8/12/2019 19680019396

1/81

N S T E C H N IC L N O T E 0-4427

LOAN COPY: RETURN TOKIRTLAND AFB, N MEXAFWL [WLIL-2)

A MODIFIED MULTHOPP APPROACHFOR PREDICTING LIFTING PRESSURESA N D CAMBER SHAPE FOR COMPOSITEPLANFORMS I N SUBSONIC FLOW

by John E . LamurLangley Reseurch CenterLangley Station Humpton vu.

N A T I O N A L A E R O N A U T I C S A N D S P A C E A D M I N I S T R A T I O N W A S H I N G T O N , D. C J U L Y 1 9 6 8

8/12/2019 19680019396

2/81

TECH LIBRARY KAFB, NMI111111IIIIIllll lllllI11IHllllIll

A MODIFIED MULTHOPP APPROACH FOR PREDICTINGLIFTING PRESSURES AND CAMBER SHAPE FOR

COMPOSITE PLANFORMS IN SUBSONIC FLOWBy John E. Lamar

Langley Resear ch CenterLangley Station, Hampton, Va .

NATIONAL AERONAUTICS AND SPACE ADMINISTRATIONFor s a l s b y the Clear inghouse for Federo l Sc ient i f ic and Technica l In format ion

Spr ing f ie ld V i rg in ia 22151 - C F S T I p r i c e $3.00

8/12/2019 19680019396

3/81

A MODIFIED MULTHOPP APPROACH FOR PREDICTINGLIFTING PRESSURES AND CAMBER SHAPE FORCOMPOSITE PLANFORMS I N SUBSONIC FLOW

By John E. LamarLangley Rese arch Center

SUMMARYThis report presents a modified ve rs ion of Multhopp' s subsonic lifting-surface

theory which has been p rogramed in two part s fo r the IBM 7094 electronic data processingsystem o r Control Data 6400 computer s ys tem along with a discussion of the c ha ra ct er ofits resul ts . The first part is used to find both basic and additional loadings over a givenplanform with known mean camber surface and the second part is used to determine therequired mean camber surface for a given planform and se t of loadings.

Fo r the loading program, various aerodynamic characte ristics a re determined onboth simpl e and composite planforms when the spanwise loading is symmetrical. Studiesa r e conducted to determine when these answe rs a re most valid, and some re sul ts fo rdelta, sweptback and tapere d, double delta, and variable-sweep wings a r e compared withother the ori es and experime nts to determin e the accur acy of this method.

Application of thi s method is then made in predicting the ae rodynamic effects ofchanging the outer-panel sweep of a variab le-sweep wing, of incr eas ing the Mach numberin the subsonic reg ime, and of incorpora ting t w i s t and camber in a wing. The first twoapplications also have experimen tal data presen ted for comparison.

A study is also undertaken for the mean cambe r surfac e program t o determine whenits bes t res ul ts a r e obtained. Subsequently, comparisons between the present and othermethods are made fo r a two- and three-dimensional ca se to aid in the evaluation of thepre sen t method. One application is made for a highly sweptback and tape red planform.

INTRODUCTIONIn the pr oc es s of wing design, one is required to determine eithe r the load distribu-

tion from a given planform and mean camber sur face or the mean camber surfa ce fro m apre scri bed planform and loading. Fro m previous investigations a considerable amount ofinformation of both a theoretical and experimental nature is available to aid the designeri f the wings have delta or simple sweptback planforms. However, planform s now being

8/12/2019 19680019396

4/81

considered (which in many cas es have the divergent require men ts of supe rson ic o r hyper-sonic crui se and subsonic loiter o r fe rr y capability) are much more complex, involving,in many instances, cranked leading and trai ling edges. Planfor ms of th is type are classi-fied as being of a composite arrangement .

Fo r thes e planforms not as much general information is available, due both to thei rnewness and to the lar ge number of planform variables. Fi rst -or der solutions to thes eproblems have been found in recent y ea rs fo r the sonic (refs. 1 and 2) and superson ic(refs. 3 and 4 ) speed regi mes which ar e applicable to arb it rar y wing planforms. Refer-ences 3 and 4 have been programed fo r the high-speed electron ic computer s o that thedesigner does not have to perfo rm the lengthy mathem atical manipulations requ ired toobtain a solution for each wing considered.

In the subsonic speed regim e, the basic theor eti cal method of Multhopp (ref. 5) hasreceived wide acceptance as one of the most accurate methods for predicting aerodynamicloading data; consequently, it has been selec ted as the approach to use in solving thesetwo design problems for composite planforms. Some reaso ns for this selection a r e

1) The chordwise pre ss ur e distributions (and other sectio n data) as well as overallaerodynamic characteri stics ar e determined

(2) The spanwise locations of the lift-producing sing ula riti es and control points ar emo re concentrated along chordwise rows near the tip to insu re an adequate representationof the spanwise load distribution

3) The method can as Multhopp mentioned i n r efe ren ce 5 be used fo r wings withplanform kinks at locations other than just at the plane of symmetryA computerized solution of the loading determination problem fo r arb it rar y wingshas been developed from the basic Multhopp method by Van Spiegel and Wouters in refer-ence 6 and is referred to as the modified Multhopp method. Ce rta in changes in and exten-sions to their work have been made in the present repo rt to in crea se the accuracy inrepr esen ting the actual chordwise pr es su re distribution and finding the influence of eachof the se dis tributions on each point where the boundary conditions are to be met. Thesemodifications resu lted fr om adding additional pres su re modes (up to 10 may be used) andby replacing the polynomial approximation technique used in the chordwise integrationwith the Gaussian quadra ture method.

In addition, the formulation p rocedure and all the modifications made to the basi cMulthopp approach a r e desc ribed as well as the techniques used in its solution for boththe load and mean camber surface determination problems. After this a study of theresulting answers is made to determ ine thei r sensitiv ity to the number and location ofthe points where the boundary conditions are met o r calcula ted. Then applications of theprograms are made to an as sortmen t of wings, including composite wings and the r esu lt s2

8/12/2019 19680019396

5/81

are compared with other the orie s and, where possible, experiment. All the predictionsmade f ro m thi s modified Multhopp method have been obtained with the aid of the IBM 7094electro nic data processin g sy stem and Control Data 6000 series electronic computers.

A listing of the computer prog ram s used to find the su rfac e loadings (Langley pro-gr am A0313) and to find the mean cam ber sur fac e (Langley prog ram A0457) is pre-sented along with a desc rip tion of t he input data re qui red, sam ple input and output data.and pertinent comments in a "Supplement to NASA TN D-4427." A la rge portion of t he setwo programs is given over to the computation of the geome tric repre sentat ion requi redby la ter p art s of th e prog ram fr om the input quantities supplied. Two additional prog ram s(Langley pr og rams A1590 and A1591) which are useful in obtaining geomet ric input datafo r either program A0313 or A0457 a r e al so present ed in the supplement along with briefdescriptions and sample ca ses of each. A request form is included at th e back of thispaper.

SYMBOLSAny convenient sy st em of m ea su re can be used as long as the linear dimensions of

the planform a r e based on a semispa n of unity.Aa.c.

b

bvwbvvCA

cB,ocD,icD,iiCL

aspe ct rat io, b2/Saerodynamic cen te r, in frac tio ns of Cref, refe renc ed to leading edge of ref-

-'Cm 1erence chord (positive aft),8CL 4

-wing span, se t equal to 2Multhopp's integrating coefficientsaxial-force coefficient, Axial fo rc e

qwsrefroot bending-moment coefficient, "1 (CzC)basicq qSref 0induced drag based on the spanwise distribution of circulation

induced dr ag based on axial force, CYCL - CAWing lift

qwsreflift coefficient,

3

8/12/2019 19680019396

6/81

overall lift-curve slope

Cm

AcPCS

C

-C

Wing pitching moment %efabout 4itching- moment coefficient, q2 f C refCr efove rall pitching-moment- curve slope about

4inc rem enta l pr es su re coefficient, Pupper - Plower - - P

qco q,leading-edge-suction coefficient fo r ze ro de gr ee s of leading-edge sweep,

drl

streamwise chord at y , ~

mean geometric chord,c2dyIbI2dy~-b/2-b/2

streamwise half-chord at n,yaverage chord, S/b

local

local

local

Local chord load4coclift coefficient,

pitching- moment- curv e slope about local leading edge

slope of norm al-force coefficient

root chordre fe re nc e chord, may be mean geome tric chord of t ota l wing c except

when the planform has an inboard trailing-edge chord-extension or may bemean geome tri c chord of only wing oute r panel extended to plane of symm e-tr y in stream wise ti p position

x-location of midchord at q

8/12/2019 19680019396

7/81

jth chordal loading functionf jK(@,y;a,q) subsonic kerne l function

Kj (@ y;q) subsonic kerne l function integrated with jth loading a cr os s the chordinfluence of jth loading function at qn on con trol point at yv and

x = (c/2)(1 - cos qs)tin @ s elem ents of influence ma trix when n #M Mach numberm number of span stations where the'p res su re modes are definedN number of chordal control points at each of m span stat ionsA P lifting pre s su re , Pupper - Plowerqj 77)qtj (7)

coefficient of jth chordal loading function fo r additional loadcoefficient of jth chordal loading function fo r t w i s t and/or camber load

q, dynamic pr es su reS total wing ar ea

Sref reference a rea, may be either total wing ar ea o r wing a rea of wing outerpanels extended to plane of symmetry in st rea mw ise tip position

S used to locate chordal control points and ranges from 1 o NU free -str eam velocityW perturbation velocity in z-directionX,Y ,= rectangular Cart esia n coordinates nondimensionalized with resp ect to b/2

where origin is in plane of symmetry at half root chord (fig. 1) (they areasso ciat ed with control points)

