An empirical method for rapidly determining the loading distributions on swept back wings

VLIEGTUIG BOUWKü IN u . Kanaaktraat 10 - DELFT

REPORT No. 32

THE COLLEGE OF AERONAUTICS

CRANFIELD

B i b l i o t h e e k T U Delft

t^acultelt der LuchNaari- en Ruimtevaaittechnlek

Kluyverweg 1 2629 HS Delft

AN EMPIRICAL METHOD FOR RAPIDLY

DETERMINING THE LOADING DISTRIBUTIONS

ON SWEPT BACK WINGS

by

B i b l i o t h e e k TU D e l f t LR 2033967

H. no in

T H E C O L L E__G _E O F A E R O N A U T I C S C R A N F I E L D t I I I M r •••••I • I • M i C T i i a i i l II I I

An Empirical Method for Rapidly Estimating the Loading Distributions on Swept Back Wings

-by-R. Stanton Jones, B.A. (Hons,Cantab.), D.CAe.

TECHNISCHE HOGESCHOOL VLIEGTUIGBOUWKUNDE Kanaalstraat 10 - DELfT REPORT NO. 52 J a n u a r y . 1950 STO/C/IARY

This paper describes the derivation of empirical formiilae for the loading distribution on straight tapered swept back wings, in terms of the parameters governing the geometric wing plan form. The derivation is based mainly on the results of the Weissinger method but corrections to give better agreement vri.th experimental results have been introduced,

The method depends entirely on the assimiption that the shape of the loading curve can be completely defined by the position of the spanwise centre of pressure (jF). itlthough this assumption cannot be backed theoretically the analysis shows that there is plenty of sound

evidence to support it.

The shape of the loading curve is then shown to be expressible explicitly in a formula involving y. This formula can be used to derive the loading distribution for any paxticiiLar wing falling within a wide range of aspect ratiO;, taper ratio, and sweepback angle, and the calcvilation can be completed in a matter of minutes to an accuracy comparable vri.th that of the Falkner method. In addition, the method allows the resvilts of the linear perturbation theory of compressible flow to be easily incorporated. However, it suffers from the limitation that it applies only to simple trapezoidal wing plan forms.

The formulae are

-The location of the Spanwise Centre of Pressure in incompressible flow

y = .42 + _A

10

3

(2:.,4 + 5A) tanP + 10.4V - 6.7

(12^

The Non-Dimensional Loading Coefficient at the Spanwise

Station - r\

( -6.35 + 14.13^X

1 2N2 = 1.28 (l-Ti )

'Tit.7 0

4.25 " 53.8 (n - .815^L^

-,

6..

425

...(16)

The location of the Spanwise Centre lof Pressiore in ^incompressible flow

y = '42 + — T 10-^

(4.4 + 5?0 -tanTj.^ + do,4?? - 6.7) /l -ir ...(18)

m s j

6\

-2-NOTATION

£^ = sweepback of the I/4 chord line

A = aspect ratio

^ 0 .

i.'

tip chord

A = taper ratio

^

root chord

b/2 = semi-span

r\

= spanwise position/semi span

c = geometric mean chord

C_ = mean lift coefficient for whole wing

y = spanwise position of centre of pressure

Kj^ = slope of ( T ^ 1 ./^v.tan^ lines

B ^ = intercepts of { T^I.'^^ .tan f" U n e s

on

the (ï^)

K = loading coefficient

^^L

Q s slope of K .-^.y lines

Aj, = aspect ratio for wing at Mach number M

i y . -

sweepback angle for v/ing at Mach number M

3

-1. Introduction

For a wing of any given plan form there are a n-umber of methods for determining the loading distribution and among the better

known are those due to Falkner ^~^'^»' "'^, Mxolthopp, Weissinger , and

more recently Schlichting and Thomas . However, all these methods involve quite a considerable amount of computing work and time and do not readily present a general picture of the relative effects of sweepback, aspect ratio or taper ratio on the shape of the loading curve. It was decided, therefore, to develop a semi-empirical approach based on the results obtained by applying one of these analytical methods modified in the light of such act\:ial experimental evidence as was available. The ultimate object vra.s the development of a method that v/ould be both quick to apply and would readily

demonstrate the part ployed by the various geometrical and aerodynar.iic parameters in controlling the loading distribution.

The resulting method presented here applies to straight tapered swept back wings and appears to give results that are satisfactory for most practical purposes, v/hilst the tine of computation involved in any particular case is of the order of a few minutes.

2. The Vfcissinger method (Reference 1)

The Y/eissinger method replaces the spanwise load by a line vortex at the I/4 chord position. The spanwise strength variation of this vortex is determined by the boundary condition that there can be no flov/ through the mean camber line at the 3/4 chord point. This is exactly the same set of assumptions as those adopted by Multhopp v/ho attempted an analytical treatment of the resulting

series, Hov/evcr, Weissinger preferred to determine the circulation directly at four spanwise points.

