REPORT 1167

METHOD FOR CALCULATING THE ROLLING AND YAWING MOMENTS DUE TOROLLING FOR UNSWEPT WINGS WITH OR WITHOUT FLAPS OR AILERONS

BY USE OF NONLINEAR SECTION LIFT DATA ‘

By bDET P. MARTINA

SUMMARY

The methods of iVACA Reports 866 and 1090 haae beenapplied to the ca.kulaiion of the rolling-moment and yawing-moment coejici.ents due to routk.g for unswept wing8 wii?L or

m“thout jfap8 or ailerons. The methods are k.wi? on lij7ing-line ttiory and allow the use of n.onlineur 8ection lijl data.The method precented herein permits cakukdioiw to be ti8omewhat beyond maximum lijl for wings hating no twkt orcontinuous twid and emplqing aiqfoi? 8ectiQm3which do notdtiplay large dkmtinuiiti in the lijl curv~. Ca&uiMtican be made up to maximum li$ for wing8 with diwn.iinuowtwi.d w.ch w thai produced bV partiakpan jlizps or aibwrw,or both. Two cakdded -mpti are presented in eimplijfedcomputing fornM in order to il.hwtraie the procedurtx involved.

INTRODUCTION

The.crdculationof the rolling-moment and yawing-momentcoefficients due to rolling has received extensive treatment inthe linear lift range; several of the better-lmown sourcesrworeferences 1 to 3. Methods for making the calculationsin the nonlinear lift range, however, are comparativelyfew (for example, refs. 4 to 6)and are based on the overalllift rmd drag of a wing.

Methods for calculating wing characteristics in the non-linear lift range by using lifting-line theory and nonlinearsection data (refs. 7 and 8) have given wtimatea whichagree closely with results of wind-tunnel tests of nonrollingwings, as seen in references 8 and 9. It was believed,therefore, that these methods could be utilized in the c-al-culation of the rolling derivatives and that such calculationsmight bo more accurate than those made by existing methods.

Although the application of the lifting-line method to arolling wing in the nonlinear range is implicitly contained inreference 7 and partially illustrated in referenoe 8, it is thepurpose of the present report to outline the procedure ofcnlculrttion by means of several illustrative examples. Innddition, some new considerations regarding the applicationof these methods to the calculation of the rolling character-istic are presented and discussed. Because of the supple-mentrq nature of this report, the reader should be reasonablyfmnilkw with references 7 and 8.

SYMBOLS

The term ‘(section” as used herein denotes the sectioncharacteristics in tkree-dimensional flow.

Ac.c,

Clp

c.c.c%

E

E’

FK, K,

Rva.bcc,zc%

(Cdo)o

c1

c1=1

c~

c4BU2

(%2)Qcl($)

P

aspect ratio

* @ Coefficientwing rolling-moment coefficient

1( )pbcoefficient of wing damping in roll, M71 i) ~

wipgliftcoefficientwing yaw@moment coefficientcoefficient of wing yawing moment due to rolliqg,

eflecti~e edge-velocity factor for symmetrical part

of lift distribution,r

1+$

effective edge-velocity factor for antisymmetrical

dpart o~ lift distribution, 1+$

factor used in altering two-dimensional lift curvmcoefficients used in obtaining succeeding approxi-

mations of lift distributionReynolds numbervelocitytwo-dimensional linear lift-curve slopewing spanlocal chord of wingroot chordmean geometric chord, b/Asection profile-drag coticienttwo-dimensional prdkdrag coefficientsection lift coefficientadditional section lift coefficient for CL= 1.0

two-dimensional lift coefficient

maximum section lift c0ef3icient

maximum two-dimensional lift coefficient

section lift coefEcientfor a particular flap deflectionangular rolling velocity, radians/see

1Superedes NAOA TN 2937,“Metbixl for Oalmdating the Rolling and Yawing Momenta Duo to Rollins for Unsw3Pt ~@s With IX ‘iVithont FbIPS IX Aikrcom by Use Of NordinExScct[on Lfft Data” by Alkt P. Marthm, 19s3.

249

https://ntrs.nasa.gov/search.jsp?R=19930092195 2020-02-08T09:50:00+00:00Z

—— —

.

REIPORT 1167—NA~ONAL ADVISORY CO~ FOR AERONAUTICS

number of intervals between points of calculationalong span

maximum thickness of wing sectionspantie coordinateangle of attick, degcorrection for induced angle of attack, degeffective angle of attack, deginduced angle of attaik, degangle of attack for zero lift, degangle of attack for two-dimensional lift curves, degangle of attack of root section, deguncorrected induced angle of attack, deginduced angle-of-attack multiplier for asymmet-

rical distributionmagnitude of discontinuity in absolute and induced

anglea of attack, deggeometric angle of twist, negative if washout, degapproximate average angle of twist,

angle of twist due to rolling motion, degangle of twist at wing tip, degtaper ratio (ratio of tip chord to root chord)ratio of actual two-dimensional likmrve slope to

theoretical value of #/90area multiplier for asymmetrical distributionsinterpolation multipliermoment multiplier for asymmetrical distributions

Superscript:* denotes value at end of flap or aileron

CALCULATION PRO CED~E

Inasmuch as complete theoretical developments of themethod used herein are given in references 7 and 8, only theitems pertinent to the purpose of the present report will begiven. The method is based on lifting-line theory and usesthe effective edge-velocity factors E and W to correct forthe effects of aspect ratio, following the concepts of reference10. Since .?3and E’ aa used herein are in themselves onlyapproximations for elliptic tigs, it should be realized thatthe results given by the present method at zero lift will notagree exactly with the values which would be obtained byusing lifting-surface solutions for a given two-dimensionallift-curve slope. However, such di.flerencesare expected tobe insignificant.

The calculations for wings having no twist or continuoustwist are discussedin the following section and are illustratedby means of example calculations for an untwisted wingdesignated as wing A. The calculations for wings withdiscontinuous twist require a slightly di.thrent procedureand therefore are discussed in an additional section. Thelatter calculations are illustrated by means of cixample cd-culations for a wing with partial-span flaps designated aswing B.

WINGS WITE NO TWIST OR CONTINUOUS TWIST

me rolling motion of a wing produces an antisymnmtriclinear aerodynamic twist distribution which gives rise tothe rolling derivatives. In the method of calculation usedherein, the aerodynamic twist produced by the rolling motionis treated as a wing twist for purposes of obtaining thedistribution of q. Components of the inclined force vectorsare ,then found, from which the rolling derivatives aredetermined.

Use of section data.-The two-dimensional data at theReynolds number appropriate to each wing station are simplyplotted against an effective angle, a,=ai!?. Either thesection lift coefficients cl or the load coeflicienta cw/b cnnbe used, although the use of the latter is believed to resultin considerable savings in computing time. The load co-efiicienta were used in the example calculations for wing A

-and are presented in iigure 1. 4A single drag curve was usedinasmuch as no account was taken of the spamvise variationin Reynolds number due to taper; the curve is presented infig-me 2. The curves in figures 1 and 2 are based on datataken from reference 11 for the NACA 65-006 section at aReynolds number of 3.OX 10°; however, the drag data were

.28

24

.20

7:;

.08

04

0 4 6 8 10 12 14 16:0002468 10 12 14 16

ae, deg

l?mxmml.-Section loading curv~~ in the example czdoulatione for

.20

.16

.!2

50

.08

.04

0 2 4 6 8. 10 12 14 16ah d.9Q

FIGURD2.-Section drag curve wed in the example calculations forwing A.