5

8/12/2019 19680019396

8/81

x-location of leading edge of Cx-location of leading edge of Crefcenters of pre ssu re

a angle of attack, deg

a i

Xcref

ycp, xcP

induced angle of at tack , radiansloca l angle of a ttack, dz =-w radiansa2 d x U

P Prandtl-Glauert compressibility factor, d l - M2YAe9

K

Ax

local nondimensional lift coefficient, cz c/2b

wing tip skew angle, deg

angle fo r locating con trol points along span (se e fig. 2(a))angle used to locate pre ssu re doublets chordwise, 0 at leading edge and a

at trailing edgeratio of distance of leading-edge bre ak from plane of sym metry to b/2outboard leading-edge sweep angle, deg

--ip chordOverall root chordtaper ratio,

rat io of distance of trailing-edge bre ak from plane of symmet ry to b/2

rect angular Car tesi an coordinates nondimensionalized with respect to b/2along X- and Y-axes, respectively, and associated with pressure doubletlocations

angle used to locate chordwise rows of control points (see fig. 2(b))inboard leading-edge sweep angle, deg

8/12/2019 19680019396

9/81

@ inboard trailing-edge sweep angle, degoutboard trailing-edge sweep angle, deg

Subscripts :CP ce nter of pre ssured distributedl e leading edgen,v suffixes numerating spanwise stations, v stat ion being influenced and nm - 12. . . o , . .- 1sta tion doing the influencing; n,v range fro m -0 value taken at CL = 0P pivotte trailing edgeX x di re ctionY y-direction1 result due to t w i s t and/or cambe red wing at ze ro root chord angle of a ttac k

BASIC FORMULATIONThe det ermina tion of surfa ce loadings on both plane and warped wings and the de ter -

mination of the mean cam be r surfa ce can be made from a number of the or ies alre ady inuse. A l l the theories shar e a common featu re in that the wing is replaced by a mathemat-ica l model which produc es a si mi la r downwash field to that of t he wing. However, themethods differ mainly in the potential used to produce the re presen tative pertur bations ofthe flow field around the wing. These potentials are ei ther of the velocity or accelerationtype (ref. 7). The o rig inal Multhopp method and its presen t modified version use t heaccele ration potential approach in conjunction with the lineariz ed Eu ler equations to relatethe pr es su re difference ac ro ss the wing to the downwash field on the wing surfac e. Thisre su lt s in the following equation which is derived in detail in referen ce 6 fo r the incom-pressible case:

7

8/12/2019 19680019396

10/81

8/12/2019 19680019396

11/81

trailing edge. The spanwise positions, which resul t fro m an equia rc angular displace-ment about the plane of sym met ry, can be repr esente d by the following equation:

V77yv = s in + l (5)

where m is the total number of spanwise stations fro m tip to tip and v ranges fromrequired by the Multhopp qua dra ture formu la fo r the integrand evaluation when approxi-mating the value of t he spanw ise inte gra l (ref. 5).111 o ~ - in inc rem ent s of 1. (See fig. 2(a).) These positions are those2 2

At each spanwise location, the contro l points are also located for equiarc angulardispl acements defined by

where N is the total numbe r of chordwise con trol points and ranges fro m 1 to N.fig. 2(b).) Note that the control points never reach the leading or trailing edge. Fro mthi s, the control-point locations a r e found to be det erminable by the following equation:

(See

which loca tes them aft of the local leading edge. (See fig. 3. )In ord er to solve for either the loading (inv erse problem) or mean camber surface

(direc t problem), equation (4) s used. The loadings a r e found by 1) inverting the matrixin equation (4) nd thus solving fo r the qj rln) rr ay and then 2) using these values inthe equations given in appendix B to dete rmine the lo cal and overall aerodynamic loadingdescription. The mean camb er surface is found by the di rect mat rix multiplication ofequation (4) long with the proc edure described in the me an cam ber surfac e determinationsect ion of this report.

4,

Modification Made to the Original Multhopp MethodFrom a study of Multhopp's original method, sev era l fea ture s a r e see n to need

modification in or de r that a computerized solut ion of it could be made to the best advan-tage. Aside from the logarithmic singularity correction term whose best form is dis-cus sed in appendix A they are as follows:

1. The method fo r solving fo r the coefficients of the loading te r m s in the invers eproblem. In Multhopp's repo rt, a scheme of i te rat ion on the downwash is used, this isimproved in referen ce 8 with a sche me of subm atrix inversion and in reference 6 with ascheme of whole mat rix inversion which is also used herein.

9

8/12/2019 19680019396

12/81

2. The chorda l integra tion tha t su ms the influence of the pr es su re doublets on con-tr ol points not at the sam e span station. In Multhopps repo rt, it is developed from aMacLaurin ser ie s then graphed for use. Van Spiegel and Wouters (ref. 6) follow thesche me developed by Van de Vooren (ref. 9) with which the value of t he in teg ral is deter-mined, by (a) approx imating the loading functions with polynomials, (b) evaluating them atdi scr et e locations, and (c) multiplying the re sul ts of part (b) with predetermined coeffi-cients. In thi s repor t, the polynomial approximation of r efer ence 6 is replaced with aGaussian 20- t o 30-point numeri cal integration scheme .

3. The number of chordal loading functions and, there for e, con trol points. In9Multhopps re po rt they are limit ed to two, cot 5 which gives lift with no pitching

moment about c/4 and cot - 2 sin 9 which yields no lift but a pitching moment aboutc/4. Fr om t he se , th e known downwash and the ma tr ix of influence coeffic ients, one isable to so lve for t he coefficients of t he loading t e rm s which are the local circulation andpitching moment about local c/4. Gar ner and Lehrian (ref. 10) have extended to four thenumber of chorda l loadings (and consequently, contro l points) tha t may be used but employcombinations of te rm s ra th er than sep arat e loading modes. Van Spiegel and Wouters a ls oextend the number of cho rdal loadings and control points t o four but se par ate the loadings

pr es su re doublets supporting that loading contribution ove r the chord. In thi s rep ortthey have been extended to 10 with the addition of si n 49, si n 59, s in 69, s in 79,s in 89, and si n 99 fo r additional flexibility.

2

into cot -, s in 9, si n 29, and s in 39, whose coefficients are the str ength of the loca l2

Multhopps method with the aforementioned modifications is hereafter re ferre d toas the p resent method and has been programed to handle a wide varie ty of compos iteplanforms, with a bre ak in the leading and trailin g edges as well as the variable-sweepwings. The program at present is capable of handling se ts of chordwise and total span-wise stations of 1 by 41, 2 by 41, 3 by 41, 4 by 41, 5 by 39, 6 by 31, 7 by 27, 8 by 23, 9 by21 , o r 10 by 19, with the number of chord sta tions given first. This is not a theoreticallimi tation but is nea r the upper lim it of machine stora ge present ly available. Also, withthe se t s of st ations l iste d above, almost any planform of low to moderate asp ect ra ti o canbe handled. . 1

The distribution of lift obtained by the p resent method is dependent on the subson icMach number (se e appendix A) at which it is computed. In this report, the Prandtl-Glauert relations (ref. 11)have been used to account for t hi s change but a r e valid only aslong as ACp va rie s linearl y with a.

10

8/12/2019 19680019396

13/81

LDAD DETERMINATIONBecause of the nature of the numerical solution, different combinations of N and

m for the same planform can result in different answers for the aerodynamic chara cter -istics. Hence, a study was undertaken to tr y and optimize th eir combination so that rea-sonably accurate results are obtained. The re su lt s of this study are presented i n the nextsection. Then comparisons of o ther theore tica l methods and experiment with the presentmethod a r e made to gain confidence in the a n s w e r s obtained with it. And lastly, illustra-tive examples a re given to show some applications fo r which the p rese nt method can beused.

Optimization of N and mFo r any given planform and pair of N and m, a se t of chordal loading coefficients

q.(q ) ar e obtained fro m the solution of t he inverse problem fo r the flat plate and areJ nused to determine the loading cha rac ter ist ics of the planform. However, since a different

set of q.(q ), and there fore loadings, will be found fo r each pai r of N and m, somecriterion is needed by which the most appropriate se t of qj(qn)crit erion, suggested by Multhopp (ref. 5), is to examine the chordwise load distribu tion bypaying par ticu lar attention to the leading-edge suction for ce developed. The contentionbeing that when the overall leading-edge suction for ce , which is assumed equal to theaxial fo rce and given,in coefficient form fo r an unswept planform by

J n can be selected . One

is such as to cause the quantity

to be equal to the induced dr ag coefficient given by the equation

then the cotangent loading (eq. (2)) can be considered valid. This contention is based onthe conclusions of a study made by Munk in reference 1 2 where he found that the termcD,i w a s independent of the chordwise loading distribut ion. In the discussion that fol-lows, the res ul t of equation (9) is called the suction value and the res ul t of equation (10)is called the circulation value.

11

8/12/2019 19680019396

14/81

Upon comparing graphically the two coefficients jus t d iscussed for many combina-tions of N and m fo r two rectangu lar planforms of different aspect ratio, as is donein f igu res 4(a) and 5(a), it is noted that where as the circulat ion value is essentially invari-ant with N and m, and is plotted as a straight line without symbols, the suction valueshows a marke d dependence upon N and m. If values fo r the aerodynamic ch aracteris-tics at each pair of N and m whe re the two solutions are first equal or have a mini-mum difference are plotted as solid symbols, it is see n that the aerodynamic cente r andCL, values for the A = 2 planform (figs. 4(b) and 4(c)) do not vary appreciably fromresults obtained when N = 8 at the higher values of m. However, in the aerodynamiccenter and CL, grap hs (figs. 5(b) and 5(c)) fo r the A = 7 planform, it appears that thevalues predicted do not nec essa rily ag ree with those at the higher N and m. One rea-son suggested for the disagreement is that not enough values of m were available to beused at the higher N values (sto rage limitation) so tha t the resulting answers found byconvergence could have been established. Note that the aerodynam ic cent er is the moresens itive of the two; hence, a major part of the di scus sion that follows is concerned withit.

