c r " ' . . . . " : " ~

R, & ~ . No, 2763 (xa,2a@

A.R.C. Sechica! Repo~t

I~AINiSTRY OF SUPPLY

AERONAUTICAL RESEARCH COUNCIL

REPORTS AND MEMORANDA

Aerodynamic ~orces on

~?~inSs Osclla'dn£ in a

Air Stream

Rec~a~gula~

Supersonic

By

W. E. A. AcuN, A.R.C.S., B.Sc.,

of the Aerodynamics Division, N.P.L.

Crown Copyrigh~ Reserved

LONDON" HER MAJESTY'S STATIONERY OFFICE

1954

P~:cE 7s 6d ~ T

Aerodynamic Oscillating

Forces

in a

on Rectangular Wings Supersonic Air Stream

W. E. A. AcvM, A.R.C.S., B.Sc.,

of the Aerodynamics Division, N.P.L.

Reports and Memoranda No. 2 7 6 3"

August, 9 5 o

1. Summary.--The aerodynamic forces on rectangular wings of various aspect ratios describing simple harmonic oscillations of small amplitude in a supersonic air stream are determined. Linearized theory is used and numerical solutions are derived by the method of 'Relaxation'.

The problem is formulated in section 4 and in section 6 it is reduced to one of finding a series of conical flow solutions. Only a few terms of this series need be determined since the process converges quickly for the range of values of the frequency parameter considered. This range is believed to cover most of the practical supersonic flutter values.

Moment coefficients for a range of lVIach numbers and various frequency parameter values were calculated and they are tabulated and plotted at the end of the report. The coefficients are referred to tile leading-edge axis position but can be referred to any other axis by the usual formulae.

For a range of Mach numbers in two-dimensional flow, the aerodynamic damping for pitching oscillation can be negative for certain positions of the axis of pitch oscillation and this implies instability (R. & M. 21401 and 219414). The results of this report show that aspect ratio has a stabilizing effect for axes less. than about 0.7 of the chord downstream of the leading edge, but has the opposite effect for axes nearer than this to the trailing edge.

2. In t roduc t ion . - -The l inearized tw.o-dimensional t heo ry of a th in aerofoil oscillating in a supersonic s t ream predicts t ha t for certain combinat ions of Mach number , f requency parameter , and positions of the axis of oscillation, the damping m o m e n t on a pi tching aerofoil m a y be negative, t ha t is it will give rise to instabi l i ty (R. & M. 21401 and 21941').

In this report the case of an oscillating rec tangular wing is considered but the m e t h o d of solution can be applied to more general plan forms.

The same problem has been considered in the U,S.A. by Garrick and Rubinow ~ but their t r e a t m e n t of the wing tips is unsa t i s fac tory in t h a t it ignores the interference of the upper and lower surfaces. E v v a r d a has given a me thod of solution for wings of a wide class of plan forms involving the evaluat ion of a n u m b e r of integrals. Watk ins * has obta ined an approx imate solution for the rec tangula r wing re ta in ing only terms of the first degree in the f requency para- meter , and gives some numer ica l results. Refs. 2, 3 and 4 use the m e t h o d of de termining a source dis t r ibut ion over the wing to sat isfy the b o u n d a r y conditions.

In this coun t ry Temple and Stewar tson have used the Laplace and Four ier t r ans form methods to derive formulae for the forces on oscillating rec tangula r and del ta wings~,L \¥. P. Jones

* Published with the permission of the Director, National Pl~ysical Laboratory.

1

(R. & M. 2655 ~) has suggested an iterative method of solution based on the use of known solutions for the steady case.

Of these Watkins (lot. cir.) is the only one to give numerical results and these are restricted to small values of the frequency parameter. The method of this report gives results which probably cover most of the practical flutter range.

Some allowance may be made for two-dimensional effects, which are important a t the lower values of M, by replacing the two-dimensional thin-wing theory derivative coefficients by more accurate values if these are known (e.g., by Jones' theory, R. & M. 27491a).

No experimental data are available for comparison but Bratt of the National Physical Labora- tory is considering the possibility of measurements of pitching-moment damping on rectangular and delta wings.

I t seems probable that the distorting wing may be treated by the present method and this possibility is being considered.

3. List of Symbols Used.

¢

tl

V

Vo C

S

A

X , Y , Z

T =

M

# p =

k = L( )

Z 0

O~

17

Velocity potential of the disturbance due to the wing

Are space co-ordinates

Time measured from some instant

Velocity of stream

Speed of sound

Chord of wing

Semi-span of wing

Aspect-ratio -~ 2sic Non-dimensional co-ordinates

V/ctl Non-dimensional time

Mach number = V/Vo

Mach angle, sin/~ = 1/M

2= X frequency of oscillation

pc/v Frequency parameter

¢' exp (i~ sec2l, cot f,X)

,~ sec

A differential operator defined in equation (5a)

Y/X z /x

Is such that the leading edge of the wing is depressed by CZo

Angle of incidence of wing

2

R a n d 0 are d e f i n e d b y v = ' R c o s 0 , ~ = R s i n 0

e = log R

¢O,,(n = O, 1, 2 , . . . ) } potentials of various conical flows (see equations 13 and 14)

q~,b(a,b = 0, 1 , 2 , . . . ) i

L, b

A~

L

(see equations (13) and (14))

Pressure due to disturbance

Lift due to disturbance

Pitching moment due to disturbance. ( ~ > 0 when it tends to raise the leading edge and depress the trailing edge)

l~, It, l~, #,, m~, m~, m~, m~,, al,, ~1~, alo, ~l~, am~, am~, ~mo, ~m~, i~, i~, i~, i~, rG, rG, rG, rG are defined in equations (29) to (31).

4. Statement of the Problem.- -The linearized theory of supersonic flow leads to the equation

- ° F + 2V axl at1 ~ Vo" + . . . . . (1) LOxl 2 ayl 2 Ozl J

for the velocity potential, ¢, of a small disturbance caused by a wing oscillating in an air stream of velocity V, where V0 is the speed of sound, and xl, Yl, zl, are co-ordinates relative to fixed rectangular axes (xl increasing in the direction of flow), and tl represents time.