METEOD FOR CALCULATING THE ROLLING AND YAWING MOMJ3NTS DUE TO ROLLING FOR UNSWEPT WINGS 251

extrapolated to higher angles of attack on the basis of resultsin reference 12 for the NACA 64AO06section.

Determination of the lift distribution,-The effective angleof attack at any station y is given by

where180 py

e*——— —TV

()r-l Clcai=~ ~ Jk

m-1

(3)

(4)

andAa E’–E

6== [(a– aJk-(a- af),.k] (5)

The evaluation of equation (1) by the method of reference7 is one of successive approximations, in which a distributionof c# is initially assumed. From this distribution and withthe values of the multipliers j?m~given either in table I forr= 10 or in reference 7 for T=20, a distribution of ai is

obtained by using equation (4); equation (5) can then beowduated. With all the components on the right-hand sideof equation (2) determined, it is possible to iind a,. Valuesof c1corresponding to the values of a, are foun-dfrom appro-priate section data, after which a check distribution of c@fcan be obtained. If these check values of cfi/b do not agreewith those initially aasumed,new valuea are sssumed and theprocess is repeated until agreement is obtained. In the

TABLE I.—INDUCED-ANGLK-OF-A~ACK MULTIPLIERS 6.JFOR ASYMMETRICAL LIFT DISTRIBUTIONS

WITH r=10 1

[“’AI (3.’-4

-a 9511 -a .sm -0.55-78 -a mm o

Zg

-a WI 9 =L 7WI 4s79 o –214s o 1 0.mu

-. m 8 -m. 4e3 nl. S47 -34. 40s o –2 .573 2 .W

-.m 7 0 -47.3s4 mm -29.&24 o 8 .E-57a

-.aml 6 –& 610 0 -35. C80 7h m –a. ‘5xf 4 .m

o 6 0 -4.377 0 -w m nom 5 0

.m 4 –L 716 0 -3.SW o –% m 6 –.mm

,m 3 0 -L 1S3 o -2 ?&z o 7 –.m

.Ea91 2 -.644 0 –.Em o –’2573 ,8 –.m

.9511 1 0 -.2% o –.W o 9 –.U511

1 2 3 4 6 km .%,b

— —

o.9s11 am 0.6s78 am o21T

lVNu@ofkat @phh*titb dmofmatlti ddqtiuofkathtimti beumdwith vofueaof m at rfght sicfo.

following examples, methods are’ indicated by which thedHerences between the check and initially assumed valuesare utilized to obtain the succeeding assumedvalues so thatagreement is reached in a minimum number of approxima-tions. Equation (6) corrects the effective angle to accountfor the E’ factor that is used with the antisymmetric com-ponent of Ioading when the section data are plotted againstae= aoE.

Determination of the rolling-moment and yawing-momentcoefficients.—After the distributions of cfifl and c%c/bare de-termined (distributions of c%c/it are obtained by using theplots of section data), components of these distributionsalong the x- and z-axes in the wind-axes system give thefollowing expressions:

c~c Clc .—. — ——-- C; b ‘s (“-a’) b ‘m(“-a’)

For the angles usually encountered,

so that equations (6) and (7) become

(6)

(7)

(8)

(9)

It is easily seen that, for%= 0°, equations (8) and (9) reduceto the usual expressions for the section contributions to thelift and drag of a nonrolling wing inasmuch as the cmtribu-tion of c% to the lift in equation (9) is usually negligible.

Spanwise integrations of equations (8) and (9) yield theyawing-moment and rolling-moment coeflkients as folIows:

VaJma of the numerical integrating-multipliers u= are givenin table II for r= 10 and r=20 in addition to the area mul-tiplied ~m. The ~mmultipliers maybe used to fid 6’L and

()CDby similar relations, that is, CL=A 51 ~ v., and SOm=l m

forth.

252 REPORT1167—NAmONALADvlSORYCO~E FOR AERONAUTICS

Application of the method .—In order to illustrate the useof the method, an example is presented for wing A at anangle of attack of 12°, rolling at such a rate that the wing-tiphelis angle p6/2V is 0.01 radian. The calculations are madefor r= 10. The pertinent dimensional data are given intable III and the lift distribution is calculated in table IV.

The initially assumed load distribution in the linear liftrange can be most rapidly obtained by using methods withwhich the render is probably already familiar. For example,nearly exact initial assumptions can be made very quicklyby proper use of the numerous lifting-surface solutions which

TABLE11.-~NG-COEFFICD3NT hlULTIl?LIERSFOR ~TGS

are available in chart form and thus effect appreciable savingsin computing time. For the sake of presenting a numericalprocedure, however, the method of referenca 13, which hasbeen shown to give very good results in most cases, was usedin this report. By combining the equation of reference 7 forthe- initial approximation with equation (33) of reference 8and equations (13) and (18) (modified for i?) of reference 13,the following expression is obtained for the initially assumedload distribution:

(%).*a.=*F%+lsyam[c1 (=) AaO(c–7) AaiNp

A+l.8+~+AE’+4 1 (12)WITHOUT DISCONTINUOUS TWIST

r-xl r.=lo I TABLE LU.-GEOMETRJC CHARACTERISTICS OF WING A

m

..-. ——---0.M472

-----------.m164

-----------. 1W9

—-------.UmEa

-.-----.---.W

-----------.W&3

—---------. 16W3

-----------.Wlrx

-. . .. —----.W73

------.—--

c“m

Tc. m

-+~ -. --—

–.02218 -.!’-–.o1246–.Oam _;-–.Ol!ws–.02118 -.----–.m .–.cmm _!-0arm +-.am9.02m ---–-.Olf$f.1.a2618 ._;-.0E44Sculls -.1--.m7@3.m ------

-aa

–.E310-.m-.?071–.6’373-.4340-.soul–.1E64o.1s64.31W.4540

:%.axo;W;

.W77

19181716

;:13121110987

:4321

: :1.3J

.047s4

.0m78

.07403

.04233

.0J331

.04m

.1K343

.0&B3

. Kc343

. 04BX!

. WI

:%%!.cBo76.04734.01616.Olms

.---. —------a03m

--- .---—----. 02fm

------------–. 6LW0

-..---..-—-–. Olml

------------0

------------.Olm

----—------.ftmm

------------.0!2469

---—------.m78

---—------- (Alter mdn@ef~nmr tlplftlpls reunde-!L)

.

TABLE IV.-CALCULATION OF LD?T DISTRIBUTION FOR WING A

[.+.W*O.01 reman]

L:~@@@@@@—

Obcckcm ~? (m+a+dp) ~* @-%-@ ?