The theory is applioablo to all plan foras, including those which have camber and tv/ist. nevertheless, it is usual to consider the loading in tvo paxts, viz., the 'basic loading' due to camber and tvri.st at zero total lift and the 'additional loading' due to angle of attack. Only the 'additional loading' will be considered here.

r^"""' I J* . . .

An empirical method for the basic loading is in the course of preparation,

4 -3» Ranp;e of aerofoils considered

In Reference 6 the Y/eissinger method has been applied to straight tapered wings within the'following

range.-Aspect ratio range 1.5 to 8 "^j

Sweep back range 0 to 60 (^

Taper ratio raxigs 0 to 1.5 I

Wings m t h five taper ratios were considered. They were

->.,= 0, .25, ,5, 1.0, and 1.5.

For each of these taper ratios graphs were drawn of

-(a) the variation of the spanwise centre of pressure y, v/ith sweepback angle f' for each aspect ratio A. (A typical set for X = .5 is shown in Fig.l),

(b) The variation of the sparavise loading

coefficient

fL

ccL

Jj .

with sweepback for

and the loading curvet

each A at the fovir spanv/ise positions

-ri = 0, .3827, .7071, and .9239.

4. Loading curve shape parameter '•

From the curves ^d) cross plots v/erc obtained of A versus

Ju for various spanwise centre of pressure positions y, at each of

the taper ratios /'^ , ( A typical set for X = .5 is shown in Fig,2),

These A-''p lines were then used in conjunction iTith the graphs (b),

cC.

against r[ v/ere derived foi- various

wings whose geometric shapes are defined by £"^, A, 'A_, for constant values of y. The process was e.pplied to numerous values of y from y = .415 to .480, (See Fig,3 for a typical set of results),

In every case it was found that all v/ings ( T , A, X ) which had the same spanv/ise centre of pressure position, y, had practically the same non-dimensional spanvri.se loading curve Hence, all the m n g s which form the locus of any one of

t}ie A'^i? lines (such as in Fig. 2) have a loading curve whose Lihape may be entirely defined by y,

cCl

CCL f^r], CCT

^. L J

/The

-5-The loading curves v/hich have been dravm to illustrate this result in each case (e.g. Fig,3) have been taken from the extreme limits of the A-v^ T lines, and the discrepancies between the

loadings represent the maximmi discrepancies that occur for v/in^s of similar y vrLthin the range considered,

The worst discrepancy encountered was only foimd to be of the order of 2 per cent v/hich compares favourably with the sort of accuracy to vj-hich experimental readings of pressure plots may be made,

On the basis of this analysis it v/'as decided to accept y as the loading curve shape parameter.

5, y'•^ A relation

The problem nov/ reduces itself into the empirical solution of the equation

y = f (p, A , A ) . .-(1)

Cross plotting from the C-^ A lines we next obtain-the set

of curves y against A for various .P at each X , A typical set

for A = ,5 is shov/n in Pig.4» It can be seen that these j'x^ A

curves arc all not far off straight lines. We therefore assume that the y'-«> A curves nay all be represented by straight lines v/hich pass through the point .42 on the y axis. The cases v;here the assumption is clearly weak fall outside normal practical bounds. Thus we note

that,-1, v/ings vri.th "h- - 0 are not likely to be met

v/ith when P ~ 0 if A is above about 4.0»

2, Wings v/ith 'X betv/een ,5 and 1.0 are not

frequently used v/hen V is lev/ and A is

bclov/ about 2.0, or v/hen V is high and

A is also high.

We nay also note that the Weiss^inger method is known to underestimate the load at the tips. We can, therefore, expect values of y to be slightly greater than those derived from the Weissinger method.

6. Relations betv/cen y. A, J"' and "X.

Using the straight line representation of the y.-w A curves

we now obtain their slopes for every value of t' considered at each

taper ratio (see Table l), Thus,

^ -\*

{%^'

<^>

where y-\ = '^^ ^ ° ^ ^ i ^ *

A

-6-Plotting these slopes {-rr) against ïtan P • v/e obtain Fig.5

which suggests that we nay reasonably drav/ straight lines through

the points obtained for any one taper ratio ?< , The points obtained

for ^1 = 0 do not lend themselves to this simplification as readily as the points obtained for the other taper ratios. The equations of these straight lines (Fig,5) nay be generally expressed by

(

''iz\

_{aAJ^ = ^ X t a n P + B ^}

where K-^ is the slope at a particular taper ratio A and Bx is the intercept on the ('r^] axis. Thus the individual equations

become

S I o

- '^^^^

"^^^^"' " •<^°^5.