From a study of fig ure s 5(b) and 5(c), the c urv es fo r N = 2 and 4 seem to showthat a particul ar relationship must exist between N and m in orde r for the answers toreach a level of convergence. Whatever the relationship is, it leads to resu lts which a r eclose to those given by the solid symbols. Fro m the N = 4 curv e in figu re 5(b), itseem s that a good resu lt f ro m the point of view of convergence would occur at m = 37, asol id symbol. Now if th is relationship between m and N is valid, then its multiplica-tion with lower N values to determine new m numbers should lead to result s whicha r e as good as those found from the original N and m values.

An application of th is p rocedure would be that of multiplying the rat io of (37/4) byan N value of 2, thereby yielding an m number of approximately 19. When the cu rve sof aerodynamic center and CL, at N = 2 and m = 19 a r e examined in figures 5(b)and 5(c), th is combination of N and m is seen to give r esu lts which approximate wellthose obtained for N = 4 and m = 37. This occ urs even though N = 2 and m = 19 isnot a solid sym bol which indicates that even though the r ati o was obtained fro m a combi-nation of N and m that was a solid symbol, appl ications of it at lower values of Ndo not neces sari ly res ult in combinations of N and m which a r e other solid symbols.It should be noted that had the N = 2 and m = 13 solid symbol been selected as thenumbers to be used in determining the rati o of m to N, the m value which would haveresu lted from the multiplication of the rat io with N = 4 would be 26. This combinationof N and m does not give ans we rs which approxim ate as well the re sult s found fromconvergence.

12

8/12/2019 19680019396

15/81

Hence, s e t s of N and m can be calculated that will resu lt in answers fo r aero -dynamic center and CL, which approximate well those found at a soli d symbol providedthey have been obtained fr om a soli d symbol which has a larger N value than for t hecombination sought. In addition, from the re su lt s of both figu res 4 and 5, the relationshipbetween N and m is see n to indicate that an aspect ra tio dependency a lso exi sts,requirin g the use of mor e m for a given N at the higher aspect rati os. This is dis-cussed in more detail later.

Examining again the variation of the aerodynamic center with N and m fo r theA = 2 planform in fig ure 4(b), it is noted that a pa rt of the aerodynamic-center movementcan be corr elat ed with the suction for ce coefficient which is seen implicitly i n figure 4(a).In general, as the suction incr eases a decre ase in CD,ii/CL2) a corresponding forwardmovement in the aerodynamic cent er is observed. This simple relationship implie s thati f any two s e ts of N and m give ri s e to the sa me suction coefficient, then their aerody-namic centers will also be the same. However, this is not genera lly tru e. Look, forexample, at a value of CD,ii/CL2 of 0.20 and note all of the d ifferent combinations ofN and m that will give it and then the corresponding aerodynamic cen ter s. Some dif-feren ces a r e see n to ex ist between the aerodynamic cent ers predicted by the differentcombinations of N and m.

(

There ar e severa l possible reasons fo r this difference. The first one is that theterms (which give

relationship assumed between the leading-edge suction and the aerodynamic center is nota completely valid one since it depends not only on the qo(qn)cotri se to the leading-edge suction) but also the q2(qn)sin 2s te rm s.

The second reason is that since in the solution for the qj(qn)z

se t the number ofcontrol points is req uired to be equal to the number of unknown coefficients , the flat platerepr esen tat ion of the planform may not be made at enough control points. Th is could beremedied by an overdetermined solution (Le., one where th er e a re mor e control pointsand number of equations than unknown coefficients).

The thi rd rea son follows fr om the second in that, with the boundary condition onlyspecified at a maximum of 10 points chordwise, the var iat ion of w/U that r esu lts bymat rix multiplication using the qj(qn)ditions does not ass um e the boundary condition value except at the control points.where it is either higher or lower and because of the logarithmic singulari ty correctionterm (eq. (A9)) it is singular at the leading and trailing edges. This variation in w/Uwhen acted upon by the chord loading distribution should also give rise to an axial forceheretofor e unaccounted for . It is called the distributed axial for ce and is computed by

se t determined fro m the specified boundary con-Else-

13

8/12/2019 19680019396

16/81

l l l l l l I l l

Numerical r esu lts have not been obtained fo r CA,d sin ce eve ry point along the chord ateach spanwise position must contain the influence of every loading function at every otherspanwise location in or de r that the quantity w/U can be determined. This impliesevaluating integra ls si mi la r to those used in finding qj(qn) but at an extremely largenumber of locations. In addition, the singu lari ties at th e edges of the chord would be dif-ficult to evaluate without a coordinate transforma tion. (Note tha t if the mathematic al pro-f i le were truly flat, then all the axial force generated would be concentrated at the leadingedge in th e fo rm of a suction force, since w/U would be everywhere equal to a.) Thus,only one solution f o r the axial-fo rce coefficient has been obtained and is derived solelyfr om the leading- edge suction.

From figures 4 and 5, it app ear s (a s one would probably expect) that fo r the mostpart at the higher N and m values the answers for CL, and aerodynamic center a r ein what appear to be converging situations. However, it has a lso been found t h a t one neednot always choose the maximum N and m to achieve the se ans wer s but may by properinsight and judicious selectio n obtain equally as valid answ ers at lower values of N andm. Thus fo r planforms which have zero sweep on the leading edge, it is suggested thatpai rs of N and m be sele cted for the particular planform at the desired Mach numberin a manner so as to emphasize its lar ges t dimension; that is, planforms having largevalues of PA would requ ir e a pair of N and m where m w a s optimized a fter asuitable value fo r N w a s established, and planforms having low values of PA wouldrequire that a large value for N be used with a corresponding reduction in m.Selec ting pa ir s of N and m by this procedure should give ans wer s that will be in o rbe very close to those in the region of converging resu lts. This can be accomplishedapproximately by using the following expression:

m = (4 o 5)PAF)A simil ar type of analysis which leads to equation (12) should be made for wings withsweepback using the leading-edge-suction equation recen tly developed by Ga rne r in re fe r-ence 13.

In an attempt to se e if the suggestions given would be applicable to arbitrary plan-fo rm s, aerodynamic-center and CL, graphs were prepared f or two double delta plan-fo rm s which are shown in figu res 6 and 7. It appear s that even with taking N = 8 theformul a does not give quite enough m (only 20) to predict the aerodynamic center at thehigher m values (figs. 6(a) and 7(a)). Consequently, fo r low-aspect- ratio wings withsweepback, an N value of from 6 to 10 should be used with as many m values as themachine storag e limitation will allow.

In addition to the over all aerodynamic cha rac teri stic s which have already been dis-cussed, the influence of di ffer ent combinations of N and m has been examined with14

8/12/2019 19680019396

17/81

reg ard t o th eir influence on the span load distribution.examination, the reaso n becomes apparent why the induced dr ag parame te r, cD,i/CLZ,w a s stat ed to be esse nti ally invariant with different combinations of N and m. Thereason is that the span loadings them selv es appear fo r the most p ar t to be independent ofN and m.

(See figs. 8 to 11.) From this

Comparisons and Illustrat ive CalculationsThe following section contains a number of comparisons with other theoretical

methods and experimental da ta which have been used to ver ify the re su lt s obtained withthe present method. After the verifications have been made, illustrative calculations a r emade to show the p resent methods' u sefulne ss in computing sweep, Mach number, tw i s t ,and cam ber effects. Additional illust rati ve calculations of the pre sent method have beenmade and a r e given in refere nce 14.

Comparisons with other theoreti cal methods.- In ord er to in sure that the computa-tiona l proce dure s presently employed give equivalent re su lt s to those obtained by Multhopp(re f. 5) and Van Spiegel and Wouters ( ref . 6), tab le I has been pre pared t o give a tabularcomparison of CL,, aerodynamic center, and cD, i/cL2. The indications fro m tab le Ia r e that all th re e methods give res ult s fo r CL, and Co,i/CL2 within 2 percent ofeach other and the aerodynamic centers a r e within 0.3 percent of t he r efe ren ce chord ofeach other when at a constant value of N . When the cD,i/c L2 values a r e comparedwith l / r A (0.07457 for th is cas e) they are found not to diffe r fro m it by more than2.2 percent, indicating that the computed span loading distributions a r e nearly elliptical.

With the present method established as giving reliable results when compared withthe basi c and modified Multhopp proc edures (refs. 5 and 6) and with the studies previouslyconducted, which indicated that an extension in the number of chordwise pr es su re modesw a s needed fo r ce rta in planforms, comparisons of p redicted res ul ts obtained with thepresent and other methods ar e made. The following planforms ar e offered as examplesof this :

1) The aerodynamic ce nt er s and CL, values for the fou r sim ple planforms shownin figure 1 2 are compared in tables II and III, respectively, with results obtained byDeYoung and H arp er i n re fer ence 15 by us ing a Weissinger seven-point solution. (Valuesof m of 9 or 11 wer e chosen since N = 4, A = 2 , and p = 1.)

From table 11 the aerodynamic cent ers predicted by the pre sent method ar e see n tolie fro m 1.7 to 4.2 percent of the r efe ren ce chord ahead of those given by r efe ren ce 15and from table IIt the values of CL, a r e se en to vary from 1.62 to 5.35 percent belowthose computed by using the present method. These differences are attributed to themo re accu rat e accounting of t he induced camber effects by the present method.

