A Note on the Estimation of Some Low Speed Characteristics of Delta Wings

TECHNISCHE HOGESCHOOL

VUEGTUIGBÓUWKUNDE

2 8 APR. 1953

REPORT No. 68

THE COLLEGE OF AERONAUTICS

CRANFIELD

A NOTE ON THE ESTIMATION OF SOME LOW

SPEED CHARACTERISTICS OF DELTA WINGS

by

R. G. ROSE, D.C.Ae., A.M.C.T.

This Report must not be reproduced without the permission of the Principal of the College of Aeronautics.

VUEGTUIGBOUWKUNOE

? 8

APR. 1953

Cr-REPORT NO. 68

JANUARY, 1953

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

A Note on the Estimation of Some Low Speed Characteristics of Delta ^'/ings

-by-R.G. Rose, D.C.Ae., A.M.C.T.

smaviARY

The results of the Weissinger swept lifting line theory have been compared with other methods and experimental results. On the basis of this comparison empirical corrections to the Weissinger theory results have in some cases been suggested. Charts have been prepared which enable an estimate to be made of

the lift curve slope at zero lift, the location of the aerodynamic centre, and the rolling and yaiving characteristics at low incidence. The estimation of C^ and a„ are also briefly considered.

lanax C^. "^ Lmax

Great accuracy cannot be claimed for the resulting

diagrams, but it is considered -chat they provide a. rapid method of estimating the above mentioned characteristics of delta v/ings, both cropped and uncropped, to an accuracy that is adequate in

the design stage of an aeroplane.

MEP

« This report is based on portions of a thesis submitted in Jvme 1951 in part fiilfilment of the requirements f or the Diploma award of the College. It was prepared for publication by

NOTATION

Sweepback of I/4 chord line, degrees.

Aspect ratio (= b / S ) .

Root chord, ft.

Tip chord, ft.

Gross v/ing area, sq.ft.

Taper ratio (= c V c ).

Semi-span (= b/2), ft.

Longitudinal coordinate, ft.

Spanwise coordinate, ft.

Non-dimensional spanvd.se coordinate (= y/s).

Local chord, ft,

Standard (or geometric) mean chord, ft.

Centroid-of-area chord, ft.

Total lift coefficient for whole wing.

Distance of A.C, aft of the L.E. of G.A.C., ft.

Damping in roll derivative.

Average section lift curve slope, per rad.

Yawing moment due to yawing derivative.(induced drag component).

1, Introduction

In this note the description 'delta wing' includes the cropped delta vd.ng, i.e. it is taken to apply to sweptback, straight tapered wings of triangular, or modified

triangulai-planforai with the apex forward, and with the trailing edge normal to the centre line of the wing. For delta m n g s as defined above a relationship betvreen the planform parameters (viz. sweep, taper and aspect ratio) csji be deduced as

follows.-(1)

\

This expression shows that a variation in any one of the three planform parameters has an effect on the other tvro. Thus, in assessing the characteristics of such v/ings, it is impossible to separate the effects of these paraineters.

1 2 3 4

Existing^ lifting surface methods '^'''>^' of obtaining the characteristics of delta wings are both lengthy and laborious, and, in consequence, little systematic information has been

published regarding the calculated changes in the characteristics vd.th alteration in the planform parameters. The object of this

investigation v/as to analyse the exiscing data, both theoretical and experimental, to see whether they lent themselves to the

production of data sheets of an acceptable accuracy in the design stage of an aeroplane.

The method adopted was to start with the results of the

6

"'

Weissinger swept lifting line method *'and to ccsnpare them vri.th the results of other methods and with experimental results. In some cases this comparison has indicated a plausible empirical correction to the Weissinger theory results, and this correction has accordingly been adopted. Clearly, there are limitations in the results of such an analysis but, in general, it is believed that they are of satisfactory accuracy,

The characteristics that have been considered are the lift curve slope at zero lift, the location of the aerodynamic centre, and the rolling and yawing characteristics at low incidence. The estimation of C^. and a^ has also

Lmax Oy b e e n b r i e f l y c o n s i d e r e d .

t a n A.1 _ 3(1 - N

A(1 + >

-4-2, L i f t Gvtrve Slope a t C.^ = 0

Jj

De Young has calculated the lift cvirve slope for a wide range of sweptforward and svireptback straight tapered wings vising the Weissinger method. Together vd-th the relationship given in equation 1 use has been made of these results to prepare a 'carpet' in which lift curve slope is plotted against sweep-back angle for a range of taper ajid aspect ra-tios. This carpet is shown in figure 1.

D

A comparison of various methods of estimation includes but a few figures for the delta configuration, and to substantiate to c\irves of fig, 1, values given by them have been compared vd.th as many theoretical and experimental results as -^rere available, Lift curve slopes as predicted by the Weissinger method and by lifting surface theories are tabiolated in table 1 and are plotted in figure 2 in the form of an accxiracy curve. In tlie r^ajority of cases excellent agreement is seen. ^^Then compared vd.th

experimental values (table 2 and figure 2) the ViTeissinger theory gives lift curve slopes agreeing to within 5 per cent for y/ings of aspect ratio greater than 1.5. Por si:ialler aspect ratios the disagreement is somewhat greater, out the values can still be predicted to within about 8 or 9 per cent. Some scatter among the experimental res-ults is to be expected since they were drawn from several reports describing tests made imder a variety of conditions and at Reynolds numbers varying f ran 0.3 x 10 to 4.1 X 10 . Little infonnation is available concerning scale

17 effect on the lift curve slope of delta wings, but, f ran tests on wings 10 per cent thick, it wo\jld appear that an increase of 2 to 5 per cent is to be expected over the Reynolds number range from 0.5 x 10 to 1 x 10^.

3. Characteristics at the Stall

In this section an analysis has been made of wind tunnel data to illustrate the effect of changes in planform, The data have been drawn from a variety of sources and sane

scatter of the experimental results is inevitable. Further the data were by no means comprehensive, so that the deductions can only be regarded as tentative. However, it is felt that the trends suggested are broadly speaking correct and significant.