(3)

(4)

(If) ^ = •°°575 tanP- .0015.

f H ) ~

'^°^'^

tanP* .0008.

id^]

= ,0093 tanr+ .00375.

\aA ƒ

1.0

ÏIAJ

-

•°^''^ tanpH- .0060.

1.5 (5) (6) (7) (8)

Plots of K-x against X , and B-s against X and X' (see Fig. 6) nov/ lead to the equations

K

^ = .0041)- + .005 A,

and

B-^ =-.0067 + .01047N'

Thus, substituting in the equation for i"^ j ^^^Q have

But

flf) = ('OOhl^ + ,005X) t a n T

- .0067 + .0104^2 ,

= .42 +

_&)r

(2)

(9)

(10)

(11)

Thus

= .42 +

A.

10'

(4.4 + 5 > ) t a n P

+ 10.4^2 > 6.7

₍₁₂₎

/ T h i s . . .

-7-This is the final formula v/hich is the parametric relationship betj/een the geometric shape of the aerofoil and the shape of the

non-dimensional loading curve.

7. T}ie accuracy of the empirical formula

The process of derivation of this formula is undoubtedly crude but its results can be readily checked against the initial data and its accuracy can be assessed insofar as one accepts the

accuracy of the initial data. This v/as done for some forty different wings, covering the entire range of aerofoils considered, and the comparisons between the original Weissinger results for y and the results obtained by means of the empirical formula are shovm in Table II.

In the no.jority of cases the empirical formula gives results v/hich are sliglitly larger than the Weissinger values but this v/as implicit in our assumptions and deliberately introduced to allow for the fact that the Weissinger method appears to under-estimate y slightly in certain cases,

In the fev/ instances where the empirical formula under-estimates the value of y the difference is small.

8. The suitability and importance of y parameter

There is some doubt as to the suitability of y as a parameter, mainly because its total variation is only from about

.400 to .500, and in this range a wide variation of wing plan form is possible, from a triangular slightly swept vri.ng of lev/ aspect ratio to a rectangular highly swept wing of large aspect ratio.

Hov/ever, the empirical formula appears to be capable of estimating y to vri.thin .003 (see Table II), v/hich I'opresents

3 per cent of the total y range, v/hile an error of .003 in estimating y only corresponds to an approximate error of 3 in the angle of

sweepback. Thus the formula might be considered as aoo-urate enough for most purposes,

9. Shape of the loading curve

Now that we have obtained a value of y v/hich may be readily computed for any wing, v/e require to relate y to the actual shape of the loading curve and so relate the loading curve to the parameters governing the geometric layout of any wing. The shapes of the loading curves used were derived from Reference 6j they represent the actual Vfeissinger results and no attempt has been made to correct them to fit experimental curves.

-8-Reference 6 only considers four spanv/ise stations but

from the mean curves of loading distributions (e.g. Fig.3) the loading i oC,

c o e f f i c i e n t

K = v/as determined a t ten spanv/ise p o s i t i o n s

L h .3

for seven types of loading (i.e., for seven values of y ) .

If K , the loading coefficient, is now plotted against y for each of the ten spanwise positions considered, we obtain points that may easily be represented by a set of straight lines (see Figs. 7 and 8 ) .

It may be shov/ti that the value of y for elliptic loading is very close to .425. Thus, if the K axis is taken at y = .425

in Figs, 7 and 8, the intercepts of the K '^ 7 lines on this axis

should give an ellipse v/hen plotted out aga.inst r]. It wa.s found

that this was almost exactly true. V/e shall therefore take the K axis as passing through y = ,425 a.nd this defines a fundamental loading curve,

The K '-V- y lines may be represented by the general equation

K

"n

= K - + Q^ (y - .425) (13)

where K is the intercept of the lines on the K axis and Q is the

slope aK _21

öy

We have already discoveixïd that the v a r i a t i o n of K_ Vvdth

y

r\ i s very n e a r l y e l l i p t i c a l and i t may be v/ritten

9/TÜ ^ ''^y^i^

K - = 1.28 (l - ^2)2^

(14)

The slopes Q are shov/n in Table' IV and plotted in Fig. 9

where it may be seen that they form a straight line from r\ = 0 to

T) = , 7 , but thereafter the curve appears to be parabolic. Hence, we find Q is very closely given by

% (-6,35 + 14.13 ^ ) ^ . . . .7 {4.25 - 53.8 (j) - ,815)^) T1^.7 (15) /Substituting .. .

