C.P. No. 309 (18,412)

A.R C Technml Report

C.P. No. 309 (18,412)

A.R.C. Technical Report ,’ *z .

‘-

MINISTRY OF SUPPLY

AERONAUTICAL RESEARCH COUNCIL

CURRENT PAPERS

Problems in Computation of Aerodynamic

Loading on Oscillating Lifting Surfaces

H. C. Garner, B.A., old W. E. A. Acum, A.R.C.S., BSc.,

of the Aerodynomrcs D~vrsron, N.P.I..

LONDON HER MAJESTY’S STATIONERY OFFICE

1957

Price 2s 6d net

C.P. No. 309

Problems in Computation of Aerodynzmu Mug on Osc~llatlng rsfting Surfaces

-By- H. C. Garner, B.A., and W. E. A. Acwn, A.R.C.S., B.Sc.,

of the aerodynamics Dlvlsion, N.T.L.

Jot-7 Annl. 1956

The linesrued theories of osclllntlry vr~ngs In subsonic cszd supersonic flow are formally dw2ussed. The computability of the suhsoru.c problem is considered whenMach number, frequency and aspect ratio are arbitrary and 111so when any one of these parsmeters takes extreme values. Methods of treating aerod..ynwc flutter problems care briefly outlmned. The most general subsouc problem for a rectangular ~JlM, already solved numerically on a desk machine, should now be examined v%th a view to mochsnized computation. Specml theories, particularly suited to hzgh-speed computation, should, If possible, be extended to srbltrary frequency and Xach number.

1. Introduction

e

2

2. General Equatrom m Subsouc Flow 2

2.1 Dlfforentul Cn_uation snd Boundary Condltlons 2 2.2 Intebyal Equat-Lons 3 2.3 Approxunatlons to the Kernel 4

3. CcmputatumslProblens 5

3.1 Use of the Exact Kernel 5 3.2 Low-frequency Theories 7

b. General Equations 111 SupersolucFlow 8

5. Aeroelastlc Problems 9

5.1 Subsonic Flow 9 5.2 Supersoluo Plow 10

6. Concluding Remarks 11

7. References (1 - 35). 12

PublIshed with permission of the DIrector, National Phyelcal Laboratory

-2-

1. Introduction

In considcrmg the perturbation of a dorm stream of velocity U due to an oscillating iving, it is usual to neglect the squares of increments III velocity, of wing thickness and amplitude of oscillation. Viscous effects are also ignored, so that the problem is specified by the planform of the wing, the mode of oscillation, the non-dimensional frequency and M, the Mach number of the stream. It is required to evaluate the locsl phase and amplitude of the aerodynamic loading. According to the linearised Meory the perturbation velocity potential, Q, satiifies the equation (Ref. <, p.326)

and the pressure at any point is given by

P -p 2 00

-p, (y?J. . ..(z)

a% aP aa* hIa a% --- - --- ---_ - -- --- = 0 a2 ua ata '

. ..(I) u axat

When the stream is sonic (M = 1) the differential enuation (1) simplifies snd certain analytical solutions exist2 (Mangler, 1552). In supersonic flow (M > I), uhen the differential eauation becomes hyperbolic, there is a class of solutions in closed form (8.4). NO general solution of the problem exrsts when the flow is subsonic: even when M = 0, (1) reduces to Laplace's equation, but remains intractable.

An important feature of current theoretical research on subsonic oscillatory flow is the diversity of methods applicable to a given problem. The choice of method. will depend on individual exporiencc, facilities for computation, the required accuracy and the intended application. The various methods all involve further assumptions or approximations, so that (I) 1s never solved sxactly for &I < 1. Several sim$ifying assw@ions are considcrcd in 3.2; the various numerical processes and approximations me discussed in 8.3. Most methods use collocation, conditions of tangential flow being satisfied at, say, n positions on the ruing. This involves arbitrary choice of the n positions ati of n complex functions to represent the phase and amplitude of the pressure. The main ci';ort of computation lies in expressing t&se boundary conditions as a set Of n complex linear swltaneous equations, vi-u& are then solved by routine methods for the unknown coefficients of the prcssurc distribution.

The precise nature of the oscillatory motion of tic ~klg 1s notimportant. Whether the problem to be solved is that of a pitching wing or an oscillating partial-span control, the matrix of the sismltsncous equations is usually the same. Moreover, it IS not highly significant vjhether the antisymnetrical (rolling P

roblcm is symmetrical (pitching) or .

2. General Eouations in Subsonic Flow

2.1 Differential Swation and Rouddary Conditions

The differential equation for the perturbation velocity potential, @ , is given in equation (I), and is subject to the following four boundary conditions:

( a) On the planform S in the plane s = 0

am az(x, Y, t) azb, Y, t) -c = ---..------- , u ______---_- as at ax '

where z = z(x, y, t) represents the surface of the tin&.

. . . (3)

b)/

-3-

(b) In the wakr

am/at + uaa/ax = 0. . . . O+)

This expressrs the fact that the prea and. follow from eqwtmn (2)

-wre is contmmus across the wake, cou led \mth the observation that

m(x, y, z, t) = - m(x, 3‘, -z, t . P

(0) xt the trallmg cd&e the veloczty is fll?lte.

(a) Itimltely f,v from the mng and i~a!ce the dmturlmmz tends to r.0??0; @ IS such t&t It represents an outgow d~turb,,mce [see footnote to equntmn (7)].

Cqmttlon (1) my h- smpld'~ecl by the substitution

. ..(5)

21, then satisflos

aaf a% ‘r' 8+ rPw= (1 - &[a) 2 + --- + --- t __----____ ?i, = 0.

a2 a+ ha P(l - hIa) . ..(6)

Fk?n so, the d~fferentml equation does not provide R solution to the problem specified above, because of the nature of the boundar.~ cordltions, (h) ia particular. nlrect numerlcnl nethods, such as relaxation, do not appc-x to have been attempted".

2.2 Intefld Equations

As shoTa mKef. 3, g.3, the upwnsh at 3 point on the mng maybe cxprcssod 11). term of 1, the dmtr>hutlon of ltit over the w"I"P, by the equatmn"

1:(x, y, 0) _______--- =

U r-

. ..(7)

:* ra = ca + /s” (Y - y’ 1” + Daz ‘j

suwices a nn3 h dcnotc vnlucs on the upper and lower mrfaccs respectlmly. An alternative integral cquatzod+ (‘ii. P. Jones, 1951) is

___-_-______---__-_------------------- "The aims sign m the term exp[- wi.~/up"] 1s implied by the

boundary cond~tlon (a) in 3.2.1. '+This approach m bemgC%mmncd m the Nathenntlcs Divlslon, N.P.L.

-b. -

where E; = (x - x') and the mtegml i.s now taken over the w&e, Yl, as well as the :"ng, S.

Equation (7) may bo wvrltten

where v = d&J anl d is any convenient lenF;th. kernel K defined by equations (7) and (9)

The form of the 1s mkward for calculation

because of the inflrute range of mtegratlon. This dlfflculty has been elirmnatcd by E'atkzns, Runyan and "oolsto$ (-19x), who have shovm that

1 idx,, Y,) = --; exp(- wxo)~ $ Ki(uIyol)

- yo - gi [IJVIY,I) - L,bIY,l)l

0

. ..ilO)

where x0 = (x L xi )/a, yo' = (y - y')jd, K; and k are Pessei fun&lots, 31143 L, IS a modlfled itruve functron (Ref. 6, p.329).

To solve the Integral equattlon (Y), It LS necessary to use a collooatlon method. Then, grven 14 and v, K may be computed for

the approprrate values of x0 and y. on a desk machine. Pitching derlvatlves for a rectangula wmg of aspect ratlo A (wmg span/mcCsn chord) = 4 at Ifi = 0.866 z-e determined in this way in Ref. 7. Eight loading functions and eight collocation pomts are taken on the half ivlng (of chord d): without further approximations the equations have been solved %!hen v = 0, 0.3 ati 0.6. The results appear satisfactory, but the method of calculation 13 too lengthy for routine use -Lf only desk machines are used.

2.3 Apzroximatlons to the Kernel

Ii, as given by (IO), IS still a rather cumbersome function of four variables, so approximate forms of the kernel are sought. These are obtained by considerrng extreme values of the parar,eters V, M anil A.

notation) For v small, Lance' (1954.) has show that (111 the present

Kix 0, Yo) = a; exp( -in,) i 1

x0 iv i - --- - -------_-__- + __________m

L- JC yyxo" -1 ,py; Jzg---a--T + P Y.

v= ” - -- log --------(Go + ,py; - x0)

* 2 2(1 - Id R

va

!

ix 1 - J. + -- - -..

(

X0 - -- Y

M 7 - _--------- + O(V3/P6) 2 7 /s2 YI,?

)! u i p-y," J

P . . ..(ilj

The/

-5-

The assumption that v=1og v,v=, . . . are negligible leads to the OO-C2l led "low-frequency" theory in Ref. 9. ~anco8 has also consiaered tne case v ? 00 and gives the equation

Mpaexp(iv[IvPxo - * --

K(x,, yo) zz Pay//p I)

- --~-----------------------------’ + o(V-1) . ..,(I.?) l/x,” + py$/xqj + pyg - !ko)

X may sundarly be varied. when xo 4 0.

Then 111 -> 1, K vanishes The general expression for K, when x0 > 0, is

given inRef. 5, equation (47a). of v

The correspondir~ exlmnsion in Rowrs 1s found in Ref. 0, ecuation (13). Neither of these ap~xYximat10~

to the kernel is thought to be of prxtioal use in computation. M = 0 elves the case of mcomprossible flow. The kernel then becomes5

X(X ", Yoo) = ;; exp(- lVXo 1 J- -_"_- 1:,b IYO I) - ;T;+ (v lye / ) - L, (v IYo) )3 L lYol 0

_-- x0 iM/xa

- __--______ Y&'Xo" + Ycy

cxp(1ux,) k 0 - Yo" ---------- sa

exp(ivx,) 0 .

+ v_“_ i^

\ . ..(I31

Y: 0 exp(lvh)&~ .

-- 1 Methods, whxh have been proposed for dealing with wngs for

whrch the aspect ratio is very large or very small, have been revie~cc? by EcHmus'~ (19-q). Host of these theories on the oscillatmg fum.te wing we restrlctecl to either lifting-lme theory, incompressible flow or unpept mmg:;. Apad from the very gcnrrjl treatmnt b:r ikissnerll (1954.1, only the slerdcrwlng theory of Xerbt and La&ihl'2,13 (1353) end the recently published strip theory by E&ham14 (1955) reed bo mentloncd hem.' Beth those mcthods ap$y to wiugs of acbitmzy svce$ and taper without restriction on v or subsonic IV:,:. In Ref. I?, A 'is ncglectcd so that the first term of the differential equation (6) dlSappW?S; this approximation is not thou&t to be valid vhen A > 4. Ref. I)+, basod on two-dmcnsional thcnry, xould not be expected to apply who?1 A c 6 or when v 1s too hi. Thus no rclmble routine method of calculation exists for wmgs of present-day aircraft.

3. C!om~&~tional Problem

3.1 Use 0,:' the Exac<Kernel

Allon' (1953) has used the exact kernel in enuatmn (?) to reduce tho computation requrred to evaluate the dovMwash so that the major portion of the work lies in evaluating certain "mf'luence functions". I, II, J and JJ.

1 r>.- L- 1(x, Y, a,) = -

l-7

- 1 + cos $4 , ,. ---~ ---.._--__________-- ):

dGJ: - ---

x 0 1 + co9 $)" + 4Y= )

fxp[- $4.X,* - 1 + cos +)" + 4Ya](l + cos $) &,...(I41

where P 1s the usual angular chordmse co-ordinate,

x = lx - X~(Y')j/C(Y'), y = pls - Y'l/C!Y')

anal the frequency R.mmcter vii c(y’ )

a = -- ----- . I. P d II/

+Sea also the slmder-wng theory of D. Idaceleby (J. AC. Sol., July, 1956).

-6-

X 11(x, Y, h$,X/L) =

i 1(x,, y, Xl) exp

-ci3 [

T

2; (x0 -xl a, . M

-. . ..(15)

J is obtained by replacing t!x factor (1 + cos $) m (11 ) by (COS $ + CDS 2 $), and JJ by replacing I bjr J m (15,. t Th.x I and J are functmns of three paxmetcrs, and II and JJ of four. The chmf dlffxulty here lx s equatmn (33).

m the infmte range of lntegratmn m

Aiternatlvely, if the form of E in equntmn (-10) is used, only 'cdo functions I and J are roqared., given by the follming equations (ilef. 3)

I = ii+1 +z +I a 3 4-l J = t, ji+j,+j,!-j,i

. . . (16)

-\

Gp(l i cos 9) d$ I (P = 1, 2, 3, d, . ..(17)

G = 1

UY IL . --

C(Y~ 1

dY 4Y' \ p *--a--+P > , Ih UY C(Y’) -L --. I----

( Y

- ___-___--__________--- e;rO t

- 1 - . -- . ----- 1P d

/ rilp PP d _.

G, = -

/* dt

Since G, and G, are mnd~pendent of $, .4,

ii = GI, ia : G2, j = j = 0.

1 a . . . (I 9)

. . (ID)

-7 -

iv c(y') ia = ---

c i

x IV c(v') ----- 71.

cos-i (I -2x) sln$ exp ---

2M.x d I --=-- I (2x - 1 + co9 6)

w a

-lvd@- 1 + co9 gQ?l _I I

d$ - * (# + sin p) sin qi x 0

r

J

iv c(ylj exp G;; --;-- [(2x - , + COS 6) - M q~----------‘- - 1 + cos $)" +&Yaj $3

11 2ru c(y) ?r w J4 = - ---. ----- i (1 + oar $) slna$

C(Y’ 1 exp i ---. -----)(2X - 1 + cos 6;

M.lx a 0 2p= a

In these equations the infinite r-&c of integration no anger appears & all the integrations are amenable to numerical treabnent + .

:...(20)

Both of the above methods use the exact kernel, so that, provided enough collocation points are used the integral equation may be solved as precisely as required. Finite integrals, simlar to (201, would amso from equation (13) in the special case M = O?

3.2 Low-frequency Theories

Vita the a2)proximation that ualog u,u", . . . in (11) nre negligible, it is possible to achieve routine numerical solutions for compressible subsonic flow. Two widely-used mctl~cds are Multhopp s t 16

(1950) theory, developed in Ref. ?a and Fal!ffler's17 (19!J) vortex-lattice theory, developed. by Miss Lehrmn (1953).

Multhopp's theory involves the calculation of the influence functions -.

1 71

J

(2x - 1 + cos $) i = 1 + - ----- -_--- ---_-_ -------- __.___ -_-.--__ - (1 + cos&3#

7c 0 4(2x - 1 + co9 $1' + LJa

(2x-l +.os$$) --.-e7-------------------- (,o,$b + cos 2$)d$ ' q2x - 1 + co9 $i"+ !&YB

i . ..(21)

i

x $3 = -mj(Xo, y)=o *

-.I These functions, Curtis1g (1952).

tihich depend on X and Y only, have been tabulated by If more terms of (II) were include4 extra functions

would have to be tabulated to account for them. The use of two collooation points on each chor&?ise section is onvissged, but, if more were requrred, furthor functions k, 1, etc. nould bc introduced by replacmg the factor (1 + cos 9) or ~(COS 6 + cos 26) zn (21)

_____________-___---____________________-

*This line of approach has been developed. in Structures Department, R.A.E, Farnborough.

-a-

by (cos 2$ + cos 3$), (00s 3$ + cos l&), etc. The possibzlity of computing these in the low-frequency case has been considered byAlwaJj!O (19541, who has expressed these influence functions in terms of simple itc??ative operations sultablc for an electronic digital computer, the computation time being 1-2 seconds.

The use of a vortex lattice to evaluate w does introduce some unknown rrreducible error, but gives good comparisons with the results of Multhopp's theory, as shown inRef. 9. In the spscaal case of incompressible flow, Miss Lehrian21 (lgfd+) has extended the vortex-lattice theory to arbitrary frequency. The most general itiluence function 1s the downwash at a point (x , yJ,) due to an oscdlatory doublet-strip of width 2d with the rm -pomt of its leading: edge as a ongin:

w&, Yi, VI 1 - -___- - ----- = -- ( - X

U 4-x 1’ Yi + I) -H(-X$,Y -1) 1

+ iuexp(- .ivX,) exp(- iyX)IH(::, Y - 1) - H(X, Y$ + I

with

H(XI, Y1) = -', I KP

+ -- - -L-L x Yi XY

,x1 = x,/cl, Yt = Yi/d. 1 i 1

Thzs quantity arises from the integral over the wake in equation (8) ‘and has br:en tabulated by tho Nathematlcs Division, N.P.L.22 (1952). +4 treatment of the problem for mcdorately smallvsJ.ues of V, suggested by W. P. Jonesb, is to appoximate to equation (8) by replacing the factor

;e;kexp[- $,) by ;;;{$} ,

lPua Mu It can bo shown that the error 111 w is then of order ---- log -- ,

P4 P' Subjeot to this error, results applloable to compressible subsonic flow can be obtain& from an equivalent problem in lncompresszble flow. Recent calculations on this basis have been carried out for rectangular Vrmgs23.

Comparisons with the results in Ref. 7 suggest that at tigh M Jones' approximation 1s only val3.d for a very restricted range of Y, but +&ls may perhaps be improved by an Lteratrve procedure. The Structures Department of R.A.E., Farnboroughhave developed amethod of hancllmg the numerical lvork on a high-speed computing machine, the influence function V/U being evaluated from an expansion in V. The possiblo usefulness of the vortex-lattice theory as a general routine method should not be overlooked.

4. Genre-al Equations in Supersonic Flow

The integral equntion for linear35303 supersonic flow analogous to equation (7) is,

TJ

-I U

-Y-

N 4X ,YtO) = - '- f /1(x', y'iK(x - x', y - y')&'dy', u t?xJ si

ihxe

Y;(x-x',y-y') = 211m exp Z90

- R = J:y - 3")' + za, j3 = v'?-=? and S 1s that part of ti;e wing XISI~C the forlJcwd Mach cone of' (x, y, 0). g3.w a general treatment of equation (23j.

Watkins and Berman2'!+ (1955)

Another Integral relation which holds for M > 1 1s 1950)

25 @ward,

Q( x, y, z, t) = - J- idu',-y', t - Ta) + w(x', Y', T - Tb)

--------------------------_--- iQayl, 2X s SJ VqT - xoa - p"(y - y'ja - fizza

1

whore (s - ;:’ )M

- ---_ -- --- &c - x’ )ap(y - ,i,,- @,a

1 ,

Ta = --------- * ------___---------------------- p aa,

, . ..(&I P&

(x - x')M LG-Tv= p"(y - y'j' -/P-2 Tb = _------_- - -----__-_-__________----------- ,

62 aal P a%a I -1 an'3 S horc rcpxscnts that part of the plnno z = 0 mtorceptcd by the forwsrd Mach cor10. This ml1 give the pessure dlstrlbutlon in closed form ln the case of vung s wLth e~lircly nupersoluc edges.

The d~fforentlal cqustlon (1) still holds, though sinoe

M > 1 It LS now h.yperbolic, snd the boundary conditions are slightly chang& A treatment f some woblems using the differantuil equuatlons is gwon by Stewartson 28 (1950):

The methods adopted for solving such oquatlons differ widely zuxl bear little relation to those use3 in subsonlo flow? In many cases, such 3s the solutions obtalned for rectangular, delta ad arrowhead wings (Refs. 27 to 3O),'tho computing consists merely in substxtuting in forrrmlae for the requued quantltles. The derwatlves found In these papors sre given as power series In V; terms contalrdng log v no longer appear. Indeed analytical treatments of -pwt1cu1er planforms are commoner than general numerical methods.

5. Aeroelastic 3robluns

5.1 Subsonic Flow

T]I~ ct~octpmic loading of rigid 1Lftlng surfaces zn subsonic fla< is normally solved by a collocatxon methp_d, which relates certain loading functions 1 to dowwash functions w by a awtrix equation With complex coefficients

iT = r,i. The/ . ..(25) ~F-------'------------------------~-------------------------". See, however, the work of iilohardson, dwzussed in the Addendum.

- 10 -

The dowwash is no longs prescribed by equation (31, when the wing is supposed to deform under IDaad; additional term 111 a *p/az,

structural deformation then gives sn which is llncarly dependent on the aerodynsmic

loFadlz?g and expressible 1.n ma&.x form as

5 = ET, . ..(26)

so that instead of couation (25)

w + 7 = n-i.

Equations (26) ad (27) lead to the formal solution

. ..(27)

T = (A - E)-' F . . ..(28)

In general, the elsmonts of the matr~ A depend on the froquenq parameter u; m a flutter problem, separate cdculatlons would be neccbsary for salccted values of u, The task of evaluating the separate matrices A rammains the most formidable part of the computation.

In the spccwZt case of low frequency, the aerodynamic problem reduces to that of Multhopp'sY theory. by Richardson3' (19s.)

A slgluflcant advance is mdc who chooses loakng functions T at, sv, N

spectiic chardwlse loc:tions instead of the basic two-dlmensronal load dlstrlbutions thzt lead to the influence coefflclcnts in equation (21). Tho real sn3. imaginary parts of his ird'lucnce coefficients are linear functions of 1, j, etc. and il, jj, etc., rzspeck.vely. By careful choice of the chordw~sc locations, R~cbardson wrivcs at approximate influence ooefflclcnts

The accuracy of these amoxhmatc values increases ~3 N increases and is stated ko appear reasonable for N = I+.

5.2 Suoorsonic Flow

The theoretuxl study of aeroelastlc effect3 In supersonic flow has madly ld. to andyt~al treatment of particular planforms. l?or eXZJllple, solutions i'or rectwular32 ard trmnguLsr33 wings are extensions of those for rlgd wngs in Refs. 27 ad 2Y.

A gcnsral treatment of tne flutter problem for wmgs with subsonlo or supersoluc led- edges is proposal by Plncs, Cuandji snd Nmrq-ers (1955). The wing LS represented by a grid. of square boxes ad the lnfluencc of one bux B on another is detormaned by the quantity

R-11 =

. ..(30)

R' = J(x - xl)" - (Ma - I)(, - 9'11.

These/

-11 -

These pressure influence coef'f'iclents are evaluated bY a?proxlmatlon to the exponential ‘and. cosine terms, but an Axact expression lnvolv~ng mf'm%te ser~cs of Bessel functions and arc sines, due to C. E. Vatkins, 1s given zn Ref. 35. E?uatlon (30) only agplles when the box 1~s COmpletCly w%th%n the forvrard Nach cone of the recelvlny poltit. When the leading edge IS subsomc, lSv~ard's~5 method is used to gjlve a correction term. The procedure of Ref. 3!t. 1s well, suted to high-speed computmg maoinnery, but ~.s only valid for M >d2. Rectangular or dumorul-shaped boxes arc sugecsted means of extendin& the range of Siach numba.

6. Conclud;mR l7ew.rks

Fractwal methods are requrred for estimating the forces on oscll1ati.q wings oE the type used on modern high-speed -craft, that 1s of medium aspct ratlo and posszbly high sncepback. In su~erson~ flow, snalytrcal formlao exist for a wade range of particular planforms (§.&j. In subsonic flow, yract-ical methods exxlst only when the frequency pzameter u is sm,&Ll (B. j.25. possible (g.3.1 .

Extanslon to higher frequencies 1s thcoretlcally

Tho method described in Ref. 3 appe<szs to bo the most prormsz.ng; Ln fact, calculations usrng desk machmes have been carrwd out for a rectangular wing of aspect ratlo A = 4 at M = 0.866 u&th v = 0.3 and 0.6, the computation tlmo bcug four months for each frequency. Thus thcory7zll probably cover the praotzcal range of LJ unless &i IS very near unity, when the assumptrons of lmnearucil theory are invalid and accurate solutions arc of doubtful merit. The method is uneconomic, unless tho calculation can be mecharucad ?artvzularlythe evaluation of the mf'luonce functions in ewatlone (141 $0 (20).

Simp1Lfication.e. result from c>nsidermg extreme values of the parameters u, A ard 1.1 (5.2.3;. Equation (12) for u -> OQ probably only a:7plics to values of v outsldo the practical rarye. High-as?ect-ratlo theorjr (A > 6) an?l low-aspect-ratio theDry (A < 4) we uw.ppllcablc to the t.Ype of vings mentioned abovc. The approxxlmatlon to the kernel when &I = 1 LS not thought to be of practical Use. The low-speed case (14 = 0) appears to bz sntlsfactorily covered by vortex-llttrcc theory though It could be treated by methods slrmlar to those a? Ref. 3. The possiblllty of extending vortex-lattice theory to cover the practroal range of UN/(1 - X2) remains to be studled ($3.2).

In many respects lfr~ng elastlczty does not affect the basic aerodynxnic computa*t+on 111 subsonic or su>rrsonlc flo%V. 1:owever, the low-frequency subsonic theory qf Rcff. 31 (B.5.1) and the box-yrld method of Ref. & for M 3 6 (b.5.2) both appear to lcad to calculations veil sultcd to programning on computing WcblerY. Though appronmate as desk computatrons, these methods :ihen mecharuzed Sh3uld

become accurate v&thin the fromovork of lincar~ed theory. Exte:slons of Ref. 31 to h+-er frequencLes and of Ref. 3!+ to Eiach embers bcluw

v? we among the most import.mt problems 113 thcorctlcal flutter research.

Kefcrences/

- 12 -

5.

1

2

4

5

6

7

8

9

Author(s)

L. Howarth (Fditor)

IL w. lb.ngler

w. I?. J0hC.S

C. E. Watkins, H. L.Runysnand D. S. Woolston

G. N. vatson

P. E. A. Aoum

G. N. Lance

H. C. Garner '

w. Eckhaus

Ii. G. KUessner

References

Title. etc.

IvIcdernDevel'o~ents in Fluid Dyearmcs. High Speed Flow (2 Vols). Oxford University Press. 1953.

A method of calculating the short period longitudinal stability derivatives of a wing m hncsrxxd unsteady compressible flm. R.A.E. Report A.oro.2@, A.R.C. 15,316. Juno, 1952. 'Jo bo publmhod as R. & Iti. 2924.

Theory of liflmg s~Jrf3cos osc11JatmE; at fplcral frequencies in a stream of hi& subsonic !kch number. With an Appctiix by Hiso D. E. L&man. A.R.C. 17,m!+. 3Cth August, 1?55.

Oscillating :.in,as LT compressible subsonic flm,. A.R.C. 14,336. October, 7951. To be published as R. 6: MI.285

On the kernel function of the integral equation relating the lift and dmmwash distributions of oscillating finite wings u1 subsonic flovi. N.A.C.A. Technical Note 3131. January, 1954.

A Treatise on the theory of Bessel functions. Cambridge University Press. 2nd. Edition. 1948.

Forci?s on a rectangular wing oscillating in a stream of high subsonic Mach numbor. A.R.C. 18,630. August, 1956.

The kernel function of the lntegrel equation relatirq5 the lift and dowlvrash distributions of osc~ilatmg fwte wings s.n subsonic 170~. J.Ao.Sci. Vo1.21, p.635. September, 13%.

Xulthopp's subsonic l:fting surface theory of wmgs 3m siow p1tdling 0sc111at10ns. R, & K. 2885. July, 1952.

On the theory of oscillating aerofoils of finite span in subsonic flmr. N.L.L. Report F-1 53. July,13%.

A gcnerel method for solving problems of unsteady surface theory in the subsonic range. J.Ae. Sci. Vo1.21, 1~17. Janwary, 1954..

Aerodynenic forces on oscillating low aspect ratio wings in compressible flofl. K.T.H. Aero T.N.30. January, 1953.

13/

&.

13

14

15

16

17

18

Author(s) . Tltie. etc,

The oscillatin& v&ng of low aspect ratio, results 2nd aumliary functions. K.T.H. Aero T.N.31. January, 1954.

Stszp theory for oscillating swept wings in compressible subsc~~~ flua. N.L.L. Report F.159. October, 19553

mc a?pllc‘%tlon of l.lfxl~$$ surface theory to the calculation OF the ac~rxlynamic forces

Et&w on a w,ng of flnlte aspect ratlo oscillating hz+xmolucd.ly in arbitrary elastic Dl0d.W. Hawkm A12-craft Ltd. Daslgn Dept. Report 1196. October, 1953.

IZethcds for calculating the lift distrlbGtion ;t ~~s~8~,!9ub~~~~r;:f;c~~.surf~~ theory).

W. Eckhaus

D. J. hllcn

H. Multhopp

V. M. Falkner

IDI-IS E, Lohrian

19 8. R. Curtis

20 G. G. AWay

21 Dorzs E. Lehr~.sn

22

23

a.

26

- 13 -

The solutron of lifting plane poblems by vortex lattice theory. R. 0 M.2591. September, 1947.

Calculet~on of stabdlty dsrxvatlves for oscdlata win&s. A.R.C.q5,695. Fcbruzry, 1953. To be publxhed zs R. S M.2922,

Tables of Multhopp's Influence functxons. N.P.L. Mathematxs Divxion Report Md21/0505. Nay, 1952.

Ii& speed automatxc computation of Etulthopp's lnfluencc functions, December, 1954. (Unpublxhed).

Calculation of flutter derlvatlves for v?lss of general plan-form. A.R.C.76,445. Janusry, 1954. To be publish& as R. SC 16.2961.

St&f of Xnthcmatlcs Downwash tables for the onlcdat~.on of Division, N.P.L. aerodynmic forces on osc.~llat~ng wings.

N.P.L./Acro./229. ~dy, 1952.

DorisE. I&riJn Calculatd derivatives for roctan@ar TIws oscdlating in compressible subsonx flo?!.

A.R,C. 18,588. July, 1956.

c. E. watlam ana On the kernel function of the integral equation J. H. Berman relating lift and dovnwash d3stributlons of

osclllatlng wings in supcvsonx flow. N.A.C.A. Tcchnxal Pate 343% May, ll35.

J. C. Evvara Use of sourbe dlstrlbutlons for evaluatmg theorehxl aercdynanncs of thxn fxxte wings at supersonxc speeds. N.A.C.A. Report 951. 1950.

IL Stewartson On the l~~z-slzd potontlal theory of umteady s~p0rsoru.c matron. Q.J.Xach. & Ap>. Maths. Voi.3, ~9182. 1950.

27/

It?.* Author ( 3)

27 C. E. Wr.tkins

28 H. C. Nelson

29 C. E. Watkins and J. H. Berman

30 H. J. CunnFngham

31 J. R. Richardson

32 H. C. Nelson, R. A. Rainey snd C. E. Watkins

33

34

35

36

37

38

C. E. Watkins snd J. H. Berman

S. Pines, J. Dugundji and J. Neuringer

s.Pinesana J. Dugurd.ji

H. L. Runyan and D. S. Woolston

J. R. Richardson

Ta Li

- 14. -

Title. etc.

Effect of aspect ratio on the sir forces ard momerts of harmonically oscillating thin rectangular rungs in supersonic potential flow. N.A.C.A. Report 1028. 1951.

Lift anl moment on oscillating triangular anl related wings with supersonic edges. N.A.C.A. Technical Note 2494. September, 1951.

Air forces and moments on triangular ard related. wings with subsonac leading edges oscillating in supersonic potential fiow. N.A.C.A. Report 1099. 1952.

Total lift and pitching moment on thin arrowhead wings oscillating in supersonic potentiel flow. N.A.C.A. Technical Note 333. May, 1955.

The calculation of flutter coefficients for elastic modes and control Surface movements using the MulthoppGarner theory. Fairey Aviation Co. Ltd. Tech. Off. Rep.136. A.R.C. 18,261. October, 1954.

Lift and moment coefficients expanded to the seventh power of frequency for oscillating rectangular wings in supersonic flow and applied to a specific flutter problem. N.A.C.A. Technxal Note 3076. April, 19.54.

Velocity potential and air forces .asso~ated with a triangular wing in supersonic flow, with subsonic leading edges, atd deforming harmonically according to a general crupdratio equation. N.A.C.A. Technical Note 3009. Septanber, 1953.

Aerodynamic flutter derivatives for a flexible wing wit" supersonic and subsonic edges. J.Ae.Sci. Vol.22, p.693. October, 1955.

Aerodynamic flutter derivatives of a flexible wing with supersonic edges. Aircraft Industries Association ATC Report ARTC-7. February, 1954.

Method for calculating the aerodynamic loading on sn oscillating finite wing in subsonic and sonic flow. N.A.C.A. Tech. Note 3694. August, 1956.

A method for calculating the lifting forces on wings (unstead subsonic end supersonic liftrng surface theory 3 . Fairey Aviation Co. Ltd., Tech. Off. Rep. 165. A.R.C. 18,622. April, 1955.

Aerodynsmio influence coefficients for sn oscillating finite thin wing in supersonic. flow, J. Ae. Sci., Vol. 23, p. 613. July, 1956.

- 15 -

ADDEVDUM ---

Three reFOI'tS (Refs. 36, 37 and 38) have como to the notlce of the authors su-~ce the paper was wItten.

Runysn and Woolston(36) have applied the form of the kernel derived 1n Xof. 5 for general frequency parameter, V) and subsonlo Xach number, I!$? to wmgs of arbitrary plan form. This paper appcc\rs to offer a good chance of prolldmg a practloablc routux method of calcu1at1on by c0110cat10n. The assumed loading uwolves the replacement of a contrnuous ohordwlse loading by dlsorete loads determined sunllarly to thoso used xn vortex lattice theory (Refs. 21 and 23). It 1s assumed that the upwash on the wing 1s represented accurately enough when the kernel 1s oxpandcd and 6th and higher powers of the frequency parsmeter are neglected. A modlflcatlon of the method to the case M = 1 1s also glvon. Tho results of calculations on a rectangular l~lng of aspect rat10 A = 2 are presented for the complete subsonlc range and V = o.l&L+. Cdlculatlons by the method of Ref. 7 for the ssme wng arc bolng made at tho N.P.L. for M = 0.866 and IJ = 0.3 and U = 0.6. The mechanlzatlon of these calculations 1s being consulered by tho Mathcmatrcs Dlvlslon of the N.P.L.

w regard to the general eqdatlons of suporsonlc flow,

Hlohsrdson 37 has presented a unlfled approach for both subsonlo and "Uuporsonlo Mach numbers. In contrast with Rofs. 27 to 30 he has suggcstod a collocation method for any plan form 111 suporson flow. Ref. 37 1s an Important step towards the extenslon of Ref. 31 to hlghcr frcqucnolos. The cvaluatlon of tho kernel function 111 Appendu I of Ref. 37 1s being programmed for high-speed computing machuxry 111 the Structures and Mathcmstlcal Sorvxcs Departments of the Royal Axcraft EstabllshJncnt.

Another lnportant problem z.n flutter research, the extcn-lo of Ref. 34 to Mach numbers below J?, has been carried out by Ta L1736T, who lnoludos the case of a subsonx leading edge. He has obtalnod hlzhly callsfactory results for two-dunenxonal flutter cocfflclents when M & 1.1.

, .c

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