(9 P)—

ea~ –0.M IL 4S d-O.04 4.m a cdml

a’x –.46 IL&f -.06 7.66 , 14s1

297 –.34 lLc6 –. a5 a 74 .1377

Mrlltfplfen @d

/, I I 2X8/’t-?–.an –.sfm

7 I 6I

5 Yrevei-m

Of@

-ml-mlo

=1=23L766 -43.6i9

-s3. m 12L347

0 -47.234

–a m o

01-2WII0-a w a om–. m I .14%

acm. lmo=-FEE-.6s3 I .lm .193

.!aal 2Q3 I –.ls I lL82 I -.03 I $$2 I .21s3 I

+=

o -4-377

–L716 o

0 –L ES

–. M4 o

0 –. 2%9

o I -am I 7L620 .23E23

.21E3

.Ixn’

3-6210 l12mlo l&2a l!2a09

.Es3 I . KG9

.fw . Isal .1484 4.34 I .~ I 1246 I .W”l a13 I ,lfi!o

.’W I .Om .ml 7.23 I .s5 I 12 u I .13k I &24 I.owl

b-=1 -+ f$b-+1

METHOD FOR CALCULATING TEE ROLLING AND YAWING MO~S DUE TO ROLLING FOR UNSVVEPT WINGS 253

where cl(=) is the section lift coefficient for the geometricangle of attack in question. This loading is entered in col-umn @ of table IV in normal order and in column @ inreverse order. The loading is entered in this manner inorder to shorten the size of the computing form. Themechanics of computing are all self-explanatory and it willbe noted that the correction to the antisymmetric part of theloading is made in column @ as obtained by using equation(6). I?or the sake of brevity only the M calculations havebeen shown in the table. The check load coefficients incolumn Q, read from the section plots, will usually not agreewith the assumed values for the first approximation. Theprocess is repeated until agreement is obtained betweenthe assumed and check values of cfi/6. The manner usedin detmmining the succeeding assumptions is dependent ona number of factors such as the linearity and the slopes of thesection loading curves and on whether r= 10 or r=20.Various methods for obtaining succeeding assumptions arepresented in the appendix. When calculations are beingmade for more than one angle of attaok, the first assumptionfor the second angle of attack can be based on the solutionfor the previous rmgleof attack by hl.ing the value of c~/6corresponding to an angle

(13)

dain which it can be assumed that ~= 0.7inthe linear range.

Once the values of a, have been de&miued for two angles ofattack, plots of a. against ar can be made for each section.Values of clc/b corresponding to the extrapolated value-sof CY,

will usually give a fairly accurate first assumption and thusminimize the amount of computing required. If onlylimited calculations are being made, it is recommended thatany calculations in the nonlinear range be based on theresults of a calculation in the linear range in a mannersimilar to that just described. In general, this procedure willeliminate a rather arduous solution since the load distribu-tions may change very rapidly in the nonlinear range.

After the induced anglw of attack and the lift distributionare determined, then the prof.k-drag distribution can bedetermined. Inaamuch as each section is assumed to beacting two-dimensionally, the section drag coefficients are

obtained at the section lift coefficients or effective angles ciefor the proper values of Reynolds and Mach numbers. Thecalculations are oarried out in columns @ to @ of table V.The calculations leading to the rolling derivatives are carriedout in columns (3Jto @ and at the bottom of the table.

WINGS WITH DISCONTINUOUS TWIST

The discussion in this section is limited to the case wherer=20; a simihr method could be devised for T=1o. It isemphasized that the method of altering the two-dimensionaldata, which is subsequently described, applies only up to andincluding maximum lift and therefore precludes the calcula-tion of Ctnand CnPbeyond mtium lift.

.

Alteration of two-dimensional data.—The two-dimensionaldata to be used for wings with discontinuous twist must bealtered in order to avoid a &scontinui@ in the spanwisedistribution of maximum lift codkient at the end of theflap or aileron since no such &scontinui@ exists in the

TABLE V.—CALCULATION OF THE ROLLING DERIVATIVES FOR WING A

@ o @ o @ @ @ @ @ @ @

~C40c ~oe

(td%e m (J2) ; O.sil!x@ (tib% ~ m~x(e,–ai) Y

~X(c,–aJ

@z@ Oxo (table IV) @)x@

-0.661 4.WI am a Kc7 o.W16 –a u &cm –7. 14 –o. Olcm a Waol –a 6426

-, w 7.6s .C#3 .21L5 . Olm —.46 3.96 -4.41 –. a567 .1484 –. 6644

-,68-9 8.74 .m .26%2 .m –. 34 !aS7 –s.31 –. m .lm? –. &an

-. 2-WI &w .u312 .2740 .OzM –. 18 !2.W –3. 11 –. 0776 . !2W –. 6??8

o am . mm .3127 .W39 o 36a –3. (C4 –.@ .2339 –.8578

,309 9.U .UW4 .2740 .0m4 .18 am –2. 86 –. 0764 .Z2m -. m

.E$a a 14 . 0a70 .2392 .m .34 x 1.5 –2 a –. (W2 .1939 –. 6449

.m au . 07a5 . !m6 .0149 .46 4.34 –a ss –. 0578 . lEM –. W2

‘1661 h 24 .Mr32 . 1%7 . 5)16 .5.5 7.26 –&n –. 0107 .@58 –. 64a

@

(tab;:II)

–a 03078

–. 02489

-. OMa

–. Olm

o

.01636

.04ES6

.024s9

.63076

__— -—.. . ..——._———— .

254 REPORT 1167—NATIONAL ADVISORY COMMITTEE FOE AERONAUTICS

physical flow. The maximum lift-coefficient values arealtered (see ref. 8) by the relation

Clm== (CZJo+F(AcJ*@) (14)

where the spnnwise variation of F is given for several casesin figure 3. An illustration for deriving the F factors forany wing is given in iigure 4. The quantity Aci*- is theincrement in cl- at the discontinuity due to the deflectionof the flap or aileron for the proper local Rejmokls number.The values of c1 and ao are then altered according to theequations

.

cl=~=aro+~(m—~lo) (ct_)o

(15)

(16)

The data altered in this manaer are Aown plotted in figure5 for wing B with 60-percent-span split flaps deflected 60°.I?or purposes of comparison the unaltered two-dimensionalsection data oroas-plotted from reference 11 are also shownin figure 5.

.The two-dimensional drag values as such are not altered,but the values of (c%)o corresponding to the valuw of Cl.are replotted against either c1or ~. The drag data used inthe example for wing B are shown in figure 6 plotted against

I I.iA

6.4–––––––– ‘::5—-— Elliptic ~~—.-— Elliptic, 12.7__-—— Elfiptic 6.4-—---— Elfiptic 6.4——--—— .40 9.0

1.2

I .0

.8

.6

.4 \ \

F \ \’. \ & “...

.2 .

0 -

.

-.2.

\ ~, ‘, \-.4

,1

-.60 ., .2 .3 .4 .5 .6 .7 .8 .9 , ~

2Vb

FIcwm 3.—Factor for altering two-dimensional dab for several wingshaving discontinuous twist.

cl. The data for the unflapped sections were taken fromreference 11, cross-plotted to the values of” R shown infigure 5. Since no drag data were available for the NAOA&&210 section with split flaps deflected, the data for theNACA 23012 airfoil section (ref. 14) were used inasmuch asthe lift curves were similar to those for the NACA 64-210section up to maximum lift. Since the data of reference 14ware for R=3.5X 10°, no account was taken of the Reynoldsnumber variation across the flapped portion of the span.The manner of alteration just described is necessarily rwbi-trary and further experimental work may indicde a dif-ferent promdure; however, it should be recognized that thedrag contributions depend on the diilerences in a, betwemthe right and left wings and therefore are not criticallydependent on the absthte values of the drag polars.

.- cb~mdhd Of ref. 7),,-

-—-—.

+ %8)-@&_-----\————— _-~-? .

cl \,

,,

C1(8)(method of ref. 8)-’’”,;’%) - %qa

\,

,/

“S61 m& Otm=lo ,,.G2c101~ .— -—. - ____

o .Q.* IJt~ *I..-””

i- A’

‘l=+p--0 I ,0

F,

Inlxmrd I OutbmrdI !

Fmum 4.-Sohematio illustration of the odcuktion of the faotor Fused in altering the two-dimensional data for a wing having dis-continuous twist with the discontinuitieslomteclat &2v*fi.

METHOD FOR CALCULATING ‘ITD1 ROLLING AND YAWING MOMENTS DUE TO ROLLING FOR UNSWDPT WINGS 255.

2,2

2.0

1.8

1.6

1.4

1.2

Ct

1.0

.8

.6

.4

.2

Q

o 000000 ooo 4812 16Cq’Jordae

CIUIID 5.—Unaltered two-dimensional and altered section lift oumw ueed in the example calculations for wing B. Two-dimensiomd ourves orow-P1OW from data of referenca 11.

2,0

1.8

1.6

I .4

I.2

c1

Lo

.8

.6

.4

.2

!170 J74 J78 .182 .186 0 .004 .008 .012 .016J70 J70 J70 no Jm .174 .178 J62 .186 0 0 0 0 0 .004 .008 .o12 .016

Fmmm 6.-Section drag uurvea used in the example oalculatione for wing B.3085fiKi-U&18

256 REPORT 1167-NATIONAL ADVISORY CO~ E FOR AERONAUTICS

Determination of the lift distribution.-l?or wings with dis-continuous twist the induced angle of attack computed bymeans of equation (4) is modified by a term which is pro-portional to the magnitude of the disccm$nuity amdacts as-acorrection factor to account for the inability of a limitedtrigonometric series to represent adequately the lift dis-tribution of a wing with partial-span flaps or ailerons de-flected. , This induced angle, according to reference 8, is

(17)

where a~~is the uncorrected induced angle of attack givenby equntlon (4), 3 is the magnitude of the discontinue@, andvalues of the correction factcra a# taken from reference 8are given in table VI. These correction factcrs depend onlyon the spamvise position of the discontinuity and apply to adefiected flap or rderon extending from the point of dis-continuity to the right wing tip. Values for any spanwiseextent of the deflection of the flap or aikron, or both, can beobtained by combinations of the values presented. Ii’orexample, if the flap deflection is symmetrical and extends over60 percent of the wing span, the values of aJ3 would be

obtained by subtracting the valuca of aJ8 corresponding to2g

2@–0 6 &om those which correspond to ~= –0.6. The6“

2$ 2ytabular values of aJISfor ~=x are for the flap side of the

()2$ 0 therefore th~ corresponding sectiondiscontinuity -@ ;

lift curva should be used with these values. The wdue of 6‘k obtained as shown in figure 5.

If the discontinuity is located at a station other than2~=cos k~b

~ the v~ues of (czc/b)* are interpolated from the

I calculated vahw of clc/b as follows:

(18)

values of the interpolation multipliers V. are given in tableVII for various spanwise positions of the end of the flnp.Multipliers for positions of 2y*/b other than those tabulatedcan be calculated from the relations given in reference 8.

Application of the method,-The method k applied to~ B tith 60-pmcentipm split flaps deflected 60° at~= 10° and rolling at such a rate that the wing-tip heli..angle pb/2V is 0.01 radian. The geometric characteristics

TABLE VI.-&TiGLE CORRECTION ~ AT 5TATION ~ DUE TO DISCONTINUITY IN INDUCED ANGLE OF ATTACK AT

STATION +

–a ‘%77-. !3511-. m-.8910-. W:.

-. m-. m-..5578-. 60XI-.4640-.4Wo-. m-. 3UQ-. m-. 1W4-.10330. lam.IM4.Xcc.Wm

:%!.4640.5WI..5$78

:%%.7071.Sxlo.&w.E91O.m.9611.W77

-0.%377

0.4340

-: %-.0134

-: %%’-.am-.am.@m.anl

-. Cm8–: ~o

-: E-. m17-. m

.UM2

-: %-. m

.C9x16

XH-. m-. ml

:%-. m-. ml-. CW

.m

. mlo000

-cl 9SU

-o.0m4.4660.n14.cE12

–.o161-.03s

–: E-.DIM–.ml.ml. ~16

–. Cc39–. 0x17–. @xc

:%%–. m–. lx03

.0015

–: MO–. m–. C4xc

.aws

–: w–. mlo

%%–. mlo0000

-0. FMO –cL&&l

— —

am –a rmn-. U324.2648 –:%%.47&s –.cws.WI .4849.W72 –.Z3a

–.W .0517-.am .0164

.0120 –. o117

.0349 –.ao”a-.c059 .0124-.m17.W31 –:E.cm3 –.CCL5

-.m –. m-.aaz-.LTM :%.am –.m7.IlxO –.ms.0313 –. Co20

-. m.W3s –:%.m –. ao4.0W2 –: ~

-. m–. ml

.m –: E

. ml –. ml-. m o-. ml :%-. ml

.0xi2 –. ml

. ml ~ ml

: 00 00 0

o.mo4–. W118

.Ccz?

–: $%–. m

–:%%.@76.04ea

–. 0182–. an4

.Wa

–:%–. W2–. (DI3

.QrB

. U)1O–. 0319–. 0034

.0M7

. alll

–: E–. CaxO

.axu

–: w–. ml–. ml

.m

. Ixolo

Io

-a m

-a cm.0336

–: %?J

–: R%–. cm9.3164.48m.Wal

–:%’–.ano–.me.0107

–:%%–.C012.W34

–:E–.am–:~

–:w–.mo.aw.awl

–.ml–.ml

. ml–. ml

8

-a 4640 –aw -a w

I I

o

0.ml.ml

–. m02-m

–:%–. W4

.m

.OWJ

-: %%-. mu

.c056

. W76

.0128-. 042a-.0491

.EJxlo

.6491

-: %X-. C@76-: g

-: %%-. M@3–. m

. m14

–: 1%–. 0m2

-: %%–. mlo

aw

o-. mlo. ml

-. W-. w

:%H

-: E-. mu. Cmlo

-: %%-. m-. ma. ml.0167

-.0447.0775.M117-.X339.cwo

-:E-.m.002.5

-:%-.m2-:~

-:%%-.ml.CO12

o

aww a4640

—+o 00.ml 8

-. ml .ml. ml –. ml.axQ –. Wr3

-. m .ml-. LW37 .m

. ml .m

.c033 –. l-mm

.MM –.W33–. m .6W4-. Wza .C015

. Wll –. m

. (016 –. am

.Coll –: LWIWI-. 0?32-. @Es .Wo

.W –. m

.0170 –. m43–. 6467 yOJ-.0712

.37M –. C062

.W36 –: ~

.6412

. CM9 .EW3-.0132 –. 6664–. 0254 .04@–. m .Mi71

.CB251–. m

.ac6 –: g;-. Ix12-. m .Ma3

.W –. W7-. 0x13 –. w-. cOs2 .0324

. ml –mlI

Io.6s78 a 7071

I

o 00.ml, :

-. ml o.ml –. ml. ml -: ~

-. ml-. Im4 . mlo.m -: E.m –. W4

-. m .Wxa–. m14 _: ~

.W14

.m -. am-. 13m7

-: %$’ .Ixn4-. ac4 .W19

.@312-. Wlo-: R

-: %–. 0107 .W2

. W19 .m

. M70 -. m

.0190 -. w-. OW .5374-. m .01E3

-. W’x-: %?.! -.037’6;% .2s02

.6110-. m .W7-. w _: ~

-: M -. m-. Co62 . CC18.03?3 -. cmxu

O.em

o0

:.ml.ml

-. ml-. mo.ml

-: E-. m

%iiJ

-. m.m

:%%-. Wll-. W4.5.m.U)15

-: W!-.0124.m.0117

-.0164-:%7

.5161

.Wb9

-; ~

o.afuo

o0

:–. ml-: ~

. ml

–:%–: ~

–:%–. CW17-. W5

:%!-. m–. KM

.@xa

.Cc2a

-: M-. W31

.5)17

–: E!-. oral

awl

-: %%-.0661

.6216–: %8

–. mu

awl

o0000.ml

-. ml-. W@o. ml

-: M-. w

:!%.0010

-. W3-. M16

.CcW4

-: %%’-: ~

:%-. M1O-. ml

.ml

.CQ20

. CQll-: g

. Olm-. mlz-.1114

.EW

.Ww

0.m

o0

I-.ml-: g

.ml

-: 8%-: ~

-: R-. am-. W

:%%-. m-: %2

. W17

-: %!-. m

, mlo

-: %!-. Cor3

.0315

.m28

-; ~

,0119-.0768

.66C0

hD3THOD FOR CALCULATRW THE ROLLING AND YAWING MOMENTS DUD TO ROLLING FOR UNSWEFI’ ~GS 257

of tho wing are given in table VIII and the calculation of theBpaxIwise lift distribution ‘is given in table IX. The deter-mination of the initial assumption for the lift distribution isnot given herein inasmuch as the procedure is fully illustitedfor this wing in reference 8 for the symmetrical case. Fortho ftsymmetrical case the approximate antispn.metricloading given by equation (18) of reference 13 (modified forE’)

w++%mTLia’4

should be added to the assumed symmetrical loading givenby the procedure used in the esample of reference 8. Anysuitable method for obtaining the assumed loading may beused, however.

TABLE VII.-INTERPOLATI~N MULTIPLIERS P. FOR

()OBTAININGVALUESOF ~ AT THE END OFTHE FLAP

Mom) O.am11—LT0.4W

o-.1640

.Eiva

-: !%o00

I

0.aco3

!-.1274

.7W3

. 47%9-.117600

:

O.w

80

-. 04n.9112

–: W!

:0

O.m

:0

-: g

-: %!%o00

~

oMm O.m

o0 :0 0c-

–:B& ;

-: &–: (EZo .19510 –. 0421

The valuea of &in column @J of table IX are those for

‘y*~=–O.6 horn table VI; whereas the values of 5 in

2Y*column @ are the negative of those for ~=0.6 and were

.obtained by the procedure outlined in the previous section.The values of ~ at + 2y*/b are found horn the flapped andunilapped section curves (fig. 5) at values of cl’ correspond-ing to the values of (clc/6)*. After tke approximate valuesfor 6 are determined, the check span load distribution canthen be calculated. The check vidues of (cJc/6)* are theninterpolated from equation (18). When convergence isestablished (i.e., column @ minus column @ iE zero), thevalues of af at & 2y*/b can be found. (The method given forcase I of the appendix was used to obtain convergence be-tween the assumed and check values of clc/b, columns @ and@, respectively.)

The drag distribution is calculated in colunms @ to @ of

table X. Note that two values are given at ~= +0.60.

The values at ~= YO.60 + Ocorrespond to the flapped sides.

The ~olling-moment and yawing-moment components are .calculated in cohmms @j to @ and the coefiicienta andderivatives are calculated at the bottom of table X. Themultipliw u. used in table X are tabulated in table XI.If the values of CL and CD are desired, the numerical inte-grating area multipliers of reference 8 may be used.

TABLE VIII.-GEOME’I%IC CHARACTERISTICS OF WING B

Taw ratfo,x Q4M Roots20i10n NAOA64-210

&%&b~ A9.ml Tfp sa2ifen NAOA04-210

m ml Mck@-_q&r:!!$

-2 al

%%~?%%~%nhar. R 4.44?%LW

Zdge+ekftyfactOr, E LW4

2! c tT :

Az R

a,; et * v. -T

o L6LWJ o.1E37 h flsxlf$ 0.10 0 0 0 0.IX03

. IW4 .@12 . 14%3 h42 .10 .fBKa –. 14 0 .W116

.W .8146 .IzQ 4.87 .10 .1517 -. 33 0 .ml——

.4540 .7276 . llaa 4.35 .10 .24%3 —.EO –. o-in –. w—

.G876 .0473 . 10%’ &m .10 .2s2 —.7a .9112—

.ala4

.m .a7a7 .0614 344 .10 .491a —.‘w . we .ft578——

.Wa3 .6146 .a317 3@ .10 .6E$ –L 23 –. Oall –. 0117.—

.E160 .40a4 .0733 278 .10 .7om –L 5a o .Om—

.W1 .4293 .Wsl 267 .10 .m@.2 -1.77 0—

–. WG6

.E377 .m .6W 222 .10 .W8 –L M o—

-W-5

+16 .W .1016 a.aa .10 .aiwl –. 7.5

For tnpeti wfoEBwith stmfght-lfae elanranb fmm mat to mnstructlon tfp:

:+(1+ ~mx2#iTJet z

(Alter vofam of r/r+nenr tlp to allow formundlng.)

(Use value of dG before roundfng tip.)

-. 029a I 2fu7 I .6910

–.4458

I2040

.6542 L 411 I it%

RDPORT 1167—NATIoNAL ADVISORY COhlMJZTEE FOR AERONAUTICS

TABLE IX.-CALC~ATION OF LD?T DISTRIBUTION FOR ~NG B

[O._mlitfiWd~W; -.lO.W*-O.Ol radian]

@ @ @ @ @ @ @ a @ @ @

I I l-l–@* @ @ @

4-“)&–.988 o am 915.651 –166.985 I’J –7.019 o I -1-401 4S 0

-.230 O.WO1] * 4.41.

-.951 0 .0482 –329.859 463.533 –122.749 o –7.438 o –1.792 o -.701 0 .ml z2b

-.891 0 Wa2 o –180.336 315.512 –96.737 o -7.073 0 -1.920 0 -.819 .0710 ,61

–.809 –. 0511 .0330 –26.374 o –125.246 243.694 -81.067 0 –6.680 o -1.977 0 @33 -.16

–.707 . 1s70 .1053 0 –17.020 o –97.524 202.571 –71.139 o -6.391 0 –2.026 .1103 -2.39

–.588 .9112 .1827 –7.246 o –12.604 o -81.392 177.054 –64.735 o -6.228 0 ,1071 z 76

–.454 –. 0477 .21cQ o –5.166 o –10.126 o –71 .296 160.761 -60.725 0 -6.192 .214ci 4.41

–.309 o 2401 –2.956 o –4.022 o –8.596 o –64.817 150.611 -58.514 0 .2431 4,23

2333 0 –2.24 1 0 –3.322 o -7.604 0 –60.768 145.023 -57.612 .20M 4,70

0 0 .2707 –1.468 o . –1.804 o –2.865 o -6.950 0 -58.533 143.239 .2707 6,83. —

.156 0 m o –1.153 o –1.518 o -2.564 0 –6.530 o -57.812 .2035 ● 4,81

.309 0 .2431 -.810 0 -.946 0 –1.319 -o –2.340 o -6.288 0 .2401 4.a,

.454 0 .2146 0 –.646 o –.800 o -1.176 0 –2.192 o -6.192 .211-m 4,62

.588 0 .1671 -.467 0 -.530 0 –.691 o –1.068 o -2.092 0 .lon 2.87

.707 0 .1103 0 –.368 o –.441 o –.604 o -.981 0 -2.026 . 10K3 -2, m

.809 0 m –.261 o –.291 o –.366 o -.528 0 -.903 0 0$30 ,12

.891 0 .0710 0 –.192 o –.225 o –.297 o -.452 0 -.819 .Wo2 .E3

.951 0 .ml –.118 o -.130 0 –.161 o -.361 0 —ii’ —,01 2.71

.968 0 .fQol -0 –.060 o –.069 o –.090 o -.133 0 -.230 .0273 &07

?$ (y):.mx@,4=(’:)”

“m

4=12.07

0.6 =a 16s2

2V

i-b- -988

H--

.951 .891 .809 .707 .568 .454

- -t-

.309 .156 0

Ikl 2 3 4 5 6 7 8 9 10

. a.. z@)x@.