9

-S u b s t i t u t i n g f o r K_ and Q t h e complete l o a d i n g formula b e c a n e s -cC^ cG, 2 s i = 1,23 (l - -n )2 +

{-è,:>5 + 14.13 ^ ) n ^ . 7

4,25 - 53,8 (TI - . 8 1 5 ) '

•n>^,7

y - .425 .(16)

where y may be calculated bytthe empirical'formula already given',

10, Accuracy of the loading; forraula

For y = ,44 we have the follov/ing

comparison,-Tl

0

.383

.707

,923

« ,

f.,.,

Cyz,

^c

11'"^,

Empirical

formula

1,185

1,168

,959

.551

We is s i n g e r

1.185

1,160

.955

,55b

This agreement is typical. Thus, providing the value of y is correct this empirical forr.iula a.llov/s us to reproduce almost exactly the corresponding Weissinger loading curve,

We have already noted that y may be in error by approximately cC.

.003 cind it is found that the maximum error in

cG

_{'L -!}

v/hich this is likely to cause is just over 1 per cent from the'mean Weissinger value, This is witliin the originally estimated maximum difference between loading curves v/ith identical y values (i.e., 2 per cent maximum difference). Hov/ever, it should be remembered that V/eissinger loading curves do not alv/ays agree with experimental results and no allowance lias been made for this fact except in the estimation of y. 11, Collected fom.ulae

The loading distribution on any straight tapered swept back wing nay be derived v/ith an accuracy suitable for most practical purposes by the application of the following formulae,

/(formulae) ... - _ _ _ _

Once y is obtained, values of —^ nay be read directly

CCT

from the K .•^-^.y lines Pigs, 7 and 8, The fonuula is mainly of use where partial areas of the loading curve are

1 0

-cC,

cC.

y = .42

10-(4,4 + 5 A ) t a n r + 1 0 . 4 ^ 2 - 6,7 . . . ( 1 2 )

- ^ = 1.28 (1 - r i ^ ) " +

cCl

(-6.35 + 14.13^)^^ -j

4.25 - 53.8 (TI - .815)'

r i ^ . 7

y-.425i.

(16)

nay also be obtained by reading d i r e c t l y fron the K . " ^ . y l i n e s

(Pigs, 7 and 8 ) ,

12, Conparison with experinental results and Falkner's calculation

A conparison between the loading curves obtained by the empirical method suggested above and an experinental pressure plot (^_9Jr^ shown in Fig. 10, (Reference 8).

The wing has A = 7,51, ^ = 23 and /^, = ,243, and there is reasonable agreement except at the root, A comparison with the results of Falkner' s calculation was made for tlie follov/ing v/ings,

(See Pigs. 10 - 14) and References 37, 38, 39),

1 2 3 4 5 6 7 8 9 10 11 12 13

£,

0 0 0 0 2 8 , 4 2 8 . 4 4 8 . 4 • 5 2 . 5 ' 5 2 , 5 7 1 . 4 59 45 45

'k

6.0 5.87 5.87 2 , 5 6 5.89 5.89 2.56 •2.31 2.31 1.01 5,b 5 . 8 . 5 . 8

X

1.0 .323 . 3 2 3 .323 .323 .323 . 3 2 3 0 . 0 0 . 2 5 .25 .25 Mach No, 0 0 . 9 0 0 . 9 0 0 .9 0 0 , 8 0

A

, wing CJ vTing F i g . No, 10 11 11 11 12 12 12 13 13 13 14 14 14

The agreement v/ith Falkner' s results appears to be quite remarkable and the only really serious discrepancies occur for v/ing No.8, i.e., the delta v/ing and for v/ing No, 13,

A point of particular interest is that the empirical fonaula agrees better with the Falkner solutions v/hich have been corrected by an auxiliary solution than with the uncorrected Falkner solution,

-11-Palkner has introduced this auxiliary solution to allow for the effects at the centre section of a sv/ept v/ing and it can be seen that it considerably alters his standard 126 vortex 6 point solution.

13» Compressible flov/

Using the results of the linear perturbation theory, R. Dickson has shov/n that a v/ing in compressible flov/

A,r, r\r, 7\ f may be represented by a v/ing in incompressible flow

•A. . P J A , where -o -o

tanP,

A =./ 1 - M^ /i,, and tan f .. ,.,_^iL^. • (l7)

o - 11 o r)

./I - ir

Substituting in the fon:iula for y, v/e have

,42 + 10^

(4.4 + 5 .) tanTj. + (10.4X^ - 6.7)/l^ M

(18) and fron this the position of y and hence the nev/ loading curve

nay be calciilated for any desired value of M below the critical (v/ithin the assumptions of the linear perturbation theory).

J* —

In his report Dickson' suggests that y remains constant independent of Mach number. According to the above equation,

however, it v.dll be seen ttiat this is only so when

(io,4i4F - 6,7), = 0, , (19) i.e., when A = .415>

N e v e r t h e l e s s , f o r wings v/ith t a p e r r a t i o s betv/een about ,55 and . 4 5 i t v / i l l be n l ^ a r t h a t the shape of tlie l o a d i n g curve v / i l l n o t be s e r i o u s l y modified by v u i - i a t i o n s of liach ironber and i n

j.>Avl--i onln\- t h e r e w i l l be no lai-ge inovji.ient of aei-odynfimic. c e n t r e , Fui'thor inspt-;ot:-i on of t h e fCJi-i.i'iT.n Riin;;ests t h a t v/hen

/N ^ , 4 1 5 i n c r e a s i n g Mach nw.iber d e c r e a s e s y and t h e aerodynamic c e n t r e moves forward. S i m i l a r I j r v/hen 'h ^ .415 i n c r e a s i n g Hach number i n c r e a s e s y and t h e aoi-odynauic c e n t r e moves backwards.

/ I 4 . C o n c l u s i o n s

TECHNISCHE HOGESCHOOL VLIEGTUIGTOUWKUNDE Kanaalstraat 10 - DELFT

1 2

-14. Conclusions

1, The method evolved applies to the additional loading of non-yawed straight tapered swept back v/ings at snail angles of incidence. This linits its use in viev/ of the present trend of sv/ept back v/ing design (i.e. cranked wings), but it is felt that an extension of the nethod to cope v/ith these other plan forr-is should be feasible,

2, The empirical forr-iulae derived for the estination of y and the shape of the non-dinensional loading curve is sufficiently accurate for most practical purposes, both at low speeds and at i'lach nunbers \irLthin the linits of the linear perturbation theory. Any particrilar case can be estimated in a few minutes,

3, The available data indicates that the accuracy of the empirical f orn'ola is better than that of the Weissinger nethod and conparable with that of the Falkner nethod, but until nore direct experiiaontal evidence is available this cannot be relied upon unconditionally,

1 3

-H IN<! co o ^^ I O co o co L A • T -1) (< o • ,--II K L A , II f< L A CM • II (^ O II -< O ^ MO O C3 • Q -4-O O • CO o o o • i;. co T -O . 1 CJN O o , o 1 1 ^— T -o o • 1 1 LTN UA r-o o • CM L A O O , 0 0 o o , 1 CM r^ o o • O T -1 O MO O o . MO CM O O • O 1 L A ^— MO cr\ O O • r-M3 o o , CM CM f A O O • 1 LA O O • O CM LA CM T -o . L H CO O O • L A -d-O O • 0 0 • V

-8

CTN r A O CJ . O I A CO ^— o • o ^A • ^ -O , L A r-o o • -:!• O O . K^ I A O O • LA ^ L A r^ CM O • I A CM v -A l O • L A CM V -O . CvJ CO O o • 1 o vo

TABLE II y for 40 wings

X

.542

.442

.418

0

.25

.5

1.0 i

r

0.9

31

46.4

45

45

20

20

20

30

0

0

30

60 15.5 0 60 ' 60 45 45 30 30 30 0 0 20 60 60 45 45 45 45 30 30 30 0 0 0

15

15

15

t A •

4.47

4.66

3.45

2.76

4.5

2.35

4.4

6.6

4.3

2.1

5.25

6

2.5

5.0

5.8

1.5

3.5

1.5 4.5 2.5 4.5 8.0

3.5

7.0 4.8 1.6 3.0 1.7 3.35 5.0 4.1 2.0 3.9 6.3 2.7 4.2 5.1 1.9 3.5

5.5

Weissinger

y

.425

.44

.442

.41

.405

.405

.395

.39

.400

.400

.38

.43

.44

.42

.41

.44

.464

.43

.454

.43

.44

.454

.4225

.424

.436

.454

.48

.44

.464

.48

.472

.436

.454

.472

.43

.436

.44

.43

.44

.454

Empirical

Formula

y

.4248"

.4395*

.4436

.4136

.4096

.408

.3975

. 3894' .402 .406 .3848 .43045 .4407 .42032 .4113 .4388'* .4638 .4311 •4535* .4313 .4403 .4562 .4219* .4239* .4347* .452* .48 .4423„ .4639* .4842 .4737 .438 .455 .4774 .4356* ,4389 .4318 ,/i/,l8 .4542 Experiment y .433 .4^44 .45

Indicates where the empirical formula slightly londerestimates the Weissinger values.

1 5

-, P H H co ü • CO -d-• -d-vS -d-• -d-LCN ^ . •4 • f ^ -d-. CM -d-• T - -4-•

ï>. 1

f / /

1 '^

L A r A CA . L A h A o . T -O 1 -CTN • r -L A r-CM • ^— r^ • 1 ^ ^^^ • o o . (^ o . r -CM T -•s— L A CO v -u^ CM • O I A . •'4-r-^ • O O ^— . L A O . •r-L A O • T - 4V -• co T -LO, OJ CM L A >JO CM • O I A • o o CM • CM V -• -d-• LO> •ï— • VX> V -• Ln ^P • r -• r^ ^— « ON r -• 'T— CM cO (^ • K ^ T -• O "J— • T -co O • \s\ o • ^ A O 'S— o ON ON • ^ O VO • o • UN -d-O • T~-o\ <J\ . UN UN ON . o\ cr^ • CO CO • L A 4-0 4-0 • ^-o r-_• • a • •;— -d-ON . CO •Xi . . KN CO . ON r--• -d- r-• T - r-• o o CO • CM cr\ • 4-CO • ON r^ • -d- r-. o r^ . r-VO • CJ ^ • o UN CO • O 1 ^ , UN VO • CM VD . LA UN UN • CM UCN . CfN •4-• ICN . ^ A CM ON . VD UN . O UN . CO -d-• KN -d-• Q_ .-d-. r-NN . o -d- 1 rA ,' . 1 VD ON .

1 6 -ÏABLS IV

•n

0

.1

.2

.383

.60

.707

.8

.85

.923

.96

%

- 6 . 24

-5.00

- 3 . 64

-0.80

+2.00

+3.64

+4.20

+4.20

+3.64

+3,1

; :

-17-RgiFEREWCES

The following references have been used to check the results of the empirical

formulae.-No. Author Title, etc.

Calculation of Loading Distributions

7

J, Weissinger J,A. Shortal and _ B, Maggin J,C, Sivclls and R.H, Neely R,A, Mendelsohn and J.D. Brewer N.H, Van Dorn and J. DeYoung J. DeYoung B,H. Wick H.E. Murray V.I. Stevens

The lift distribution of swept-back v/ings. Dec. 1947.

N.A.C.A. Tech. Memo 1120

Effect of sweepback and aspect ratio on longitudinal stability

cliaracteristics.of vTings at lov/

speeds. Jvlj, 1946.

N.A.C.A. Tech. Note 1093

Calculation of v/ing characteristics by lifting line theory using non-linear section line data. April, 1947.

N.A.C.A. Tech. Note 1269

Comparison betv/een the measured and theoretica.1 span loadings on a moderately sv/ept-for\7ard and a moderately sv/ept-back semispan wing. July, 1947.

N.A.C.A. Tech. Note 1351

A comparison of three theoretical methods of calculating span load

distribution on sv/ept wings. November, 1947.

N.A.C.A. Tech. Note I476

TheorcticQ.1 additional span loading characteristics of v/ings with arbitrary sweep, aspect ratio and taper ratio. Dec. 1947 N.A.C.A. Tech. Note 1491 Chordwise and spanwise loadings

measured at lev/ speed on a triangular vfing having an aspect ratio of two and an N.A.C.A. 0012 airfoil section. June, 1948.

N.A.C.A. Tech. Note 165O

Comparison v/ith experLment of several • methods of predicting the lift of

v/ings in subsonic compressible flow, October 1948.

N,A,C.A. Tech, Note 1739 Theoretical basic span loading

characteristics of v/ings vdth arbitrary sweep, aspect ratio and taper ratio, Dec. 1948.

N.A.C.A. Tech. Note 1772

Title, etc.

Some theoretical low-speed span loading characteristics of svrept vdngs in roll and sideslip. March, 1949. N.A.C.A. Tech. Note 1839

A method for deterr.iining the camber and tvdst of a surface to support a given distribution of lift, with applications to the load over a sv/eptback y/ing. 1945. N.A.C.A. Tech. Report 826

The determination of span load distribution at high speed by use of high-speed

wing-tunnel section data. Feb. 1944.

N.A.C.A. A.C.R. 4B22 (A.R.C. 81I4).

Preliminary notes on aerodynar;iic centre of sv/ept-back vdngs.

N.P.L. Report Sept. 1944 (unpublished) Pvirther notes on aerodynamic centre of

sv/ept-back v/ings.

• N.P.L. Report Sept. 1944 (unpublished) Calculation of aerodynamic loading on a

sv/ept-back vdng. Jan. 1944 A.R.C. 7322 (\m:§ublished) Comparison of simple calculated

characteristics of four sv/ept-back vdngs. Feb. 1944

A.R.C, 7446 (unpublished)

A general solution of the problem of loading on vdngs with discontinuities of incidence. April, 1944

• A.R.C. 7629 (unpublished)

Calculation of induced caiabers on four vdngs. May, 1944

A.R.G. 7667 (unpublished)

Construction of Tables for calculating the aerodynamic loading of m n g s , May, 1944.

A.R.C. 7732 (unpublished)

Effect of sv/eepback on the aerodynamic

loading on a 'Y' vdng. J-une, 1944

A.R.C. 7786 (unpublished)

Characteristics of a sheared 'V wing, Dec. 1944.