Let c be the chord of the rectangular wing and

xl = Xc cot [~, y~ = cY, z = cZ, t, = c T / V

where s ine = 1/M, M = V/Vo. (0 < ~ < z/2). (2)

Then in these non-dimensional co-ordinates X, Y ,Z , T, equation (1) becomes

M 2 ae¢ 2M e ae¢ U¢ + ~2¢ 326 b - U + - - - _ . cot # 3X a T ~ Y~ 0 Z "2 OX ~

( 3 )

The motion is

where

Let

Then

where

now restricted to be simple harmonic, so that

¢ = 4' e% = ¢' e? ~

= pc~V, and ¢' is a function of x~, Yl, zl only and hence of X , Y , Z only.

¢ ' = #(X, Y,Z) exp (-- i~ sec2# cot/~X) . . . . . . . . . . . (4)

= . . . . . . . . . . . . . . . . . . . . ( s )

k = ~ see/,, and

L(q~)- ~2q5 ~2# U~b - - ~ y----~ + 0 Z----~- ~X. 2 •

. . ( S a )

This report deals with a thin rectangular wing situated symmetrically in the stream (Fig. 1). The origin of the co-ordinates was taken to be at the intersection of the leading edge and the wing tip and their directions as shown in Fig. 1.

The Mach cone of any point (Xo,Yo,Zo) is that part of the cone

(X--Xo)2 = ( Y - Yo) ~ + ( Z - Zo) 2

which lies in the region X > Xo, i.e., downstream of its vertex.

The wave front is the envelope of the Mach cones of points on the leading edge. Fig. 3 shows the shape of the wave front, the closed curve ABHFEGA being its section by a plane X = con- stant, downstream of the leading edge. The circles B H F K and AJEG are the section of the Mach cones of the intersections of the leading edge and wing tips. On the wave front # ---- 0, and moreover it is known that if (1,re,n) are the direction cosines of the normal to the wave front at any point on it, then the derivative of ~ in the direction whose direction cosines are ( - - l,m,n), (the co-normal) is also zero at that point.

The boundary conditions on the wing are known since ~#/~Z is known on the wing by the motion which it is prescribed to have. I t is assumed that these conditions may be applied not in the displaced position of the wing but on its projection on the plane Z = 0, since the displace- ment of the wing is assumed to be small.

The motion of the wing is defined by Zo, and c~, where CZo is the depression of the leading edge and c~, the angle of incidence. The motion of the wing is assumed to be simple harmonic so that

Z 0 ~ Z ' e iP t l and Ct ~ cX' eiP~l .

Then the boundary condition on the wing is given by

0 ¢ _ ~zl Vet - - C~o - - x # (compare R. & M. 21401, equation A. 18)

= - - o~' e% V - - Zo' e% ip - - x~ipcd e % .

In terms of the transformed variables this may be written

~ - - Vc exp (iZ s e c ~ cot l~X)[i,tZo' + c((1 + iX cot t~X)l ~Z

and hence O#/OZ may be expanded as a power series in X.

• o

This fact is used later.

. . ( 6 )

I t follows from the symmetry of the boundary conditions tha t # is symmetrical about the plane of symmetry of the wing, and hence we need consider only half of the region inside the wave front.

In tile region inside the wave front, but outside the Mach cones of the points of intersection of the leading edge and the wing tips, the solution for # is that associated with an infinite wing of the same section performing the same motion. I t will be assumed that these two Math cones do not intersect upstream of the trailing edge, and then in the regions whose sections are ABKJ and E F K J (Fig. 3), # is given by the two-dimensional solution

( s t = , , , ~ . /Xo , o,., o [ k { ( X - - X o ) " - - Z2)I/~J d X o . . . . . . . . (7)

where J0 is the Bessel function Of order zero, and (O#/OZ)xo. o denotes the Z-derivative of at the point X = X0 on the wing. The negative sign refers to the region Z > 0 and the positive to Z < 0. (See R. & M. 26557, equation 38).

Since the boundary conditions are tile same on the upper and lower surface of the wing it follows that #(Z) ---- - - # ( - - Z ) and hence # -= 0 for Z = 0 in the region Y > 0, since the contin-

4

ulty of pressure across the surface, (Z = 0, Y > 0), implies the cont inui ty of #. Thus it is necessary to consider the region Z 7> 0 only.

5. The Equations of Conical Flow.- -The potentials considered in the report can be expressed in the form

# --- X " f ( Y / X , Z/X) . . . . . . . . . . . . . . . . (8)

= X ' f ( v , ~) where ~ = Y/X, ¢ = Z / X ,

and ~ is a posit ive integer.

Then, by forming the required derivatives, it may be shown tha t

- 0 Y - - -~ + aZ ----~' - - 0X --="~ - - t~-~ + o¢ 2 - - <

+ ~ ~ + 2~¢0~ a¢ 0¢2J)" .. (~)

In part icular if n = 1

+ 2v~- Ov 0¢ O¢-Ji"

When the independent variables are t ransformed by put t ing ~ = R cos 0, ~ = R sin O, it follows tha t

L@) = X "-2-' + - - - - 1 of + ~ oy R DR ~0 '2

. ( n - - 1 ) f + 2 ( n - - 1 ) R of R~ oy) • OR ~ f

_ X , - 2 1 8 ~ f 1 ) : f 1 Uf n ( ~ - - l ) f } . . . (10) - 1 ~ (1 - R ~) + ( ~ + 2 ( . - 1)R + R 0¢~

The unit circle B ' H ' F ' K ' in the (~, ¢)-plane corresponds to the circle B H F K in Fig. 3 and the interior of the first t ransforms into the interior of the second.

Let ~ = log R, then

{ ( 1 - e ~0) + + - I" L@) = X ''-2 e -20) 0y ( 2 , - - 1) Of Oy , ( n -- 1)e2°f .. (11)

The uni t circle in the (~, ~)-plane transforms into the semi-infinite strip, 0 < 0, - - ~ ~< 0 ~< in the (0,0)-plane.

Solutions of type (8) are such tha t along any straight line passing through the origin they vary as the n th power of the distance from the origin. In part icular if n = 1, the derivatives of # are constant along such lines ; this is the form associated with s teady flow.

6. Reduction of # to a Sum of Solutions of Conical Type . - -# must satisfy L@) = k2#, and # = 0 on the wave front. Also O#/8Z is prescribed as a function of X on the wing and this function can be expanded as a power series in X. Since all the conditions imposed on ¢ are linear it is permissible to consider each power of X separately and then form a solution by adding suitable multiples of the functions thus obtained.