15

8/12/2019 19680019396

18/81

TABLE I.- COMPARISONS O F RESULTS OBTAINED FOR THIS PLANFORMWITH THE PRESENT AND THE MULTHOPP METHODS

O F REFERENCES 5 AND 6

-A z o o1 = 0.25,m = l l0.0427

.0415

.700

A =45OX = 1.0,m = 90.0398

.0376

Present

Ref. 6

Present

A = 4.000

I

TABLE II.- AERODYNAMIC CENTERS FORA = 2 PLANFORMS O F FIGURE 12(a)

WHEN N = 4

1 Aerodynamic center for -x = 1.0m = 11

A = Oo,A = 0.25m = 11

0.191.245

k = 45O, A = 45O,i = 1.0, A = 0.250.156- ::2:n = 9 m = 9I .

.192

cLa0.0564

.05690.0571

.0580

.05800.O5836

.05834~~

a.c.0.270

.2710.279

.278

.2770.262

.263

~

.081290.08069

.08093

.080870.08090

.08086

TABLE m.- LIFT-CURVE SLOPE FORA = 2 PLANFORMS OF FIGURE 12(b)

WHEN N = 47 ift-curve slope for -x = 1.0m = 11

A = 45O,X =0.25m = 90.0432

.0425

16

. . .. m I ...

8/12/2019 19680019396

19/81

TABLE 1V.- DOUBLE-DELTA NONDIMENSIONAL LIFT COMPARISON

K

0.25.50.75

1 oo

Aspect ratio

1.888101.415001.249201.118303.90020

aN = 8; m = 23.

82.6378.077 5 OO40.00

cos(n,/2)ef*

0.7559.52 12.4800.5939.9221

Acos(*c/2)eff

~~2.49792.71482.60251.88304.2297

LaA(Present)

40.02086

.02234

.02253

.02309

.01545

0.02108.02040

Comparisons of the span loadings ar e al so made between the present method andref erence 15 for two of the planforms in figure 1 2 and ar e presented in figures 8 and 9for various combinations of N and m. The span loading dist ributions computed by thetwo methods a r e in good agreem ent fo r most se ts of N and m.

2) A se ri es of double-delta planforms, whose gene ral planform par am et ers aregiven in table IV,have been used to obtain values of C Ltable IV. They are compared with the values of C /A predicted by Spencer in refer-ence 16. From table IV, the C L A predicted values show that a maximum differenceof 6 percent or -9 percent exists between the re su lt s of the two methods fo r the planformsexamined. The difference is attributed to the method used in refe rence 16 because it doesnot consider the distribution in lifting pre ssu re over the su rfac e as the p resent methoddoes but is a refinement of a mathematical expression relating A and sweep to the valueof C L ~or wings which have broken leading edges.

A which are also presented i na/a/

(3) The thre e predicted span loading cu rves, given in figure 13 for a sweptback andtapered planform, come fro m ref eren ces 15 and 17 and the pr esen t method. The span

17

8/12/2019 19680019396

20/81

.. .....

loadings are see n to diff er, in t e rm s of ycp, within 0.9 percent of the semispan fro m eachothe r. The CL, re su lt s of the present method and Campbell's method (ref. 17) agre equite w e l l and are about 5 percent gr ea te r than the r es ul t of DeYoung and Ha rper (ref. 15).For the aerodynamic center, the results of the present method and Campbell's method(ref. 17) are 3.2 and 2.1 perc ent, respec tively, of the re fer ence chord ahead of that pre-sented by DeYoung and Harper (ref. 15).

(4) The two sp an loading curves (obtained fr om ref. 8 and the pres ent method) shownin figure 14 for the A = 3.436 double-delta planform have both been computed usingmodifications of the Multhopp approach, The curve of re fe renc e 8 is based on the twochordal loading function combinations used by Multhopp with a pr oce ss of submatrix inver-sion of the mat rix of influence coefficients to a rr iv e at the local nondimensional lift andpitching moment. The re su lt s of CL, and cD,i/cL2 are within 2 and 1 percent,respectively , of each other, whereas the aerodynamic cen te rs differ by about 3.7 percentof the referen ce chord. The reas on for this la rg er discrepancy in aerodynamic centerthan found for CL, or cD,i/cL2 can be trac ed primarily to the different localpitching-moment t er m s computed by each procedure. The reason for the differences inlocal pitching moments is not clear.

5) Thre e span loadings ar e presented fo r the double-delta planform of figure 15 anda r e computed fro m the following methods: Campbell's (ref. 17), subsonic kernel function(ref. 18), and presen t. Since the subsonic kernel function and the present method a r e basi-cally iden tical (in the reduced frequency cas e) except fo r the handling of t he integrationac ro ss the spanwise singularity, it is not s urp ris ing that the s pan loadings should be quitesi mi la r. Consequently, the spanwise cente rs of pr es su re ycp differ only by about 2 pe r-cent semispan. Fu rthe r, although the values of CL, are within 1 percent, reference 18gives a more stable pitching moment resulting in a more r earw ard aerodynamic center.

Reference 17 presents a span loading which has more of its loading outboard, t he re-fore, a la rg er value of ycp. Also, its CL, is lower and its aerodynamic center isahead of the others.

In figure 15(b) the sa me trends in the spanwise variation of the cen ters of pre ssu reare see n to occur in each of th e two lifting surf ace the orie s ove r the first 75 percent ofthe semispan . However, for q > 89 percent of the semispan, the subsonic kern el func-tion indicates that the chordwise center of p re ss ur e (xcp) li es ahead of the wing. Thelocal center of pre ss ur e fo r Campbell's method is the centro id of the two-dimensionalloading and occu rs at the local 4 4 .

The difficulty experienced by the subsonic kerne l function approach can be attributedto an improper location of collocation points used in that solution. The control pointswere at $ = 0.2, 0.4, 0.6, 0.8; y .15, 0.35, 0.55, 0.75, and the semispan of the regionwhich contains the control point w a s se t equal to 0.10. However, since th er e is not an18

b/2

8/12/2019 19680019396

21/81

optimum location procedure specified with this method, it is unclear what number andlocation of collocation points would yield the best res ults . Pe rha ps more reali sti c resu ltscould have been achieved by locating points far ther outboard than 0.75. A cosine distr i-bution would have made an interesting comparison, but the collocation points cannot belocated very n ear the tip.

A simple case w a s attempted using only the outboard panel extended to the root, withthe collocation points located the sam e as previously mentioned in te rm s of x/c and' and with a region around the contro l point of semispan equal to 0.10. In this ca sereasonable resul ts were found which indicate that the planform shape should be of pr i-mary importance in selecting these points.

(b/2

Concerning the distr ibution of loading along the chords (fig. 15(b)), it is seen that,in general, both lifting surfa ce theorie s have the sam e overa ll shape and differ mostly inmagnitude until the subsonic ker nel function becomes influenced by the locations of thecollocation points, that is, when xcp is ahead of the planform. Comparing the pre sentmethod with that of the two-dimensional dist ribution, the effects of induced camber can beseen concentrated mostly inboard of the leading-edge break. The stre ng th of the two-dimensional loading of any spanwise station w a s chosen to be that predicted by Campbell'slifting-line solution. Thi s was accomplished by requiring that the total circulationstrength be equal to the integral of the two-dimensional d istributed loading tim es a suita-ble multiplier.

Comparisons with experim ental data.- Since experimental data ar e genera lly lackingfo r the planforms previously examined, the ability of the presen t method to predict theaerodynamic cha rac ter ist ics of thes e wings cannot be appraised. Ther efor e, the followingdiscussion will rela te to th ree wings fo r which experimental data ar e available and com-parisons with theoretical resu lts will be made,

1 ) The two predict ed span loadings presented in figure lS(a) for the A = 2 deltaplanform agree closely and estimate reasonably the experimental values found in refer-ence 19. Also, the CL values predicted by the two theoret ical methods, presen t andvortex lattice (ref. 20 , are in good agreement with the experimental result computed fromthe CL, of the fo rce data, since they differ by only about 5 percent. (F rom the vari ationof CL with a determined from the integrated press ure measurements, the CL valueat a = 4.30 is lower than that found by the force tests. The difference is attributed tothe lack of enough pr es su re or ifices nea r the leading edge result ing in a poor integration.)

A comparison of the th re e aerodynamic c en ter s shows that the value given by thepresent method is ahead of both the experimental location and the one found by refer-ence 20. (In genera l, for the lower aspect r ati o wings, solutions obtained using highervalues of N and m resu lt in a mo re aft location of the aerodynamic center . (See

19

8/12/2019 19680019396

22/81

fig s. 6(a) and 7(a).) Fr om an examination of the loca l loadings (figs. 16(b) and 16(c)), itis seen that although the experimental local ce nt ers of pr es su re are better predicted byref ere nce 20, ove r most of the chord (x/c of 0.1 to 0.7), he present method gives betteragreement with the experimental pre ssu re distribution. The disagreement between thelar ge pre ss ur es predicted ne ar the leading-edge singulari ty and those measu red may haveresulted from th e thickness effects pres ent in the experimental data. The singularity inpressure at the leading edge also ca uses the local ce nt ers of p res sur e, predicted by thepres ent method, t o be ahead of t he experim ental value.