A.R.C. 8255 (unpublished) Lifting plane theory of vdngs v/ith

discontinuities of incidence. May, 1945. A.R.C. 8638 (unpublished)

Siiiiplification of vdng loading calculations by lifting plane theory.

A.R.C. 9211 (unpublished)

Title, etc.

Calculation of compressibility effects on loading of a sv/ept-back vdngo Dec. 1945

A.R.C. 9261 (unpublished) Proposed definition for sv/eopbackc

Dec, 1945. A.R.C= 9237 (unpublished) Theoretical work required for the

assistance of designers of aircraft

v d t h F^'^^pt-br-ck I'sn-n'^B, "R'p>\. 1 9 4 6

A.R.C. 9409 (unpublished)

The use of equivalent slopes in vortex lattice theory/. March, 1946

A.R.C. 94''f6 (to be published as

A.R.G. Reports and Memoranda No. 2293) The accuracy of calculations based on

vortex lattice theoi-y. May, 1946 A.R.C. 9621 (unpublished)

The calculation of aerodynamic loading on surfaces of any shape. 1943 A.R.C. Reports and Memoranda 1910 A note on the present position of

calciilations by vorte:: lattice theory. May, 1946

A.R.G. 9637 '(unpublished)

Tables of Multhopp functions for use in vortex lattice theory. Dec. 1946 A.R.C. 10,220 (xinpublished)

Calculated aerodynaiuic char.acteristics of two infinite v/ings vdth constant cliordc May, 1947.

A.R.C. 10,628 (unpublished)

The solution oi lifting uxcune problems by vortex lattice theory, Sept. 194-7-A.R.C. 10,895 (unpublished)

The soluticm of lifting line theory of problems involving discontinuitiest Oct. 1947

A.R.C. 10, 922 (unpublished) Notes on the stalling of swept-back

wings. Oct, 1947»

A.R.G. 11^009 (unpublished)

Tables of Multhopp and other fiinctions for use in lifting line and lia. bing plane theoryc Feb. 1948

A.R.C. 11,234 (unpublished)

-20-No, 37 38 39 40 41 42 43 Author V,M. Falkner t I « I W. Eisenmann P. Jordan YJ.F, Jones H. Schlichting and M. Thomas Title, etc.

Calculated loadings due to incidence of a number of straight and swept-back vdngs, June 1948.

A.R.C. 11,542 (unpublished) R.^ H ^^~<i*

A comparison of two methods of

calculating wing loading with allownance for compressibility. Nov. 1948

A.R.C. 11,944 (unpublished)

Experiments on sv/eptback and delta v/ings in E.A.E. high speed tunnel. Feb, 1947 R.A.E. Tech. Memo Aero 47

(VII International Congress of Applied Mechanics)

Calculation of spanvdse lift distribution on vdng. Feb. 1947

Volkenrohde R. and T. 239

High speed swept-back vdngs. April 1948 Volkenrohde R. and T. 1014

Lifting plane theory vdth special reference to Falkner's approximate method and a proposed electrical

device for measuring dov/nwash distributions. May, 1946

A.R.C. Reports and Memoranda 2225 Calculation of lift distribution of

swept v/ings. Dec, 1947 R.A.E, Report Aero 2236

Wind Tunnel Tests on Sv/ept-Back \7ings 44 . 45 46 47 48 D.H. Williams, A.H, Bell and E. Smyth

D.H. Williarüs, A.F, Brown and C.J.W. Miles M. Gdaliahu D.H. y/illiams, A. P. Brov/n and C.J.W. Miles B. Regenscheit

Tests of aerofoil N.A.C.A, 23012 in the compressed air tunnel. 1939

A.R.C. Reports and Memoranda 1898 Tests on a tapered vdng (N.A.C.A. 23O12)

wi.th and without sweepback in the compressed air tunnel. May 1945 A.R.C, Reports and Memoranda 2151 A summary of the results of some German

model tests on vdngs of small aspect ratio. Nov. 1946

R.A.E. Tech. Note Aero I767

Tests on some 'General Aircraft' wings, T/ith and vdthout sweep back in the

comiiressed air tunnel. Jan 1946 A.R.C. 9321 (unpublished)

Tests made on rectangular vdng with vdng tip aileron, Sept. 1946

Volkenrohde R. and Ï. 223

2 1

-No.

49

50

51

52

53

54

55

56

Author

W, Krlïger

G, Thiel and J, Weissinger W, Jacobs Th, Schwenk H,J. Luckert Puffert and Bolkow A.W. Quick A. Busemann Title, etc,

Six component measurements on a cranked swept-back vdng.

Volkenrohde R. and T. Jan.15th, 1946. Six component measurements on straight

and 35° swept back trapezoidal

v/ings vdth and vdthout split flaps» July, 1946.

