1、, -. -i.,. .4.,- ,. - -. .TECHNICAL MlilMORANDUIUIS,.NATIONAL ADVISORY COMMIil?TE13FO# AERONAUTICS# .,. ,.,-.,- ,!.%;m“,G33:NI?JIIALAIRFOIL THEORY.,!,.,By H. G. Kiissner,.J- Iiuftfahrtforschung, vol. 17; No. 11/12, December 10, 1940Verlag Ton R. Olde,pbour:.i.,.”J.iProvided by IHSNot for ResaleNo re
2、production or networking permitted without license from IHS-,-,-/:y,$. NATIOitAL ADVISORY COMMITTEEw.303 AERONAUTICS“l!E”CHNIC”AL”MEMORANDUM No. 97”9-”-”GENERAL AIRFOIL THEORY*By H. G. KiissnerOn the assumption of infinitely small disturbancesthe author devekclps a generalized integral equation ofai
3、rfoil theory which is applicable to any motion and com-pressi-ole flv.id. Successive specializations yield vari-ous simpler inte$ral equations, such as Possiofs,Birnbs.ums, and i)randtl is integra,l equations, as well asnew ones for the wing of infinite span with pericdic down-wash distribution and
4、for the oscillating wing with highaspect ratio. Lastly, several solutions and methods forsolving these integral equations are given.:. INTRODUCTIONThere are a number of airfoil theories which holdtrue in two or three dimensions, are stationary or nori-stationary, and allow or disallow for the compre
5、ssibilityof air. All these theories have one thing in common:They are, strictly speaking, valid only for infinitelysmall disturbances; hence the airfoil must be assumed asinfinitely thin “and with infinitely small deflections froma regulating surface , the generating line of which isparallel to the
6、direction of flight. Then the regulatingsurface itself:,can be approximately considered as theplace of the wing and the ar”ea,of discontinuity emanatingfrom its trailing edge. Up to the present time, a planehas been commonly chosen as a regulating surface, butthis restriction is not necessary.Follow
7、ing the temporary interest attaching to thev“ortex theory, the introduction of Prandtlls acce.le.r.?t.i.u .-potential rnadq the ol,d,o,$eg$iql,teo.ryapplicable to air-”foil theory. The particular advantage of this method over,.”*lAilgemeine Tragf15chentheorie .II Luftfahrtforschung,vol. 17, no. ll/1
8、2,.December 10, 1940, pp. 370-78.:,. - Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-2 NAGJVTechuical MtirnbrariduiNo. 979. .the vortex method is that the C_9_rnRE.S,S,.$,$.Yof air canbe taken into account. BY th.is,method the wing iS re-placed by
9、an arrangement of acoustic Ir?i.a.tors.11 Thesole essential restriction of thzs theory consists in theassumption of moderate fields of sound, that is, smallinter.ferenees. Then the classical wave equation(1)is ,applicable for the velocity potential and the sound.pressure of a quiescent source distri
10、bution. Their solu-tions have already been explored in all directions. Itsapplication to the moving airfoil is achieved with theaid of the well-known Lorentz transformation, the soleinvariant,Ptrig the speed of radiation c, which, in thecase in question, is equal to the velocity of sound.The setting
11、-up of the integral equations of the air-foil theory is a preliminary task, which is definitelyachieved by the subsequent expositions. But this pre-liminary work alone accomplishes little without attackingthe purely mathematical main problein, namely, the solutionof these integral equations without
12、entering into newdiscussions every time regarding the method of derivationand its physical significance.2. THE VELOCITY POTENTIAL Ol?THE ELEMENTARY RADIATORThe wave equation a75 = O has, as is known, a verysimple solution fors spherical wave that sprciads ,out ra-dially at speed c from its source. T
13、he solution reads:,(-:) , 2)(Do*f t,“ .,., where r denotesthe radius, t the time, and f an arbitrary function. Such,a spherical wave is producedby an elementary radiator of zero order, ,wich representsa“simple point source. The velocity potential of radia-tors of higher order fo3.lows from (2) by pa
14、rtial deriva-tion alongqny coordinate directions.The airfoil which is to manifest a pressure differ-ence between its two surfaces is best replaced by a su-Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-.NA,CA Technical Me.mor:an*dti-No:.9?9 3.perpos