Thus let ¢ , be a solution of (5) satisfying all the boundary conditions for ¢ except (6) which is replaced by a ~ , J a Z = - - X~/n! on the wing. (n = 0,1,2, . . .).

The solution in t h e region where the flow is two dimensional is given by

O :oo, < . . . . . . . . . .

and hence # . mus t assume this value on the arc BK (Fig. 3).

a~ / Y , Z L e t {~n = Xn+l f, t,o ( Y ' Z ) .3 7 k2 Xn+ j: ' I~X X )

5 Y Z +k iX '~+ f " ' 2 ( x - ' X T ) + " ' ' . . ( l a )

= ~ , , o + k2¢,,~ + k4¢%,2 + . . . say . . . . . . . . . . (14)

where the f ' s and # ' s are functions which will be de termined later, and impose tile conditions

L(#o,o) = 0

L(¢,.,.) = ¢,~,~

• ° . . ( i s )

Then equat ions (15) ensure tha t L@,,) = k2¢. i.e., #~ satisfies (5).

I t is possible to satisfy condit ions (15) by functions of the type given in (8) since both sides of any equat ion of (15) contain tile same power of X, which may be removed leaving a differential equat ion wi th ~, ¢ (or 0 and 0) as independent variables, connect ing two successive f ' s .

BY (12)' # ' (X 'Z ) = j o ~.r - - ~ [ ( X - - X ° ) 2 - - Z 2 1 +2~742I ( X - X ° ) 2 ""

_ ( X - Z ) "+1

(~ + 1) l + k~P,,+a + k~P,,+~ + . . .

where P , + 8 , P~+5, • • • are homogeneous polynomials i n X and Z of degrees n + 3 , n + 5, . . . .

Hence #,,(X, Z) -----X "+1 (1 - - Z / X ) n + I k~X,+8 k,X,,+5 (,~ + 1)! + O,,+~ + G+5 +

where Qn+ a, Q~+ 5, • • • are polynomials in ¢ -= Z / X .

In order to ensure ¢~ assumes the value given by (12) on the arc BK it is necessary to impose the condit ions

((~ + 1 ) ! '

on the arc B ' K ' in the (~, ¢)-plane.

f ~ , @ , ¢ ) = 0 , , + d ~ , * ) ,

f . @ , ¢ ) = G + 5 ( ~ , ~ ) • • •

6

On D ' H ' and the arc ]3 'H ' , f,,, o ,, f,,, 1, f~, 2, • • • are all prescr ibed to be zero, so t ha t # , = 0 on D H and BH.

The b o u n d a r y condit ion on the wing is satisfied by mak ing 3~, , ,0_ X ~ , and ~ # . , 1 _ 0, 3Z n ! OZ

~#'~'2-- 0, . . . on Z = O , Y < O . aZ

af~,_~ = ~f,,,__~ . . . . = 0 . a¢ a~

In te rms of f ' s and ~ and ¢ this implies Of,, 0 _ 1 a~ n ! '

Thus for any ~ the de te rmina t ion of 4,, is r educed to the solution of an infinite chain of differ- ent ial equat ions in ~ and ¢.

Howeve r it is p roved in Appendix I t h a t in the case of the rec tangular wing

I X

¢~,,, + 1 = ¢,, d X where X = M ( Y , Z ) is the equa t ion of the wave f r o n t . . . (16) M(Y, z)

Thus it was necessary only to calculate 4o and then de te rmine ~1, ~ , . . . . . by repeated integrat ion.

In the par t icu lar case ~ = 0, the previous results b e c o m e : - -

~o = ~o,o + k2~o, 1 + k4~o,2 + • • •

= X f o , o(I,],~) -J r- X3]~ ,1 (~ ,~ ) --~ XSk4 fo ,2 (T] ,¢ ) + . . . . . . . . . (17)

and a Z 1 on the wing, i.e., .Ofo.0 _ 1, 0fo. l - - o, afo, 2 - - O,

a¢ a¢ . . ( i s )

On the arc 13K (and in the region where the flow is two-dimensional) ,

i ~ - z ' ZZ) ,, ~]

= f: I C - • . . } d X o

Z 3 = (X - - Z ) - k2 X 311 - - 3 ( Z - f + 2(-~-) }

• / ~ (

X 5 Z 2 . _ / Z . ~

Hence on B ' K ' , f0,o = 1 - - ¢ , _ - - 1 { 1 _ 3 ¢ 2 + 2 ¢ 8 } , fo,1 12

] ; , 2 = V 6 - 6 i 3 - 1 0 + i s - s ~ 5 , . . . . . . (19)

The condit ions (15) become (by equat ion 9)

7

~2 " O~,,fo, n "fo, o + ~" 3~ 2 3

0:~f0' 1 [ 02-~ a7o, + o,1

and so on.

V~2 ~-f0,0 2 ' L --6-@-V 2 + 2 ~ ¢ + = 0

=fo, o

(20)

There are, of course, sets of equations corresponding to (20) with (R,O) or (e,O) as independent variables and the solution is required in the region V2 + ¢2 ~< 1 (or thecorresponding regions in the (R,O) and (e,0)-planes).

7. D e t e r m i n a t i o n o f r o o, f0 1 , f 0 2 , • . . . . - -The equations (20) are all elliptic partial differential equations the solutions' of whic}{ are to be found in a semi-circular region B ' H ' D ' K ' in the (V,¢)-plane (or on the e,0-plane in a semi-infinite strip) on every point of whose boundary f or its gradient are known. This is the kind of problem which can be tackled by the method of Relaxation (see Appendix I I for a brief description) and this method was used in the present case. (See Appendix II.)

The values of fo,0, fo, 1, fo, 2 are shown in Figs. 5 to 7 as functions of e and 0. From the rate at which the fo,, , 's decreased with n it was concluded that fo,3 was negligible to three decimal places.

Since #o, 0 (and hence f0, o) corresponds to the case of a rectangular wing in steady supersonic flow it was possible to check the solution obtained with the analytical solution (see for example Ref. 8, appendix (B)). The difference was found to be not greater than 0.004 (i .e. , less than ½ per cent) at the points for which the values were compared. All these values were on the line 0 = ~, since the analytical solution applies only to the surface of the wing.