^

3.1. Maximun Lift Coefficient and Stalling Incidence

Prom the results given in table 3 the variation of ^Lmax ^^^ "^C with sweepback for triangular wings of zero

lmax

taper are plotted in figures 3 and 4.

For wings of moderate sweepback and fairly large aspect ratio (3.6 say) it wovild appear that C, =0.9 can be expected. i But with increase in sweepback to about éO and the consequent

reduction in aspect ratio the value of C- is increased to Lmax

1.2 to 1.3. This occurs when the angle of sweep is about 60 . At very large sweepback angiles and ver^r small aspect ratios Cj is again of the order of 0.9. Similarly, the stalling incidence is a maximum when the sweepback angle is about 65 .

3.2, Lift Curve Slope up to the Stall

Some typical lift curves for delta wings are given in figure 5, The increasing non-linearity with reduction in aspect ratio is well illustrated. Comparison of the three families of curves indicate that the non-linearity of the lift curve is relatively little affected by changes of taper and sweep. A similar trend can be noticed in the resu

rectangular vd-ngs of small aspect ratio.

20 similar trend can be noticed in the resiiLts of some tests of

3. 3. Longitudinal Stability near the Stall

The results of table 3 have been plotted in figure 6 21

after the manner of Shortal and Maggin. Thus, figure 6 is a chart from which the longitudinal stability characteristics near the stall can be predicted. Owing to the limited extent of the available data concerning heavily swept delta vdngs of low aspect ratio the suggested margin between stable and unstable configiira-tions near the stall can only be regarded as tentative.

4. Location of Aerodynamic Centre

In this analysis of resvilts concerning the location of the aerodynamic centre of a delta wing, the leading edge of the centroid of area chord has been chosen as the reference point

and distances are quoted as fractions of this chord length. For '' the sake of canpleteness, definitions of this chord, the standard mean chord and relations between the two are given in an appendix to the report,

The results of reference 7 have been used to prepare the carpet shown in figure 7 fran which a first estimate of the position of the aerodynamic centre for any delta wing can be made. The values of x' _ given in figure 7 have been

-6-compared with those of as many experimental results and other

theories as possible. The comparisons are given in figure 9 and

table 4, and it is seen that in nearly every case the value of

X. „ has been \anderestimated. This is to be expected since an

assumption in the derivation of figut^e 7 is that the locus of local

aerodynamic centres follows the quarter chord line of the vd.ng,

9 19

In practice it has been shown ' ' that at the root the local

aerodynamic centre is well aft of the quarter chord point and at

the tip it is slightly forward.

An analysis of these errors led to the construction of

the set of curves given in fig\jre 8, Thus, figures 7 and 8 can

be used to estimate x, „ , An error curve for this modified

A.O.

estimate is shown in figiire 10, and it is seen that, in general,

the position of the aerodynamic centre can be predicted using

figures 7 and 8 with a probable error of the order of + .01 c,

As an exaxiiple of tlie method it is required to find the

position of the aerodynariiic centre of the following wing.

-A = 3.0 ; -A = 0,05 ; / \ = 42.2°

4

Figure 7 shows x.' _ = 0 , 3 1 8 and

figure 8 gives Z^ x! „ r ' « ^ Per cent

'•• ^^^A.C. = M ^ -5^8 = ,011

^A.C. "" 0.318 + 0,011 f: 0.33.

5« Rolling and Yawing Characteristics at low incidence

The Weissinger lifting line method has been used to

obtain the additional span loading which results from an

anti-syrametric distribution of incidence, and from such loadings the

damping derivative in roll (i? ) has been calc\iLated in references

22 and 23. From cvorvea given in these references the variation

of i m t h changes in planform parameters has been plotted in

P 'J

fifjvire 11, On comparing these values of -c with the results

of some vmpublished calculations u.sing the Falkner 21 vortex

3 point solution (see table 5 and figure 12) it is seen that

good agireement exists between the results of the two methods,

In using the resvilts of figure 11 it must be

remembered that the section lift curve slope has been taken as

22 23

-7-\

i =

_{p p^}

(i)

f i g , 11

Actvial section l i f t curve slope }

27^ „J

Comparison with experimental results (figure 12 and table 6) gives support to this suggestion which is accordingly reconmended,

Use has been made of unpublished calculations using Falkner's 21 vortex 3 point solution to predict variations with planform parameters of the damping derivative in yavr (n , due to induced drag) and of the yawing manent due to rolling (n ). These variations are shovm in figvires 13 and 14. Quantitatively, the results, -vdiich are based on linear section lift data, will

probably suffer from some inaccuracies. However, it is thouglit that the trends suggested are correct.

6. Conclvisions

In general, fair agreement has been found to exist when the results of Weissinger's swept lifting line theory have been compared with those of other theories and with a limited number of experimental results, and, where possible, this comparison has been used to derive corrections to the results of Weissinger's theoiy. It is believed tliat the resulting diagrams provide a rapid method for estimating the particular characteristics of delta wings with an accuracy considered adequate in the design stage of an aeroplane. REFERENCES No. 1. 2.

3.

4.

5.

Author V.M. Falkner R. Dickson H.C. Gamer H. Schlichting and W. Kahlert M. Gdaliahu Title, etc.

Calculations of the aerodynamic

loading of a delta wing. ' A.R.C. 9830. July 1946.

Comparison of two methods of cal-culating aerodynamic loading on an aerofoil with large svreepback and small aspect ratio.

A.R.C. R. and M. 2353.June 1946. Theoretical calculations of the distribution of aerodynamic loading on a delta Td.ng,

A.R.C. 12,222. March 1949. Calculation of lift distribution of sweptback wings.

R.A.E. Report No. Aero. 2297 Oct. 1948, The three-component characteristics

of delta wings.

8

-References. continued

No.

6.