15、ition” of.raditors of the first order (so-calleddoublets) whose axes “are normal to the airfoil.” lf.ith ndenoting the direction of the normals, the potential ofa doublet of the superposition iS:(3)The sound ,yressure of the. field of?sound is ,(4)hence it satisftes equations DXJ = O also because p-
16、constant for very small disturbances.3. THE LORENTZ TRAUSFORMATION .The arguments sofar have dealt with the radiatorat rest at infinity. j,Toreach-the pressure field of theradiator moving at constant speed V the PotentiaThe analyzed element da lies”in the zero point ofqur coordinate syste,m that is
17、at x, y, z = O. . .5. “THE GENERAL INTEGRAL EQUATION OF AIRFOIL THEORYThe location of our elementary radiator is now shift-ed to,any.point of the airfoil with the coordinates s .Y(n), zn)* Coordinate is measured parallel to the,generating ,liqe of the regulating surface; hence alongthe. x axis. Coor
18、dinate o is so chosen that, after de-velopment of the regulating surface .in a Panes andn form a system of Cartesian coordinates, and the sub-stitutionx Y zx-g Y- Y(q) z - z(m) 17)must be effected in (16).”.The element of the surface of the airfoil is da =tida. QrdinariJy Y .will.still lie a fuction
19、of andProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-?-”-,., , ,., ,NACA !Jechnical Memorandtirn No. 979 ?. .,.- . . .q because “-of the variation of doublet intensity on theairfoil. The “differentiation along the direction of thenormals “can be div
20、ided in %a a dy+ G dz z- sin a() -$ + cos a(i)%ay, dn az dn (18)where angle a(q) between the direction of the normalsand the z axis is a function of the curved coordinate n.Posting (17) and (18). in, (16), followed by integrationover the airfoil, gives the complete term of the velocitypotential of t
21、he airfoil at:,.I=x-frJJ a(x,y,z,t)=*( ,“d!ddq”sin a(n)a$+cos a() w)F1 m X1-x+f X1$ Xla+(l-pa) (y-y(q)a+(z-z (1-1)2y t.,a?t+. -v c(l-2) C(I - 2) )(19). .+(1- P2) L-(Y - Y(n).)2 + (z - z(d):On the other Band, the p,ressure jump TT on the air-foil is proportional to the intensity of the doublets.The c
22、onstant factors in (9) are precisely so chosen thatn(g,n,t) = pvY(5,n,t) “ (20)This conforms to usual practice: lift, positive upward,downwash, positive ”downward. Literature at times quotesthe more abstract? downwash, po”sitive upward, in which. case the prefix of the right-hand side of (20) must b
23、ereversed., “Equations (1.9) and (20) represent the most generalintegral equ,atio.nof the airfoil theory for small dist-urbances that can be used for computing thepressurejump - for a given down”wash. E“quatio.n (19) representsa boundary value problem. Its solutiori-rests on”the factthat. the ,downw
24、ash,on the airfoil itsel”fis given by thetype of,motion and forq change of the” airfoil in firstapproximation. Assume that q = n(x,.n,t). is a small de-flection of the 00 For in any event the downwash must disappear atinfinity. In the divergent method ,ofyriting the down-wash on the airfoil ultimate
25、ly assumes the form:b) It can be assumed that the regulating surface isa plane, say, the xy plane, for instance. Airplane wingsare usually. flat structures. ,Then a= O, y()= , andz(q) = o. Nith specialiaztion a) z = O also. This as-sumptin itself effects a substantial simplification inthe,eguation f
26、orm and has been practically always intheqce,harrnoqiq oscill.ations,of the airfoil. The preemi-hent importance of the harmonic solutions of the wave.equation in.,physicq is an es”tablishedf act. Once harmonicsolutions of the integral equatio,n,are known, solutionsfor any time rate of change of down
27、wash can be found bysuperposition whereby the rLalace transformationplays a proinent part. Proceeding to the limiting case,of”verysm?ll oscil,lati.o.and putting. .y = y(t)results1111111 II ImIn I I I., I . . .I. m m, , . ,. .,.,.- . ,-, I . . . . . . , ,. .- - .-. -.-.Provided by IHSNot for ResaleNo