After f0,0, f0,1, and f0,~ had been calculated, the values of 40,0, ¢0,1, q~0,~ were computed in the plane Z = 0 by (17) and then integrated to obtain #1,0, #1,1, #1,2, #2,o , ~°2,1, #2,2 , and so on, up to ~ , o , ¢4 ,1 , #.~, 2, all the others being negligible so far as results given in this report are concerned. Outside the Mach cones of the intersections of the leading edge and wing tips the # ' s were derived from (12).

I t is necessary to know the values of # on the wing only since the pressure is determined by this alone.

In order to determine the forces acting on the wing as a whole it is necessary to integrate spanwise, so the integrals

f L°o0 , LOyola, 5o jo ~q)o, 1 d Y , q~o 1 d Y , . . . . ¢)4 1 d Y

x ~ X - x ' x '

and f~ a ¢ ° ' 2 d Y ' x a X Y~x ¢° '"dY' . . . . j-0 ¢2, 2 d Y were calculated for Z = 0. --X

Their values are given in Table 1. The range of integration is spanwise from the point where the flow ceases to be two-dimensional to the wing tip..

8

i t is not necessary to differentiate numerically in order to obtain ~o ~o,o d Y since we have J -x aX

,) < o ,) ,:oo_.,Ioo 6'o,o(X,Y,Z) =Xfo , o ~ , ~ so that aX - - f ° ' ° ~ , X - - X ar .Y 8¢ '

3 # 0 . 0 _ 3fo, o

~Y or

,Ooo "[' I°, :o,o<",°>" 'I and it follows tha t J - x ~ ' - d Y =

and similarly for 8#0,1 ~#0, 2 3X ' 3X

j.o o,o(X,Y,o)dY x" f ° = f0,o(r,0) dr and similarly for the others. - - X - - 1

8. The Calculation of the Forces A ctir~g on the Wing . - -The pressure due to the disturbance is given by

(~¢ " ) ~p=-~ ~ + v ~

- - [~¢ a n . o ] (21) -- c ° ~ V t a n ' e x p ( - - i 2 s e c e " c o t , X + i 2 T ) ~ - x - - i a t . .

and since ~ ( - - Z) = - - ¢(Z) it follows that the lift on an element of area dA is - - 2Ap dA.

Then supposing for the moment that Ap is known all over the wing the lift, L, is given by

L 7~ I: 21j°~s -- 2Ap dyll dx I = -- 4!~ 2 cot.I: nl~ [I°_s,: Ap dY] dX.

Similarly Je' the moment about the leading edge, is given by

- / ' 4c' c o t ' . [ j _ . Ap

J~' is positive if it tends to raise the leading edge and depress, the trailing edge.

These are the forces acting on the complete wing of span 2s and chord c.

In order to determine Ap, we expand 8#/aZ as given by equation (6) in a power series in X.

Suppose 3¢ /8Z = -- Vc ~ A.X'Tn[ . . . . 0

where the A,,'s are constant (in general complex).

Then since 8qo,/OZ = - -X" /n ! on the wing,

qa = Vc E A,#,~ 0

9

. ° . ° o . ° . o o

These A. 's will be linear in Zo' and c~'.

. . (22)

and . . . . . . . . (23) a~/$X ~-- Vc[Ao a¢o/~X + A~¢o + A2#I + . . . } . .

(since so ,+ ~/ax --- ~,,,).

[ a¢o + ~ B.#~] (24) Hence, b y (21) A p = - - e V 2 t a n ~ e x p ( - - i 2 s e c ~ c o t ~ X + iAT) Ao-~- Z o ""

when the B,,'s are new complex constants .

H e n c e f [ x A p d Y = - - e V 2 t a n l t e x p ( - - i ~ s e c e t , cot/~X + i2T)

x [AoI° o 1 -~-X d Y + Z B " t - x ¢ O O .. (25)

i o z ~ dY , x ~o d Y , . . , can be expanded as power series in X by (14) and the values

given, in Table 1.

Hence, collecting up the te rms in the square bracket ,

[ Alb d Y = --5 V 2 t an ~ exp ( - - i2 sec 2 ~ cot t~X + iaT) Z C,X* X 0

where the C,/s are constants .

. . (26)

In the region where the flow is two-dimensional it follows from (12) t ha t

k~X 2 k ~ X ~ - - = so t h a t t#0 = 1 - - - - + Jo(kX) a X 2 ~ ~ " " "

k S X 8 k ~ X 5 q~0-~X " +

23 3 224 ~ 5

¢ h = f : O o d x _ X 2 k 3 X 4 2 23 3"4

and so on for all t he #~'s.

. . .

W h e n these expressions are subs t i tu ted in (24) it follows t ha t

Ap = -- O V 2 exp ( - - i~ sec 2 t* cot ~X + i~ T) t an t* Z C~'X" 0

. . (27)

where the C,/ 's are complex constants .

Over the pa r t of the wing where the flow is two-dimensional , independen t of y , it follows t ha t

Ap is known, and since it is

. . (28)

10

Then the expressions for L and J/{ reduce to sums of integrals of the form

fo~"X"eos (~ sec~/~ cot l , X) d X a n d I ~ " X ' s i n ( ~ s e c ~ eott~X) dX.

These integrals were computed by expanding the integrand in a power series and integrating term by term.

The lift and moment acting on the wing are thus found in the forms

L = qV%2e~rlZo'{A(l~ + i~l~)+ (~l~ + i~ ~l~)t

+ + + + . . . . . . . . . . (29)

and -//: = oV% ~ e'ar[zo'{A(m. + i2m,) + (Ore. + i~ Ore,)

+ g ' {A(m, + iam~) + (Ore. + ia 0ra~)}] . . . . . . . . . (30)

In these equations l,, l,, lo, l~, m., m~, m~ and m~ are the two-dimensional thin-wing derivative

= , - : : + +

and -//¢ = eV~c 8 e i~r A ? o ' l ~ . + i2r~} + ~z'{n~o + i2n~}l.

Evidently i., etc., are functions of A, the aspect ratio.

Write i~ = 1, + OlJA, and corresponding expressions for the other coefficients.

Then (29) and (30) may be written

Since it has been assumed that the Mach cones of the tips do not intersect upstream of the trailing edge, A is restricted to be greater than 2 tan/z.

All these coefficients refer to the leading edge but if it is desired to take an axis hc downstream of the trailing edge, the corresponding coefficients may be calculated by using the equations

zh = Zo + ho:, J/~, = ..//¢ + hcL, Lh = L, . . . . . . . . (33)

where cz~, is the depression of the new axis and .~,, is taken in the same sense as J [ about it.

