1、.fI .ffk “ .7TNATIONAL ADVISORY COMMITTEE- j-4/, deriratlon of the integral equation which involves thiskernel functio7 h originally performed elsewhere (see,for example,N.IC=i Technical .emorandum 979), is reproduced as a_appendix. Another appendix gi_,es the reduction of the formc_the kernel fu_lc
2、t_on obtained herein .for the three-dlmensionalca.,e to a known result of Possio Jor two-dimensional flow. Athird appendix contains some remarks on the evaluation of thekernel _/unction, and a Jourth appendix presents an alternateJorm oJ expression Jor the kernel functlo_x.INTRODUCTIONThe analytical
3、 determination of air forces on oscillatingwings in subsonic flow has been a continuing prol)lem for thepast 30 years. Throughout the first and greater part ofthis time, efforts were directed mainly toward the determina-tion of forces on wings in incompressible flow. These effortshave led to importa
4、nt closed-form solutions for rigid wingsin two-dimensional flow (ref. 1), to solutions in terms ofseries of Legendre functions for distorting wings of circularplan form (refs. 2 and 3), and to many approximate, yetuseful, results for wings of elliptic, rectangular, and tri-angular plan form (see, fo
5、r example, refs. 4 to 12).Although these results for incompressible flow play ahighly significant role in applications of unsteady aerody-namic theo_-, the advent of higher and higher speed aircraftduring the last 15 years has brought a growing need forknowledge of the effect that the compressibilit
6、y of air mighthave on unsteady air forces, or for analytically derived un-stead)- air forces based on a compressible medium. Thetransition to results for a compressible fluid from those foran incompressible fluid is not likely to be accomplished byi Supersedes NACA T,N“ 3131, 1954.394619-56applicati
7、ons of simple transformations or correction factors,such as the well-known Prandtl-Glauert factor for stcadyflow. This (lifi3culty is associated with the fact that the timerequired for signals arising at one poi,_t i,l the medium toreach other points gives rise not only to changes in magni-tudes of
8、forces but also to additional phase lags betweeninstantaneous positions, velocities, and accelerations of thcwing and the corresponding instantaneous forces associatedwith these quantities. In order to ohtain results for thecompressible case, it therefore appears necessary to dealdirectly with the b
9、oundary-value problem for this case.The boundary-value problem for a two-dimensional wingin compressible flow has been successfully attacked from twopoints of view. First, by consideration of an acceleration orpressure potential, Possio (ref. 13) reduced the problem to thatof au integral equation re
10、lating a prescribed downwash dis-trit)ution to an unknown lift distribution. The kernel of thisintegral equation, which is a rather abstruse finntion, wasreduced to a form that, except at singular points, couhl beevaluated. Schwarz (ref. 14) later isolated and determinedthe analytic hchavior of the
11、singular points of Possios resuhsand made fairly extensive tables of the kernel function.These tabular values were used by various investigators(for examples, refs. 15 and 16) to obtain, by numericalprocedures, initial tables of force and moment cocflicicntsfor oscillating wings in compressible subs
12、onic flow.The second successful approach to the solution of theboundary-value problem for a two-dimensional wing (seercfs. 17 to 19) is achieved by a transformation to ellipticcoordinates followed by a separat ion of variables that reducesthe boundary-value problem from one in partial-differcntialeq
