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

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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.

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 efficiencies of any specific camber SYMBOLS shapes, actual calculations are presented for a aspect ratio wing having a circular-arc camber line, in order const ant , f1-5Iy to illustrate the design procedures and to indicate . -l ro the order of magnitude of the lift-drag ratio wing span of flat reference wing attainiible for R particular set of operating projected wing span of cmil)ered- conditions. span wing Consideration is given herein only to the design wing total-drag coefficient of the basic cambered wing. In the design of a wing induced-drag coefficient coniplete aircraft, the entire vehicle configunttion wing profile-drag coefficient must ol course be considered and the parasite section drag coefficient drag of the fuselage and other components intro- wing lift coeficient duced into the design calculations. However, the wing lift coefficient for (L/D)mor procedures outlined herein specifically for wing section aerodynamic-force (lift) co- design are easily adapted to a complete aircraft efficient design by proper inclusion of rill the coiiiporient section aerodynariiic-force coeffi- drag coefficients. In addition, the relations dis- cient for wing (L/D)mnl. cussed t~pply equally well in principle to the wing pitching-monieii t coefficieii t design of more complex wing forms such as those about arbitrary axis through having compound camber, or bninched tips (ref. point P l), as well as to the design of low-drag hydrofoils. wing chord Because of the restricted allowable span length of mean aerodynan~icchord of flat most hydrofoil systems, the use of cambered-span wing hydrofoils appears advitntageous as a means for wing root chord reducing both the induced drag and support-strut wing total drag force drag of conventional systems. The curvature of wing total drag force intensity such hydrofoils might, also allow some reduction (force per unit length) in free-surface effects. The theoretical basis and wing induced drag force intensity, practical application aspects of cambered-span Pwr wings are discussed in reference 1. spcmwise-camber depth (fig. 3) When the use of cambered-span wings is con- :ierodynaiiiic-force loading intensity sidered for specific aircraft designs, a knowledge (fig. 7) THE AERODYNAMIC DESIGN OF WINGS WITH CAMBERED SPAN HAVING MINIMUM INDUCED DRAG 3 r wing circulation distribution ro circulntion of wing root section G constant, J1Lsec rc~y Y nondirnensional coordinate, -L -l ro b'/2 h flight altitude z 6 noncliniensional coordinnte, - k induced drag efficiency factor b'/2 L wing lift force e wing twist function L' lift-force intensity (force per unit P atmosplieric density length) atmospheric density at sea level M wing pitching moment P8 1 M' section pitching moment cr atmospheric density ratio, p/psl S 1 dz constant, ~ - 7 slope of camber-line tangent, tan- - m b c0B- span ratio, blb' dY 2 9 NA Subscripts:

P dynamic pressure c condition at cruise 7' radius of curvature ol circultir :ire 44 about quarter-chord point of section S wing area of flat reference wing .f flat-span wing S' wing tirca of cambered-span wing, L condition at landing ?nux niiixinium 0 in plme of syiiiiiietry P arbitrary nxis location S tirc-sp:in coortliiinte (fig. 2) arc-tip coordinate 7' t riiii-axis 1ocat ion St inoiiient about 17-:ixis V flight velocity IF W weight force Parentheses ( ) are used tliroughout to denote W effective downwasli velocity the independent vnrirtble of the function preced- 700 downwasli velocity at center of ing the parentheses rind should not be interpreted span r to nietiri iiiultipliccitiori. Exciniple : -(s) denotes X ('artesian coordinate in free-stream 1'0 tlirec t ion 1 that - is tt function of s. XP longit udirial coordinate of iirbit r:iry ro asis Y, 2 ('iirtesiiin coordinates ol sprinwise THE DRAG POLAR OF CAMBERED-SPAN WINGS ccmiber line z(y) diininiy variable of integriition In reference 1 tlie relation between the lift Y' coefficient and the induced drag coefficient for 2' distance of line-of-action ol Do' above Y-:txis :in?- cainbered-span wing is expressed :is Zp z-coordinate of arbitrtiry asis ZT coordinate of trim axis CY geometrical angle-of-attack fuiict ion CY' effective angle of attack where the lift coefficient PL and aspect ratio A CY'* effective angle of attack for are based on the span length and wing area of a (UD)ma.Z flat, elliptical-planforrn wing arbitrarily chosen as ffi induced angle-of-attack function a basis for deterniining the induced drag of the a0 geometrical angle of attack of wing cambered wing. The factor k is an efficiency root section f actor which determ ines the m axini uin effective d aspect ratio kA of the curved wing and is shown P camber fwtor, __ b'/2 to depend only upon the geometrical form and 4 TECHNICAL REPORT R-15 2-XATIONAL AERONAUTICS AND SPACE ADMINISTRATION

length of the camber line when the wing possesses In equation (2), k is a dimensionless constant the optimum spanwise distribution of circulation for the optimum lift distribution and each of its r three main factors also dimensionless, so that - (s) corresponding to minimum induced drag. is ro the expression for k may be simplified to the form The value of k is given by (4) where In this equation + is the ratio of the span b of the (5) reference flat wing to the projected span b' of the curved wing, and Y=~Y is the nondimensional b 12 abscissa coordinate of the camber line (fig. 1). Thus, if a flat-span, elliptical-planform wing of The factors NA and B are constants for any par- span b and area S is arbitrarily specified, the ticular camber-line shape and apply only to the corresponding lift and drag coefficients of any condition of optimum circulation loading for mini- cambered-span wing can be determined and the niuni induced drag. The factor B depends upon total drag polar of the curved wing can be ex- the form of the optimum circulation distribution, pressed as and NA depends upon the circulation-downwash ratio, which is also purely a function of the arc (3) curvature. By use of equation (3), the aerodynamic effi- where CDois also based upon the wing area S. In ciencyparaineter LID and - ~- can be established, effect, then, the cambered wing is aerodynamically ( equivalent to a flat wing of area S, aspect ratio within the limitations of linearY) airfoil theory, for kA, and profile-drag coefficient CDo. The value of any cambered-span wing. This efficiency index the profile drag coefficient CDoof course depends can then be directly compared with that obtain- upon the physical surface area and forin of the able with optimally loaded flat-span wings. cambered wing at the corresponding lift condition. In the following considerations, the factor $ is PROPERTIES OF CAMBERED WINGS HAVING MINIMUM INDUCED DRAG assigned the constant value 1.0, which means that the coefficients of the cambered wings are THE OPTIMUM CIRCULATION DISTRIBUTION being based on a flat wing of equal span, b=b'. For any curved wing forin having a spanwise The significance of the factor $ is fully discussed camber line specified by the symiiietrical, dirnen- in reference 1. sional function ~(y)(fig. 2), the span must possess r a specific circulation distribution -(s) if the ro induced drag is to be u niiniiiiuni at a given lift. This optiriiuni circulation distribution can be

1 1 ~~ FIGUREl.-Geoiiietry defiiiitioiis of flat aiid cari~hc~rc~d- span lifting lilies. FIGURE2.-Cambered-span liftiiig line.