In the actual computations only a finite number of terms of any of the infinite series were retained, the rest being negligible.

If more accurate values of the two-dimensional coefficients are known, from theory (or even from experiment) (e.g., those taking thickness effects into account), more accurate estimates of

11

. . . . ( 3 1 )

. . . . ( 3 2 )

coefficients, and $l,, ~l~, ~la, ~l~, ~m~, ~m~, ~m~, ~m~ which are defined by these equations may be regarded as corrections for the effect of the tips.

L an d J / / might be obtained by using these in (29) and (30), or possibly by multiplying l,, etc., by tile ratio of the more accurate two-dimensional values to the thin-wing two-dimensional values.

9, N u m e r i c a l R e s u l t s . - - T h e coefficients dl,, . . . dma were computed for 2 = 0.2, M = 1.2 (0.2)1.83 Z = 0.4, M = 1.2(0.2)1.8; and ,t = 0.6, M = 1.4(0.2)2.0.

The coefficients 1 , , . . . m~, also appeared in the course of the calculations.

The results are given in Table 2, and are plotted on Figs. 8 to 15.

In the case of a wing oscillating about a fixed leading edge (i.e., z0 = 0) the damping moment is given by

- - e V%3~'Z(m~ + Om~) = - - e V % ' ~ ' 2 A ~ . . . . . . . . (34)

so that positive values of ~ imply negative damping.

The Values of ~ have been plotted in Figs. 16 to 18 and from these it will be seen that with the leading edge as axis the negative damping predicted by two-dimensional theory occurs in the finite case but is reduced by aspect ratio.

If the matter is referred to an axis distance hc downstream of the leading edge it follows that

This equation defines (~,)h, etc., where in particular

= ma + h(l~, - - ~ ) - h21~ + IOma + h(Ola - - O m ~ ) - h ~ dl~-J/A by equations (33).

In order to discover the effect of the position of the axis on the damping effect of the tips the quant i ty --[dm~, + h(dla - - dm~)-- h 20l~] was plotted for various values of M and X (Figs. 19 to 21).

The damping moment for a wing pitching about the new axis is

so that if --[dma + ~/(dla -- din,)-- h ~ Ol,] is positive the tip effect increases damping and vice v e r s a .

Figs. 19 to 21 show that the damping effect of the tips becomes negative for axes near the trailing edge, the point of change from positive to negative being almost independent of M and varying only slightly with 2.

Compar i so~ w i t h Other R e s u l t s . - - I n order to obtain a check on the process the computations were also carried out for the case M = ~/2, ~ = ½, with the slight difference that the two- dimensional mid-chord derivatives were computed instead of the leading-edge derivatives. These were compared with the mid-chord derivatives given by Temple and Jahn (R. & M. 2140a), and found to agree to three decimal places.

A further comparison was made with the result given by Temple "~. The case considered was

12

again M = V'2, ~ = ½, the wing having a simple vert ical oscillation (~' = 0), and aspect ratio 2. The results agreed numer ica l ly though the signs were different, bu t it is beleived tha t Temple 's result as given has the wrong sign.

Comparison of a case with Watk in ' s result 4 tends to the conclusion tha t this is sat isfactory for small values of ,~.

A P P E N D I X i

,-X P r o o f t h a t Ore+ 1 = j O,n d X

M(Y, z)

Let X = M ( Y , Z) be the equat ion of the wave front.

-X I f O 1 ~-- j O o ( ~ , Y , Z ) d~ ,

M( Y, Z)

t hen 0 o 1 _ ( x boo b z OM-gZ ( G Y , Z) d~ e - - - - b M O o [ M ( Y , Z ) , Y , Z ]

OZ

= ,x( (e, Y , z ) - ~ - . . dM

since O0 = 0 on the wave front.

b"01 b Z 2

f X b20o (~,Y,Z) d ~ - - - - g-U d llsI bM boo [M(Y,Z),Y,Z~ o Z bZ

Similarly b°'O1 (X b2~} 1 - - ~ "M ( L Y , Z) d~ b y 2 bM boo EM(Y,Z),Y, zl b Y b Y

also aO, _ O o ( X , Y , Z ) b201 - - b o o ( X , Y , Z ) bX ' bX" ~X "

Hence L(O1) --k201 = b201 + b20~ - - a20~ - - k~O~ b y2 a Z ~ 0X 2

_ fx (b~oo a®o - - M \ b Y 2 + bZ ---~2 - - /GY,Z

b M boo ( M , Y , Z ) b Y b Y

8M 8¢o ( M , Y , Z ) 86)° ( X , Y , Z ) b Z b Z bX "

13

(A.1)

]But L(4o) = k2#o, hence

f x ~24o

L ( 4 d - = - - - - M(Y,Z) o X 2

( M , V , Z ) - - a m (M,Y,Z) ~ Y ~ Y ~Z ~Z

~4o (X,Y ,Z) ~X

~4o ( M , Y , Z ) - - ~M OX ~ Y

~4o (M,Y ,Z) - - aM a Y aZ

ae (M,Y,Z) ~Z

Hence L(41)-- k241 is equal to the scalar product of the vectors (-- 1, - - OM/OY, - - aM/aZ) and (O4o/OX, a4o/OY, a40/aZ)M(y.z),y,z, but the first of these is along the co-normal to the wave front, X - - M ( Y , Z ) = O, and the second is the gradient of 40, i.e., L(41) --/~241 is equal to the derivat ive of 40 along the co-normal.

But since 40 is a solution this gradient is zero and hence L(4~) -- k°-4~ = 0, and similarly L(4~)-- k242 = 0 and so on.

Moreover ~4~ _ r | x a4° (~ ,Y ,Z ) d~ by (A.1) ~Z JM(Y,Z) ~Z

Now for Y ~< 0, Z = 0, (i.e., on the wing), O#o/aZ = -- 1, and therefore

8Z M(y,0) ( - 1) df ---- - - X + M ( Y , O ) = - - X ( Y <~ O)

041 X on the wing. so tha t OZ - -

Repeat ing this process it follows tha t ~4,/OZ = - -X" /n[ on the wing.

For Z = 0, Y ~> 0 , 4 0 -= 0, and hence 4 -= 0 in the same region, and so on for all the 4~'s.