Volkenrohde R. and T. 523

(A.R,C, 11,741)

Pressvire distribution measurements on a yawed sv/ept back wing of constant chord, July, 1946,

Volkenrohde R, and T, 431

Measurements on trapezoidal swept- back vdngs, July, 1946

Volkenrohde R, and T. 525 Lift distribution on yawed vdngs,

Volkenrohde R, and T, 270

Three component vdnd tunnel tests on swept back v/ings and a complete model, July, 1942

GDC 15/98.T.

Plight mechanical properties of sv/ept back vdngs at normal speeds, Apr, 1943 Volkenrohde R, and T, 473

Swept back wings at high speeds, Apr,1943 Volkenrohde R, and T. 342

Stability and Control 57

58

59

60

V.M. Falkner and H.L. Nixon H.A. Soule R.S. Swanson and S.M. Crandall J.G. Lowry L.E. Schneiter

Wind tunnel tests on the stability of tv/o swept-back wings. Oct. 1948 A.R.C. 11,854 (unpublished)

Influence of large amounts of wdng sweep on stability and control problems of aircraft, June, 1946

N,A,C.A, Tech, Note 1088

Ijifting-s\jrface-theory aspect-ratio corrections to the lift and hinge-moment pararaeters for full-span elevators on horizontal tail surfaces. Feb. 1947,

• N.A.C.A. Tech. Note 1175 Investigation at lov/ speed of the

longitudinal stability characteristics of a 60 swept back tapered low-drag vdng. Aug. 1946.

N.A.C.A. Tech. Note 1284

-22-No.

61

63

64

Author

B. Maggin

C V . Bennett

T.A. Toll and

M.J. Queijo

W. Letko and

J.W. Cowan

Title, etc.

Low-speed stability and damping-in-roll

characteristics of some highly sv/ept

wings. Nov. 1946

N.A,C,A. Tech. Note 1286

Plight tests of an airplane model v/ith a

42 swept-back vdng in the Langley

free-flight tunnel. Oct. 1946

N.A.C.A. Tech. Note 1287

Approximate relations and charts

for low-speed stability derivatives

of swept vdngs. May 1948

•N.A.C.A. Tech. Note I58I

Effect of taper ratio on low-speed

static and yavdng stability

derivatives of 45 sv/eptback wings

vdth aspect ratio of 2.6l. July, 1948

N.A.C.A. Tech. Note I67I

Sv/ept-Back ITings with Flaps

65

G. Brennecke

66

61

68

69

70

71

W. Kr^'ger

G. Brennecke

M. Petkin and

B, Maggin

77, Letko and

D, Peigenbaum

S, Fischel and

M,P. Ivey

G. Lovrry and

L. Schneiter

Sv/ept-back vdngs vdth counter split

flaps. Majr, 1946.

Volkenrohde R. and T.

66

'Wind tunnel investigations on 35 swept

back vdngs v/ith different high lift

devices. Oct. 1946

Volkenrohde R. and T. 311

Investigation of sweepback vdth different

high lift devices.

Volkenrohde R. and T. 122

Analysis of factors affecting net lift

increment attainable with trailing

edge split flaps on tailless airplanes.

Sept. 1944

N.A.C.A. A.R.R. L4II8 (A.R.C. 9444)

Wind tunnel investigation of split

trailing edge lift and trim flaps on a

tapered vdng vdth 23 sweepback,

July 1947

N.A.C.A. Tech. Note 1352

Collection of data for lateral control

vdth f-all span flaps. Apr, 1948

N.A,C,A, Tech. Note I4O4

Estimation of effectiveness of flap

type controls on sv/ept back wings.

Aug. 1948.

N.A.C.A. Tech. Note 1674

-23--No.

Author Title, etc.

Induced Drag and Dpy.Ti-.Yash of Sv/ept-Back V/ing^

72

V.M. Falkner

73

74

75

76

P.E. Piorser, M.L. Spearman and V7.R. Bates S. Katzoff and M.E. Hannah

The effect of vdng tv/ist on the induced drag of sv/eptback vdngs,

Sept.

^^kk

A,R.C. 8012 (unpublished)

Addendum to 'The effect of v/ing tvdst on the induced drag of sv/eptback vdngs'. A,R.C. 8012. Oct. 1944

A,R,C. 8137 (unpublished)

The calculation by lifting plane theory of the downwash behind a wing,

Sept, 1948

A,R,C, 11,778 (unpublished)

Preliminary investigation at low speed of dov/nwash characteristics of small-scale sv/eptback vdngs, July 1947 N.A.C.A. Tech. Note 1378

Calculation of tijnnel-induced upwash velocities for swept and yawed vdngs. Nov. 1948

N.A.C.A. Tech. Note 1748

Miscellaneous .

77 R,T, Jones Effects of sweepback on boundary layer

and separation, July 1947. N,A.G,A. Tech. Note 1402

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