I THE AERODYNAMIC DESIGN OF WINGS WITH CAMBERED SPAN HAVING MINIMUM INDUCED DRAG 5 determined by either the conformal-transforma- optimum nondimensional circulation loadings and tion or the electrical-analog technique discussed corresponding values of NA and B are presented in reference 1. These methods can also be used in figures 4 and 5 as functions of p for circular and to determine the value of NA (eq. (ti)), and inte- semielliptic arcs, respectively, in order to illustrate gration of the nondimensional circulation loading the loading variation with camber for simple arc T/r, determines the value of B (eq. (6)). For forms. The distribution r/rofor the case p=O families of camber lines such as circular-arc seg- in each figure corresponds to the loading for a flat ments or semiellipses, the optimum distribution wing, and it is evident that the effect of camber r/r,, and the constants NAand B can be expressed is to increase the relative loading of the outer as functions of the camber factor p which describes portion of the projected span. the particular arc members. (See fig. 3.) The THE EFFECTIVE ASPECT RATIO When the values of NA and B have been determined for a given camber line z(y), the value of the efficiency factor k can be calculated by equation (2), for #=l.O. Since the effective aspect ratio of a cambered-span wing is then equal to kA, where A is the aspect ratio of the reference elliptical wing of equal span, the induced drag of the curved wing will be less than that of the flat wing for equal lifts when k>l. This result, of course, is dependent on the assumption that the curved span is optimally loaded at all values of the lift coefficient. The variation of the span FIGURE3.-Definition of camber factor 0. efficiency factor k with degree of camber p is

I / I .o .65 I- B /’- /-

-r .6 ro

0 .I .2 .3 .4 .5 .G 7 .0 .9 1.0 Y FIGURE4. -Variation of the optimum nondiincnsional circulation distribution and c.fficicbncy constants of circular-arc camber lines with degree of camber. 656-793 G-l -2 6 TECHNICAL REPORT R-15 2-XATIONAL AERONAUTICS AND SPACE ADMINISTRATION

e I1*65Fwm .550 .2 .4 .6 .8 1.0 P

.2 .3 .4 .5 Y FIGURE5.-Variation of the optimum nondiniensional circulation distribution and efficiency constants of semiellipse camber lines with camber factor B. presented in figure 6 for the complete fniiiilies (0 5 p 1) of circular-arc segiiients and semiellipses for the condition 1.O. += These equations in turn can be used to determine The following considerations of cainbered-span the wing efficiency parameter LID as a function of wings are divided into three parts. The first part presents a developnient of the relations CL . Consider now an optiiiially loaded, ctiiiibered- necessary for designing wings which will possess a span wing of projected span length b' equal to that niaxiniuin value of LID (for the imposed structural of the flat wing and producing mi equal lift zit the restraints) for specified flight conditions. The 1 second part, presents actual cwlculwtions for a wing same dyiiwiiiic pressure -pP. This caiiibered 2 with circular-arc cainber RS :in illustration of the design procedure, and as an indication of the wing can have any desired coiiibimition of wing magnitude of the efficiency obtainable with cam- twist, chord distribution, and section profile varia- bered wings. The third part presents relations tion, subject only to the requireinent that at the for determining the wing pitching monients at prescribed flight conditions under coilsideration design flight conditions. the wing possesses the optiiiiurii circulation loading r(S) for which the iiiaxiiiiuiii efficiency factor THE DESIGN OF CAMBERED WINGS k applies. If the wing aren S of the reference flat Consider first a flat reference wing of elliptical wing is used as the basis for determining the force planforin which is specified by stating its span b, coefficients of this cambered-span wing, there wing area S, and constituent section profiles. results The aspect ratio is tlien also deterliiined, A=b2/S. I, The force coefficients :ire given by (9) I; PL=- (7) 1 CL2 ,jpv2s C,,S-- c,= n-kA THE AERODYNAMIC DESIGN OF WINGS WITH CAMBERED SPAN HAVING MINIMUM INDUCED DRAG 7 1.5 ---- I I I / / / I.4 / /' / / 1.3 / / C ular- k / / / / 1.2 / /

/' / I. I / I I / I / I .- _/ I.c -

5 B FIGUREG. -V;iri:ttioii of t.he eWcicsiicy factor k of circular and st:~riic~llipsearcs Lvith the cattiI)or factor p.

These coefficients now determine the lift-drag cambered wing. 'I'lie induced drag of a flat, ratio LID for the citinbered wing. elliptically loaded wing is of course independent A coniparison of equations (8) and (10) indicates of the physical wing area and hence of the physical that the cambered-span wing can attain a higher aspect ratio for a given span and lift, as can be seen total aerodyntuiiic efficiency than tlie flat wing at from the fundamental relation equal lift duesif k> 1.O, provided of course that cyD, is not proportionrttely larger than CD, ,. Because tlepcnds criticdly upon the wing section profiles :ind pliysicd wing surface area, 'I'liis fact is discussed in sollie detail in reference 2, tlie ability to reiilize the full induced-drag efficiency pages 32 and 33, :ind in :tppenclis A of this report. gains offered by sptinwisc c:iiiiber obviously rests upon the possibility of constructing curved wings DETERMINATION OF THE WING SHAPE FOR MAXIMUM LID with relatively small surftice areas or low drag The first step in the design procedure is the profiles or both. As is discussed subsequently, a specification of the nondi~iiensionalcamber func- considerable latitude exists for designing wings 2 tion 6(y) where 6= - and y= y-. This func- that possess low values of CDoat specified flight b'I2 b'l2 conditions, since the value of k depends only upon tion then becomes the corresponding tlitiiensional the at tttiniiient of the optiiiiuiii circulation loading camber line z(y) when the projected span length r(s) and is independent of the pliysical iiieans for b' is given. Obviously, the particular value of b' obtaining this distribution. to be used for a given design depends entirely upon It should be noted that the use in this analysis criteria associated with the requirements of the of a given flat wing :is the basis for deterniining specific aircraft mission, and hence no general the coefficients of the cambered wing is purely procedure can be stated for determining its selec- arbitrary and is valuable priiiiarily for efficiency tion. Once the camber line 6(y) or z(y) is specified, comparison purposes. In reality, any area S r. the values of IC, NA,and B, and the function -(s) could be used to define the force coefficients of the ro 8 TECHNICAL REPORT R-15 2-XATIONAL AERONAUTICS AND SPACE ADMINISTRATION