Obviously 41, 42, . . . are all zero on the wave front.

Thus the 4 , ' s obta ined in the wing satisfy all the boundary condit ions and L(4~) = k24,.

A P P E N D I X II

A brief description of the Relaxation Method

As applied in the present case this me thod uses two facts

(i) The derivatives of any function may be expressed to any required degree of accuracy by a linear combinat ion of its values at a discrete set of points, provided tha t it satisfies certain not very restrictive conditions.

(ii) When the expressions thus obta ined are subs t i tu ted into the linear differential equation, and its boundary conditions, the result ing set of l inear equations can be solved (again to any required degree of accuracy) by a me thod of successive approximation.

14

Iff(x) is any function of X, where values are known at the points (x0 4-nh), (n = O, 4- 1, 4- 2, . . . and a table of its values is formed, and hence a table of differences, as below

f ( X o - 2h) ~-2 ~4_~ -- 3/2 ~ 3 3/2

f ( x o - h) 0% -- 112 ~ 3112

f(xo) do (~112 (~31]2

f(xo + h) ~3/2 ~3312

f(Xo + 2h) d22 ~'2

then hf'(x0) = 1((~_112 + $112)-- 1/6 X ~((~13_112 + $3112) + 1/30 )x( 2((~ - 1 1 2 1 5 + da/2)_. . . . . (A.2)

h~"(Xo) = d20- 1/12 d% + 1/90 b60 -- . . . . . . . . . . . . . (A.3)

and so on.

These formulae may be derived by differentiating Stirling's formula (see Ref. 9, p. 64). For a rigorous discussion of the subject and in particular of the conditions under which these expressions converge see Steffensen (Ref. 10, pp. 60:71).

In these expressions the first terms may be regarded as being an approximate value of the derivative, and the remaining terms being the corresponding correction.

Consider the values of the dependent variable at a lattice of points covering the region in which the solution is required, as in Fig. 5. Substituting from (A.2) and (A.3) into the differ- ential equation, and its boundary condition when this contains a derivative, a set of linear equations (one for each point) is obtained, connecting the values at points of the lattice. These are now solved by adjusting the values until the conditions are satisfied to the required degree of accuracy.

For a full account of this method see SouthwelP 1 and Fox TM.

As Q tends to -- o~ the differential equations tend to the Laplace equation O2f/Oo~, + ~2f/002 = O, it is possibleto infer the manner in which f tends to 0 as 0 tends to -- ~ by considering the solutions of this in an infinite s t r ip . This avoids the necessity for considering very large values of -- q.

No. Authors

1 G. Temple and H. A. Jahn

2 I . E . Garrick and S. I. Rubinow

3 J .C . Evvard . . . . . .

R E F E R E N C E S

Title, etc.

.. Flutter at Supersonic Speeds, Part I. Mid-chord Derivative Coefficients for a Thin Aerofoil at Zero Incidence. R. & M. 2140. April, 1945.

.. Theoretical Study of Air Forces on an Oscillating or Steady Thin Wing in a Supersonic Main Stream. N.A.C.A. Technical Note 1383. July, 1947.

.. A Linearized Solution for Time-dependent Velocity Potentials near Three-dimensional Wings at Supersonic Speeds. N.A.C.A. Tech- nical Note 1699. September, 1948.

15

4 C .E . Watkins . . . .

5 G. Temple . . . . . .

6 K. Stewartson . . . . . .

7 W . P . Jones . . . . . .

8 J . C . Evvard . . . . . .

9 E. Whit taker and G. Robinson

10 J . F . Steffensen . . . .

11 R . V . Sonthwell . . . .

12 L. Fox . . . . . . . .

13 W . P . Jones

14 W . P . Jones

Effect of Aspect Ratio on Undamped Torsional Oscillations of a Thin Rectangular Wing in Supersonic Flow. N.A.C.A. Technical Note 1895. June, 19i9.

Modern Developments in Fluid Dynamics. High Sfleed Flow. Vol. I, Chapter IX. Oxford University Press, 1953.

On the Supersonic Flow over Laminar Wings at Incidence. A.R.C. 12,112. (Unpublished.)

Supersonic Theory for Oscillating Wings of any Plan Form. R. & M. 2655. June, 1948.

Distribution of Wave Drag and Lift in the Vicinity o f Wing Tips at Supersonic Speeds. N.A.C.A. Technical Note 1382.

The Calculus of Observations. Blackie and Son.

Interflolation. Bailli6re, Tindall and Cox, London: Williams and Wilkins Co., Baltimore. 1927: Chelsea Publishing Co., New York. 1950.

Relaxation Methods in Theoretical Physics. Oxford University Press. 1946.

Some Improvements in the Use of Relaxation Methods for the Solution of Ordinary and Part ial Differential Equations. Proc. Roy. Soc., A, 190 (1947), 31-59.

Aerodynamic Forces on Biconvex Aerofoils in a Supersonic Stream. R. & M. 2749. May, 1950.

Negative Torsional Aerodynamic Damping at Supersonic Speeds. R. & M. 2194.

16

T A B L E 1

VaIges of Ce,/ain fnlegrals

- x ~ (X,Y,O) d g ~ + 0'49994X

• :c - ~ . ( X , g,O) d g ~_ _ O. 1657 X"

"° ~ ( X , J_ ~ Y,o) es~ ~_ + o. , lee x~

f° - x #o,o(.,.Y, Y,O) d Y ~ + O, 74997Xe a fO

. - x q~o,;(X, ]7,0) d Y = _ 0 "06225 X 4

£ e ¢~,,o(X,g,O) d:F ~ + 0 . 4 1 6 6 7 X a fO

o-x ¢~a(X,Y,O) d Y ~ _ 0 . 0 1 6 6 1 X 5 1,

- x ¢~,.,(X,Y,O) d Y ~. + O.O00417Xr [o . -.r ~IS,°(X, Y,O) d Y ~ + O. 14582X~ / o = _

' q5~.1( X , I7,0) d Y 0 . 0 0 3 4 6 X 6

f [ ,~ .~ ~x, <o> eY = + o.oooo628 x~

£. #~,o< >o~ e:~ _- + o.oaTsox~

f[.~ <.o(X, <o) eY ~ + o.oo76a7 xo

S: ~.o(X, Y,O) d Y _~ + 0.001288X7

x ¢'~,o(X,Y,O) d Y ~ + 0"0001885X,

fo ®~,o(x,Y,o) e Y ~ + o.oooozaaxo

.1°_~. % ( < <o) gy ~_ _ o.ooo589x, -0

J - x ' - - 0 . 0 0 0 0 8 5 X ~ ¢~ 1(< Y,o) dY

I.