7.

^

8.

9.

10.

11.

12.

13.

14.

15»

16.

17.

18.

19.

Author

J. Yfeissinger

J. De Young

N.H, Van Dom

and

J, De Young

S,B. Bemdt

and

K. Orlick Ri5ckemann

V.M. Falkner

S.B. Bemdt

Lange/'facke

Lange/iVacte

H. Voepel

R.R. Hills, R.C. Lock, and J. G. Ross. R.C. Lock, J.G. Ross, and P. Meiklem, Jones, Miles, and Pusoy. L.P. Tosti B.H, Wick Title, etc,

The lift distribution of sweptback wings.

N,A.C,A. Tech. Memo 1120. March 1947. Theoretical additional span loading characteristics of wings vd.th

arbitrary sweep, aspect ratio and taper ratio.

N.A.C.xl. T.N, 1491, Dec. 1947. A comparison of three theoretical methods of calculating span load distributions on swept wings.

N.A.C.A. T.N. 1476, Nov. 1947. Comparison between theoretical and experimental lift distributions of plane delta wings at low speeds and zero yaw.

K.T.H. Aero. Tech. Note 10. Dec. 1948. Calculated loadings due to incidence

of a number of straight and swept-back wings.

A.R.C. R. and M. 2596. June 1948. Three component measurement and flow investigation of plane delta wings at low speeds and zero yaw.

K.T.H. Aero. Tech. Note 4. June 1948. Test report on three and six component measurements on a series of tapered wings of small aspect ratio.

(Partial Reporti Triangular Wing). N,A.C.A. Tech. Memo 1176. May 1948. Test report on three and six component measurements on a series of trapezoidal wings of small aspect ratio.

(Partial Report: Trapezoidal Y'ing). N.A.C.A. Tech. Memo 1225, May 1949. Tests on wings of small aspect ratio, R,A.E, Library Translation No. 276.

Oct. 1948. Interim note on wind tunnel tests of a model delta wing.

R.A.E, Tech, Note No, Aero, 1869 Feb. 1947, Vfind Tunnel tests on a 90 apex

delta wing of variable aspect ratio. (Sweepback 36 ). Part II. Measure-ments of down\'ra.sh and effect of high lift devices.

R.A.E. Report No, Aero. 2284. Aug, 1948, A,R,C, Current Paper 83.

Experiments in the C.A.T. on svrept-back wings including two delta wings. A.R.C. 11,354. March 1948. Lov/ speed static stability and

damping-in-roll characteristics of some swept and unswept low aspect ratio wings. N.A.C.A. T.N. 1468. July 1947. Chord-vise and span:.ïise loadings measured

at low speed on a triangular wing having an aspect ratio 2 and N.A.C.A. 0012 aerofoil section.

References, continued 20. W.S.D. Marshall 21. Shortal and Maggin 22. J.D. Bird 23. J. De Young 24, E.C. Polhamus

The distribution of pressure over the surface of wings of small aspect ratio. College of Aeronautics Report No. 52.

Feb. 1952. Effect of sweepback and aspect ratio on

longitudinal stability characteristics of wings at low speeds.

N.A.C.A. r.N. 1093, May 1946. Sane theoretióal low speed span loading characteristics of swept wings in roll and sideslip.

N.A.C.A. T.N. 1839. Dec. 1948. Theoretical antisymmetric span loading for wings of arbitrary planform at subsonic speeds.

N.A.C.A. T.N. 2140. July 1950. A simple method of estimating the

subsonic lift and damping in roll of SY/eptback wings.

N.A.C.A. T.N. 1862. April 1949, >

-10-APPECTDC

Mean chord definitions and relationships

The following formulae apply to any straight tapered Bweptback wing which is uncambered and untwisted, and has zero dihedral.

4

1 i

4

\

s.

Xc.

8i^C^-

A:

( c * ^ -Tl S. b/2 = s PLANFORM PARAEvJETERS

Let A = t a p e r r a t i o = c V c = 1 - T , s a y . The s t a n d a r d mean chord (S.M.C.) for g e o m e t r i c mean c h o r d (G.M.C.Jis d e f i n e d a s ,

- _ g r o s s wing a r e a span (1A) s b = 0 ^ 2 (l +?\) (2A) The c e n t r o i d - o f - a r e a chord ( c . A . C . ) i s d e f i n e d a s , ? 1 c dn o = ' - 1 . _•^1 (3A) o dn

J_1

/s

m e e

Since the wing is assumed sjrametrical in planform about the centre line,

Hence c = > 1

J o

ni

c dri (l-TiT)2dn

ni

= o o [11 c = c c dri

(1-x ^ f )

(1 - f )

(1-TIT) dn V o 3 ^0 (1 + A + ;^^) (1 -f-^^) (4A) A r e l a t i o n s h i p between t h e l e n g t h s of t h e s t a n d a r d mean c h o r d and c e n t r o i d of a r e a c h o r d can e a s i l y be o b t a i n e d f r o n e q u a t i o n s (2A) and (4A).

Thus,

i _ = 3(1 +>>f

c 4(1+ '^ + A^)

(5A)

These two reference chords are located at the same span-wise station, namely at the centroid of area of the half wing, Their spanwise position is therefore given by

f.s 1^1 ens dy

•^cA.

• ^ = ;;r

3 s -n(l-rix)dn (J o c dy ÜO

n

'C.A, ^1 (1 •Jo

s(i-f)

1 + 2A - TiT;)dri 3(1 +h) (6A) /TABLE 1 , , .

1 2 -T^3I£ 1

LIFT CURVE SLOPE VALUES GIVEN BY LIFTING SURFACE TKEORY

REF, NO.