corresponding to minimum induced drag can be ditions. Thus the optimum distribution r/r, can determined. be used in conjunction with the landing require- The next step is the selection of the area S of the ments to determine the minimum allowable chord flat reference wing of span b (=b‘) to be used as a distribution of the wing. basis for the coeacients. The choice of equal span In terms of the nondimensional coordinate lengths as a basis for comparison rests upon the r=y/(b’/2), the lift force at any fight condition fundamental importance of the span in determining the induced drag. (See eq. (Al) and ref. 1.) While the value of the area S is purely arbitrary, it is convenient to use the value of S corresponding becomes, when equation (6) is used, to that of the “optimum” flat wing which best satisfies the requirements of the specific mission b‘ for which the curved wing is being designed. Then L=pvro B (13) the aerodynamic efficiency of the resulting curved wing can be directly compared with that of the for the design flight and landing conditions. For optimurri flat wing. landing, L= WLand V= VL,so that Specification of design requirements.-In order to design the wing for practical flight operations, certain basic design requirements must be specified. These requirements will determine, to some extent, the design procedure. For the purposes of Here ro,Lis the circulation in the plane of sym- this analysis, however, the design requirements are r taken as the speed VLcorresponding to the desired metry of the wing at landing. Since -(s) can be landing speed, the load WL corresponding to the ro total weight supported by the wing at landing, the established by the methods of reference 1, the design cruise speed Vc,and the initial cruise weight dimensional circulation distribution r(s)along the W,. These criteria are of course arbitrary but arc span can be determined as serve as a basis for development of the general design relations. Since the landing speed in general is low compared with the cruise speed (depend- ing upon the particular aircraft mission), the land- The aerodynamic-force loading intensity F‘ ing speed then determines the minimum physical along the camber line (fig. 7) is given by area of a cambered-span wing. Under certain conditions, take-off speed and weight rather than ~’(s)=~vr(~) (16) landing speed and weight may be the governing factors. The corresponding section “lift” coefficient varia- Determination of minimum value of chord tion el (actually cz determines the aerodynamic- function.-A cambered-span wing is usually designed to possess its niaxiniuni value of LID at sonie specific cruise flight condition, and the physical form of the wing is therefore such that the optiniuin circulation distribution for minimum induced drag is obtained. That is, the wing is designed priniarily for optiiliurn efficiency at cruise. It is assuiiied for this analysis, however, that by a proper use of wing flaps or other variable chord- wise ctiiiiber devices, the optiniuiii nondiniensiorial r circultition distribution -(s)ro can also be obtained (approsiniately) at landing conditioiis. Then the FIGURE7.--Acrody11ar1iic-force loading iiiteiisit y and its vaiues of NAand B applyto both basic flight con- relat.ion to lift intensity. THE AERODYNAMIC DESIGN OF WINGS WITH CAMBERED SPAN HAVING MINIMUM IN,DUCED DRAG 9 force intensity) is given by Determination of wing profile-drag coefficient and optimum chord function.-When a wing section profile is selected, the section drag polar cd(cI) and the section aerodynamic-force curve cl(a’) are specified. The wing profile-drag coefficient is then given by and equations (14), (15), and (17) give the chord distribution for the landing condition as

where cd is the section drag coefficient corresponding to the operating value of c; of the section. The drag coefficient as given by equation (21) This equation, upon specification of the function does not, of course, include any of the effects c~,~,determines the necessary chord distribution such as tip-drag and profile-drag changes which and minimum allowable physical surface area of may occur with the curved, three-dimensional the wing as set by the landing requirements. wing. If it is anticipated that such effects will Obviously, the minimum allowable value of the be significant, an additive term can be applied in “effective” wing area S‘ is determined by the equation (21) to account for them. Assume now maximum allowable value of c~,~,where that the wing is constructed entirely of similar profiles all of which are operating at the same value of el; the total wing profile-drag coefficient is then

It also follows from equation (18) t,hat if each section profile is operated at the same value of c~,~,the chord distribution c(s) is uniquely deter- S‘ mined by the camber-line geometry z(y) since x r -(s) is a function only of the arc shape. The total wing-drag coefficient then becomes ro Although equation (18) determines the minimum value of the allowable chord length which still satisfies the landing requirements, this distri- In this equation the profile-drag term appears as bution is not necessarily the optimum one corre- function of el, where cEis the section “lift” sponding to the rnaximum value of LID for the a coefficient corresponding to any given flight condi- wing at a given lift coefficient CL, since the profile tion. Since, however, tbe wing is being designed drag is critically dependent upon the actual for inrtxinium LID at cruise, cd must now be section chord length. Thus, if the wing is to expressed as a function of C, (the total lift coeffi- operate at the nisxirnuni possible value of LID at a given lift coefficient, the chord distribution must cient) so that the function (C,) can be deter- D be optimized so as to have minimum profile drag. mined. This optimization procedure is considered next. The relation between cz and CL is established If the optimum chord distribution is less than that by substituting the general weight expression required for landing, the limiting distribution as given by equation (18) must of course be used. 1 - pV2S The general form of equation (18) for any flight w=cl, 2 condition for which the induced drag is to be minimum and in which W, p, and V are specified into equation (20), and after rearrnngernent the is desired relation is obtained as Sr 10 TECHNICAL REPORT R-15 2-NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

Since the factor in brackets is a constant, the equation (32) can be written as relation between the section and total wing-lift b' coefficients can be expressed simply ns X'=T Gc, (33) c,=mCL where where (34) m=- sr- b' Po Substituting these values into equation (28) yields GB- 2 as the drag polar

S I b' (27)

The individual factors r/r, and c are of course functions of the arc-span coorcliniite s but their where it should be noted that the fact,or in paren- ratio is a constant when each wing section is theses is the value of el for which cd is to be evel- operated at the same value of cl. (See eq. (20).) uated; it is a functional relation and not R niulti- The total wing-drag coefficient can be expressed as plication factor. The equation for LID as a function of eo, for a constant value of C,, is given by

In order to determine the chord distribution that will make CD a minimum for w, given value of C,, it is necessary to find the optimuni value of the root chord ea. The optimum chord-length distribution at that C, will then be given by Now if the section drag polar of the selected profile r is used, a plot can be made at constant C, of the c(s)=c, (s) (29) r0 variation of LID with c,, and the value of c, for which LID becomes a niaxiinuni determines the In order to find this optiniuni value of ea,the wing optimum chord distribution for that operational profile-drag coefficient value of C,. When this procedure is carried out S' cDo=cd(mCL) x (30) for n range of C, values, a plot of L/D against C, will deterniine the optimum cruise lift coefficient C,* corresponding to the absolute niasimum value must be espressecl as a function of eo. From of wing LID, denoted by and the cor- equation (27), responding value of ea. Then, from equation (30) S 7123- the profile-drag coefficient at cruise is given by b' c,B - 2 CD,=cd(mCL*) S'- S (37) and substitution of c(s) from equation (29) yields where S' of course depends upon the correspond- s=ytc(s)ds ing optimum value of c,. Since the foregoing procedure deterniines the optimum chord distribution not only for the opti- ra muni cruise condition (C,* and but for the entire lift-coefficient range as well, the wing Since the second integral is a constant for any dimensions for niaximuni LID at any other value particular camber line when it is optimally loaded, of CL are also specified. THE AERODYNAMIC DESIGN OF WINGS WITH CAMBERED SPAN HAVING MINIMUM INDUCED DRAG 11

It should perhaps be emphasized that the rela- effective downwash velocity w(s) is shown in tion between LID and CL as given by equation (36) reference 1 to be given by for the optimum value of e, at each C, is not the variation of a particular wing LID as the angle of w(s)=wo cos r(s) (42) attack is increased. It merely gives the relation between LID and CL for the optimum wing form In this equation wois the niaximuin downwash at a given CL value, and for each value of CL the (occurring at the center of the span) and r is the wing form will be different (different chord slope of the camber-line tangent at point s. The distribution). effective downwash at a point s of the camber line The optimum cruise altitude.-If cruise at the is the component of the total induced velocity at most efficient lift coefficient CL is to be txttained, point s normal to the camber line (fig. 8). In the the wing inust fly at the altitude where the opti- case of symmetrical arc forms the total induced mum equilibrium condition velocity is w,and is constant along the arc. From equation (5) 1 w,= cL*zucpstvc2s (43) is satisfied, since the cruise speed V, is specified. (Alternately, of course, the cruise altitude could be specified and the cruise speed calculated. If and if equations (13), (42), and (43) are used both the altitude and speed at cruise are specified, with the condition W=L, it follows that the necessary CL for cruise is determined and the foregoing procedure gives the maximum value of W w(s)= 2-- cos 7 (s) (44) LID at this C> value.) The subscript C denotes PV BN.4 the cruise values of the variables and cC is the [;I density ratio pc/psl. The density ratio is of course a function of altitude h and uC=u(hC). The opti- Then from the expression for the camber line, mum cruise altitude is then given implicitly by z(y), there follows dz tan r=- d?i (45) (39) whence 1 cos r= - -~ tlirough the altitude-density function ~(h.).The dl+W air density at cruise altitude is This equation defines the relation cos r as a.

Pc = UcPs 1 (40) function of the coordinate y, and since In equation (39), Wc is the initial cruise weight supported by the wing. In practical aircraft op- (47) eration with conventionally fueled engines, J4’c will decrease as fuel is used up and if the aircraft is to fly at CL* continuously, the altitude must increase in such degree that

wc=Constan t UC The effective downwash distribution at cruise,- The effective downwash distribution along the * cambered span at cruise must now be determined WO so that the geometrical twist necessary for opti- FIGURES.--T>iagram of induced velocity and its relation mum loading of the wing can be calculated. The to effective downwash. 12 TECHNICAL REPORT R-15 2-ITATIONAL AERONAUTICS AND SPACE ADMINISTRATION

cos is also determined as a function of s (the and the section chord line) at cruise in terms of arc-span coordinate). Thus, the effective down- the camber and design lift coefficient CL* wash distribution corresponding to minimum in- CL* duced drag becomes known for any flight condi- a(s)=(Y’*+2 -cos r(s) (54) tion. For the cruise condition, use of equation ?rkA (44) yields If the geometrical twist function O(s) is defined as wc w(s)= cos r(s) (48) e(s) =ff (8) - (55) pCvc [~-JBN~ where a0 is the geometrical angle of attack of the Wing twist function for minimum induced root wing section (at s=y=O), it can be seen that drag.-The induced angle-of-attack distribution the twist for an optimum cambered-span wing (~~(8)is defined by constructed of similar profiles will be negative, since (Y decreases as increases, as indicated by equation (54). This relationship means that the (49) wing must possess geometrical “washout” for minimum induced drag at cruise. where w(s).. is the effective downwash distribution Final wing form.-With the determination of along the arc span. At cruise conditions, sub- the twist function, all the information necessary stitution of equations (39) and (40) into equation for the aerodynamic specification of the total wing (48) gives form for maximum cruise efficiency is available. The camber line z(y) specifies the frame upon which the chosen section profiles are to be arranged. For simplicity, the camber line can be assumed to pass through the quarter-chord point of each The induced angle of attack at cruise is, therefore, section or else the location of the profiles relative to the camber line can be specified. The optimum f chord distribution c(s), as determined by equation (29) when the optimum root chord length is specified, follows directly from the optimization procedure for maximum LID. Finally, equation (55) for the twist function specifies the angular This equat,ion can alternately be written as arrangement of the section profiles along the camber line, and a. gives the geometrical angle of attack of the center wing section.

ADDlTIONAL DESIGN CONSIDERATIONS by using the usual linearizing assumptions and Although the design relations presented have equation (4). included the principal factors involved in optimiz- The required geometrical angle-of-attack dis- ing a wing form for specified operating conditions, tribution a(s) at cruise is given by several other special factors which are of importance must be considered in a complete wing a(s)=a&) Sa’* (53) design. These are briefly discussed in this section. where a’* is the section angle of attack correspond- Section profile selection.-A small effective ing to the optimum section lift coefficient cl= wing area S’ is part.icularly desirable for cambered mC’. Under the foregoing assumption of geo- wings, especially when use is made of the thick, metrically similar profiles for all sections, a’* is laminar-flow profiles, such as characterized by the of course constant. Substitution of equation (52) NACA 65,-618 wing section. Since the section into equation (53) gives the geometrical angle of drag coefficient, of such profiles is very small and attack (measured as the angle, in the plane IJOIT~~~ is essentially independent of the section lift co- to the camber line, between the direction of V efficient over a wide lift range (the drag bucket), THE AERODYNAMIC DESIGN OF WINGS WITH CAMBERED SPAN HAVING MINTMUM INDUCED DRAG 13

use of such profiles operating at a relativeiy high value of c1 and with a small chord lenqth can result in a substantial lowering of the wing’profile drag at cruise as compared with that of more conventional sections, provided that the full extent of laminar flow can be realized and that the cruise Reynolds number is not so large that the and drag bucket has become very narrow. In addi- L’=pcV$c cos 7 - tion, the relatively large thickness ratio of these -C1QcC cos 7 (59) airfoils provides considerable depth for housing the wing-spar structure. The attainment of a Under the given cruise conditions, equation (59) small value for S’, however, depends upon the leads to the condition maximum value of cl that can be attained for c lc= Cons tan t (60) +. satisfaction of landing or take-off requirements. It thus appears highly desirable when such pro- for any specific section and use of this condition files are used to make full use of such high-lift in equation (57) will ultimately determine the 4 devices as multiple-slotted flaps in order to in- optimum chord length, angle of attack a, and crease to a value sufficiently high for low-speed twist angle for each section, corresponding to the operation. value of (L’/D‘)maz. The section data used In order to evaluate various wing-section pro- should of course correspond approximately to the files for optimizing the wing (L/D)muzvalue at anticipated flight Reynolds number of the section. cruise conditions, the function Angle-of-attack variations.-The optimum cambered-span wing is designed primarily for cruise conditions, and the geometrical twist provides the angle-of-attack distribution necessary for the optimum circulation loading at cruise speed. For lower speeds an increase in section angle of can be optimized for a series of different profiles, attack is necessary if a constant lift is to be main- and the profile having the largest value for this tained. Because of the camber of the span, parameter can be determined. The final profile however, an increase in the effective angle of selection must of course also be based upon the attack of the center wing sections obtained by c~,~~~value and upon a sufficient thickness ratio pitching the wing will have less effect on the out- to meet structural requirements. board section angles of attack, due to the effect The optimization considerations itre based upon of camber, if the wing is rigid. A possible solu- the restrictions that the wing be constructed tion to this problem lies in the use of trailing-edge entirely of similar profiles all operating at the flaps. By a suitable programing of flap deflec- same value of cl. It is possible, however, to tion, the section lift coefficient can be varied design the cambered-span wing so that it will while maintaining a constant section geometrical + possess the milximum possible value of LID when angle of attack (corresponding to the cruise the restrictions of operation at constant c1 and condition). In this way, the wing could always similar profiles are removed. When every sec- operate at essentially the same flight attitude for 4 tion of the wing is simultaneously operating at take-off, landing, and cruise. In addition, it its own value of (L’/D’)maz,the total wing will would be possible to maintain an approximately then have its maximum value of LID for a given optimum circulation distribution at all flight lift. For example, if the cruise conditions Wc, conditions without need for pitching of the entire Vc, and hc are specified, the value of (L‘/D’)muz wing. The use of such flaps is also quite com- patible with the need for high values of c~,~~~ can be determined for each section by use of the and small values of S’ at cruise. relation Take-off and landing requirements.-The design procedure outlined above is based upon the speci- (57) fied landing weight and speed. Tnasmuch as the 14 TECHNICAL REPORT R-15 2-XATIONAL AERONAUTICS AND SPACE ADMINISTRATION

take-off weight is usually considerably greater than increase the effective value of the aspect ratio the landing weight, for the same the take-off over that of flat-span wings. Various other areas speed must therefore be greater than the landing include efficient subsonic cruise at extremely high speed. The take-off speed could be equally well altitudes (where the low density requires high C, used as the criterion for determining the minimum values), high endurance aircraft, and cargo aircraft chord distribution. However, the desirability of operation with increased payload. Possible sta- maintaining a reasonably small value of S’ for bility and control applications are also suggested optiniuni LID operation may require some com- in reference 1. promise of the minimum landing or take-off speeds, unless means for obtaining very high section lift ILLUSTRATIVE DESIGN OF A CAMBERED-SPAN WING coefficients, such as external boundary-layer control, are utilized. For special missions where In order to illustrate briefly the application of cruise efficiency is critical, use of jet-assisted talie- the foregoing design procedure, calculations have off is quite feasible as a means for overcoming any been carried out for a cambered-span wing which take-off problenis which might exist. satisfies an arbitrary set of prescribed operating Structural considerations.-The preceding de- conditions. The conditions selected are typical sign outline has been based solely upon aero- of current operational requirements met by con- dynamic considerations without regard to struc- ventional-winged transport aircraft. No attempt tural factors. The structural factor of priniary has been made in this illustration to determine importance is the wing structural weight as coni- the absolute optiniuni wing forin as regards pared to a flat-span wing producing equal lift. (L/QmaZ,and the win: chord distribution used In a complete design analysis, the effects of struc- corresponds to the niininiuni value as set by the turd weight niust of course be considered. The prescribed landing condition. The wing section effect of the span ratio $ on weight properties, as chosen for the illustration is the NACA 65,-615: well as the possibility of utilizing lightweight profile. However, the use of laminar-flow profiles aeroelnstic wing structures for weight reductions, is in no way mandatary with cambered wings, is discussed in reference 1. although such profiles possess some advantages Special missions.-In addition to the possibility because of their low drag coefficients. It is of using cambered-span wings for improvement of assumed for purposes of illustration that the the efficiency of conventional aircraft, there are a section profile-drag coefficients obtained in wind- number of special areas where application of such tunnel tests can be realized in full-scale, practical wings may be of significant value. Since the operation. In reality, extremely smooth wing priniary advantage of cambered wings lies in their surfaces would of course be required for extensive high effective aspect ratios for a given span length, areas of laiiiinar flow. The results of this study, operational missions where high induced drag is a therefore, may be considered as a realistic, i~1- serious problem are of particular interest. There though not conservative, indication of the general are several such missions. In the operation of level of aerodynamic efficiency of the particular STOL aircraft, the very low take-off and landing cambered wing for the given operating require- speeds require operation at very high CL values ments. L with very high D,. For such airplanes, operating It should perhaps be emphasized that the rela- conditions often severely limit the allowable span tive efficiencies of the flat and cambered wings of length and hence the geometrical aspect ratio. this illustration are valid only for the particular C In addition to the very high induced drag en- set of operating conditions imposed. The particu- countered by such aircraft, it is predicted in lar ctimber line used was quite arbitrary and use reference 2 that at very large values of CL (or for of a more optimuni cmiber form might result in sniall values of A), the airfoil lift can be appre- significant increases in wing efficiency. In any ciably reduced by the large downwasli velocities. specific application, the most optiniuni camber Use of spanwise camber could help in alleviating forin can only be found by a series of design both problems. In situations where the physical trials based upon the particular operational span length is limited, camber can be used to requirements. I'

THE AERODYNAMIC DESIGN OF WINGS WITH CAMBERED SPAN HAVING MINIMUM INDUCED DRAG 15

BASIC DESIGN CONDITIONS the value of r is given by The cwinbered wing arbitrarily chosen for this 1 b' i+p2 illustration has a circular-arc camber line with r=-- - (63) b' 22 P @=OS and ;=5S feet. Thus, when optimally & loaded, the wing will have the following values and substitution of equation (63) into equation for its constants: k=1.32, NA=2.561, and B= (62) gives the dimensional cnniber line z(y). For 1.619, RS shown in figures 4 and 6. The basic 6' the present configuration where p=0.8 and -=5S flat wing chosen for comparison purposes and as a 2 basis for coefficients, has the following c'naracter- feet, the function z(y) is as shown in figure 10. b Since most of the wing design variables are istics: elliptical planform, -=5S feet, A=S, 2 espressed as functions of arc length s instead of 'y 5=1,683 square feet, and the wing section is the (fig. 2), the relation s(y) is needed. This function I NACA 65,-618 throughout. This same profile is obtained by use of the arc-length espression is used for the cambered wing. The prescribed speed and weight conditions are <&

which, when applied to equation (62), yields

These operating requiretnents and the span and wing weti of the flat reference wing are typical of t.he operritirig specifications of an ncturil transport Equation (65), when plotted for the present case, aircraf t wing. gives the curve shown in figure 11. This curve allows direct conversion from one system of THE SPANWISE CAMBER-LINE GEOMETRY coordinates to the other. The equation of n circular-arc spanwise camber It is also necessary to determine the function line is given by cos ~(s).The relation for ~(y)is given by

dZ 7 (y) =tan -1 - --

where I' is the riidius of curvature of the arc. (See fig. 9.) For any values of p and semispan b'/2,

where is given by equation (63). Use of equation (65) allows the determination of ~(s), and the function cos ~(s)follows directly. Alternately,

l The function cos ~(s)for the present camber line FIGUREO.-Geometry of a circular-arc camber line. is plotted in figure 12. 16 TECHNICAL REPORT R- 15 2-NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

I, ff

Y. ff FIGURE10.-Circular-arc camber line used in the illustrative calculations. b’ j3= 0.8, x= 58 feet. 88

8 -

28 0 4 Il 56 60 FIGURE11.-Relation between arc length coordinate s and span coordinate y for a circular-arc camber line of j3=0.8. THE AERODYNAMIC DESIGN OF WINGS WITH CAMBERED SPAN HAVING MINLMTM INDUCED DRAG 17 I.c

\ \ .a

.e \

cos r

4 \

~ 0 8 16 24 32 40 48 56 64 72 80 88 s. ft

FIGUREl2.-Variation of COS T with arc length for a circular-arc camber line of 0=0.8.

THE CHORD FUNCTION C(8) Thus, in conjunction with the section drag polar Although the foregoing procedures allow the for the NACA 65,-618 section (see fig. 14), the optimum chord distribution to be determined for wing drag function CD(CL)follows directly from the cambered-span wing, the chord distribution equation (28). The “polar” for the present wing to be used for the purposes of this example series is presented in figure 15, where the function

corresponds to that set by the landing require- (QL) is also plotted. From this latter curve ments listed previously. Thus, the chord distribution used is not necessarily optimum. the value of CL for maximum LID is The chord function is obtained by substitution of the landing conditions of equation (61) into C’* =0.3 15 (69) equation (18), with pL taken as 0.002378 slug r The polar of figure 14(a) is for a Reynolds per cubic foot. The function --ro (8) is given number of 9 x lo6, while the cruise flight Reynolds r number (based on the wing root chord) is approsi- indirectly in figure 4 as -ro (y), and it is assumed mately 27 x IO6. Although experimental data on the SACA 65,-618 section does not exist for that by use of a suitable slotted-flap arrmgeinent Reynolds numbers above 9 IO6, there is evidence the value ~~,~=3.0is opcrationally attainable x that similar profiles operating at the design cl value with the NACA 65,-618 profile. Values of of 0.528 have even lower values of the drag coeffi- 4 as high as 3.2 are obtainable with double cient for Reynolds numbers up to 35X106. (See slotted flaps (see ref. 3), and the chosen value ref. 4, fig. 14, p. 18.) With increasing Reynolds of 3.0 is not unusually opt.iinistic, even without a the use of synthetic means of boundary-layer numbers above 9 x lo6, however, the extent of the low-drag bucket decreases slowly and for high control. The resulting chord distribution c(s) values of c1 (c1>0.8) the drag coefficient (for is shown in figure 13. sections near the wing root) will probably be THE WING DRAG POLAR FOR MINIMUM INDUCED DRAG slightly greater than shown in figure 14(a). Use of equation (19) yields the effective physical Still, since most of the wing will be operating at area of the wing S’ as 1,340 square feet. Equation local Reynolds numbers considerably below (27) gives the value of ?n as 1.675, and the relation 27X106, where the section drag coefficient is between total and section lift coefficient becomes actually lo~ver than indicated in figure 14(a), the total wing drag Coefficient variation should be very close to that based on figure 14(a). 18 TECHNICAL REPORT R-15 2-XATIONAL AERONAUTICS AND SPACE ADMINISTRATION

-Elliptical distribution

i i N , \ \

I I, i I II 8 16 24 32 40 48 56 64 72 s, ft FIGURE13.-~1inimum-chord-leiigth distribution for circular-arc wing. B= 0.8.

(:I) ])rag polar. FIGURE14.- S&ioil covfficic>iit:: for the, SACA 6&-618 -.4-0 -4 0 profile. of, deg THE OPTIMUM CRUISE ALTITUDE b) Lift curve. From equations (39) mtl (69) there follows FIGURE14.-Coritinried.

cc=O.Y45 (70) of either of equations (49) or (52) the induced whence wngle-of-at tack distribution may be calculated. hc=5,600 ft (71) These two functions are presented in figure 16, for the present wing. At this altitude, the Mach nuinher for cruising THE TWIST FUNCTION e(s) flight is 0.416, which is considerably below the 'The value of a'* (eq. (.53)), corresponding to the critical value at the section operational lift optiinuiii section force coefficient cl= VI(;*= coefficient (see fig. 14(c)). 0.528, is obtnined from the section lift curve shown THE EFFECTIVE DOWNWASH AT CRUISE in figure 14(b) and equals 0.4". Equtitions (53) From equation (50) and figure 12 the downw:isli and (55) can be used to calculate the wing geo- distribution w(s) niay be determined, and by use iiie trical angle-of-a tt ack and twist distributions THE AERODYNAMIC DESIGK OF WINGS WITH CAMBERED SPAN HAVING MINIMUM INDUCED DRAG 19

cz (e) Critical XIach nuniber. FIGURE14.-Concluded. CL

.ET I I I I I I " " '/' "- (b) LID variation. FIGURE15.-Coiicluded. .7 +i NACA 65,-618 section, fully determine the i physical form of the wing for cruise at the intixi- 6 mum value of LID (for the particular chord function being used). As can be seen from figure .5 13, the variation of the chord length along the arc span is different from an elliptical distribution, c, .4 especialll- along the outer GO percent of the iirc span. From figure 17 it is seen that the opera- tiontil angles of iittack nt cruise are very sinall, due priinarily to the high :1iiiount of camber of the

I., basic section. Only R very slight ailiount of

Ill wnshout is required, the total twist tinioiinting to

114 0111~-0.85O. The LID variation of the flat reference wing is plotted in figure 15(b), where the inasimum value .02 .03 is seen to be 37.2 at a lift coefficient of npprosi- CD mately 0.35. The degree to which the efficiency (a) Drag polar. of the flat wing can be improved is indicated in FIGURE15.-Total drag polar and LID variation for the cambered wing. this same figure by the curve labeled 'LT\.4ininium- area flat wing." This LID vnriation is for n a(s) and e@). These functions are plotted in flat-span wing which possesses a minimum chord figure 17 for the esn.niple wing. distribution based upon the same operiitional THE FINAL WING FORM conditions as the cambered-span wing. This The relations shown grnphically in figures 10, minimum-area flat wing is :issunled to hiive a 13, and 17, together with the specificat,ion of the proper twist dist.ribution so that, tiiiniin~~ln 20 TECHNICAL REPORT R-15 2-NATIONAL AERONAUTICS AND SPACE ADMINISTRATION 4-

0 16 32

FIGURE16.-Downwash 1 and induced angle-of-attack distributions for the cambered wing. induced drag is obtained. The effective area of operating conditions. Neither of these wings can this wing is only 1,000 square feet. It is evident be said to possess maximum efficiency in the sense that appreciable decreases in the profile drag are of absolute values obtainable. It is possible that theoretically possible when the physical wing by use of other airfoil sections and other spanwise- area is reduced. camber forms, and with more optimum chord- The LID variation for the cambered-span wing length distributions, even higher efficiencies could is shown as the top curve in figure 15(b), where it be obtained with both the flat and cambered-span is seen that a maximum value of 50.0 occurs at wings. CL=0.315. The cambered-span wing (at least for An additional point of interest in figure 15(b) is the conditions and assumptions of this example) is the value of LID for CL values greater than CL*. more efficient than either the reference or mini- Even when wings are designed for operation at C, mum-area flat wings for a given lift. Although values considerably above CL*,the LID of the the profile drag is a significant part of the total cambered-span wing is still considerably greater drag at the value of CL for that occurs in than the of the flat-span wing, at least for this particular example, it is of much less inipor- the assumed conc1it.ions of this specific example. tance at the higher values of CL where the induced drag predoniinwtes. This difference means that with the specified value All the cotnparisons cited are based on the use of of T4>, the cainbered wing could cruise much more the SACA 65r618 airfoil section and merely intli- efficiently at high altitudes than could the flat cate the relative efficiencies of a specific cambered- wing. As ct increases beyond 0.7, however, the span wing (circular arc, P=0.8) and a specific cambered wing beconies less efficient than the basic flat-span wing under a particular set of specified flat wing. This efficiency loss, of course, is purely THE AERODYNAMIC DESIGN OF VINGS WITH CAMBERED SPAN HAVING MINIMUM INDUCED DRAG 21 I .6

\ 1.2 \

m vQ --- .8 0

0 b, 0 n

- .4 m -0al u, -m -.8

-1.2 1 56 64 72 00 80 s. ft FIGURE17.-Geometrical angle-of-attack and twist distributions for the cambered wing. the consequence of the relatively small section Idevelopment of section profiles, each operating chord length of the cambered-span wing, resulting with its optimum value of el and chord length c is from the completely arbitrary choice of a niini- considered first. Then the equation for the special muin chord lcngth to satisfy the ltintling conditions. case of a wing which is constructed of siinilar ‘I’his chord tlistrihtion may in no wily be the profiles tliroughout, with each section operating optitiiurii distribution for 1:trge (> v:tlucs. At high itt the sanie cz value, is tlcrivetl. It is tissumed values of (\ the sectional operational lift coefficient in both of these cases that correspontling values cl=tn(\ csceetls the estent of the 1:itninar flow of the functions cL(s),cd(s), cdz(s), C,,~,~/~(,S),c(s), capabilities of the particular section being used, and w(s) have been determined by the foregoing and the profile drag becomes exceedingly luge (on methods and that the corresponding flight dynamic 4 the saine order of or larger than the induced drag). pressure at cruise, ::pVz, is also known. The value For flight at high CL values the cambered-wing of is, of course, determined from the section profile drag can be considerably lowered by in- pitching-moment curve em,c,4(cz) for the cl value 3 creasing the section chord length to its opt hum at which the section is operating. value, by nieans of the optimization procedure The equations are developed for two pitch-axis previously outlined. This procedure of course locations. The first location is for the IT-asis of would also aid in reducing the value of re- the wing camber line and the second is for an quired for landing, iirbitrary axis location. The derivations that follow are based upon the assumption that both DETERMINATION OF THE WING PITCHING types of wings are so constructed that the catiiber MOMENT line z(y) passes through the quarter-chord point The pitching-nioment equation for the most of the section profiles. For wings which are not general case of a wing composed of an arbitrary constructed in this manner, proper account must 22 TECHNICAL REPORT R-15 2-XATIONAL AERONAUTICS AND SPACE ADMINISTRATION

be taken of the displacement distance of the local line-of-action of Do‘ above the Y-axis, being profile quarter-chord point from the spanwise positive in the upward direction. The positive camber line. direction of the resulting pitching moinent is as shown in figure 18. The pitching moment of PITCHING MOMENT OF THE MOST GENERAL WING FORM this section about the Y-axis of the airfoil is The section pitching-moment contribution.- given by Consider first the section witfh the quarter-chord My‘ = -L’x +D,’ Z’ +Dl 2 (74) point located at the point (y, z) on the span line and since of a cambered-span wing. This profile lies in a plane which is normal to the camber line at (y, e) DO’z’=D,’[z’- Z+Z] (75) and thus makes an angle T with the vertical, where the nionien t becomes T is the slope of the tangent line at the point. (See fig. 7.) At the design operating conditions the My‘=Mc/4’ COS +D’z (76) section mill produce a pitching nionient of inten- , sity Mc/4/ about the spanwise camber line. This Consider now the case where the pitch axis is monient is a function of el, the operating force located at any arbitrary position having coorcli- coefficient of the section, where nates with respect to the Y-axis of the caniber 0 line zp and z,, as shown in figure 19. The section F‘ c -- moment about this axis is l- 1 3pv% M,’ = -L’[X- x,] +Do’[2’ - z,] +D i’ [: - z,] = -L’x+D,’[z’-z]+L’T,+D’[~-zp] The component, of this moment about an axis through the quarter-chord point and parallel to =Mc/J’ COS T + Id’s,+ D’ [z- zp] (77) the Y-axis of the camber line is Mc,4/ cos T. The force system creating this monient is shown in The subscript P denotes the nrbitrrwy axis posi- figure 18 for the particular case where the section tion and z, is defined as positive when the pitch chord is alinecl with the direction of the free-stream axis lies above the Y-axis of the airfoil. flow. For the coordinate system shown in this When the section force coefficient cl(s) (eq. figure with the origin at the center of the cainber (72)) is specified, along with the chord line, the section pitching moment about the hori- function c(s), the pitching rnonient of each section zontal axis through the quarter-chord point is can be determined for either of the axis locations

Mc,4’ cos T= -L’2+n0’[Z’-z] (73) Z-axis mliere L’=F’ cos T and z’ is tlie distance of the I L, Z-axis I i‘

-V i

FIGURE18.-Force and moiiieiit system for a n-ing sectioii FIGURE 19.-3Iomc~iit relat.ioiis for an nrbitr~ry asis at an arbitrary spanwise location. location. THE AERODYNAMIC DESIGN OF WINGS WITH CAMBERED SPAN HA4VING MINIMUM INDUCED DRAG 23

of equations (76) or (77) since line z(y), 1 cos r= Jl+KT (S6) The section iiiotnent, inoinent coefficient, and and then as a function of s from the relation chord-length distributions are continuous functions of the arc-length coordinate s. , Wing pitching moment about Y-axis.-The total pitching moment of the wing, with regard to the Y-axis, is obtained by integating the con- From equation (87), z can also be determined tributions of the sections along the arc span, as a function of s. Since all factors of the inte- grand of equation (85) are variables in the general case being considered, this integral cannot be further simplified. and substitution of My' froin equation (76) yields Wing moment about arbitrary axis.-The P wing pitching moment Mpabout an arbitrary axis Sl P(xp,zp) follows directly by substitution of 3dy= f-sl [MC/?'cos r+D'z] ds (80) equation (77) into the integral

In this equation the factors are all functions of s. The niotnent MC,,'is given by equation (78), where c,,,,~/*is II function of the sectional force Thus, coefficient cl. The drag force D' is niatle up of two components, the profile drag Dof nnd the local induced drng D,'where nor=CdqC (81) and D,'=P' tan ai (82) As hiis heen shown previously Alternately, equation (89) can be written as II" =c 1qc (83) M,=A~,+~S~~~~~c,c cos T~S and

W tall 01. --v - (S4)

PITCHING MOMENT OF A WING CONSTRUCTED OF i where tu is the induced velocity component norninl to the lifting line (the "effective" down- SIMILAR PROFILES wash, as given by eq. (42)). The pitcliing-tnoinent relations for a canibered- .I Thus, equation (80) can be written in the span wing which is constructed entirely of similar espantled form profiles, all operating at the saiiie value of the section force coefficient c1can now be derived. Wing pitching moment about Y-axis.-The pitching moinent about the Y-axis of a wing with similar airfoil sections along the span can be obtained directly by specializing the general form of The function cos T(S) can be determined as a equation (85). Under the design restrictions, it function of y from the geoinetry of the camber follows that the chord distribution c(s) can be 24 TECHNICAL REPORT R-15 2-XATIONAL AERONAUTICS AND SPACE ADMINISTRATION written as moment coefficient can be defined on the basis of r the wing area S and mean aerodynamic chord c(s)=eo - (91) r0 of a flat-span reference wing where co is the wing root chord and r/rois the P=l . (98) nondimensional circulation distribution for mini- Cm, Mp zpv2sz mum induced drag. Thus, equation (85) becomes The lift of the cambered-span wing can also be expressed in coefficient form

Pitching moment about an arbitrary axis.-The pitching moment about an arbitrary axis P is and since C, is related to the section force-coeffi- obtained directly by spec,ializing equation (go), cient distribution c ,(s) and chord distribution 4s) by n c,=s-y:,CLC cos rds ( 100)

the variation of Mp and hence Cm,pwith C, can be determined. It should perhaps be emphasized that the Since the following relations are valid : function Cm,p(CL) does not represent the variation of Cm,pas the angle of attack of a particular wing (94) is varied, as with conventional airfoils. Rather, it gives the pitching-moment-coefficient variation with the optiniuni wing forni corresponding to the (95) masimum wing LjD at each value of CL. That is, the physical wing fori11 is different for each value of c,. VARIATION OF C, WITH CL FOR SIMILAR-PROFILE WING The pitcliirig-inonient coefficient of the wing

--- b' -w, constructed of siniilar section profiles all operating 2 VB at the same vttlue of czis also given by equation (%), but with the value of LWpbeing obtained equation (93) becomes froin equation (97). In this special case, however, equation (100) reduces to the simple form b' 6' w, & A%=My+qc0 [g czBsp+- cdGzp+- - cfBzp 2 2v 1 p -62 L-- m (97) b where m is a constant (eq. (27)). Thus, the Here, the monient Myof course applies to the rase individual terms of equation (97) can be directly of the siniiliir-section wing, its given by equation evaluated for each value of (;,, using the corre- (92). sporitlinp section force coeffickit cf= mPL. VARIATION OF C,,P WITH CI. FOR GENERAL WING FORM LOCUS OF TIUM AXIS Equation (go), in conjunction with equation As can he seen froiii equation (90) for the most (85), gives the wiiig pitching moment .Mp about general foriii of the pitching moment, there are any arbitrary axis. In order to express this an infinite number of axis locations for which equation in dimensionless form, a pitching- Alp will be equal to zero. An asis for which THE AERODYNAMIC DESIGN OF WINGS WITH CAMBE RED SPAN HAVING MINIMUM INDUCED DRAG 25

Mp=O will be defined as a trim axis. A knowledge center of gravity of a complete aircraft configura- of the locus of such axes is obviously important tion so as to optimize the longitudinal trim for determining aircraft component arrangements requirements at cruise. which xvill possess satisfactory trim properties at CONCLUDING REMARKS cruise flight conditions. For a given set of cruise flight requirements General relations needed for the design of and the corresponding optimum wing form, the arbitrary cambered-span airfoils which will possess value of My (eq. (90)) can be calculated and the the theoretical miniinutn induced drag for specified values of the two integral factors determined. flight conditions have been developed. These Then, the trim condition Mp=O leads to the relations, however, allow the determination of the equation optimum wing form not only lor minimum induced drag, but also for the maximum attainable value of lift-drag ratio by optimizing the chord J -s, I_ --. distribution of the wing. The procedures developed can be used for the determination of the optimum wing design when a specific spanwise camber line and section pro- r which defines the locus of the trim axis, zT= zT(xT). Since My and the integrals are constants in this file are given. The camber-line shape and wing section profile that will be best for a particular set equation, the zT value for any xT location is thus determined for the trim condition. of flight requireinents depend, of course, upon the For the case of a wing constructed of similar specific mission involved, and only by a series of profiles operating at the sanie value of et, the locus comparative designs can the best overall wing zT=zT(xT) is obtained froiii equation (97), forni be determined in any particular case. The design procedures prrscntetl, Iiowevrr, are in a form which tillows suc-h efficiency coiiiparisolis to be easily made. It should perhaps be explicitly emphasized that cambered-span wings will not in all cases possess superior aerodynamic efficiency compared with By use of' the equations developed herein, the the optiniuni flat wing of equal projected span. wing pitching nioment of tiny optimally loaded The relative efficiencies of flat and curved wings cambered-span airl'oil cm be calculated for the will depend ('rifically upon the ratio of the induced design flight condition. In the geneid case, thc drag reduc*tion to t tic in(-reased profile drag of optiiiiiiin iioricliiiieiisioii~ilriiwilatioii lotitling r/r, the c:inil)eretl wing, and the tiiagnitilde of this may Iitive to be tlctcwiiilictl by thc elec~trical ratio (-an be dctcrtiiined only hy cxrrying out a nnalog iiietliod and hence iiiiist be expressed in series of coniptiixtive design wn:ilyses according graphicti1 l'oriii. ( 'onsequently, thc cnsiiing de- to the niethods presented herrin. sign analysis and iiionient deterinintitions will The design procedure presented does not specifi- also have to be carried out graphically. Even in cally include such possible effects as interference the particular cases where r/r, can be obtained and tip sepiirittion drags. The method is based analytically by confoimial trtttisforin:itions, the on the assuniption that the two-dimensional sec- 3 resulting functions may result in extreniely tional force coefficients are closely approximated complex integrals, so that niachine solutions are by the profiles of the three-diniensional wings. necessary. For large span wings or wings with moderate In the particular case of the similar-profile spanwise camber, the procedure outlined should wing, the calculation of the pitching-moment be quite valid within the limitations of linear variation with lift coefficient is considerably lif ting-line theory. The procedure also includes simplified because most of the integrals are then the assumption that the section profiles selected const ants. for wing construction have a sufficiently large The locus equations (102) and (103) can be mininium thickness that they will provide ample used to determine the proper location of the room for housing the wing spar structure. 26 TECHNICAL REPORT R-15 2-XATIONAL AERONAUTICS AND SPACE ADMINISTRATION

The results of the illustrtitive comparison of a optiniuni flat wings will be smaller at low lift cambered wing with a flat wing of equal span coefficients due to the predoniinance of the profile (although neither wing has been optimized with drag. At higher lift coefficients, however, where respect to niaxiniurn lift-drag ratio) indicate that the induced drag becomes a signscant factor in gains in operational efficiency can be secured with wing efficiency, the higher effective aspect ratio cambered-span wings, as compared with the effi- of cambered wings becomes very important for ciency of conventional wings currently in use, wings of limited span. when use made of special profiles to minimize is LANGLEYRESEARCH CENTER, the wing profile drag. The magnitudes of the NATIONALAERONAUTICS AND SPACE ADMINISTR.41710N, gains attainable with cambered-span wings over LANGLEYSTATION, HAMPTON, VA., June 5, 1962.

APPENDIX A

THE RELATIONSHIP OF LIFT TO INDUCED DRAG FOR OPTIMALLY LOADED AIRFOILS

It can be shown (ref. 1) that for any optimally and 1 loaded lifting line of projected span 6' and pro- f (cambered line)=- ducing a lift force I,, the induced drag is given by BN.4 (A5) From equation (4),when $= 1.O (corresponding n --[--If L2 (AI) to the condition where the projected spans of the b' 2--4 straight and cambered wings are equal), where y. is the free-stream dynamic pressure and f k= T-' BNa (A6) is a constant that depends only upon the shape of the spanwise camber line z(y). and the j value for any canibered forin, when For a straight lifting-line seglilent, the circula- d'=l.o is used to determine ki is then giver1 by tion distribution is elliptical -when the line is 1 0.318 (cambered line)=-=-- optimally loaded and f rk k When k>1.0, the cambered span will possess less (A21 "P induced drag than the optimally loaded flat span. Thus, for wings of equal projected span, the Therefore induced drag depends only upon the total lift 1, 1 and spatial distribution of the wake vorticity f (straight line) =-=O.3187T (A:3) (whose effect is nieasured byJ), and is independent of wing area S or the associated aspect ratio For a cambered-span airfoil (or any other nonplanar lifting system), the induced drag is so long as the wing loading is optimum. i The need to introduce a reference area S and (A4) aspect ratio A arises in defining coefficients. ?

REFERENCES

1. Cone, Clareiicc D., Jr.: The Theory of Induced Lift 3. Abbott, Ira H.. and Voii I)oenhoff, Albert E.: Thcory of and Alininiuni Induced Drag of Nonplanar Lifting Wing Sections. AIcGraw-Hill Book Co., Inc., 1040. Systems. SA4SATR R-139, 1962. Abbott, Ira H., Van I)oellhoff, Albert E.. and StiverS;, 2, cone,clarellcc, I)., jr,: A Theoretical ~~~~~~ti~~ti~~~of 4. Vortex-Sheet Deformation Behind a Highly Loaded Louis S., Jr.: Summary of Airfoil Data. NACA Wing and Its Effect on Lift. SAS.4 TS D-657, 1061. Rcp. 824, 1945. (Superscd(~sSACA WR L-560.)

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