ,~ Angleofattack ,:~ _. D Drag, absolute co.&cient .&==g -. B Augle of :downwash .’ Prosle drag, absolute~coefhcient&,,,=$- ’ -2 An& of attack, i&&e aspect ratio ,_’ ., ) ‘. -- DO Auele of-attack. induced Angle of attack, absolute (measured from zero- Induced drag, absolute’coefficieut i&=$ lift position) -’ 1 -z II*‘-, e)4 Night-path angle Parasite .drag, absolute ooeEcient C-=3 cross-wind force, absolute coefficient C&=$ ‘. , - REPORT No. 865 METHOD FOR CALCULATING WING CHARACTERISTICS BY LIFTING-LINE THEORY USING NONLINEAR SECTION LIFT DATA By JAMES C. SIVELLS and ROBERT H. NEELY Langley Memorial Aeronautical Laboratory Langley Field, Va. I . ------- _-.,,.___-_,_. National Advisory Committee for Aeronautics Headquarters, 1724 F Street NW, Washkgton 25, D. C. Created by act of Congress approved March 3, 1915, for the supervision and direction of the scientific study of the problems of flight (U. S. Code, title 49, sec. 241). Its membership was increased to 15 by act approved March 2, 1929. The members are appointed by the President, and serve as such without compensation. JEROME C. HUNSAKER, SC. D., Cambridge, Mass., Chairn~an ALEXANDER WETMORE, SC. D., Secretary, Smithsonian Institution, 1Tke Chairnzan HON. JOHN R. ALISON, Assistant Secretary of Commerce. EDWARD M. POWERS, Major General, United States Air Force, VANNEVAR BUSH, SC. D., Chairman, Research and Development Deputy Chief of Staff, Materiel. Board, Department of Kational Defense. ARTHUR E. RAYMOND, M. S., Vice President, Engineering, EDWARD IT. CONDON, PH. D., Director, Xational Bureau of Douglas Aircraft Co. Standards. FRANCIS W. REICHELDERFER, SC. D., Chief, United States DONALD B. DUNCAN, Vice Admiral, Deputy Chief of Naval Weather Bureau. Operations (Air). CARL SPAATZ, General, Chief of Staff, United States Air Force. R. M. HAZEN, B. S., Chief Engineer, Allison Division, General ORVILLE WRIGHT, SC. D., Dayton, Ohio. Motors Corp. 'THEODORE P. WRIGHT, SC. D., Administrator of Civil Aero- WILLIAM LITTLEWOOD, M. E.! Vice President, Engineering, nautics, Department of Commerce. American Airlines System. THEODORE C. LONNQUEST, Rear Admiral, Assistant Chief for Research and Development, Bureau of Aeronautics, Navy Department. I HUGH L. DRYDEN, PH. D., Director bf ~Ael;&autical Research JOHN F. VICTORY, LLM., Executive Secreiary JOHN W. CROWLEY, JR., B. S., Associate Director of Aeronaufiical Research E. H. CHAMBERLIN, Executive Oficer HENRY J. E. REID, SC. D., Director, Langley Memorial Aeronautical Laboratory, Langley Field, Va. SMITH J. DEFRANCE, B. S., Director Ames Aeronautical Laboratory, Moffett Field, Calif. EDWARD R. SHARP, LL. B., Director, Flight Propulsion Research Laboratory, Cleveland Airport, Cleveland, Ohio TECHNICAL COMMITTEES AERODYNAMICS OPERATING PROBLEXS POWER PLANTS FOR AIRCRAFT SELF-PROPELLED GUIDEI) MISSILES AIRCRAFT CONSTRUCTION INDUSTRY CONSULTING Coordination of Research Needs o.f Military and Civil Aviation Preparation of Research Progmms Allocation of Problems Prevention of Duplication Consideration of Inventions LANGLEY MEMORIAL AEROXA~TIC~L LABORATORY, .4mcs AERONAUTICAL L.~BOR~TORY, Langley Field, Va. Moffett Field, Calif. FMGHT PR~PuI~SI~N RESEARCH LABoRBTORY, Cleveland Airport, Cleveland, Ohio Conduct, un,der unified control, SOT all agencies, of scientific research on the fundamental problems of $igAt OFFICE OF AERONAUTICAL INTELLIGENCE, Washington, D. C. Collection, classi’cication, compilation, and dissemination of scientijic and technicnl information on aeronautics II REPORT No. 865 METHOD FOR CALCULATING .WING CHARACTERISTICS BY LIFTING-LINE THEORY USING NONLINEAR SECTION LIFT DATA By JAMESC.SIVELLS and ROBERT H. NEELY SUMMARY different from the geometric angle of attack by the amount A method is presented for calculating wing characteristics by of the angle (called the induced angle of attack) whose l$ting-line theory using nonlinear section lift data. Material tangent is the ratio of the value of the induced velocity at; from various sources is combined with some original work into the lifting line to the value of the free-stream velocity. The the single complete method described. Multhopp’s systems of efiective angle of attack is thus related to the lift distribution multipliers are employed to obtain the induced angle qf attack through the induced ,angle of attack. In addition, the ef- directly .from the spanwise lift distribution. Equations are fective angle of attack is related to the section lift coefficient developed *for obtaining these multipliers for any even number according to two-dimensional data for the airfoil sections in- qf spanwise stations, and values are tabulated for 10 stations corporated in the wing. Both relationships must be simul- along the semispan for asymmetrical, symmetrical, and anti- taneously satisfied in the calculation of the lift distribution symmetrical lift distributions. In order to minimize the com- of the wing. puting time and to illustrate the procedures involved, simpli$ed If the section lift curves are linear, these relationships may computing .forms containing detailed examples are given for be expressed by a single equation which can be solved analyt- sym.metrical lift distributions. Similar forms-for asymmetrical ically. In general, however, the section lift curves are not and antisymmetrical lift distributions, although not shown, can linear, particularly at high angles of attack, and analytical be readily constructed in the same manner as those given. The solutions are not feasible. The method of calculating the adaptation of the method -for use with linear section l<ft data spanwise lift distribution using nonlinear section lift clata is also illustrated. This adaptation has been-found to require thus becomes one of making successive approximations of less computing time than most existing methods. the lift distribution unit1 one is found that simultaneously The wing characteristics calculated from general nonlinear satisfies the aforementioned relationships. section lift data have been found to agree much closer with ex- Such a method has been used by Wieselsberger (reference 1) perimental data in the region of maximum lift coeficient than for the region of maximum lift coefficient and by Boshar those calculated on the assumption of linear sect& lift curves. (reference 2) for high-subsonic speeds. Both of these writers The calculations are subject to the limitations of lifting-line used Tani’s system of multipliers for obtaining the induced theory and should not be expected to give accurate results -for angle of attack at five stations along the semispan of the wings qf low aspect ratio and large amounts qf sweep. wing (reference 3). Tani, however, considered only the case of wings with symmetrical lift distributions. Multhopp INTRODUCTION (reference 4), using a somewhat different mathematical treat- The lifting-line theory is the best known and most readily ment from that which Tani used, derived systems of multi- applied theory for obtaining the spanwise lift distribution of pliers for symmetrical, antisymmetrical, and asymmetrical a wing and the subsequent determination of the aerodynamic lift distributions for 4, 8, and 16 stations along the semispan. characteristics of the wing from two-dimensional airfoil data. Multhopp’s derivation, in slightly different form and nomen- The characteristics so determined are in fairly close agree- clature, is presented herein and tables are given for the men t with experimental results for .wings iyith small amounts multipliers for 10 stations along the semispan (the usual of sweep and with moderate to high values of aspect