1 ^

1 9

1

^^

1 .[ REPORT NO. AUTHOR,ETC. : UNPUBLISHED 1 GALCULATIOB ARC 12,222 H.C. G a m e r ARC 9,830 V.M. F a l k n e r 1 KTH-Aero T.N. 10 1 1 1 r ARC 11,542 V.M. F a l k n e r » WING NO. AO BO CO A2 B2 C2 j ASPECT RATIO A 1 . 0 1 2 . 0

1 3.0

4 . 0 1.0 2 . 0 3.0 4 . 0 1.0 2 . 0 3.0 4 . 0 1.0 2 . 0 3 , 0 4 . 0 3 . 0 2.31 2.31 2.50 1.67 1.00 1.34 • 0.89

0.54 I

4 . 0 3 . 0 2.31 ! : TAPER 1 RATIO

i A

' 0 0 0 0 0.10 0.10 0 . 1 0 0 . 1 0 0 . 2 0 0 . 2 0 0.20 0 . 2 0 0 . 3 0 0.30 0 . 3 0 0.30 0.143 0 0 0 0 0 0 . 3 0 3 0 . 3 0 3 0.303 0 0 . 1 3 8 0 . 2 6 8 S'.TEEP-BACK

^4

71.57 56.32 4 5 . 0 0 36.87 67.83 50.87 39.30 3 1 . 5 3 63.43 4 5 . 0 0 33.70 26.57 58.23 38.92 28.33 22.00 37.0 5 2 , 4 5 2 . 4 5 0 . 2 61.0 7 1 . 6 5 0 . 2 61,02 7 1 . 4 0 36.87 3 6 , 8 7 36,87 THEORY DIE TO,-F a l k n e r 21 V o r t e x , 3 P o i n t S o l u t i o n . Garner Method (c) F a l k n e r , 21 V, 3 P n t . . S o l n . F a l k n e r 126V 6 P n t . Soln. (Uncoi'r,) F a l k n e r 126 Vortex 6 P n t . S o l n . ( c o r r e c t e d f o r d i s c o n t y . a t c e n t r e l i n e ) F a l k n e r 126 Vortex, 6 P o i n t . ( u n c o r r . ) JJ 1 THEORY 1 . 2 5 3 2.180 2.865 3.360 1.336 2.320 2.990 3.515 1.386 2.370 3.100 3.660 1.400 2.420 3.130 3.630 3.038 2.402 2.518 2.62 1.97 1.31 1.84 1.32 0 . 8 4 3.47 3 . 1 4 2.76 mm PIG, 1 1

1.191 1

. 2.182 1

2.830 1

3.339

1.350 2.286 2.935 3.500 1.425 2.360 3.085

3.595 1

1.4i)-5 2.402 3.130 ^•

3.640 1

3.030 1

2.410 2.410 2 . 5 4

1.90 1

1.191 1,825

1.305 1

0 , 8 2 5 3.339 j 3.025 2.65 j

TABLB 2

EXPERBIENTAL Vz^JiUBS OF LIPT CURVE SLOPE - WBfG .\LOHE

REF. NO. 9 and 11 12 13 14 15 and 16 17 , 18 ] REPORT NO, AUTHOR, ETC. KTH - Aero T.N.4 and 10 S . B . B e m d t and K.Orlik-RUcke-mann 1 NACA TM 1176 Tiange/Wacke t t NilGA Wi 1225 Lange/Zfacke 1 RAS T r a n s l a t i o n 276.H.Voepel RAE TIM y\ERO.l869 and Rep. 2284 Lcxik & o t h e r s i^RC 11,354 J o n e s , M i l e s , and Pusey 1 NACA. TIM. 1468 L . P . T o s t i WING NO. AO BO CO A2 B2 C2 FD ED DD CD DT-f DT^ DTi ET 1 2 3 ^ 1 A l A1 Zi2 11 12 13 14 15 16 ' ASPECT RATIO A 2.50 1.67 1.00 1.34 0.89 0 . 5 4 3 . 0 2.0 1.33 1,0 1.33 1.33 1.33 2 , 0 4 . 0 3.0 2.31 3.87 3 . 0 4 2,38 2.31 3 . 0 2 . 0 1,0 0 . 5 2 , 0 1,0 TAPER R'vTIO A 0 0 0 0 . 3 0 3 0 . 3 0 3 0 . 3 0 3 0 0 0 0 0 . 5 0 0 , 2 5 0.125 0 . 3 3 0 0 . 1 3 8 0 . 2 6 8 0 0.143 0 . 2 6 8 0 0 0 0 0 0.20 0 . 5 0 a'ffiEP-B.\CK

4

5 0 . 2 6 1 . 0 7 1 . 6 5 0 . 2 61.02 7 1 . 4 4 5 . 0 0 56.32 66.03 7 1 . 5 7 3 6 . 8 7 5 3 , 4 7 60.25 3 7 . 2 3 6 , 8 7 3 6 . 8 7 36,87 3 7 . 8 3 6 . 0 5 2 . 4 4 5 . 0 5 6 . 3 7 1 . 6 8 0 . 4 4 5 . 0 4 5 . 0 R.N.xlO fBased AEROFOIL on SfIC) SECTION 1,02 1.02 1.02 1.33 1.33 1.33 1.35 1.66 2,03 2.35 2.03 2.03 2 . 0 3 ' v i . 7 2.1 2 . 4 2 . 7 3.è 4 . 0 2 . 4 4 . 1 0 . 3 0 . 4 4 0.62 0.87 0.41 • 0 . 5 3 FPA 104-5106 (syiimietrical^ a t 4 0 / 0 c ' NAa\ 0012 1 1 1 NACA 0012 r 1 N.iCA 0012 S q u i r e ' C ' ( t / c j ^ = 1 0 % a t 357o 0. Symmetrical ( t / c ) = 1 0 % a t 3 5 % -1 ——— . .h ., , EXPT, mom FIG. 1 i 1 2,59 1.92 1.31 1.85 1,31 0,80 2.807 2,140 1.525 1. 202 1.86 1.83 1.69 2.50 3.A4 3.15 2,69 3 . 2 3 3 . 0 3 2.65 2.39 x-lat ^ l a t e 2.863 3 " T h i c k n e s s ="4 2.19 Nose Rad. =8 ( 1 . 3 6 T . E . T h i c k . = / ' i | 0.62 T.E,Angle I 2 . 6 7 = 9 . 8 ° 1 i l . 5 5 i 1 — ~ 2 . 5 4 1.90 1.191 1.825 1.305 0 . 8 2 5 ' 2.830 2.182 1.560 1.191 1.80 1.825 1.75 1 2,407 3.339 3.025 2.65 3.27 3.06 2.698 2.41 2 . 8 4 2.18 1.191 0.61 2.36 1.43