T A B L E 2

Leading-Edge Derivative Coefficients

2 = 0 . 2

2 = 0 . 4

2 = 0 . 6

2 = 0 . 2

, / ,=0 '4

Jt = 0 " 6

M

1"2 1"4 1"6 1"8

1"2 1"4 1"6 1"8

1.4 1.6 1.8 2.0

M

1.2 1.4 1.6 1.8

1"2 1"4 1"6 1"8

1.4 1"6 1"8 2'0

5L

--0'156 --0.042 --0.019 --0.011

--0.504 --0.156 --0.074 --0.043

--0.311 --0.155 --0.093 --0.062

+0.126 i +0"041

+0.020 +0.012

+0.390 +0"150 +0"076 +0.045

+0"287 +0.154 +0.094 +0.062

--2.034 --1.006 --0.629 --0.441

--1"445 --0.904 --0.593 --0.423

--0"756 --0'538 --0.396 --0'304

+2.803 +1.999 +1.585 +1.328

+2"292 +1.880 +1.537 +1.303

+1".712 +1.465 +1.264 +1.111

dl.

--2" 074 --1.016 --0" 634 --0" 443

--1"584 --0.944 --0"612 --0"434

--0.840 --0.578 --0.420 --0.320

+2.846 +2.013 -}-1.591 +1.332

+2.442 +1.933 +1.563 +1.318

+1-820 +1-520 +1.297 +1.133

5le,

+3.201 +0.705 +0.270 +0.131

+2-568 +0.654 +0.258 +0.127

+0"575 +0"239 +0.120 +0.068

--1.696 --0" 020 +0.293 +0 ' 372

--1" 127 +0" 043 +0.310 +0"379

+0.137 +0" 337 +0" 390 +0"395

dm~

+0.116 +0.031 +0.014 +0.008

+0"365 +0"115 +0"055 +0"032

+0.227 +0"115 +0"069 +0.046

--0.083 --0-027 --0.013 --0'008

--0"241 --0.097 --0-050 --0.030

--0.179 --0.099 --0"061 --0"040

+1.325 +0.666 +0.417 +0.293

+0.862 +0.585 +0.389 +0.279

+0.469 +0.345 +0.257 +0-199

--1.349 --0.989 --0.788 --0.662

--0.979 --0.901 --0.753 --0.643

--0"779 --0"699 --0.615 --0.545

~rn~

+1"357 +0" 674 +0"421 +0.295

+0.972 +0"617 +0.404 +0.288

+0.535 +0"377 +0.276 +0.211

~ba

--1"382 --0-999 --0-793 --0"665

--1.089 --0.940

! --0.772 i -0.655

--0-858 -0:740 --0.639 --0.561

~m~

--2"381 --0"528 --0"202 --0-098

--1"858 --0"484 --0.192 --0"095

--0" 420 --0" 176 --0" 089 --0" 051

Tl¢i~

+1. 102 +0"011 --0. 196 --0.248

+0" 653 --0' 039 --0"210 --0.254

--0" 114 --0" 231 --0. 262 --0" 264

18

i V t • I

I i I

O~

-'< 2s ' >

The axis oPz, is perpend;cul&rl~ oug oF P.he paper lchrough 0

FIG. 1. Plan of wing.

A , , ,

£a . - - . . . . . . , 3 "

I

E

2 s

B

H

F

.FIG. 3, Section by any plane. X = constant > 0,

cD

M e a n ~s.~t::ion oF_wfin. .9_

,, t

-~a (3 J FIG. 2. Mode of vibration of wing.

et i

, /

- I

FIG. 4. The (~/,~)-plane.

155 ZOZ 262 3 3 8 4 3 ~

14.3

122.

97

66

34

O'

154.

119

2..31

192.

145

29.1

2.34¸

172

36t

27~

,41

0

97

4 8

0

112.

56

0

12.4

bO 60 57 5"I ,42. 3e

2 9 2.,B 24 20 t4

0 0 0 0 o 0

557 627 704 7~5 868 94~ {(~oo

7i2 550 ~ 6o ¢ 664- 74S 741

43':) 4"78 5{5 '545 563 5~5 5oe

31~ 404 4~Iz 434 423 389. 2~3

319 334 34.1 536 3t2 2.9~. l~,4

26~ 2"/I 269 25'5 2.21 R;") 34

Z I 4 2L5 201 189 155 9~" o

169 167 157 138 IO7 6o o

1 2 9 125 {t6 9 9 73 3 8 o

93 . B9 81 ~ 48 24

t4

7

0

FIG. 5. V a l u e s of fo ,o x 10 3.

o

o

o

c

~ 7 ~

- - 14- - t~

- 1 3

-12.

__9 ¸

- 6

- 3

0 .... 0

- 1 7

-15

- 12

- 8

. . - 4 .

- 2 - 4 - . ~ Z - - 4 3 - , . 5 0 - 5 8 - 6"1' - 75 - - 81 - , 5 3

.- 22

- - 19

- - iS

- - I 0

- - 5

0 0

--3O ,,° , ,

--24 ° .

- - 18

- - t l

- 42. - 4 9 - S g - 6 3 - 6 9 - - 7 Z - 69

--.3'~ --45 --.~(3 - S 4 - 5 6 - 5 3 - 4 2

- 3 ~ - , : 3 9 - 4 2 . - 4 3 - 4 - 1 - 5 3 - - 1"7

- - , 3 0 - - 3 3 - 3 4 - 3 3 - Z 9 - 1 9 - - 4

- Z5 - 26 -26 -2~ ~- I~ -'I0

-- zoi- 20 -2G - I? - 13 -- 6

- t6 - ~5 --t5 - 12 - 8 --

- - 12 - t~ - I O i - - 8 .--.,,5 - - Z

-- 8 ~- 8 - 7 -- S --3 --~

-- Z -- ~. --,2. -- 2 - I - - o I |

~3 0 ~ 0 O, 01 o

FT

e = -~ E = -T

FIG. 6. Values offo,~ × I0 s.

o

o

0

o

Q

c

c

e = o

20

O=-/T

5 6~ 9 12

5

4-

:3

2

5

.4.

