(T. 1881.)

AND MEMORANDA, No. 910.(As. 136.)

A THEORY OF THIN AEROFOILS.— BY H. GLAUEBTor THE ROYAL AIRCRAFT ESTABLISHMENT , PRESENTED BTTHE DIRECTOR OF RESEARCH,

o

1924, \"i, ,1

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3 8006 )005™

A THEORY OF THIN AEROFOILS.

BY H. GLATJEKT of the R.A.E.Presented by THE DIRECTOR OF RESEARCH.

Reports and Memoranda No. 910. February, 1924.(Ae. 136.)

Summary.—The present report develops a theory of thin aerofoilsin two dimensional motion and simple integral expressions are obtainedfor the angle of incidence and moment coefficient at zero lift. A graphicalmethod of integration is developed which can be used to determine thecharacteristics of any thin aerofoil. The method is applied successfullyto three aerofoil sections and results are also derived for a tail-piano andelevators.

1. Introduction.—The general theory of aerofoils in two dimen-sional motion at small angles of incidence indicates linear relation-ships between lift coefficient and angle of incidence, and betweenmoment and lift coefficients. Also the slope of these curvesdoes not show any considerable variation between differentaerofoil sections and so the problem of determining the character-istics of a given aerofoil section is essentially that of determiningthe angle of incidence at which the lift vanishes and the corre-sponding value of the moment coefficient. In addition there isthe yet unsolved problem of determining the drag of an aerofoilin two dimensional motion. When these characteristics areknown, the behaviour of any finite aerofoil or system of aerofoilsin three dimensional motion can be deduced by Prandtl's theory.

The present report develops a method of determining theangle of no-lift and the corresponding moment coefficient for athin aerofoil of any given shape. A treatment of this problemhas been given by Munk in Report No. 142 of the NationalAdvisory Committee for Aeronautics, but the analysis is notquite free from errors. The present report adopts the generalprinciples used by Munk but presents the analysis in a morelogical order.

2. General Analysis.—The basis of the two dimensionalaerofoil theory initiated by Joukowski depends on the possibilityof converting a circle in the z plane into the desired aerofoilshape in the £ plane by means of a suitable conformal trans-formation. The present analysis will be confined to the case ofa thin aerofoil in which it is possible to replace the aerofoilapproximately by a curved line which will be the mean of theupper and lower surfaces of the aerofoil. This assumptionsimplifies the analysis considerably and has been found to leadto very satisfactory results for ordinary aerofoil shapes.

* (5)22530 Wt 122—10176/1085 500 8/24 A

The starting point of the analysis is the conversion of thecircle G (see Pig. 1) into the straight line A'B' by means of theconformal transformation : —

where z and £ are the complex variablesz =x +iy^=^+i-n

and a is the radius of the circle C whose centre is at the origin 0.The straight line A'B' obtained by the transformation extendson the real axis between the points £ = ± 2a.

Now consider a thin aerofoil represented by the curved lineA'S'B' which deviates only slightly from the straight lineA'B'. The corresponding curve in the z plane, related to theaerofoil by the transformation (1), will be a curve S deviatingonly slightly from the circle C and intersecting it at the pointsA and B. The equation of this curve S may be taken to be

where r is a function of 9 and represents the difference of radiusbetween the circle C and the curve S.

Since the hypothesis

the corresponding point of the aerofoil is£ = 2a (cos d + i r sin 8)

showing that the ordinate -q of the aerofoil at the abscissa£ == 2a cos 9 is

t = 2 a r sin 9... ........................... 3^To solve the problem we require also the transformation

from the circle C to the curve S. Now suppose that the pointz = a e1'* of the circle corresponds to the point z' = a (1 -f- r) eieof the curve, where

^ =0 +e.....,...........,.,...,...,,....(4)

The transformation is then found to be

(5)L £" i

wherer = 2 An sin n 9 \

andO f71"

• sin n & d 9,.........................(7X,

Tiie trailing edge of the aerofoil is given by 9 — TT and socorresponds on the circle to the position <f> — TT + e0 where

e0 = A! - A2 +A3- ......

To determine e0 we have formally9p-

e0 = - (Aj sin 9 +A2 sin 2 9 +...)(sin0 — Sin2# +...)dOTTJO

and so e0 is the imaginary part of

i.e.,

giving finally

eQ = - \ r tan -. d8 =TT

I 0

3. Lift of the aerofoil.—The circulation round the aerofoil isdetermined by Joukowski's hypothesis that the flow must leavethe trailing edge smoothly. This implies that the rear stagnationpoint on the circle must be situated at the point (f> = TT + eflcorresponding to the trailing edge of the aerofoil. If now theundisturbed flow is inclined at angle a to the chord of the aerofoilor to the real axis as in Fig. 1, the complex potential of the flowpast the circle will be

Va2 iKw = Yze^— — e~M— 5— logs........ ....,....(9)

where K is the circulation which must be determined so thatthe rear stagnation point occurs at the point <f> = TT + e0.

Now at the surface of the circle

dz

and vanishes for <f> = TT + e0 ifK =47raVsin(a + e0).. ..,,................(11)

Thus finally the lift of the aerofoil isL = p V K = 4 TT a p V2 sin (a + e0)

and since the chord of the aerofoil is 4a and the angles a and e0can be regarded as small, the lift co-efficient is

kL=7r(a+€0) (12)

The no-lift angle of the aerofoil is, therefore, a = —e0, wheree0 is determined from the shape of the aerofoil by means ofequation (8),

The slope of the lift coefficient against angle of incidence ihTT per radian or 0 • 055 per degree in two dimensional motion.The corresponding alue for the rectangular aerofoil of aspectratio 6 is found to be 0-039 per degree by the method of reportR. & M. 824. This value refers to a thin aerofoil and may beexpected to vary slightly with the actual shape of the aerofoilbut it represents a good average value for ordinary aerofoils.

4. Pitching Moment of the Aerofoil.—From equations (10)and (11) we deduce that the velocity at the point (f> = 8 -I- eof the surface of the circle is

= 2 V/sin (a + </>) + sin (a + 6o)

= 2. V< sin 9 + (a + e) cos (9 + (a + e0) V ..... (IS

For the corresponding point 9 on the surface of the aerofoilwe have

3)

where

X z jo2r y _Lt, = 3 i —^

Thusr/y'

- i 2 (« — 1) A,? e-'r'e

= 1 — (w — 1) A,j sin «0

and

de— ] _ —

(V

= 2 e~!'*/isi

,| =2sin0(l-r)|dz'|

sin 0 + r e~l6 V

givinc!fe

5

and

The pitching moment round the origin is now obtained as

M0 = I "2a2 p q2 sin 0 cos 8 dBJo

-= -2fi'2 p Y2 /sin 0 4 2 (a 4- e) cos 0 + 2 (a 4- e0). o I

<fe -+ 2^sin

wheree = — S An cos TI 9

de^ = 2«Ansin^

and on integrationMo = 2 a2 p V2 (27TO — TT A2 2 TT A2)

= 2 TT a2 p V2 (2a + A2)To obtain the moment round the leading edge we must subtract(2aL), and so obtain

M = 2 TT a2 p V2 (2a -f- A2) — 8 TT a2 p V2 (a 4- e0)= 2 TT ffi2 p V2 (A2 - 2a - 4 e0)

giving the moment coefficient

Now put

u« =- - A2 — f sin 20 (iff8 .0

1 jv"= T^S ri cof, ti d9 .............. ......(15)

4/7 I

and then finally

A'm- -i*L-| €„+,*„.... .,..,.....,,.,..(18)

Thus the slope of the moment coefficient against the liftcoefficient is 0-250, but there will actually be small variationsin this value with the thickness of the aerofoil. The value

of the moment coefficient at no lift is (,«e — - e0) and is deter-mined from the shape of the aerofoil by the two integrals (8)and (15).