1 4 -TABLE 3

EXPERBIENTAL RESULTS FOR STALLING i\ND

PITCHIIMG-MOMSIMT CURVES AT THE STALL (WING ALONE)

REP. [NO. 12 13

h4

1 15 18 ;17 19 i REPORT NO. 1 AUTHOR,ETC. NAGA TM. 1176 [ Lange / ' Wacke N.\CA TM. 1225 Lange 'Wacke RAS L i b r a r y T r a n s l a t i o n 276 RAS. T.N. Aero. 1869 Lock & Others NACil TN, 1468 L . P . T o s t l ARC 11354 J o n e s , M i l e s , and Pvusey | IwiNC NO. FD ED DD CD D T i DT-4: DT| ET 1 2

3

11 12 13 14 15 16 17 ! 18 .61 j

^.1 1

A1 1

A2 NACA TN.I650I B.H. ?/ick j 3 ASPECT j R/VTIO A 3 . 0 2.0 1.33 1.0 1.33

I1.33

1.33 2.0 4 , 0 3.0 2.31 3 . 0 2.0 1.0 0 , 5 2.0 1.0 0 . 3 3 j 0,176 i 3.87 3 , 0 4 2 . 3 8

2.31 1

2.0 1

j TAPER R/ITIO ! 0 0 ! 0 0 . 1 2 5 . 2 5 0

1.500

.33

0 .138 .268 0 0 0 0 . 2 0 . 5 0 . 5 0 . 5 0 0 . 1 4 3 1 .268 0

0 1

SfffiEP-iBAGK

1 y\°±

4 5 . 0 56.32 t 66.03 7 1 . 5 7 60.25 5 3 . 4 7 36.87 3 7 . 2 3 6 . 8 7 36.87 36.87 4 5 . 0 5 6 . 3 7 1 . 6 6 0 . 4 4 5 . 0 4 5 . 0 7 1 . 6 8 0 . 4 3 7 . 8 ; 3 6 . 6 3 6 . 0 5 2 . 4 5 6 . 3 IRNXIÖ (Based on SHC) 1.35 1.66 2 . 0 3 2.35 2 . 0 3 2.03 : 2 . 0 3 1.7 2.1 2 . 4 2 . 7 0 . 3 0 . 4 4 0 . 6 2 0 . 8 7 0.41 0 . 5 3 0 . 9 3 1.31 3 . 6 4 . 0 2 . 4 4 . 1 1.8 i^EROFOIL SECTION NACA 0012 1 1 1 NACA 0012 r • NACa\ 0012 S q u i r e ' c ' (t/c)=1C|S max , a t 35% c P l a t a a t e Thickness = 3 / U n . Nose R-^d-i u s = I " TE.Thick-n e s s = l / l 6 * TE. Angle = 9 . 8 ° Symnetri-c a l ( t / c ) =10% ma?: a t 3^oC Ii\CA 0012 i , i l ^ L 1 °C |G^jR^T3NG| SYI.I-""^•^l ^max AT STALLj BOL j

_ __ i J ^A 1

_{1 • • !••• ' ••" • !} 0 . 9 1 3 1 2 5 . 6 1 S t a b l e 1.165138.0 ! M a r g i n a l

1.030|39.0 1 '

0.94o|38.0"^ j

1.20 1.21 1.15 0 . 8 6 2 0 . 8 6 8 0 . 8 8 0 . 8 6 0 . 9 3 1.267 1.04 0 . 8 7 1.00 1.35 0.99 0 . 7 8 0 , 8 9 5 0 , 8 8 0.925 1.13 1.24 s 3 0 . 8 j M a r g i n a l 3 6 . 0 [ s t a b l e 3 8 . 0 1 S t a b l e

i

21.0 j S t a b l e 2 0 . 6 2 0 . 6 20.6 s t a b l e r t i 2 5 . 5 i S t a b l e 34. 5 1 S t a b l e 3 4 . 0 U n s t a b l e 34.0 U n s t a b l e 24.0 ' s t a b l e i 32.0 1 S t a b l e 36.0 1 M a r g i n a l 3 8 . 0 i U n s t a b l e

Q \

0 j

1

Cv 1

0 !

?' I

\7 1

<7 i

V' 1

A i

0 1

i , ! i

20.0 ! stable i v |

i i —

20.4 j • \^^ '

19.8 ' h ? 1

32.4 1 • 1 ^" i

39.0 j *

« R e s u l t from p r e s s u r e d i s t r i b u t i o n - e x p e r i m e n t a l + C.p c o n s t a n t w i t h a a f t e r t h i s v a l u e . majc

TABLE 4

EXPERJJSHTiMi Vi\LUES OF THE AERODYNAI/IIC CENTRE LOCATION ( x ^ Q ) , /iND DATA POR ERROR CURVES

REF. NO, 1 5 a n d 16 12 11 a n d 9 18 19 17

i

!REPORT NO. AUTHOR, ETC. RAS. TM. A e r o 1869 c& R e p . 2 2 8 4

Lock & Others NACA m. 1176 Lange / Wacke KTH-Aero TN.4(&10 NACft. TN. 1468 L . P . T o s t i NACA TN. 1650 B.H. Wiek ARC.11,354 J o n e s , Müe^ (i Pusey

wmG

NO. 1 2 3 FD ED DD CD AO BO 11 12 1 5 Al -(il Ü1 ASPECT RATIO A 4 . 0 3.0 2.31 3 . 0 2 . 0 1.33 1.0 2 . 5 0 1.67 3.0 2 . 0 2 . 0 2.0 3.87 3 . 0 4 2 . 3 8 2.31 TAPER RATIO

A

0 . 1 3 8 . 2 6 8 0 0 0 0 0 0 0 0 . 2 0 0 0 . 1 4 3 .268 0

S^JfflEP-B.igc

36.87 36.87 36.87 4 5 . 0 0 56.32 66.03 7 1 . 5 7 5 0 . 2 6 1 . 0 4 5 . 0 5 6 . 3 4 5 . 0 5 6 . 3 3 7 . 8 3 6 . 6 3 6 . 0 5 2 . 4 RNxlÖ (based on Ff.T!) 2.1 2 . 4 2.7 1.35 1.66 2 . 0 3 2.35 1.02 1.02 0 . 3 0 . 4 4 0.41 1.8 3.6 4 . 0 2 . 4 4 . 1 /iEROFOIL SECTION S q u i r e ' C ' ( t / c ) = ^0^ max a t 35/0 c. NACA 0012 1 FPA 104-5106 ( t / c ) = icTo max a t 40% c F l a t P l a t e 1 NACA 0012 Symmetrical ( t / c ) = lOfo max a t 35% c ^A.C. ( E x p l . ) . 3 ¥ ) . 3 0 5 .286 . 3 4 4 . 3 7 4 .393 .411 .363 . 3 8 3 .340 .350 .306 .390 .331 . 3 0 3 .271 .367 (Pran P i g . 7) .329 .292 .267 .336 . 3 4 2 .339 .343 .336 .342 .281 .342 .331 .292 .267 .341 ^'^A.C. a s %of <5 3 . 3 4 4 . 4 5 7 . 1 2 2 . 3 8 9 . 3 5 4 . 7 2 11.65 1.2 2 . 3 8.9 1 4 . 6 0 3 . 8 1.5 7 . 6

1 6 -TilBLS 5

DiUflPING-m-ROLL DERIVATIVE ( - £ ) . VALUES GIVEN BÏ LIFTING SURFACE THEORY

REPORT NO. AUTHOR,ETC. Unpublished C a l c u l a t i o n s ASPECT RATIO A 1.0 2 . 0 3 . 0 4 . 0 1.0 2 . 0 3 . 0 4 . 0 1.0 2 . 0 3.0 4 . 0 1.0 2 . 0 3.0 4 . 0 TAPER RATIO >> 0 0 0 0 0.1 0.1 0.1 0.1 0 . 2 0 . 2 0 . 2 0 . 2 0 . 3 0 . 3 0 . 3 0 . 3 S?ffiEPBACK , 0

H

7 1 . 5 7 56.32 4 5 . 0 0 36.87 6 7 . 8 3 50.87 39.30 3 1 . 5 3 6 3 . 4 3 45.00 33.70 26.57 5 8 . 2 3 38.92 28.33 22.00 THEORY DUE TO P a l k n e r j 21 V o r t e x , 3 P o i n t S o l u t i o n . ( - ^ P THEORY .0842 .1505 .2037 .2468 .0904 .1690 .2287 .2758 .0944 ,1761 .2422 .2943 .0955 .1799 .2499 ,3065

) 1

FROM FIG. 11 I •'"• "•'"•••1 i .080 .150 .205 1 .247 i .090 .165 .228 1 ,280 .095

.175

1

.241 ! .296 .097 . 1 8 0 .250 .308 1 TABIE 6

EXRERE,EJINTAL V.ILUSS OF DAIgHMG-IN-ROLL DERIVATIVE {-i ) (WING ALONE)

REP. NO. 18 24 REPORT NO. AUTHOR,ETC. NACA TN.I468 L . P . T o s t i NACA TN.I862 E.C.Polhamus WING NO. 11 12 13 14 15 16 5 6 15 19 ASPECT R/iTIO A 3 . 0 2 . 0 1.0 0 . 5 2 . 0 1.0 4 . 0 3.0 2.31 1.07 Ti.\PER RATIO 7v 0 0 0 0 0.20 0 . 5 0 0 0 . 1 5 0

°

SV.EEP-EACK 4 5 . 0 5 6 . 3 7 1 . 6 8 0 , 4 4 5 . 0 4 5 . 0 3 6 . 8 7 3 6 . 5 5 2 . 4 7 0 . 0 RNxlÖ^ (Based on SMC) 0 . 3 0.A4 0 . 6 2 0 . 8 7 0.41 0 . 5 3 AEROFOIL SECTION F l a t P l a t e 3 ' ' T h i c k n e s s = 5 3 " Nose Rad. =6 TE. Thick. ='''(• TE.Angle=9.8° NACA 0012 (.approx.; NACA 0012

. H' __

EXP. .166 .112 .036 .012 .140 .105 . 2 2 8 .229 .150 . 0 7 5 FROM PIG. 8 .205 .150 .080 .040 . 1 7 4 .100 .247 . 2 3 5 .170 . 0 8 8

COLLEGE OF AERONAUTICS

REPORT No. 68.

FIG.

O X cr O

z

Jn 10 Ü3

g

O •¥• EXPERIMENTAL RESULTS THEORETICAL RESULTS ( S E E TABLES 1 & 2 TABULATED VALUES) FOR

'2

4 0 3 0 2 0

0

It J

0

o

l-O ^ t )

LINE OF PERFECT CORRELATION,

5 % ERROR LINES

2'O 3-0 4-0

(è~iA GIVEN BY OTHER THEORIES &

EXPERIMENTAL RESULTS.

CORRELATfON OF LIFT CURVE SLOPE BY WEISSINGER

THEORY WITH VALUES GIVEN BY OTHER THEORIES &

COLLEGE OF AERONAUTICS REPORT No. 68.

•MAX

VIi£GIUiGBOüWK.L"Ni)£

FIG. 3.