(2) The pre sen t method pred ict s quite closely the experimenta l span loading of th eA = 8.02 sweptback and tape red wing of f igu re 17 as found in refe renc e 21. Thi s is seennot only by the distribution but in the ycp location. Also, the agreem ent between thevalues of CL and aerodynamic cen te r is good. Fr om figu res 17(b) and 17(c), it can beseen that whereas the local chordal centers of p re ss ur e a r e accurately predicted, thepre ssu res ar e, in general, underpredicted over the chord from x/c of 0.1 to 0.6 andoverpredicted from 0.6 to 1.0. Because of the singula rity at the leading edge, the theo-reti cal p re ss ur e coefficients in thi s vicinity a r e in exce ss of experimental values.

(3) In figur e 18, the th eore tica l and unpublished1 experimenta l loading and momentdistribution a r e given for a variable -sweep wing in one sweep position. The predictedoverall CL at Q = 3.14' is over 10 perc ent higher than that found experimentally fr omfor ce data and thi s difference is mentioned later. The predicted aerodynamic cente r is7 percent of the r efe ren ce chord ahead of the exper ime ntal value; however, th is is not alarg e discrepancy when one considers that the refe rence chord in this c ase is based onthe oute r panel extended to the root. (See fig. 18.)

The experimental values of span loading which a r e shown in figure 18(a) agr ee wellwith those predicted by theory; however, there is a slight ove rpredict ion by the theoryover the outerm ost pa rt of the outer panel which can also be seen in the graph pre -sented in figu re 18(b). Thi s theo reti cal overprediction of local Cn, lead s to the over allCL being higher than the experimenta l value. The effect of the leading-edge shed vorte xand different amounts of induced angle of att ack genera ted by the theo reti cal and expe ri-menta l loading distribution undoubtedly account for some of the disagreement seen infig ure s 18(a) and (b). Good agre ement is found fo r the Cma da ta obtained fr om theoryand experiment. The xcP graph is just th e result of the Cn, and Cm, cu rv es andhence shows the local cen ter of p re ss ur e slightly off of that predic ted by theory.

Cn,

At the fou r different spanwise stat ions shown in figu re 18(c), the chordal loadingsa r e compared and the agreement is found to be reasonab le at most locations.

Illustrat ive calculations. - Effect of varia ble sweep: Changes in the aerodynamiccharacterist ics due to increasing o r decreasin g the ou ter panel wing sweep (from the

'Tests perf ormed in the Langley high-speed 7- by 10-foot tunnel.20

8/12/2019 19680019396

23/81

strea mwi se tip position) can also be predicted with the p resen t method. For example, theplanform in figu re 18 has its sweep varied fr om 15O to 40 at a Mach number of 0.23 andthe resul ts, shown in figure 19, are based on the area and mean geome tric chord of theoute r panel extended to the root when A = 30. Those result s presented are from theprese nt method and unpublished experimental data me asured on a semispan pressurewing at the Langley Resea rch C enter.

The agreem ent obtained between theoretical and experimental res ul ts is only fair inthat the theory shows the prop er tre nds and si mi la r increments, but not absolute levels.Experimental values for CD,i/CL we re not available fro m force data. A s reportedin references 22 and 23, the variations pre sented in fig ure 19 are typical of variable-sweep wings.

Effect of Mach number: The effect of increasing the subsonic Mach number on thewing-alone aerodynamic chara cteri stic s is accounted fo r by us e of t he Prandtl-Glauertcompressibility rule as mentioned ea rl ie r. (See ref. 11.) In general, the re sul ts obtainedappear to be questionable above a Mach number 0.8 because of the inapplicabil ity of th ePrandtl-Glauert transformation as M -. (ref. 11).

Some re su lt s obtained by applying thi s ru le t o a highly sweptback and tapered winga r e presented i n figu re 20 along with those determined fr om Mach numbers 1 .2 to 2.8 byus e of the super soni c lifting-surface theory of r efe ren ce 4. Experimental values for th ewing-body combination ar e presented for comparison .

In general, the theoretical methods are able to predict reasonably well the exp eri-mental flat wing overall aerodynamic characteristics (which for subsonic speeds have notbeen published but a r e found in r ef. 24 for supersonic speeds) even though the experimen-tal data contained the influence of a body. The one exception is the induced dra g param -et er which does differ considerably. However, th i s is not unexpected since the re alleading edge is sh ar p and the leading-edge suction is considerably le ss than the 100 per -cent assumed by the theory. Fu rth er, no induced drag, called vortex dra g at supersonicspeeds, is given supersonically since it w a s not available separat e from the wave drag.

In the transonic region (dashed lines) the r esu lt s a r e obtained by fairi ng from sub-sonic to superson ic speeds.

Warped wing: The determination of the aerodynamic cha rac ter ist ics for a warpedwing can also be found by using the pre sen t method when the local su rfa ce slope dist ribu -tion is specified. In addition to the flat plate loading charac teri stic s, which are alwaysfound, the ba sic loading fea tu res are als o computed by the u se of the appro priat e equationsgiven in appendix B. Also determined are the zero-li ft angle of attack ao pitching-moment coefficient Cm,o, and root bending-moment coefficient C B , ~ . Illustrativeresul ts of this procedure ar e presented in figure 21 for a double-delta planform with 5O

21

8/12/2019 19680019396

24/81

of linear twist and 2-percent circula r a rc camber. (Other de ta il s of this planform andloading distribution are to be found in the "Supplement t o NASA TN D-4427 since it isone of the sam ple ca ses given.)

MEAN CAMBER SURFACE DETERMINATIONThe problem of determ ining the mean cam ber s urf ace requi red to support a pre-

scri bed loading on a given planform is easier to solve from t he ba sic formulation thanfinding the loading asso ciat ed with a prescribe d sur face shape for the sa me planform.This can be s ee n from the fundamental equation in comp ressible subsonic flow which isequation 1)and is given as follows:

fro m which the mean camber surface is found by use of t he following equation:

The determ ination of the local Ap(5,q) fr om equation 1) has been shown to involvesolving the equations inversel y fo r a part of its integrand; wher eas, to find w/U at anypoint, the individual specifi ed loading must be simpl y multiplied with the appropr iate ker -nel function value and summ ed ove r the surfa ce. Thus, if the more difficult problem issolved o r programed first (as in the pres ent study), then the oth er only req uire s, i n addi-tion to those changes mentioned, the deletion of the ma tri x inv ersi on and the addition ofthe w/U integra tion to find z/c.

Since values of w/U a r e only computed at the control points, some representationat other chordwise locations is needed in o rd er that the integra tion of equation (13) canbe carried out. A least-square polynomial curve f i t of the w/U values at the controlpoints w a s chosen to rep res en t th e variat ion of w/U along the chord between the firstand last control points. The polynomial curve f i t is specified to have a degre e of one lessthan the number of chordal control points to in sure that a good f i t can be obtained. Out-sid e the range of th e cont rol points, a linear extrapola tion of the polynomial curve f i t isused to the n eare st chordwise edge.

22

8/12/2019 19680019396

25/81

Optimization of N and mAs in the load determination section, a study w a s undertaken he re to find the rela-

tionship that N and m should have in ord er that the bes t mean camber lines could beobtained. It w a s intuitively felt that the se mean camber line s would occur when the mostcontro l points were employed. For he A = 2 delta planform shown in figure 22, fourcombinations of N and m w e r e selected and in each ca se the sam e sur face loadingdistribution w a s used. It w a s composed of only a sin 9 chordwise loading which variedelliptically from t ip to ti p and w a s adjusted to give an overal l lift coefficient of unity.All other p re ss ur e mode coefficients wer e s et equal to zero.

Figure 22(a) shows that t he computed downwash w/U values at a constant m withvarying N do not lead to a much changed polynomial curve f i t , whereas at a constantvalue of N varying m does. The curves appearing in this figure a r e integrated andappear i n figure 22(b). As would be expected, sets of N and m which have essenti allythe same curve f i t lead to the sam e mean camber line. Thus, the mean camber lines forthe sets N = 4 and N = 8 at m = 11 ar e the same. This is also true for the sets ofN = 4 and N = 8 at m = 23.

Thus it appear s that for the given sur face loading, increasin g the number of span-wise stations at which the desir ed chord loadings a r e defined is more important inobtaining the best mean cam ber lines than increasing the number of chordal pre ssu remodes involved (much above those needed to define the shape) at each spanwise station.This is a sim ila r conclusion to that reached i n refer enc e 25. (Note that the prescribedloading does not contain any leading-edge suction and hence it is not known what theeffects of including the suction te rm would have on the c amber fo r optimum combinationsof N and m.)

With the rela tionsh ip that N and m should have fo r cer tain loadings established,some comparisons with other theoretica l methods are made and follow in the next section.

ComparisonsThe accuracy of the p rese nt method may be appra ised by comparing in figure 23

the downwash distribution at the plane of symm etry of an A = 10 rectangular wing withthat of a two-dimensional wing with the same si n 9 chord loading at each span station.

Over most of the chord the three-dimens ional downwash is slightly lower than thatof the two dimensional by a constant amount. If a much higher aspect ra tio wing had beenused in the computation, this difference would become sm all er.

At the edges, where x/c - 0 and 1, the three-dimensional downwash is seen totend toward plus and minus infinity, respect ively. Thi s is caused by the logarithmic

23

8/12/2019 19680019396

26/81

singularity correction te rm , which re sults f rom the second-order singularityin the integrand, containing a te rm which inc re as es without bound at all edges. (SeeY ,)"eq. (A9).)

Fro m figure 23, it can al so be noted that by using only a sin 8 loading increasingthe number of chordal contro l points does not change the downwash dist ribu tion at or inthe vicinity of the chorda l control points alr eady computed but does give a mor e detailedrepresen tation of the distribution.

As another check, the m ean camber lines obtained with the present method for thesweptback wing shown in figure 24 a r e presented there2 fo r comparison with those pre-dicted by the mean c am ber su rfa ce solution of Katzoff, Fai son , and DuBose (r ef. 26).Thre e differences a r e apparent in the procedures used and are

1. The present method rounds any kinks that oc cur in the wing planform (i.e., at theroot) which r esul ts in a clo ser represe ntation of the is ob ars on the planform ( ref. 5) andhence gives a bett er downwash distribution than the method used in re fere nce 25 (ref. 27).

used in refere nce 26 ar e approximated hereinand even sine functions give g rea ter flexibil-

2. The uniform chord loadings ACpby a Four ier s er ie s of four odd sine te rm s - sin 8 sin 39, sin 59, and si n 79. This9se r i e s o r others like it composed of cotity in approximating arb itr ary p res su re loadings. The amplitudes of the chord loadingwer e uniform over the entire ar ea and produced a lift coefficient of 1.0.

3. The presen t method us es the least-square polynomial curve f i t and constant slopeschem e to determine the downwash values all along the chord as discussed earl ier. Thisprocedure diffe rs from that used i n refere nce 26 where the downwash distribution com-puted at selected internal constant chord lines is completed by f airin g except nea r theedges of the chord. Th er e the downwash is assumed to be equal to that required to pro-duce a constant pre ssur e all along the resulting mean c amber lines.

The curv es which are discussed in the previous paragraph are then integrated tofind the mean cam ber height z/c above the x-y plane. Despite the differences dis-cussed between thes e two methods, figure 24 shows that the agreem ent between them isgood except ne ar the plane of sym met ry. This is explained in par t by the first differ-ence listed. (See als o ref. 27.)

Application.- An applica tion of the presen t method to the highly sweptback andtapered planform shown in figure 253 is made where the li ft coefficient for th is planform

2The re su lt s of ref ere nce 26 were found by integrating from the leading edge re ar -

3Tabular res ult s fo r thi s planform and loading distribution are to be found in the

~~ -

ward and hence were adjusted in figure 24 to the sam e ze ro level for comparison."Supplement to NASA TN D-4427" sinc e it is used as the sample case.24

8/12/2019 19680019396

27/81

w a s taken as unity at a Mach number of 0.30. The chordal loading function coefficientswere specified so that there w a s 1) a ze ro cot distribution, (2) an elliptical distribu-tion of si n 9 acr oss the span, 3) a sin 29 distribution such that the local xcp variedparabolically from 50 percen t of the local chord at the plane of symm etry to 20 percent atthe break and from the break linearly back to 50 percent at the tip, and (4) a sin 39 dis-tribution which caused the slope of th e chordal loadings to be ze ro at the trailing edge.

2

Figure 25 shows that the mean camber lines have a rearwa rd movement of the x/clocation of maximum camber and a twist which incr ea se s toward the tip. The mean cam-be r line at q = 0.997 is typical of stations ne ar the ti p and should be examined carefu llybefore a direct application is made since the downwash distribution which produced it maybe overly influenced by the singu lar na ture of the downwash at the tip.

CONCLUSIONS

The results obtained with a modified vers ion of Multhopp's lifting-sur face theoryprogramed for the IBM 7094 electronic data processing system and Control Dat a 6400computer system to find the surf ace loadings and mean ca mber sur fac es have beenstudied and the following conclusions a r e drawn:

Fo r the su rfa ce loading program:1 . It has been found tha t the loading prog ram may be applied with reasonable accu-

racy to conventional and composite planforms to find the ir flat plate cha racteri stics and towings with t w i s t and/or camb er to determine the basic loading distribution as well as theforce and moment coefficients at ze ro lift.

2. In predicting the surfa ce loading, the number and spacing of control points is moreimportant when seeking an accurate overall aerodynamic center than when a solution forthe lift-cu rve slope , span loading, or spanwise location of cen ter of p re ss ur e is beingsought. It has been found that at lea st four, and preferably more , chordal control pointsshould be used in obtaining aerodynamic c enters . Thi s number should depend upon thePA of the planform being considered, since the m ore control points used (100 available)at each spanwise station necessita tes a dec rea se in the number of spanwise station s. Thismay lead to a problem when values of PA a r e large. However, in general, the numberof chordal contro l points should be kept between 6 and 10 to ins ure that the res ulting val-ue s of aerodynamic cente r lie within the region of converging resul ts .

3 . Reasonable agreem ent is generally obtained with both experiment and othe r theo-ri es concerning local and overall loading char acterist ics.

4. The effect of Mach number c an be dete rmined and the r esu lting agreement withexperiment is found to be reasonable.

25

8/12/2019 19680019396

28/81

For the mean camber surface program:1. Good agreement is achieved between the downwash required by two-dimensional

theory for a sin 8 loading and that predicted by a three-dimensional uniform si n 8loading at the plane of symmet ry of an aspect-ratio-10 rect angular wing.

2. The agreement with res ul ts from other mean cam ber predictions is found to befai rly good and the p rese nt method has the capability of approximating most chord loadingshapes by the u se of a Fourier sine series.

3. For an acceptable prediction of mean ca mber s ur fa ce s which includes no leading-edge suction, the number of chordwise sta tions should be nearly equal and never l es s thanthe number of pre ssu re loading te rm s r equired to properly define the desired shape andthe number of spanwise sta tions should be as many as can be accommodated by themachine fo r that number of chord stations.

4. The resultant mean cam ber surfa ce predicted for a composite planform whoseloading includes no leading-edge suction w a s found to have a reasonable twis t and camberdistribution inboard of the tip. In the vicinity of the tip, the mean camber sur face doesnot appear to be as reasonable due to the singular nature of the downwash at the tip.Langley Research Center,

National Aeronautics and Space Administration,Langley Station, Hampton, Va., January 19, 1968,

126-13-01-50-23.

26

8/12/2019 19680019396

29/81

APPENDIX AFORMULATING THE INFLUENCE COEFFICIENT MAT=

The formulation of the influence coefficients is accomplished by integra -ting (o r approximating) the chorda l influence that the p re ss ur e loading functions exert oneach control point from every other spanwise station. (See ref. 6.) Then, for symmetri-ca l spanwise loadings, the chprdal influence at q-n,Ltn(@s)., o give the t otal ch(@~),nd fo r anti symmetrical loadings the influence atq-n, LJv-n(@s), s subtracted from that at qn, L&(@s). The value at the plane of sym-metry, is j us t equal t o L ~ A ) ( @ ~ )hen the loading is symme trical and zero

LJv-n(@s), s added to that of qn,

jwhen antisymmetrical.

Now,for v n,

withm - 3 m - 12 2- 1 m - 3v = - - - - , . . . , o , . . .,2

where f-

and the subsonic kern el function is given by

27

8/12/2019 19680019396

30/81

APPENDIX Awhere

X = -C Y)COS @ + d(y)5 = - C ( ~ ) C O S 8 + d(q)

N ow ,for n = v,L i V GS )= bvVKj (@s,yv;qv)+ Logarithmic singularity correction

where

The logarithmic singularity c orrectio n te rm (LSC) arises because of the spanwise inte-. Its derivation is discus sed bygration acr oss the second-order singularity (Y - d 2Multhopp in refe renc e 5, Mangler and Spencer -in re fe renc e 28, and Van Spiegel and

Wouters in referen ce 6 . Its best form is r(In - V odd) (A9)-n=-- m- 12

where the prime on the summation sign means that the te rm when n = v is eliminated inthe summation proc ess and28

8/12/2019 19680019396

31/81

29

8/12/2019 19680019396

32/81

APPENDIX BEQUATIONS USED I N THE COMPUTATIONS

The following equations are used to determine the total local charac teristic s at thedesign lift coefficient as well as the overal l char act eris tics fo r wings with twist and/orcamber. Note that the qtj(qn) te rm s represent the pre ss ure mode coefficients asso-ciated with the twist and/or cambe r boundary conditions, and the qj(qn) te rm s repr ese ntthe pr es su re mode coefficients associated with the flat plate boundary conditions. Theqt-(r] ) t e rms a re a r r ived at in a manner sim ila r to that of the qj(qn) te rm s, as givenJ n

-4w( s yv values are now they the matrix inversion of equation (4), except that th elocal angle of attack for the warped wing at the flat plate c ontrol points when the ro otchord is at ze ro angle of at tack .

U

Local Aerodynamic Characteristics(a)Pr es su re coefficient

f

i+L= 1(b) Chord loading

L , l -(c) Nondimensional lift coefficient (called circu lat ion i n the "Supplement to NASA

TN D-4427")

30

8/12/2019 19680019396

33/81

APPENDIX B(d) Pitching moment about local leading edge

e ) Location of ce nte r-o f-p ressu re loading fro m Y-axis

f ) Center-of -p re ss ur e loading in frac tions of local chord

(g) Span load coefficient

Overall Aerodynamic Characteristics(a) Flat plate lift

m- 12

31

8/12/2019 19680019396

34/81

8/12/2019 19680019396

35/81

APPENDIX B(h) Pitching-moment coefficient at zer o lift

where Cm,l is determined sim ilarly to flat plate Cmi) Root bending-moment coefficient at zero lift

33

8/12/2019 19680019396

36/81

REFERENCES

1. Cri gler, John L.: A Method fo r Calculating Aerodynamic Loadings on Thin Wings at aMach Number of 1. NASA TN D-96, 1959.

2. Smith, J. H. B.: Calcula tion of the Shape of a Thin Slender Wing fo r a Given Load Dis-tribution and Planform. C.P. No. 385, Br it . A.R.C., 1958.

3 . Carlson , Har ry W.; and Middleton, Wilbur D.: A Num erical Method fo r the Design ofCamber Sur fac es of Supersonic Wings With Ar bi tr ar y Plan for ms . NASA TN D-2341,1964.

4. Middleton, Wilbur D.; and Carls on, Har ry W.: A Num erical Method fo r Calculat ing theFlat-Plate P re ss u re Distributions on Supersonic Wings of Arbi trar y Planform .NASA TN D-2570, 1965.

5. Multhopp, H.: Methods f o r Calculating the Lift Distr ibution of Wings (Subsonic Lifting-Surface Theory). R. & M. No. 2884, Bri t. A.R.C., Ja n. 1950.

6. Van Spiegel, E.; and Wouters , J . G.: Modification of Multhopp's Lifting Surface TheoryWith a View t o Automatic Computation. NLR-TN W.2, Nat. Lucht- Ruimtevaartlab.(Amsterdam), June 1962.

7 . Milne-Thomson, L. M.: Theoret ical Aerodynamics. Second ed., MacMillan and Co.,Ltd., 1952, p. 225.

8. Nagaraj, V . T.: Theoret ical Calculation of the Aerodynamic Cha rac ter ist ic s of a Dou-ble Delta Wing (by Lifting Sur face Theory). M.E. Th es is , Indian Inst . Sci., 1961.

9. Van de Vooren, A. I.: An Approach to Lifting Surf ace Theory. Rep. F. 129, Na t .Luchtvaartlab. (Amsterdam), June 1953.

10. Garner , H. C.; and Lehrian , Do ri s E.: Non-Linear Theory of Steady Fo rc es on WingsWith Leading-Edge Flow Separat ion. NPL Aero Rep. 1059, Br it . A.R.C., Feb. 15,1963.

11. Kuethe, A. M.; and Sc he tz er , J. D.: Foundations of Aerodynamics . Second ed., JohnWiley & Sons, Inc. , c.1959, pp. 200-203.

12. Munk, Max M.: The Minimum Induced Dr ag of Aerofoils. NACA Rep. 121, 1921.13. Garne r, H. C.: Some Rema rks on Vortex Drag and Its Spanwise Distribution in Incom-

pr essi ble Flow. NPL Aero Note 1048, Br it . A.R.C., Nov. 4, 1966.14. Lamar, John E.; and Alford , William J., Jr.: Aerodynamic-Center Considerations of

Wings and Wing-Body Combinations. NASA TN D-3581, 1966.

34

8/12/2019 19680019396

37/81

15. DeYoung, John; and Ha rper , Cha rles W.: Theoret ica l Symmetric Span Loading atSubsonic Speeds fo r Wings Having Arbi tr ary Plan Form. NACA Rep. 921, 1948.

16. Spencer, Berna rd, Jr.: A Simplified Method fo r Es tima ting Subsonic Lift-Curve Slopeat Low Angles of Attack for Irr egula r Planform Wings. NASA TM X-525, 1961.

17. Campbell, George S.: A Fini te-Step Method for the Calcula tion of Span Loadings ofUnusual Pl an Fo rms. NACA RM L50L13, 1951.18. Watkins, Cha rl es E.; Woolston, Donald S.; and Cunningham, Herbert J.: A Systematic

Kernel Function Pr oc edure fo r Determining Aerodynamic Fo rce s on Oscillating o rSteady Finite Wings at Subsonic Speeds. NASA TR R-48, 1959.

19. Wick, Bradford H.: Chordwise and Spanwise Loadings Measured at Low Speed on aTriangular Wing Having an Aspect Ratio of Two and an NACA 0012 Airfo il Section.NACA TN 1650, 1948.

20. Dulmovits, John: A Lifting Surface Method for Calculating Load Dist ribu tors and theAerodynamic Influence Coefficient Matrix fo r Wings in Subsonic Flow. Rep.No. ADR 01-02-64.1, Grumman Aircr af t Eng. Corp. , Aug. 1964.

21. Graham, Robert R.: Low-Speed Charac te ri st ic s of a 45O Sweptback Wing of AspectRatio 8 From Pr es su re Distributions and Forc e Tests at Reynolds Numbers Fr om1,500,000 to 4,800,000. NACA RM L51H13, 1951.

22. Baa ls, Donald D.; and Polhamus, Edward C.: Varia ble Sweep Aircraft. Astronaut.Aerosp. Eng., vol. 1, no. 5, June 1963, pp. 12-19.

23. Alford, William J., Jr.; Luoma, Arvo A.; and Henderson, Will iam P.: Wind-TunnelStudies at Subsonic and Tra nsonic Speeds of a Multiple-Mission Variable-Wing-Sweep Airplane Configuration. NASA TM X-206, 1959.

24. Morri s, Ode11 A.; and Fourn ier , Roger H.: Aerodynamic Cha racte ris tic s at MachNumbers 2.30, 2.60, and 2.96 of a Supersonic Tran sport Model Having a Fixed,Warped Wing. NASA TM X-1115, 1965.

25. Garner, H.C. : Accuracy of Downwash Evaluation by Multhopp's Lifting-SurfaceTheory. NP L Aero Rep. 1110, Bri t. A.R.C., July 1964.

26. Katzoff, S.; Fai son, M. Fra nc es ; and DuBose, Hugh C.: Determination of Mean CamberSurfaces fo r Wings Having Uniform Chordwise Loading and Arbi tra ry SpanwiseLoading in Subsonic Flow. NACA Rep. 1176, 1954. (Supersedes NACA TN 2908.)

27. Thwaites, Bryan, ed. : Incomp ressible Aerodynamics. Clarendon Press (Oxford),1960, pp. 321 and 363.

28. Mangler, K. W.; and Spencer, B. F. R.: Some Re ma rks on Multhopp's SubsonicLifting-Surface Theory. R. & M. No. 2926, Bri t. A.R.C., 1956.

35

8/12/2019 19680019396

38/81

(a) Planform variables.

(bl Reference wing geometry.Figure 1.- General planform which can be used in these programs along with its coordinate system and reference wing geometry.

36

8/12/2019 19680019396

39/81

Y = lyv= - 1 Y=O2=7T 8:- T 8=0L e f t t i p P l a n e o f s y m m e t r y R i g h i t i p

(a) Spanwise. Bv = - where v ranges from - m-l to m-l2 m t l 2 2 .

x/c =o+=OL ead inge d g e

X/,=l.O+= T

T r a i l i n ge d g eb) Chordwise. QS = sn where s ranges from 1 to N.2N t 1

Figure 2- Control-point locations.

37

8/12/2019 19680019396

40/81

S

Figure 3.- Detailed chordwise location of control points. = (1 - cos QS),

3 8

8/12/2019 19680019396

41/81

N u m b e r o f sp a n s t a t i o n s , m

8/12/2019 19680019396

42/81

(b) Aerodynamic center.Figure 4.- Continued.

8/12/2019 19680019396

43/81

/v0 20 40 6A 8

0 4 8 12 /6 20 24 28 32 36 40N u m b e r o f s p a n siai ions,m

(c) CLa.Figure 4.- Concluded.

8/12/2019 19680019396

44/81

Number of spanwise slo t ions, h-

(a) Induced drag parameter.Figure 5.- Effect of varying number and locations of control Points and chordal loading functions on aerodynamic characteristics of A = 7 rectangular planform at M = 0.

8/12/2019 19680019396

45/81

N0 2

a.c.,penE

Num ber o f spanwise s ta t ions ,mb) Aerodynamic center.Figure 5.- Continued.

8/12/2019 19680019396

46/81

8/12/2019 19680019396

47/81

U.C.p e n

I2.480

A = 1.700

N u m b e r o f span s ta t ions , m(a) Aerodynamic center.

Figure 6.- Effect of varying number and locations of control points and chordal loading functions on aerodynamic characteristics of A = 17 double-delta planform at M = 0.

8/12/2019 19680019396

48/81

0 4 8 /2 /6 20 24 28 32 36 40Number o f spun s i a i i o n s , m

b) CL,Figure 6.- Concluded.

46

8/12/2019 19680019396

49/81

a.c.1

(a) Aerodynamic center.

Figure 7.- Effect of varying nu mber and locations of contro l points and chordal loading functio ns on aerodynamic characteristics of A = 1.97 double-delta planform at M = 0.

percen

8/12/2019 19680019396

50/81

N0 2n 40 6A 8

U 4 8 12 16 20 24 28 32 36 40N u m b e r of span s t a t i o n s . , m

b) C L ~ .Fig ure 7.- Concluded,

8/12/2019 19680019396

51/81

8/12/2019 19680019396

52/81

b) Con s t a n t N.Figure 8.- Con t inued .

50

8/12/2019 19680019396

53/81

(c) Constant m.Figure 8.- Concluded.

51

8/12/2019 19680019396

54/81

,1111.. I. ,,,,,., I I I, I , , , , 1 1 1 1 1 I..._.. ..--.-....

.5

.4 'J1 4 ,043120 6A 8R e f 1 5 ,0425,04324.04295.04282- :00049.00005-.00026700044-G , l / C L 2./59 8./59/./59 7./59/7

0 ./ .2 3 .5 .6 .7 8 .9 LOQ

Figure 9.- Span loading coefficient variat ion wi th different values of N for A = 2 sweptback and tapered planform at M = 0 when m = 9.

52

8/12/2019 19680019396

55/81

Cz.

Figure 10.- Span loading coefficient variation w ith different combinations of N and m for A = 1.7 double-delta planform at M = 0.

53

8/12/2019 19680019396

56/81

.2

. /.04090.0406204074.04067

-.00306-.00247-.00268-.00293-.00304

.32336 .39533

.3/04 .4/220.3/592 .4/2//

.32/96 .420/7.32475 .42098

./6/65

.I6162.I6162

.I6162./6/62" 'l1lllIlIII ' I 1 I / ' / I / , I / I I I I II I t / I I I

0 / 2 3 4 5 6 7 8 9 LOQ

I I 1 I I I l l I I I I I I I I I It 4Ti0Figure 11.- Span loading coefficient variation with different combinations of N and m for A = 1.97 double-delta planform at M = 0.