'8 II

7

4-

2 2

t 6 ~ 9 22. 2-5 2.7 2.7 25

15 17 I~ 2.1 "25 18 12

1:5 tS le, 16 l~i " I0 3

I I 1"2 I~j 1'7 ~, 5 I

7 7 7" 6 5 I o

5 5 5 4- 2 I 0

4- "~ 4" ~ I o o

5 3 3 2 I o 0

2 2 2 I! o 0 o

I I I o 0 0 0

0 0 0 o o o o o ol o

8 =-~r 8 =-= z £~o

FIG. 7. Values of f0, ~ X 10 4.

~)--7¢

9 = 0

21

~rO,50

"Lz. , A- 0.2_

\

/ 8Lx, k= o-2

f f f / "

J f

-O.SC

5L2:,A'°'4 M 1.0 1,2 l,~- I-~ t.O 2-o

FIG. 8. Variation of I, and ~l, with M and 1.

• i

-HZ.O

+ t . 0 "

~ P - 0

- 2 . 0

@L£ A~0~ / ' IM

oO I-z, 1.4- I - 6 I - ~ z - o

FIG. 9. Variat ion of dl~ and l~ with M and ,~.

2 2

FIG. 10.

/ L

I I i , J

i

I. ~ t,8 2.6

V a r i a t i o n of 6l~ and l~ w i t h M and 4.

I !%L&, k*o.~

~-3.0

f~1.6,$~- 0-4

4 2 . 0

~'1 "@

bL&

0 t a 7~,o.6 / >

-1-0

la, ~.o4 ]

-~.0 ~.&,~.o..z

- ~ * 0 I*0

FIG. 11.

1,4 ~.2 ,.~ i.~, i.~ z.o

Var ia t ion of ~l~ and l~ w i t h M and 4.

2 3

* 0 . 4 0

- 0 " 3 C HO FIo. 12.

,£rnz, A~ 0,4.

\ 6"rex, A = 0.6

,~,,=\ < o \ \ <

O

f / / -o-i( rnz ~.=O/.Z ----

mz,A~ O.4

tq ~'Z l.z$ ~6 l,g 2 ,0

Variat ion of m, a n d dm~ with M and ,I,.

+1.0

I &m~,X=o,2

rn £,X.=o.6

rni, •= /

/

f l

/ /

I ' 0 I-Z 1,4- 1,6 I" 8 Z.O

FIG. I3. Variat ion of m~ and dm~ with M and 2.

24

Sme~,A=o.~

~ m ~ , ~

8m, ,X= 'N~

l m ~? ~, ~.: 0 .~///

i rrn~,A.=O-Z I ' ,O I -2 . I "4-

FIG. 14.

t.6 1.8 2-0

Var ia t ion of c3m~ and ma wi th M and k.

+1 ° 0

-1-0

-2"0

\

gin& X

8m,~,A~o,2

t-O

FIG. 15.

I b4 i-z 4.4- L-G i'8 2.-0 Var ia t ion of ma and (Sma wi th M and 7.

( F o r 2~ ~- 4"0, r e a d ~. = 0.4.)

25

+1"O

A

+O-.~

A~6

A~o~

~ , & : ' o ;mpIfes nega.t lve da~npm~

0 .5

'Fr~'l; z >0 implies r l e g e . e t v e dm'npin 9

"0"51,O I.Z t,~l- 1.6 I 'g 2"O

FIG. 17. V a r i a t i o n of ~ga, = ma + dma/A wi th M a n d A(~L = 0 .4) .

F I G . 1 6 .

-0 .~ r l t .o t.2 i~t 1.6 t .~ 2 . o

V a r i a t i o n of n~a = ma + $ma/A wi th M a n d A (4 = 0"2).

FIG. 18.

` F I ~ . "k m # > o l p es. nega i : i v e

- 0 , 5 I ,o l * z 1"4 I ' 6 ] 'B 2"O

V a r i a t i o n of ~a, = ma + Om~/A wi th M a n d A (~L = 0" 6).

26

-t- V~. ommpir~

-(Sma÷h[Si..a- ~m±)- m2 St_>)

-f 2 .0 •

4- IGO

M , , i * 6

0

- t'O

- , 2 ° o

~ I G . 1 9 .

Io0

Effect of axis position on damping effect of tip correction (A = 0.2).

27

-(6.~,,-,(~,l~,-,~.)-~?'~-t.~)

-~ VE da~p,n 9

-rE dampi~ 9

FIG. 20.

+ I'0

\

\ . \

\ , _¢"I= I , & ~ , \

L Effect of axis position on damping effect of

tip correction (1 = 0.4).

- ( 8 ~ . h C~l,~- ~me)-h ;~ ~.~)

M.2.0

+rE P~mpi~j

0

-VE p m D n c 3

-0.5

FIG. 21.

I '0

Effect of axis position on damping effect of tip correction (1 = 0.6).

28 (680) Wt.16 K9 11]54. H.P.Co 34./261 P R I U T ~ D I1q GREAT B R I T A I N

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December 1, 1936 - - June 30, 1939. Ju ly 1, 1939 - - June 30, 1945. Ju ly 1, 1945 - - June 30, 1946. July 1, 1946 - - December 31, 1946. January 1, 1947 - - June 30, 1947. July, 1951.

R. & M. No. t850. R. & M. No. 1950. R. & M. No. 2050. R. & M. No. 2150. R. & M. No. 2250. R. & M. No. 2350.

Is. 3d. (ls. 4½d.) is. (Is. 1½d.) ls. (Is. 1½d.) ls. 3d. (ls. 4½d.) ls. 3d. (Is. 4½d.) Is. 9d. (ls. 10½d.)

Prices in brackets include toostage.

Obtainable from

H E R M A J E S T Y ' S S T A T I O N E R Y O F F I C E York House, Kingsway, London, W.C.2 ; 423 Oxford Street, London, W. 1 (Post Orders : P.O. Box 569, London, S.C.1) ; 13a Castle Street, Edinburgh 2 ; 39 King Street, Manchester 2 ; 2 Edmund Street, Birmingham 3 ; 1 St. Andrew's

Crescent, Cardiff ; Tower Lane, Bristol 1 ; 80 Chiehester Street, Belfast, or through any bookseller.

S.O. Code No. 23-2763

R. & M. No. 276: