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The Aerodynamic Design of Wings with Cambered Span Having Minimum Induced Drag

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The Aerodynamic Design of Wings with Cambered Span Having Minimum Induced Drag https://ntrs.nasa.gov/search.jsp?R=19640006060 2020-03-24T06:40:56+00:00Z View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by NASA Technical Reports Server TR R-152 NASA- ..ZC_ L mm-4 -0-= I NATIONAL AERONAUTICS AND -----I- SPACE ADMINISTRATION LC)A~\c;i)p E AFW Wm* EP1 KIRTLANi =$= MS; TECHNICAL REPORT R-152 THE AERODYNAMIC DESIGN OF WINGS WITH CAMBERED SPAN HAVING MINIMUM INDUCED DRAG BY CLARENCE D. CONE, JR. 1963 ~ For -le by the Superintendent of Documents. US. Government Printing OIBce. Wasbingbn. D.C. 20402. Single copy price, 35 cents TECH LIBRARY KAFB, NM 00b8223 TECHNICAL REPORT R-152 THE AERODYNAMIC DESIGN OF WINGS WITH CAMBERED SPAN HAVING MINIMUM INDUCED DRAG By CLARENCE D. CONE, JR. Langley Research Center Langley Station, Hampton, Va. I CONTENTS Page SU~MMARY-________._...-.--..--.---------------------------------------- 1 INTRODUCTION __.._.._..__..___..~___.__._.._....~....~~-_--~~--~-------1 SYMBOLS___._____.___------.----------.---.--.---.---------------.--.--- 2 THE ])RAG POLAR OF CAMBERED-SPAN WINGS-_- - ...______________.__ 3 PROPERTIES OF CAMBERED WINGS HAVING MINIMUM INDUCED DRAG_._.._.___.~._.___~_._.___.._.____..__.__.-~~_-~---_---~~---------4 The Optimum Circulation IXstribution.. - -.- - -.- - - - - - - - - - - - - - - -.- - - - - - -.- - 4 The Effective Aspect Ratio ______________._._._____________________------5 THE DESIGN OF CAMBERED WINGS ____________________________________ 6 I>et,erininationof the Wing Shape for Maximum LID- - -.--_-___-___-___--__ 7 Specification of design requirements- - - - - - - - - - - - - - - -.-. - - - - - - - - - - - - - - - - 8 1)eterniination of minimum value of chord function- - - -.- - - - - - - - - - - - - - 8 Determination of wing profile-drag coefficient and optimum chord function- - 9 The optimum cruise altitude- -. - - __ - - - - - - - -. -. - - - - - - - - - - - 11 The effective downwash distribution at cruise- - .. -.-. -.- - - - - -.- - - - 11 Wing twist function for minimum induced drag._. - -. -. -.-. -. -. -. - - - 12 Final wing form____________. .~ __.~ _....__.. .._.._.._. ..____.___-.__ 12 .4ddit.ional I>rsign Considerations. .~. .. ~ . .. ~ .. .. -. .. .. -. -. .. - -. 12 Section profile srlection _.__... .~ ~.~ ~.. .~. .. .-. .. ~. .. .- ___ 12 Angle-of-attack variations. ~ .. .. .. -. .. .. .. -. ~.. -. .. -. -. ~. .. - - 13 Take-off and landing rcquircincnts.. - -.. ~ .. -. .~ .. -. __ 13 Structural considerations- - - .. -. - -.-. - - - - - - - -.-. - - -.. -. - -..-. - - - - 14 Special missions- - __________.-.. __ -. __ __ - __ __ __ -.__. __ - - __ ____ - 14 II,LUSTRATIVE TIESIGN OF A CAMBER.El>-SPAN WING-. - -.__ -.-. - --- __ 14 Basic Design Conditions ___________..____._____________________..-.... --- 15 The Spanwise Camber-Line Geometry. ______. - __.- -. -. - -.-. -. -. - . .-. 15 The Chord Function c(s) _________..._...._.___... ~-.._.__.....__._-.. ~- 17 The iVing Ilrag Polar for Minimum Iiitluwd lhg. .~ ~ ~ . .. ~. ~ - 17 The Optimum Crnisc Altitude ..__.. .. ~ . ~ . ~ . ~ . .. ~ . 18 The Efrct.ivr 1)owiiw:tsh at Cruise,.. .~ ~ . ~ . ~.. - . ~ 78 The Twist Fuiirlion e(s). .. .. ~.. ... 18 The Final Witig Form.._. ~ . ~.~ .. .. .~.~- 19 I)I~;TERMINAT1ONOF THE WING I’ITCIIING MOMENT.. .. 21 Pit.ching Moment of the Most Gencral Wing Form .___... ~.. ~ . - 22 Thc section pitching-moment contribution.. .. .- . 22 Wing pitching moment about Y-axis.. .. .. .~ ~ .-.. ~..~. .~ . .. 23 Wing moment about arbitrary axis._. .. ~ . ~ -.-. -. ~. ~ ~.. 23 Pitching Moment of a IVing Constructed of Similar Profiles.. ~.. .._ .. .____ 23 Wing pitching moment about Y-axis-. .__.____... .. .. ._._. .. 23 Pitching moment about an arbitrary axis. - ____.-.. .. .. ~ .. .~ - .-.. 24 Variation of C,”,J With CL for General Wing Form.. ~.. -. .. .._. ._-. - ~ - 24 Variation of C,, With CL for Sirnilar-Profile Wing_-. ... .. - .__... _--- ~. - 24 Locus of Trim Axis- -.___. ._.~. ~. ._. ._. ._. ._. .. -.-. .___._________ 24 CONC1,UI)ING REMARKS.. -.-~. ._.. ..__.._.._. ._.._..__________________ 25 Al’1’I~;NI)TX A-TIIE ItIC1,ATIONSIIIP OF LIFT TO INDUCED DRAG FOR OPT1M AL1,Y J,O A I>EI> AIRFOI1,S. - - - - - - - - - - - - -.. -. .. - - - - - - - - - - - - - - 26 I~EFERENCES 26 I1 I TECHNICAL REPORT R-152 THE AERODYNAMIC DESIGN OF WINGS WITH CAMBERED SPAN HAVING MINIMUM INDUCED DRAG By CLARENCED. CONE,JR. SUMMARY has curvature in a plane perpendicular to the The basic aerodynamic relations needed for the direction of flight. Reference 1 pointed out, design of wings with cambered span having a mini- however, that the practical realization of a net mtcm induced drag at spect$ed Jlight conditions are efficiency increase would depend in considerable developed for wings of arbitrary spanwise camber. measure upon the ability to construct cambered Procedures are also developed for determining the wings with profile-drag coefficients and structural physical wing .form ~equiredto obtain the maximum weights which would not greatly exceed those of value of lift-drag ratio at cruise, when the wing the equivalent flat-span wing producing the saine spanwise camber-line and section projiiles are speci- lift. Otherwise, the benefits of the increased Jied, by optimizing the wing chord and twist clistri- effective aspect ratio attainable with c:iniber butions with respect to both projile and inrlitcerl might be cancelled by increased profile drag, ant1 drags. un increrised operittioniil lift force. The application of the design procedure is illus- The tiforeriientioned efl'ective-aspect-rNtio prop- trated by determining the physical wing form for a erties of cambered-span airfoils were derived by a circitlar-arc spanwise camber line. The e$iciency theoretical treatment of the ideal vortex systems of this cambered wing is compared with that of an necessary for ininimuin induced drag, without ey"!-span, frat wing of elliptical planform which consideration of the physical wing forms needed satisjiee the same set of fright operating conditions to obtain the optinium distributions of vorticity. ass does thr cambered wing. Tn order to investigate the overall performance The wing pitching- mome nt e pations for opti- gains obt:iinal)le with such airKoiIs, the details of t tie physical wing design niust be considered. maI1 y loa rlr (1 cain br rfd-spa n wings at dpsign flight contlitions are nlso developetl for use in trim annlyses These tlet:iils include not only tierotlyniitnic factors on complete aircraft designs. but iilso the cffects ol the wing structural weight on tlie opercitioniil efficiency (ref. 1). In g~nernl, INTRODUCTION however, structurttl \veiglit effects can he t alien Tlle theoreticnl considerations of refererice 1 into iiccount after the acrotlynannic design of the indiciited that the reduction in induced drag wing is specified. The purpose of this nnaIysis, attaiimble with optiinally loaded, nonpliinar lifting consequently, is to develop the general relntions systems might allow some g:iin in wing aerody- needed for the aerodyniiniic design and efficiency naniic efficiency (lift-drag ratio) over that of flat evn1u:it ion of airfoils which have arbitrarily speci- wings when the lateral spiiri of tlie wing system is fied camber fornis (such its circular-arc spanwise limited by operational requirements. Tn purticu- segments and semiellipse curves) and which possess Iar, it wtis shown that substantial increases in ininimum induced drag at specified flight condi- effective tLspect ratio (as compared with that of tions, without regard to structural weight effects. flat wings of equal projected span) could be The methods by which the optimuni lift loading, obtained with relatively simple wing forms having corresponding to the rniniinuin induced drag of a cumbered spws, that is, wings in which the span given spanwise citinber line, can be cleterniined 1 2 TECHNICAL REPORT R-15 8-XATIONAL AERONAUTICS AND SPACE ADMINISTRATION are presented in reference 1. The relations are of the pitching-moment characteristics of such developed in a form which permits a direct wings is of considerable importance. The curva- efficiency comparison to be made between an ture of the span results in a vertical distribution elliptically loaded flat span wing and an optimally of the drag forces and this distribution in turn loaded (minimum induced drag) wing of arbitrary affects the wing pitching moment. A quantita- camber having an equal projected span and pro- tive knowledge of this effect is necessa.ry if the ducing the same lift force. Thus, in addition to aircraft is to be designed to possess desirable prescribing the wing shape necessary for mininiurn longitudinal trim characteristics. Consequently, induced drag with a given. camber form under the subsequent section “Determination of the specified flight conditions, the results also show Wing Pitching Moment” presents the develop- the relative efficiency of the complete cambered ment of the various aerodynamic relations needed wing. The results may be used to investigate for determining the pitching moment of optimally the maximuin values of lift-drag ratio attainable loaded cambered-span wings for any arbitrary with any given camber forni for any prescribed pitch-axis location. These relations are essen- set of flight operating conditions. tially confined to the calculation of the pitching Although the intention of this paper is to discuss moment of the optimum wing form at specified only the general design procedure for wings with cruise conditions. arbitrary camber, without consideration of the comparative
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