13、uations to one in ordinary differential equations of the._Iathicu type. The solutions iuru out as infinite series intcrms of 5athieu functions. Numerical results obtainedrecently by this procedure a_ee with results previously ob-tained 1)3“ the numerical procedures using the kcrncl func-tion (see, f
14、or example, ref. 20).With regard to boundary-value problems for finite wingsin compressible flow, it appears that the procedure of sepa-ration of variables eouhl be a feasible approach only forwings of very special plan forms such as a circle or an ellipse.In any case, the development of the appropr
15、iate mathe-matical functions for a particular plan form wouhl hecomeProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-2 REPORT 234-NATIONAL ADVISORT COMMITTEE FOR AI:;RONAUTICShighly involved. On the other hnnd, it appears that approxi-mate procedures
16、stroll:u“ to those used for two-dimensionalwings might afford an approach to solutions of these prob-lems which, though laborious, might be handhd by routinenumeried methods.The kernel function of the integral equation relating pres-sure and downwash for the three-dimensiomd case appearsas an improp
17、er integ,al. The purpose of this report is totreat and discuss this kcrlwt function. The improper integralis red(iced to a forln that can bc accurately evaluated bynumcrical procedures. The form and order of all its singular-ities ave determined and an expression for the kernel functionis derived in
18、 which the singularities are isolated. Specialforms of the kerncl for the sonic case (M-= 1), the ineompres-sibh, case (3/=0), and the steady ease (k-0) arc prcsemcd.A series expansion in powers of the reduced-frequency param-eter k is developed.The availability of the kernel in a form which can ber
19、apidly evaluated makes possible the use of numerical pro-cedures, similar to those used in tile two-dimensional case,to obtain aerodynamic forces for finite wings.L,LK(.ro,Yo)K (xo,Yo)kLo,L_lMPr = 3._F_o_+ z“SlI“V(z,v)x,y,z,_,:?Yo=Y-n=,_-M _-V-_ 0 , M dO4,PSYMBOLSvelocity of soundIlankel functions o
20、f second ldnd of zeroand filet order, respectivelymodified Besscl functions of first kind ofzero and first Order, respectivelyBessel function of first ldnd of zero ordermodified Bcssel functions of second ldnd ofzero and first order, respectivelykerlnl function of integral equalionsingular part of K
21、(xo,Yo)reduced-frequency parameter, ho/Vmodified Stmtve functions of zero and firstorder, respectivelyunknoult lift distributionreference lengthMath number, V/cpressureregion of x/i-plane occupied 1)y wingtimeforward velocity of wingamplit ude funct ion of prescribed downwash,w(z,y,t) =d_(x,!l)Carte
22、sian coordinatesEulers constantvelocity potentialacceleration potentialfluid densitycircular frequency of oscillationANALYSISINTEGRALEQUATIONANDORIGINALFORM Or KERNELFUNCTIONThe main propose of this analysis is to treat ill(, kernelfunction of an integral equation that relates a known orprescribed d
23、ownwash distribution to an unknown lift dis-hibution for a harlnonically oscillating finite wing in com-pressible subsonic flow. The intcgrfl equation referred tocan be obtained 1)y employing the I)randtl aeeelerationpotential to treat linearized 1)oundary-vahw wol)lems foroscillating finite wings b
24、y means of doublet distribulions.Dcrivation of this integral equation from the linearizcdboundary-value problem for a wing is a preliminary taskthat has t)een clone elsewhere (see, for exaInple, ref. 21), butit is reproduced herein as an appendix for the sake of com-pleteness.In keeping with the con
25、cepts of lineu theory, the wing isconsidered a phme impenetral)le surflwe S which lies neqrlyin the Xy-l)lane as indi(.atcd in sketch 1:V _J/Sketch 1.The _,y,z coordinate system and the surface S are assumed tomove in the negative x-direction at a uniform velocity V.In terms of these coordinates, tJ
26、m integral equation may beformally written as,s(Uwhere _(x,y) is the amplitude function of the prescribeddownwash, K(x,yo)=K(x-_, y-n) is the kernel functionand physically represents the contribution to downwash ata field point (x,y) due to a pulsating pressure doublet of unitstrength located at any
27、 point (_,r/), and L (_,_) is the unknownlift distribution or local doublet strength.I:lProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-3ON THE KERNEL FUNCTION FOR FINITE WINGS IN SUBSONIC FLO_“The kernel funelion may 1)e mathem_!ivally drfim.d by th
28、efollowing improper integral expression (see eq. (A12),appendix A) :i_, f eiz(x-M_ ;_) (lX (2)K (x, 7,.)= lim O:2 e :_(, J-= _ X2+_:!/o24-f12z2where 3 is _Iach number, 5=,1-3/r:, _=%/1“132, _ois thecircular frequency of oseilution, l“ is the velocity, andX is the variable of integration. Evalualion
29、of this inte_alconstitutes a main difIi(ulty in obtaining aerodynamiccoefficients for oscillating finite wings in compressible flow.The present analysis is therefore devoted to reducing it to aform that can be accurately evahlated by numerical pro-eedures combined with the use of tabh,s of certain t
30、almlatedfunctions. The form and order of all its singularities aredetermined, and an expression for the kernel fimction isderived in which the singularities are isolated.REDUCTION OF THE KERNEL FUNCTIONIn considering the reduction of the ,:ernel functionK(zo,yo), the integral involved can, for conve
31、nience, bewritten as the sum of two integrals, namelyf f, ei_(X- M_f _-_)_ ,Lo (V,/_) (12,Note that the end result indicated in equation (12) is in-dependent of Maeh number. The second integral in equa-tion (11) may be written in another form as- - dr= dr (13)This integral has not been redueed to do
32、sed form; however,it is nonsingular and can be readily handled by numericalmethods.Coml)ining equttions (12) and (13) gives tin, followingexpression for F,:,.,o-F,_yo=q-za)J-L - (14_r “ dr (14)By performing the differentiations indicated in equation (4),there is obtained for the first part of equati
33、on (4) the follow-ing expression :, * ) b( )lm ._-25_2- . w rri w w ,-o 0z I_0 7yo -ff Flyo -L,?e Y -vlYo T_rae-vlddr (15)All terms of tiffs expression other t.han the integral maybe evaluated at small intervals of yo flom existing tables,exempt at yo=0 where the function is singular. The integralis
34、 well bdmved and can be aeeuratdy evaluated by numericalor approximate procedures. The type and order of thesingularities at yo=0 are discussed in a later section.Evaluation of Fa.-In order to reduce the integral F2,equation (6), it is convenient to make t.he substitutionX=r sinh 0 (16)so that- 1 .t
35、_/*sinh -F_=J0 r et_,r(sinhO_McoshO)dO (17)Noting that z appears only in r and performing the differen-tiations indicated in equation (4) yiehls- l YO/x2 7:, i_a Osinh -O -_ _ ,w, / eiv0i(sinh 0-5I eosh 0) e_-_I(_,_h0-.vo_h 0)dO Xo d: (_o-i_)o-_ ,.o-lyoi3o , ,_w_“._.2“ y0 _x0 +_ Yo“ 4- ,%inh=- “_ 7
36、,Z e (,-,w,_)_ “ | o,_01B2eosh 0-(cosh O-M sinh 0) e_ol_l(sInhe-Mcosh 0)dO_._,-_)-_-,o,1-.-:_ f ,“ eosh0_=_,(_,._0-.oo._.,do (18)or, by reverting completely to Cartesian coordinates throughequation (16), there is obtained(o!e;_ =_ _-1. Expressedill terms of :“uand 7/+ equation (25) l)ccomese .f-(._o
37、+_X, , 2yo)_,_,0,v0J_ z_ 1. v_?_:+/vo_ -_ ,.,-:+/:/l,.(k. k _, -;-, -_ J2(o)+log E (27)wln, rl, hom equation (25),.f,(0)=_. 0-0-_(leos=+Sin00) (2s)k(l-sin 0) , k cos _0.f:(0)-log 2(i-3i) -log 2(1-2:11)(1 +sin 0)Although of no particular significance in applications, it isof intcresl to note that the
38、 quantities.f_ and .f2 each haveminimum ,-alues (I.f,l,_*,=_ and ,J2l,=log _:1/)atO=-r/2, which corresponds to points directly ahead of thedoublet position; and, as 0 increases from -_-/2 to + r/2, thevalues of these quantities continuously increase from theseminimum values to infinite quantities as
39、 follows:f, i/_!=lim _L (-)=li,n 12#1, co+(;_OI,+,0-,(-)f2 (-_)!=lim log .Tlms K(xo,yo) is singular for 0=_-/2 even when the distanceflom the doublet is not necessarily of zero order. Thisimplies that. the doublet l)roduoes a wake of discontinuousdownwash that extends downstream flom the doubletposi
40、tion to infinity.With knowh, dge of the singularities involved in the kernelfunction K(:ro,y0), an expression can t)e written in which thekennel is separated into a singular part and a nonsingularpart (as was done by Schwarz, ref. 14, for the two-dimen-sional ease) as followsK(Xo,Vo) =-K(.ro,yo) - K
41、 0“o.Vo)+ K (xo,yo) (30)where K(xo,yo) is dcfined in equation (20) or (22) ande-“:“ r _,z, ,/3 ,io _,ro+ ?XK(.,_,71.)=-I=, - ;, : ; _ o L ?/,*o%_“uo_ -,:ro-t-_-:/xo- 3/_ x, +.8 y, Z“_lo,“2if“ ,_xu2-F_“-ffo2 2 _ 2(1-.11) jor ill terms of e and 0, introduced by equations (21),(31)K,(_,O)=e_-;tl ,In 0
42、_2 -_- (Sill 0-3I)- d(1-sin 0) ,-log _4_ -si2 0)12 2(1-M) I (32)The term K(.ru,y_)- K(x0,yn) in equation (30) is a continuousfunction for all vahles of k, Xu, and y0 and for values of 3/intile range of 0 O,yo=O). It is to be noted, however, thateach term of K(xo,yo) possesses a simple indefinite int
43、egralwith respect to y0 or with respect to _=Y-Yo, a fact thatmay be useful in some numerical applications. Ttw mannerin which these integwals are to be evaluated is indicated ina subsequent section that deals with steady flow. Thelinfiting values at y0=0 of K(:ro,yo)-K(xo,yo) for bothsubsonic and s
44、onic flow are given in appendix C togetherwith some remarks on evahlation of the kernel function.TREATMENT OF THE SONIC CASEBecause of its special nature, tile borI. Front the form of F_ given by equation (14),tlinl F1 = hIn Ko(_,yo+,)-wrlo(w,yo-rz)3I_1 31_l( “,V / /“ L V /, +, r-,o :co,- ._rl +,il
45、_ -f dr (33)But since (see ref. 22, p. 172)_ : cos D- d_=-Ko(i)d,_ -_1-l- 7_ (34)lli l.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-ON THE KERNEL FUNCTION FOIl FINITE VINGS IN SUBSONIC FLOW 7and (see ref. 22, p. 332) f= sin _r dr=_rit,j, -,i 1+ r:
46、5 2- (-0-L(f) (as)il may l)e con(hl0) (aS)From physical considerations, the right side of equation (38)is to be considered zero for x0e obtained if the limil under consider-at.ion wore sought fiom theory of supersonic flow, _I.The integral in equation (38) cannot be completelyexpressed in lerins of
47、known fimelions. Furtwrmore, sinceit is singular at its lower lilnit, furlher treatment is requiredto reduee it to a form such that its derivatives with respectto z can be numerieally evaluated. For this purpose theintegral may be written as two integrals, namely(F_)_.,=0 (l:, y0=+ P) _ :_ (Z, y0+ P
48、) -L,(X, ry0_+ Z_)-2(43)Differentiating this restih twice with respeet to z and thenset, ling z=0 gives_ K,(CyoI)-(_Y“_ _ - i:,.,7001 _YJ,=o=Y“;E2_o I ll(k Y“ 13- Z(JY 3- (44)Differentiating equation (41) twice with respect Io z andsetting z=0 gives f(oW“_ _“_ “_- i e- “ w-“ dxj (46)k“yJJ* _eEquations (44) and (46) me combined to give baF“Then, in accordance with equation (4), there is obtainedfor K(xo,yo)_-l :For a_o0,K(.%,yo).,=,:_ “rt was to presenl the kernelfunction of the inlegral equation relating lhe downwash tothe lift distributio