METHOD FOR CALCULATING ‘mm

—

ROLLING

.,

AND YAWING MOMEINTS DUE TO ROIJJNCI FOR UNSWDPT WINGS 259

IWWw+-i ‘“=1-=1-+15]3.loial~--lam am am 7.49 ~~ d–. .aZ76

-,W36 o -. Ml 7.F&j 2 21] –. 02/ &d .7231.O@Sll .0492

.0120! .0@)21 .161 7.d .651 -.d 7.d .4951.07391 .W81

-.0117 0 -.14 &28 –.29 –.02 &w 1.025 .asl .0$37

.m . L 8.61 -1.34 –. 9. LI .@314 . lm

,3154 0 ] 3.79] &941 eb541 –.021 24 1.W41.1027I .1627

-.m .IMa -.m %24 4.33 –.01 4.m 1.825 .llM . 1210a

(030 .Oxll .W 9..52 4.27 –. 01 5.2$ LW9 . Im .2464

9.77 4.78 0 4.W 1.m .1438 .2324—.

.Ccml-.@ml -.C81 9.761 4.441 .011 &311 l.sml .11651 ..3146

0 I .31541 S.811 9.611 6.&$l .021 2911 LA .l@71 .16711

.Oxn . LWJ 9. -L 14 .02 10. 1.207 .oa14 . lKQ

o -.0117 –. 14 9.21 –. .a2 93 1.W .a31 .WS

.m21 .o12d .151 &WI .%4 .021 7.QSI .%11.0i391.O?loo I -.mM1 –. 041 ad 26s1 .04 .?.07i .nsl .W!311 .0530

2+I m II

.

.

.——— _ ._. —._-~—~

REPORT 1167—NATIONAL ADVISORY COMMJ7JTEE

.—. —~. -—.

FOR AERONAUTICS

TABLE X.—CALCULATION OF THE ROLLING DERIVATIVES FOR WING B

[0.eamnmm aaw deamtww; G-10.W. , gv=o.ol rrlab3D.1

o @ @ @ @ @ @ @ @ @ @ @

$ c1 e c+ c+TX W

acyx am

(table m) (83) F CA CL&x@ (tree Ix) 33 ‘323) (tibh) %1?$(tablo XI)

-aW3 6.467 o.mE4 aoSm am –a b7 460 –h 07 -O. W16 am -a 1410 -a OcW

-.8.51 .723 .m92 .OtW .CW@3 —.In . !LZl –276 –. 0317 .04G2 -.1368 -. W769

–. Sal .S3s .0103 .0739 .W76 —..51 .!33 –L 17 -. m .M32 -.0774

–. m LOM .0115 . @317 .m 46

-,02118

—. —.23 –. 17 –. m .CK8 -. Ola

-.707 1.1s2 .0E3 .M14

-.01272

.@m2 —.u –L 34 .94 .coil .10s3 -. m -. 0Z4’M

–. w-o LS13 .0147 .1016 .cBn49 —.34 -4,7a Ala .OWJ .1637 .6733 -. alw

-. w L 513 . lm .1016 .01769 —.34 7.!m –7. 05 -.1341 . 1LU7 -1.1727 -. WaEa

–. ‘5ss L.584 . Hal .1027 .01777 —.34 6.E4 -6. m -. S223 .1627 -L 1194 -. an16

-.454 L82Z . lm .1166 . Olwa —.B ksa -4. N –. W17 .2106 -.9390 -.02118

–. 309 LW4 . 173) .lxc .02237 —.18 4.27 -4.46 –. W9s .2464 -L W% -, mm

-. l&3 I.&a . lm .1438 .02E3 —.cm L78 487 -. lall .ZB5 -L %33 -. W300

o L ia .1731 . W-7 .02747 0 L@ –6. 89 –. 1618 .2767 -L m o

.m LW .1733 .1433 .024sS .Cra 483 –3. 74 -. m .2WJ -.9611 .KtsfJJ

.SW LSXl .1730 .MJ3 .02237 .18 434 -4.16 –. m .2431 -1.0113 .037fKl

.454 L= .1730 .1166 .OIVw .!23 444 -4.18 -.0936 .2146 -. W70 ,02118

.6s L628 .1730 .1027 . Olm .34 6.ea +34 -. Im . lm -L 0504 ,C0710

. @3-o L&? . ma .1016 .01738 .34 7.43 –7. w -. lz46 .1662 -1.1216 .K0s6

.ao+o LW .01’55 .1016 .WJIEa .34 -4.64 4.M .W7a . 1s82 .7876 .W634

.707 L2i17 .Olm .cra14 .mll!n .41 –L 14 Lb5 .ml’s . Ilm , lno .024QS

.E03 LIE7 . .Ol!ia . cs17 .C)KO .46 —.m .4$ .W15 .Wa .6-W .Olnl

.Eal .m .0110 .07a9 .m .S1 .% –. 47 –. m .0710 -. @34 .02118

.W .776 .W6 . MU .W .64 213a –214 -. m14 .0s31 –. 1L36 .W@

.ss .514 .Im70 .0E4n .Cw41 .57 h 16 -466 –. Mug .ml -.1379 #m

I.

I

DISCUSSION

Lack of eiiiher experimen$l wing-alone rolling data orsuitable two-dimensional section data prevents the makingof exact correlations of. this method with experiment in thevicinity of masimum lift. It was possible, however, to

. compare the wing-alone calculations for wing A with experi-mental wing-body results, iiice suilicient section dataexisted to allow the calculations to be tied slightly beyondmssirnum lift. The comparisons are presented in figure 7.Agreement is considered to be good when the diilerences be-tween the conditions of the calculations and tests are con-sidered. The failure of the present method to predict theincrease in CIPfor 5° <a<S” maybe inherent in the methodsince similar esperimenta~ &ds have been observed forother wings of this plan form. The differences between thecalculated and experimental variations of Cnr in the same

angle-of-attack range could be partly associated with tlmpreviously noted increase in Cb and partly due to bocly-interference effects as shown in reference 16.

Calculations which included the body-induced angle-of-attack distribution on the wing were made for wing A up to10° angle of attack. The indications were that this com-ponent of body intderence had negligible effects on tlmcalculated variations presented in figure 7. Aclditiomdcalculations for wing A using a section drag of cltan a (noleading-edge suction) yielded a C~P variation that rgroeclquite closely with the experimental variation in the lowangle-of-attack range; such a Cmrvariation would indicatethe possible existence of early separation in the eqerimental

results, a likely possibility considering the low test Reynolclsnumber and the section thickness ratio. ~he difbrencea inthe high lift range, particuhwly with regard to the U.Pvaria-tions, probably can be all ascribed to body-interference

METHOD FOR cALclJIATiNQ THD ROLLING AND YAWUS7G MOMENT% DUE TO ROmG FOR UNSWTKT WINGS 261

TABLE XI.-ROLLING- AND YAITUfG-MONfENT-COEFFICIENTMULTIJ?L~RS C= FOR JTDTGS~TH DISCONTINUOUSTWIST AND r=20 AND HAVING DISCONTINUITIES LOCATED AT +% ‘

~olma for@tlvo * aroshowq fcmnwtfve ~, reverses@ of all mnltfpltem]

\%

\’ o a 16s4

&b

\

o. lm o am. Isoi o . . . . . . . . . ..m o .m7E0.m o .O-1785.W o .m.4032 0 .M794.4540 0 .M758,Eim o .m.Gm o.Om o :%,m .Mma,‘iw :,Em3 o :%%’.Iww o .m.8910 0 .Msm.fQm o .WOJ.9611.$577 ! :%%!