VARIATION OF MAX. LIFT COEFFICIENT WITH SWEEPBACK (AND

ASPECT RATIO) FOR DELTA WINGS OF TAPER RATIO = O

(TRIANGULAR WINGS.)

4 0

FiG. 4.

35 -MAX 3 0 25 2 0 IS / . - I P Ö - — 1 „ /

r

/ ƒ

P

lA

( O ! / \ \ . , - — !EYNOL[ " ^ ^ " " ^ N » . >S No. • ^ \ O " RANG

V

\ E FO( FESTS 0 - 3 TO 5 - 0 X l O ^ (BASED ON S.M.C.) O ; 30 4 0 5 0 6 0 TO

SO A*!/

VARIATON OF INCIDENCE AT MAX. LIFT COEFF WITH SWEEPBACK

(AND ASPECT RATIO) FOR DELTA WINGS OF TAPER

1-25

Cu

l O O 0 7 5 0-50 0'25 0 - 5 0 O-25 _{T l}

P

70 m T3 O O O rf -m O n f O •n O' co > n O z > c

o

REPORT No. 68.

FIG. 6.

RATING SYMBOL STABLE V MARGINAL O UNSTABLE A SEE TABLE 3 FOR

ORIGIN OF RESULTS.

EFFECT OF PLANFORM PARAMETERS ON LONGITUDINAL

STABILITY CHARACTERISTICS OF DELTA WINGS NEAR THE

STALL

TYPE OF C J ^ ~ C L

COLLEGE O F AERONAUTICS REPORT No. 68.

36 ' M r 1 ' 1 M I M 1

RESULTS OBTAINED F R O M R E F E R E N C E 7 1 *» A \ . , 1 . 1 i_i i _ L • L^ L 3 4 1"' 1 1 1 1 1 1 |~[~pp]—1 1 1 1 1 p —1"[ 1 -l—h"] 1 • 1 ' > '^P / '^ * ' ^ /

/ ' ^r

'A r / - i l l f'^/ \

M \ \ < \\

i '7^ / '

3-^ / !\ L

32 1 1 M M 1 1 1 M M M M L L L L / L L/L 1 1 iLiNES O F C O N S T A N T ! 1 I / I 1 , M / U f ( l ^ \ \l \ \ l\ A^^A \ ASPECT RATIO ( A ) i ,[ . / T 1 / I I . , 1 1 V 1 ; \rr 1 '

P ^ P P P ^sP 'l-t lU'

P ^ P P P S 1 t-t z p

t t • t t t C ^ * t t L '^ t ^

PP • P P P P * PK 1 ut

30 - -pp - p p ^ t 4 Vf^^t'-t!^

'"TE"—FT—T^r"^

t--i-: EF t--i-: Ft--i-:Et--i-: t--i-: _pL/ ^r f

___rr_,—r_r_!—_j i x^

PP P P -^—^ T-

-tvt-PP P P -T 1- -tV- 2 -tvt-PP

" PP LLP P 1 t 1 t f \T1

28 t t - U t - t - -'^ ^ -^ LL2 L ^ t ^B ^^ -h^ k - H - 7 ^ Z^I-±Z^—E2

tP P P P =7--2-r7 7 h r

_{t ^ ^ Z .^'^ LJ -'^L 2} / ^ \^ \ ^ j \

' ' ' F -J- / b ï ^ / F^^ —^

F-^z=7T^^fA-P—^

p z b'^pbqiL7T: \r t t .

P ^ ^ . ^ y u ±. a- \X—X-'

26 " Ljri 7-^ ^tzzrzt nil .

^° t z a r t z h ? ' — r z ^ t r r t z '

-__±—^^^^±^_t-,p_==^±PPP^5b

/' / K i X / P' '

\\7X\y y '—b'^^~-r T 4-P

FIG. 7

I I I 1 I I I t 1 1 J 1 l>S

I I 1 I I 1 1 1 1 r

PPPP ptpp;itptp2'gir t r t

_ ± ± EH: ?;!h±b/±_ ±b_/.-.^= 5

-»:EEÏTESHS: EEr : t :

• T / L4-r TT / E M

_{' ^ P L r ^ t r r r r r ü i n t t t r t o t t t \t\}

i Urn 1 J U ' /

/ r i T J / f r

m' N / E

_{ƒ L / / . kk}

y / LE T f rn

E' E \W[I\\ y M

_{J L -h'r lE ' W /}

M/.LW-iT L

-rf rr / / /

_{/ M \ \ 1 \ \ \\\}

/ / i / r t t t r t t i T

/ EM / XiA-M

t t \i ' t 4P=--- z p t

' P t L.-- u. P P

_{-, —-"^ "^f / /} «-i-^'^^ZC t LLZt £1Z t t

— 7 - F — F 7 - F ^ — i F — F

:--'^-E-y :E-i--t-^ii-E—=-=P

L_i • - i ^ " L ^ t

?r 7 \ i \ E '

^ ^P J P P '^PP ^

'P^P Z^ P P 2—2D

t p j ^ -ry L

T'^r7 "E'^~ EE ~r

P P'^ P Z LP r

P 2 L -v'^ t t

ty ^ ; _ 2 - ' 3 5 L ^ P E

t /—E-t~E E-—

- t — ^ : z ^ x i ^ uT F i r F

-t - ^ZJJ^E _p^u P X

p Z t t ^ t t t

L T

t t

=: ^/..i \ F E+ TF

--/Ï-Ï-V-F—F=f

FF-- j ^ l t j i p i L t LINES OF CONSTANTtt —I^M—^ V- -\— TADC-O 13ATI/-\ ^ V \ L h

.. k E ^ P L -=t^-=>,é:

LEJE>^^ETEEEI^EEE M I

i ii i i 11 EEE

24 1 YT —\—r^*"'••^>€>-tr

_{Tl 1—trrrtr trrrrtttttttttt}

ID 2 0 3 0 4 0 SO 6 0 f\x. 7 0

FIRST APPROXIMATION OF POSITION OF AERODYNAMIC CENTRE.