5. Graphical Integration.—in denning the shape of an aerofoilit is customary to measure the ordinates and abscissae as fractions

a 22530 A 2

of the chord and to take the leading edge as origin, To trans-form the expressions to this form, we write

2 x = 1 — cos 04 a y ==f]

obtaining

where (, .,..,,..,.. ....(]7

7F (1 — X) Vx (I — X)

and

=- f0 Jo 2

where y ...............(18)

Vx (1 -x)When the form of the aerofoil is a simple mathematical

expression, the values of e0 and //,„ can be obtained by directintegration. One example of this type is the aerofoil whoseshape is defined by the equation

y = h x (1 — x} (a — x)which represents an aerofoil with reflex curvature towards thetrailing edge when a lies between J and 1. Direct integrationof the expressions (17) and (18) is carried out quite simply inthis case by the substitution x = sinz 0, and leads to the results

so that

Thus the aerofoil will have a constant position of the centre ofpressure if we take the value a = f. In report R. & M. 911.aerofoil D has been designed on this basis in the hope of testingthe validity of the result.

In general, the form of the aerofoil necessitates the use of *graphical method of integration a,nd for this purpose numericalvalues of /x (x) and /2 (x) are given in Table 1. The determina-tion of p,0 in this manner presents no peculiaritiess since y /2 (x)is zero at both ends of the aerofoil, although /2 (x) tends toinfinity. In the case of e0, however, the value of y f± (x)generally tends to an infinite value at the trailing edge x — 1,This difficulty can be avoided by performing the graphicalintegration from x =0 to x =0-95, and by estimating theadditional contribution from x =0-95 to x =1 analytically onthe assumption that this part of the aerofoil is linear. Tt can

easily be shown that this additional contribution is 2-9 y'.whore y' is the value of the ordinate y ah x =0- 95.

6. Numerical examples.—As an example of the method, thedetails of the calculation in the case of R.A.F. 15 are given inTable 2 and Fig. 2. In Table 2 the ordinates«/-, and y2 of the upperand lower surfaces are first tabulated, and the fourth columngives the mean ordinate. These values are referred to thestandard chord, which is the tangent to the lower surface of theaerofoil, and- from them the values of the ordinate y are deduced,referred to the base line joining the leading and trailing edges.This base line makes an angle 0°-2 with the chord. The lasttwo columns give the values of y /x (x) and y /2 (x), which areused to prepare the diagrams of Fig. 2. By integration, the partof e0 up to x =0-95 is found to be 0-029, while the additionalcontribution is 2-9 X 0-0030 or 0-009. Thus, e =0-038 or2°-2, so that the angle of no lift — 2°-2 referred to the baseline or — 2°-4 referred to the chord. Also, by integration, thevalue of ju.0 is found to be 0-009, and so the moment coefficientat no lift is predicted to be — 0-021.

A comparison with two sets of experimental results is shownin Fig. 3. It will be noticed that the calculated values lie closely011 the experimental curves obtained at LV = 100, but that thecurves obtained at LV =10 show a peculiar and characteristicscale effect at small values of the lift coefficient. It appears thatthe theory gives a good prediction of the actual values obtainedat a reasonably high value of LV. If the only experimentalvalues available were obtained at a low value of LV, it would benecessary to ignore all values below &L = 0-15, and to comparethe calculated values with the general run of the curves abovethis point.

This point is confirmed in the case of the aerofoils R.A.F. 14and R.A.F. 18, for which calculations have also been made.The comparison with experimental data is shown in Figs. 4 and 5.The calculated points again agree excellently with the form ofthe curves above kL =0-15, but below this point there is thesame peculiar scale effect on the model curves. It appears thatthe method of calculation gives good predictions of the valueswhich may be expected from model tests at a reasonably highvalue of LV. The method of calculation, however, deals withthin wing sections only, and might need extension or modificationbefore it could be applied to thick sections.

7. Tail-plane and Elevators.—The method of analysis can alsobe applied to the case of a tail-plane and elevator by insertingthe valuea

h xy = rzrpi from x = 0 to (1 — E)

i Jii .x"*"",fa n _ r\ X - (j!'

y = —-^—-from z = (1 — E) to 1, /_^where E is the ratio of elevator chord to total chord

!,0.

By direct integration the values of e0 and /j,0 are found to be

arc cos~W(T E) f

2E -_ _ _L _n^ _ arc cos) ' 2 Ell -E) "/

so that, regarded as an aerofoil, we have the characteristics

It is more usual, however, to express the lift coefficient of atail-plane in the form

&L = a1 a j -j- a2 a 2where ax is the angle of incidence of the front part of the tail-plane, and <x2 is the angle between the front and rear parts. Inthis form the theoretical result is found to be

2 r _ ______ ~\- < arc cos A/E — A/E (1 — E) >TT {_ J

and the numerical results deduced from these formulae are givenin Table 3 and Fig. 6. The values of a2/ai found in this way arehigher than those which have been deduced from various experi-mental tests. It seems probable that the sharp angle at thejunction of the tail-plane and elevator exerts a harmful effectwhich is not allowed for by the theory, while, on the other hand,the experimental tests refer to elevators which are divided inthe middle and therefore lose a certain part of their efficiency.No fair comparison between theory and experiment is thereforepossible, but the theoretical curve probably represents the bestpossible result which could ever be expected from an elevator011 a symmetrical tail plane.

TABLE I.Graphical determination of efl and p0.

x = distance from leading edge.y = ordinate.

ti f (/y\ /7-r1 i ('nf*\ — 1 ITT (~\ _ rv\ \lnp /I rv\y 11 \ / " J1 \ ) —— J- il 1-5- vt/) V i t / S J . —— <AJ \0

i? i/ 2 V / t/ 2 \ / \ ' I ^ /0

^ Jl (&) /2 (%)

0-025 2-09 6-100-05 1-54 4-130-10 1-18 2-670-20 1-00 1-500-30 0-99 0-870-40 1-08 0-410-50 1-27 00-60 1-62 -0-410-70 2-31 -0-870-80 3-98 -1-500-90 10-6 —2-670-95 29-2 -4-13

In obtaining e0, graphical integration should be used fromx = 0 to x =0-95, The contribution to e0 of the part fromx — 0 • 95 to x = I may be taken to be 2-9 y1; where yi is theordinate at x =0-95,

TABLE II.Calculations for E.A.F. 15.

X

00-0250-050-100-200-300-400-500-600-700-800-900-951-00

2/i

0-03810-04950-06010-06690-06690 • 06450-06070 • 05530-04810-03890-02840-0215

———

2/3

0-00360-00120-00010 • 00430-00840 • 00800-00560-00210 • 00020-00030-00230-0040

, ——

i (2/1 + 2/2)

0-01270-02080-02530-03010-03560-03760-03620-03320-02870-02420-01960-01530-01270-0095

y

00-00820-01280-01770-02350-02590-02480-02210-01790-01370-00940-00550-0030

0

2/A (x)

00-0170-0200-0210-0240-0260-0270-0280-0290-0320-0370-0580-087

—— -

Vft (*)

00 • 0500-0530-0470-0350-0230-0100

— 0-007-0-012— 0-014-0-015—0-012

0

Angle between chord and base line = 57°-3 X 0-0032 =. 0°-2.

TABLE 3.Tail-plane and Elevators,

E = ratio of elevator chord to total chord.

E.

0-050-100-200-300-400-500-600-700-800-901-00

^

4-863-302-191-721-451-271-151-061-000-951-00

— 1cm „/

4-363-002-001-531-221-00C-820-650-500-330

a1/ai

0-2820-3960-5500-6610-7480-8180-8760-9230-9600-9861-000

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Printers to the .King's most Excellent Majesty.

.Ro,

OF TRRNSFORMRTION,

KI706 H&Sp t.227 ,22 10176/1085. IO2S 8'

CRLCULRTIOMS FOR R.R.FIS.

'050

'026

Ms.

•0250 D'2 D'4- Ob O'S I'D

•100

'075

•Q5O-

'025 ——

R* M. 910.

-O'l-4

* CHLCULHTED VRUU&S..o MODE.U TEStLJiTiLVjsjo..+ MODEL TEST flT LV-joo

LIFT COEFFICIENT0 O'l O'S Q'3G\

Rg_M:_9IO.R fl.F i4 WINS SELCTJON.

« CRLCUL.HTED VBLUE&_o JODEJ-_T£_ST QT LV= 2O.

O'S

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- 2 0 2 4

Li FT CoEFFiClg.HTO O'l O'E 03 Q'4- 05

- !E5

UyEu.yoU

lL

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TOO

0-7*

35. 0-50a,

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