0.30W

a ms14.0W4.C@s3.M4al

. . . . . . . . . ..Wr316.01164.Omla. W@.M7@.m769.m769.M7m.W69

:%%

:%%

a 45io

a maw.01993.0m31.Ml18.02118.01E31

----------. 016S3.09118.ml18.m.Olwa. M118.02118. rolls.02118

:%

a5878

a Ol!m.01245.OrMu.olz45.Owiu.Olm.01M7.0E%3

-.—------.M716.01770. olm7.01272.01245. 0U45.01245.01M6.01246

am . astw

amms a orw.0Z318 .o12i6.6a318 .01245.02318 .01245. 02als .01245.02618 .01245.02i64 .01245.02ml .01245.02318 .01245.024@3 .01Z?2.WJ83 .Olms

---------- .0W37.024m .C0716.02918 --:ii%-..m454.0W8 . Olal.0z31s . ola45.0n318 .0W5

a .mo

a mm.02118.0211s.0211s.02118. Ml18.mns.02118.MU8.MU8.02U8.01M5.mlle.02116

--------- ..M3m.02118.01S%3

a w

a mrm.m?w.Mr@.m.m.mi’m.WeQ.M71w:%%.mm.m?w.M769.M7m.0U54.olm4

----------. 0U54

2f4awn

(“)

amm aciw.M3m.Wm –: %%’.m?m

:%:% .Kws.W3m

–; E:%%.mw xK&.mw.cwm .msls.m .ms34cowl .W@3.M7m.Co773 –: %’%.m’x@ .@E&$

--------- .mlss

● mr ncdho‘p U& b+nOma-yl C+-”)--%+’ .

b For nccdve ~t tbb IXCOma- (??+O). -y -0.

effects since the differences are similar in trend to those ofMethod Gnffg. Airfoil R ‘% deg

o EXP —

}

nwx‘

—-— C& ~~ : Wing-body NACA 65AO06 0.7X106 14—.. —

------- GoIc Rws#WqA NAGd63-006 3.0 13

-.46 !2 4 6 8 10 12 14 16

a, deg

FIGURE 7.—The calculated and experimental variatio~ of CI=and Cfiflwith angle of attaok for two wings having aspect ratios of 4.0 andtaper ratius of 0.60. Experimental data from the 6-foot-&rnet.8rrolling-flow test section of the Langley stability tunnel.

reference 15.Also shown in figure 7 are the variations of C,r and C=P

calculated by means of references 5 and 6, respectively, inwhich use is made of the experimental wing-body lift anddrag curves. It appears that both the present method andthat of reference 5 give reasonable estimates of the variationof Clp.The present method, however, appems to give amore realistic picture of the variation of Cmflin the I@hangle+f-attack range than that of reference 6. The edge-suction effects which are applied in reference 6 were negligiblefor this bpect ratio and therefore do not account for thediilerences shown.

Several items of interest were observed during the courseof the calculations for wing A. For example, the contribu-tions of the section lift and profile drag to the rolling deriva-tives could be separated as shown in figure 8. The prof3le-drag component of C%,is seen to be opposite and nearly tmoto three times that due to the lift at high angles of attack.The proiile-drag contribution to the damping ~ roll, on theother hand, is either zero or negligible even at the higherangle9 of attnok

The method presented herein k primarily intended for usein the nonlinear range where some flow separation is present,and the subsequent discussion briefly oovers some considera-tions which limit the use of this method. The conditionthat must be easentirdlyfuliilled is in keeping with the bmicassumption that all sections operate two-dimensionally ornearly so. Therefore any separation that is present mustnot give rise to large amopnts of spanwise flow which cancause a complete redistribution of lift and thus invalidatethe basic assumption. Lack of sufficient cxperimentrd dataon load distributions beyond maximum lift prevents theformulation of positive limits concerning these phenomena,but, on the basis of airfoil characteristics and on observed

—..—. .— —. —.. —.

.

262

.2

.1

c% o

-.1

-.2

.1

0

-.1

%

-.2

-.3

-.4~

REPORT 1167—NA’ITONAL ADVISC

2 4 6 8 10 12 14 16% deg

~Q~E 8.—The calculated rolling derivatives for wing A showing thelift and profile-drag components. R=3.OX NY.

stalling behavior, regions most likely to be amenable tocalculation oan at least be classified.

It is believed that departure horn the hvo-dimensional-flow assumption will not seriously affect the eahxdation of therolling derivatives for wings which incorporab airfoil sectionsthat exhibit gradual changes in lifkmrve slope beyondmaximum lift. On the other hand, calculations beyondmaximum lift for wings which incorporate sections havinglarge discontinuities in the lift curves are believed to havelittle significance because with such airfoils there is no known

Y COMMITTEE FOR AERONAU!HOS

way of predicting either the extent of the initial stall or theinfluence of the stalled area on the section characteristics ofneighboring sections. In addition, such wings display atendency toward asymmetrical stall under no-roll conditionswhich leads to rolling and yawing moments. The deflectionof trading-edge flaps generally produces abrupt lift-curvepeaks at masimum lift on all but the thin airfoils. Suchconditions coupled with the inability to treat the discon-tinuous-twist oasea beyond maximum lift, as previouslydiscussed, limit the feasibility of calculations beyond mmti-mum lift to a very limited number of casea with full-spnnflaps.

Although calculations may appear to be feasible on thobasis of airfoil-section stalling oharacteristicsj there is anadditional consideration which may limit the extent of thecalculations. This limit, referred to in reference 16 as thestability limit, is

(19)

that is, an increment in the angle of attack Aa cannot resultin a decxe.akein the induced imgle by an amount greaterthan the increment Aa. An idea of the meaning of thislimit can be obtained horn the calculations for wing A. Themaximum negative value in this esse ma —0.75 comparedwith —1.0 given by equation (19).

Although one has an apparent choice of performing calcu-lations with either r=10 or 7=20 for the unflapped cam, ithas bem found from experience that calculations beyondmaximum lift with r=20 rarely are required except perhapsat v-~ high aspect ratios. If such calculations me neces-sary, however, it is suggested that the results from calcula-tions with r= 10 be used as the initial approximations forthe r=20 solution, a procedure which will generally shortencomputing time.

LANGLEY AERONAUTICAL LABORATORY,NATIONAL ADT-ISORY COm~E FOR &RONAUTIOS,

LANGLEY FIEI,D, VA., January 99,1963.

.

APPENDIX

METHODS FOR OBTAINING SUCCEEDING APPROXIMATIONS FOR THE SPANWISE LOAD DISTRIBUTION FROM ANINITIALLY ASSUMED LOAD DISTRIBUTION

METHODS FOR USE WITH r-20

These methods for obtaining succeeding approximationsfor the load distribution have been extensively used andproduce convergemm in reasonably few approximation.

dc,Case I: —&P ositive and linear.—For a positive and linear

lift-curve slope, the succeeding approximation, denoted bythe supemcript 1, is given by the equation

(H=(%n.+”w. ‘A1)

FIGURE Q.—Coefficients used to obtain succeeding approxbnatiom.

r=20.

where.

“(%’).=+Z%’A(%*‘A’).

()in which the increments A ~ are the difkrences between

the check vihms and the ~~d values (column @ minuscolumn @, table IX), and K and K are constants for anyparticular wing. Values of K and Kt taken from reference8 are presented in @ure 9 as functions of AE/q. ValuW ofK, for i greater than 3 are mmll enough to be considerednegligible.

The number of terms of equation (A2) needed for anyParticular approximation depends upon the convergence ofthe assumptions; fewer terms are needed when the differences

()~7m are small or when positive differences nearly cnncel

negative differences. Equation (A2) applied to wing B at

-%=0.688 becomes~=fl( b )

()A, *

b, =+ {3.2(@–@o+(@-@)5+(@ -@)7+

0.4 [(@–@)4+(@–@)81 +0-3 [(@–@)a+(@–O)d }

For m= 1 and 2, equation (A2) is expanded as .

(A3)

A’~+),=+[(K1–QA(~),+m(?),+

+)g+~’A(%?4+=(%J‘A4)since

@_,=-A(%), (A5)

I?or m=] 8 and 19, the values of A’(%)l*m’A’(%)lQme

similar in form to the values of A’ &),andA, ~+),

respectively. For wing B, equation (A3) ~o~d be.

()b ,=& [(3.2 –0.4) (@–@),+A, ~

(1.0–0.3)(@–@)*+ 0.4(@ –@)s+0.3(@–@)d

203

.

264 REPORT 1167—NATIONAL ADTTSORY CO~ FOR AERONAIJITCS

whereas equation (A4) would become

()A’ ~ =~ [ 1.()-().3) (@-@)l+3.2(@-@)2+

b,8.3(

1.0(@–@)3+o.4(@–@).+ o.3(@–@),]

Use of the K factors has been found to establish con:vergence within three approximations when the initialassumption is reasonably close. .’ .,

Case II: ~ positive and nonlinear.-Although the factors

in fiawre 9 were derived for a linear lifkmrve slope that isconstant across the span, they can be used in the nonlinearrange. An estimation is made of- the wing lift-curve slopeat the angle of attack in question, from which a value of q kobtained. The K factors are then obtained at the propervalue of AlZ/q. In this way it is possible to use these factorsfor vnlues of q as low as 0.3 for A=3.O. From the trendsof the variations, it is seen that considerable e~polationis permissible at the higher W&S of AE/q.

dc,case ‘: z

negative and linear or nonlinear,-For a

negative lift-curve slope, either linear or nonlinear, theprocedure is as follows:

(1) First obtain values if the span load distribution withfewer sigdicant &ures than desired.

()(2) Find A ~ at each station, and work with these

.values directly. -

(3) Adjust the load distribution at those stations where

the largest values of A()

~ occur by adjusting the loadings

at m and m + 1 stations to obtain approximate convergence

()at the mth station. If large values of A ~ occur at ty-o

adjacent stations, then adjuat the loading at only one of thestations, although with some practice the adjustmentsrequired at two such stations become quite obvious.

()(4) As the values of A ~ are made smaller, the number

of significant @m in the solution oan be increased to thedesired amount.

()(5) Keep ~ vahms of A ~ either positive or negative,

if possible, for easier mental manipulation.

METHODS FOR USE WITH r-10

Although factors similar to the K’s given in case I couldbe derived for the corresponding condition with r= 10, it is

equally facile to use the method described under case IIIfor all casw with r=10. -

REFERENCES1. Pearson, Henry A., and Jones, Robert T.: Theoretical Stability

and Control Characterietica of Wings With Various Amountsof Taper and Twist. NACA Rep. 635, 1938.

2. DeYgmg, John: Theoretical Antisymmetrio Span hading forWings of Arbitrary Plan Form at Subsonio Speeds. NACARep. 1056, 1951. (Supersedes NACA TN 2140.)

3. Bird, John D.: Some Theoretical Low-Speed Span LoadingCharacteristic of Swept Wings in Roll and Sideslip. NAUARep. 969, 1950. (Supersedes NACA TN 1839.)

4. Joiwa, B. Melvilk Dynamics of the Aeroplane. The Asymmotrloor Lateral Moments. Vol. V of Aerodynamic Theory, div. N,ch. III, SCCS.19-23 and 3>85, W. F. Durand, cd., JuliusSPriwer @erUn), 1936, Pp. 70-73,76-80.

5. Goodman, Alex, and Adaii, Glenn H.: Estimation of the Damp-ing in Roll of Wings Through the Normal Flight Range ofLift Coefficient. NACA TN 1924, 1949.

6. Goodman, Alex, and Fisher, Lewis R.: Investigation at LowSpeeds of the Effect of Aspeot Ratio and Sweep on RollingStability Derivatives of Untapemd Wings. NACA Rep. WI,1950. (Supemedes NACA TN 1836.)

7. Sivells, Jamea C., and Ncely, Robert H.: Method for CalculatingWing Cbaracter%tim by Lifting-Line Theory Using NonlinearSection Lift Data. NACA Rep. 865, 1947. (SupenmdesNACA TN 1269.)

8. Sivelle, Jamw C., and Westriok, Gertrude C.: Method for Cal-oulating Lift Dk@butions for Unswept W’ings With Flaps orAilerons by Use of Nordincar Seotion Lift Data. NACA Rep.1090, 1952. (Supersedes NACA TN 2283.)

9. A’eely, Robert H., Bollech, Thomas V., Weetriok, Gertrudo C,,and Graham, Robert R.: Experimental and Calculated Char-acteristics of Several NACA 44-Series Wings With AapootRatios of 8, 10, and 12 and Taper Ratios of 2.5 and 3.6, NACATN 1270, 1947.

10. Polhamuc, Edward C.: A Simple Method of Estimathg thoSubeoniu Lift and Damping in Roll of Sweptbsok Wings,NACA TN 1862, 1949.

11. Abbott, Ira H., Von Doenhoff, Albert E., and Stivers, Louis S,jJr.: Summary of Airfoil Data. NACA Rep. 824, 1946. (Su-perscdea NACA WR L-560.)

12. McCuIIough, George B., and Gault, Donald E.: Boundary-Layerand Stalling Characteristics of the NACA 64AOO(3AirfoilSection. NACA TN 1923, 1949.

13. Sivells, Jamea C.: An Improved Approximate Met~od for Cal-culating Lift Distributions Due to Twist. NACA TN 22s2,1951.

14. ‘iVenzinger, Carl J., and Harris, Thomas A.: JVind-Turmrd In-vestigation of N.A.C.A. 23012, 23021, and 23030 AirfoilsWith Various Sizee of Split Flap. NACA Rep. 668, 1930,

15. Letko, William, and Riley, Donald R.: Effeot of an UnsweptWing on the Contribution of UnsweptiTail C2m@urations tothe Low-Speed Static- and -Rolling-Stability Derivatives of aMidwing Airpl+e ModeL NACA TN 2176, 1960.

16. Multhopp, H.: Die Berechnung der Auftriebsverteilung vonTra@geln. Luftfahrtforschung, Bd. 16, Lfg. 4, Apr. 0, 1938,pp. 153-169. (Available as R.T.P. ‘Translation No. 2302,Britieh M. A.P.)