4 - 5 .*•—V r 2\ 3.5 L ^""^

Q

_y~ <

h-a 2-5

a ^ i . e

.L"

,'' ^^ ' ::= * = = = , ^ ^ .^ .^ „ = ^ / X ;^ ^ --/ ^ ^ -" - * • .« = = ^ / • ^ ^ —' -— ^ / (^ ^ »* . - • = \ J L L '' ,' ^ ' '' = - • = = ^ J

•v,

L" /

J.^

,-' ,-' 3 < ' -^ — — ^ .^ z ' ^ ^ ^ — = ^ / /* ,«-^ / ^ ^

I

,/ /• ^ ^ •*' = ~ ^ / /" X i H -A -2 «' / ^ t ' < z' -— ^ '/o / t^ t ' _ ^ ^ ^ -3

-J'

J\

^ • •

EE

y

, ^ 3 % / •" *^ ^ •* = • " H -• ^ ' E E ^ ^ ^ • • /> ^ -' = = E ^

FIG. 8.

ADC'

A.C. 4 % 5 % 6 %

Ax'

L-:y- IS GIVEN AS A PERCENTAGE OF DC^^, A.C. A.C. A.C. O l _{0 - 2} 0-3

7 7o

'Jo

1 1 % 1 2 % TAPER RATIO ( X ) SEE EXAMPLE IN PARAGRAPH 4.

<

2

Z

i o

o

Ö:

a.

a < 2

g

ÜL REPORT No, 58.

FIG. 9.

EXPERIMENTAL & THEORETICAL VALUES OF X , _AC.

COMPARISON OF ESTIMATED X ^ c . VALUES FROM FIRST

APPROXIMATION ( F I G . l) WITH VALUES GIVEN BY FALKNER

THEORY & EXPERIMENTAL RESULTS,

FIG. lO.

<

X

u. O

I*"

•24 .28 -32 '36 - 4 0 EXPERIMENTAL & THEORETICAL VALUES OF X A . C .

COMPARISON OF ESTIMATED A.C. LOCATION VALUES FROM

FIGS. 7 & 8 WITH VALUES GIVEN BY FALKNER THEORY

COLLEGE OF AERONAUTICS REPORT No. 68.

FIG 11.

0 - 4 0 - 3 0 - 2 O l <

X

4Ó y • ^ 5 3 •!i ^ L \

y-A

_\ • J

-y

. ' \ \)\ " >' •A .^ y

^y

V" ^ 5 ^ 5 \ ^ \

r-J

s

- \

V

A-V

_\ >• " ^ ^ • v v > ^ \ ^^ A^ \, - - 1 \ \ , - ^ y \, fc t ~ N ( • ,^ ,> A < \ "

s

s.

5

CURVES BASED ON WEISSINGER

T H P f ^ O V I I C I K I ^ AKI A V / c o A r ' C

SECTION LIFT CURVE SLOPE = 2 T r

i,

'"V

^^

A

\

A

«^ - V s - r* A [ 1 1 i II ^

^"3

N •^ V s _ ^ Si ^ .:^ s V \ 1 ^ >

s*

\ s, s

c

^ _V.

k

\ K

s

.^ ^* '' s s \ s A ^ , ? ^ \ - Ï » , ^, s LINES OF CONSTANT ASPECT RATIO. lA)

^ , ^ ^ _ , \ s s s V i \ \ j s • • s s s ^ ^ s^ s V V s ( LINES OF CONSTANT TAPER RATIO. [\] ^ \ ^ \ s )• ^ s \ s. s ">, 5 \ '^i; , s - N ' ^ \ s Si, N V

J

.J *> X, "«J ' S ^ V ^ N s \ _ V ^ •<i _ s »!; ^ s, is S N s l ^ _ t

s

s s S s • ^ 3 ^ J\, ^ ^ •s-l ^ ^ s . \ r\ „ — „ —

„

lO 3 0 SO 7 0 '/4 CHORD SWEEPBACK ANGLE ( A , )

' 4

9 0

VARIATION OF DAMPING-IN~ROLL DERIVATIVE ( I p ) WITH

WING PLANFORM PARAMETERS FOR DELTA WINGS.

FIG. 12.

0 - 4 0 - 3 0 - 2 O-l FALKNER THEORY (21 V 3 P P ) i EXPERIMENTAL RESULTS ( S E E TABLE 4 ) REFS. 18 & 24. O O l 0-2 0-3

VALUES O F ( - l p ) GIVEN BY FALKNER THEORY, AND EXPERIMENTAL RESULTS

0-4

COMPARISON OF RESULTS WITH OTHER THEORIES AND

EXPERIMENTS.

REPORT No. 68. . 0 7 •06 • 0 5 ( -• 0 4 • 0 3 • 0 2 •01 LINtÜ TAPER 1

CALCS. MADE USING FALKNER 21 V. 3 PP SOLUTION

A

" • — y ^ , , / ^ ^ 4 C r 1 OF CONSTANT RATIO \>^) J ^

/1

/ / / / > \ '" '1 0 - 3 / / / / / / / /

"0

^ ^ ^ 2 0 • L I AS 1 0 - 2 ^ J O-l

TV

FIG. 13.

>\=o

/ ^ 1.75 ^JES OF c o r PECT RATIO NISTANT ( A ) V / 2 0 3 0 4 0 5 0 6 0 7 0

t\

!, 8 0

VARIATION OF YAWING MOMENT COEFF DUE TO YAWING-INDUCED

COMPT-(^^ycu?) — WITH WING PLANFORM PARAMETERS FOR

DELTA WINGS.

0 - t

LINES OF CONSTANT TAPER RATIO ( N )

VARIATION OF YAWING MOMENT COEFF DUE TO ROLLING V^O^

WITH WING PLANFORM PARAMETERS FOR DELTA WINGS.