54

8/12/2019 19680019396

57/81

A a2.000 M = OI b1.31.a.c.,percent

U 0 4 8 12 /6 20 24 28 32 36 40Number o f s pan s ta t i ons , m

a ) A e r o d y n a m i c c e n t e r .F i g u r e 12.- E f fe c t o f v a r y i n g n u m b e r a n d l o c a t i o n s of c o n t r o l p o i n t s o n a er o d y n a m i c c h a r a c t e r i s t i c s f o r f o u r A= 2 p l a n f o r m s a tM = 0 w h e n N = 4.

5 5

8/12/2019 19680019396

58/81

A 2.000 M = O

LNumber o f span stat ons, m

\ I I f h *

(b) C L ~Figure 12.- Concluded.

8/12/2019 19680019396

59/81

MethodN=4,m=3/ .06/5 .252 .460Ref /7 .06/3 263 .463Ref 15 .0583 284.

A = 6M=O

0 ./ .2 3 .4 .5 .6 .7 .8 .9 LOI

Figure 13.- Span loading distributions predicted by three different theoretical methods for A = 6 sweptback and tapered planform at M = 0.

57

8/12/2019 19680019396

60/81

i I cma 0 ; ~ c p cQ / cL zN = 2 m = / 5 .05722 -.00525 .3484 .4 0 .09270

=2, m =/5,ef 8 .05792 -.003/8 .30500 .427 .09259

0 ./ .2 .3 4 .5 .6 .I .8 .9 LO7

Figure 14.- Span loading distributions predicted by two different theoretical methods for A = 3.436 double-delta pla nfo rm at M = 0.

58

8/12/2019 19680019396

61/81

a ) Spanw ise .F i g u r e 15.- L o a d in g d i s t r i b u t i o n s p r e d i c t e d b y t h r e e d i f f e r e n t t h e o r e t i c a l m e t h o d sfor A = 1 .4 1 5 d o u b le - d e l t a p la n f o r m a t M = 0.

59

8/12/2019 19680019396

62/81

XCP,percenloco/ chrd

0 . / .2 .J

4 .5 .6 .7 .87

.9 LO

4 .5 .6 .7 .8 9X/C

(b) Local spanwise centers of pressure and chordwise load distribution.Figure 15.- Concluded.

60

8/12/2019 19680019396

63/81

ICLCav

-0 . / .2 .3 4 .5 .6 .7Q

Q =43O

,408,396 ,414

.8 .9 10

(a) Spanwise.

at M = 0.13.Figure 16.- Loading distributions predicted by two different theoretical methods and compared with experiment for A = 2 delta planform

61

8/12/2019 19680019396

64/81

N = 4 , m = / /Ref 20

0 Experimen , re f /9

XCP,percent/oca/chord

-0 2 3 .5 .6 .7 .8 .9 IIOQ

-.6-4

-.2

0

. / .2 3 .4 .5 .6 .7 .8 .9 LOX/C

(b) Local spanwise centers of pressure and chordwise load distr ibutions.Figure 16.- Continued.

62

8/12/2019 19680019396

65/81

b ) C o n c lu d e d .F i g u r e 16.- C o n c lu d e d .

6 3

8/12/2019 19680019396

66/81

MethodN =4, =37Exper iment ,ref:2/

0 ./ .2 3 .4 .5 .6 .7 .8 .9P

(a) Spanwise.

at M = 0.19.Figu re 17.- Loading distr ibut ions predicted by present method and compared with experiment for A = 8.02 sweptback and tapered pla nfo rm

64

I l l I I l l I II I

I=4P

8/12/2019 19680019396

67/81

8/12/2019 19680019396

68/81

At= 4 , m = 3 7

0 . / .2 .3

Q Experimenf ef.21

, / I I i i, p = . 5 5

II

I l

TII1 1II' I

p =.75' I1, III

IIT

/ p = . 9 6

5 .6 7 .8 .9

b ) Concluded.Fi gu re 17.- Concluded.

66

8/12/2019 19680019396

69/81

0 = 3 / 4 O

,295MethodI

A =4.303M=0.23

- 0 . / .2 3 4 .5 .6 .7 .8 .97

(a) Spanwise.

planform at M = 0.23.F igure 18.- Loading and moment distr ibutio ns predicted by present method and compared wi th experiment for A = 4.303 variable-sweep

67

8/12/2019 19680019396

70/81

local chord

0 . l .2 .3 .4

rII

=4 ,m=33Qo Experimentunpublished

.7 1.0

(b) Local slope of normal- force a nd pitchin g-moment coefficients an d centers of pressure.Fi gu re 18.- Continu ed.

68

8/12/2019 19680019396

71/81

= 4 , m = 3 30 Experiment ,unpublished

.5 8 .7 .8 .9 /.O

(c) Local chordwise loadings.Figu re 18.- Concluded .

69

8/12/2019 19680019396

72/81

Figure 19.- Effect of leading-edge outboard sweep on some longitudinal aerodynamic characteristics of variable-sweep wing N = 4, m = 37, 33, and 29 fo r A = 15O, 300, and40, respectively) at M = 0.23.

8/12/2019 19680019396

73/81

A 1.697

2.400 \

....."...._".........".~__-_______ _ _ _

~ _ ___~ ~~~ ~

__.-~ unpubl ished subsonic do fa____ Wing alone theory subsonic ond supersonic)

--- Transon ic fa i r ing______-_________--_

0 .8 I 2 I 6 20 24 28 0 4 .8 l.2 l.6 20 24 2BM M

Figure 20.- Effect of Mach number on some longitudinal aerodynamic characteristics of highly sweptback and tapered wing (subsonic theory, N = 8, m = 23).

8/12/2019 19680019396

74/81

T w i s t

3543 ---I? O HI

I

A =1(49/M = O

Tw/s f

Meun cumber-02Hei9ht -

C O I0 .2 .4 .6 .8 1.0

Figure 21.- Warped A = 1.491 double-delta planf orm at M = 0 and computer results.

72

8/12/2019 19680019396

75/81

(a) Computed w/U and least-square polynomial cur ve fit.Fig ure 22.- Effect of var yin g number and locations of co ntro l points on downwash dis tri but ion and shape of mean camber li nes for A = 2delta planform at M = 0

73

8/12/2019 19680019396

76/81

1 1 1 I I I. m1mm I .... . .II..I.I ,1 1 I 1111.1.1 .

m/ /

"0 .2 4 .6 8 LO0 .2 I .6 8 /OV C

b) Mean camber lines.Figure 22.- Concluded.

74

8/12/2019 19680019396

77/81

V CFigure 23.- Comparison of two- and three-dimensional downwash predictions from uni form sin 0 chord loading at tl = 0.000.

8/12/2019 19680019396

78/81

N = 8, =29

samespan1

3 4 .5 .6 .7 .8 .9 1.0X/C

Figure 24.- Chord loading distributions and comparison of mean camber shapes resulting from present and another method for A = 8sweptback and tapered planform at M = 0.9 for a uniform area loading a t CL = 1.0.

76

8/12/2019 19680019396

79/81

. /9 c 0

-./.f

'%c @

-.I

I

9c L- . I

vc

...

rI1'ITII'IIIAIIIiI5I

i

II7

I-I-

ITi1$III

r-345 A = 8.000M = 0.9040.65'

-1.000

N = 8 ,m = 2 9Ref 26

.2 3 .4 .5 .6X/C

.7 8 .9 LO

Figure 24.- Concluded.

77

8/12/2019 19680019396

80/81

0 . /

1I

TTII

TI

I-r

A = 1.705M = 0.30

. 4 .5X/C

.6 .7

II3?

3-+T

II

IiII1+i

-7-

fI~ II

iIi

II

7IIII

.8 .9

Figure 25.- Predicted mean camber shapes for A = 1.705 highly sweptback and tapered planform at M = 0.30 when N = 4 and m = 39.

78 NASA-L angley . 1968 L-4760

8/12/2019 19680019396

81/81

ONAL ERONAUTICSND SPACE ADMINISTRATIONWASHINGTON,. C. 20546

OFFICIAL BUSINESS FIRST CLASS MAILPOSTAGE AND FEES PASPACE ADMINISTRATIONATIONAL AERONAUTICS

' T h e aeronautical and space activities of the United States shall beconducted so as t o contribute . . . t o the expansion of human knowl-edge of phen ome na in the atmosphere and space. Th e Administration..shalZ $rovide fa the widest practicable and appropriate disseminationof information concerning its activities and the results thereof.-NATIONALAERONAUTICSND SPACE ACTO F 1958

%

NASA SCIENTIFIC AND TECHNICAL PUBLICATIONSTECHNICAL REPORTS: Scientific andtechnical information considered important,complete, and a lasting contribution to existingknowledge.TECHNICAL NOTES: Information less broadin scope buthevertheless of importance as acontribution to existing knowledge.TECHNICAL MEMORANDUMS:Information receiving limited distributionbecause of preliminary data, security classifica-tion, or other reasons.

TECHNICAL TRANSLATIONS: Informationpublished in a foreign language consideredto merit NASA distribution in English.SPECIAL PUBLICATIONS: Informationderived from or of value to NASA activities.Publications include conference proceedings,monographs, data compilations, handbooks,sourcebooks, and special bibliographies.TECHNOLOGY UTILIZATIONPUBLICATIONS: Information on technologyused by NASA that may be of particularinterest in commercial and other non-aerospace