28、 reproduction or networking permitted without license from IHS-,-,- _ -10 “NACA !l!ec”hni”calMe”mci%rid.umNo. 979in the stationary form of the fntegral equation Enc”um-bered with further r,estrictiona, this special case hasbeen, treated most so “far. ,. Jd) “Simplified “assumptions regarding the sha
29、pe of theairfoil can be made,; Allowing the span of the airfoil toincrease to infinity and assuming the downwash to be in-dependent of rI gives the same flow process in all theplanes n = const, so that the integration with respectto V can be effected, The form of the integral equation1 so obtained r
30、epresents a two-dimensional flow process,This form is of particular intere”st because it is solvedfor any downwash function under the restriction 9 = Oanii for stationary flow also with #o. The significancebf this solution in the so-called vortex fi”lament theoryw*13 be explained elsewhere. ., .,On
31、an airfoil of constant chord and infinite spanboth the ,downwash distribution periodic in q.and that independent of q can be taken into considera-tion This defines a Y distribution pe,riodic i q. The integration with res,pect to q can be effected.invarious cases. From these” harmonic solutions super
32、posi-tion affords solutions for any finite airfoil of constantchord, in motion. The method is therefore applicableonly to airfoils having a parallelogram (especially arectangle) as contour and pressu-res that no flow “passesaround the” lateral edges of the parallelogram.The ultimate aim of the simpl
33、ifications is thechange of the surface integral in (19) to a line integral,which naturally is more promising” for a solution of theintegral equation. This aim can also be reached by as-suming airfoil contou”rs representing coordinate lines insimple sys”temsof curvilinear orthogonal co-ordinates. “On
34、airfoils of elliptic plan form the “nonfocal” elliptic co-ordinates permit” integration with the aid of the Lamfunctions. For circular contour there arethe sphericalfunctions of the first a“fidseeond type. Examples can befound in the. reports by Kinner (reference), Krien8s(r.eference.4), andSchade (
35、reference 5).” e“) It is readily apparent from (19) that the assump-tion of incompressible fluid affords”a stibstantial sim-plification. in the form of the integral equation beeause .Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-y-“ -) ., NA-CA T,e
36、chni.cal Mem;ran AY is computed and this added toY. as correction factor, etc. The convergence of thismethod needs to be checked of course from one case to theother, although it should be sufficient in general fopslender airfoils.There is nothing to prohibit the application of thismethod to compress
37、ible fluid (s+0), once the generalsolution of (26) in form of an integral representation ofthe type of solution (32) found for = 0, is available.9. SYSTEMS OF SEVERAL AIRFOILS,“The general form (19) of the integral equation of theairfoil theory comprises the possibility that the surfaceintegral ddq
38、can be extended over severalspatially sepa-rated regions of the airfoil. A case in point is theProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-I NACA Technical, e”tiorsn;o.,.9,?9 25!.biplane or the wing with .spl.it;i:f:apOf .cours.e,.Kuttalsflow-off
39、 condition mus-t-bsatisf-ied for the trailingm. -I;) edges of each part of the airfoil.i ,. . . .,.-,;:, ,.A Give: a general”solution Y = o the-! superposition prin i le” .? As-.sumethe. given downy s: of the. airfoil (1) as” w ,1 t?.andof airfoil (2) as Wg:a” = O.tFirst Beitrag zur Theorie der Trag
40、enden l?lche. Z.f.a.M.M.,vol. 16, no. 6, Dec. 1936, pp. 360-61: Theorie desYlugzeugtragfliigels in Zusamnendrflckbaren Mediun.Luftfahrtforschung, vol. ,13, no. 10, Oct. 12, 1936,PP a71 313-19.3. Kinner., W.: Die Kreisf5Fmige Tragflche auf potential-theoretischer Grundlage. Ingenieur-Archiv, vol. 8,n
41、o. 1, 1937, pp, 47-80. illus.4. Krienes, Klaus: The llllliticWing Based on the Poten-tial Theory. T.M. NO. 971, .NACA, 1941.5. Schads, Th. : !Theorie der schwitigenden kreisf8rmigeni!ragflche auf potentialtheoretische r Grundlage.Luftfahrtforschungi vol. 17, no. 11/12, Dec. 10, 1940,387-4d0.6. Yrank
42、-Mises : Die ifferential und Integralgleichungender Mechanik und Physik. Braunschweig, 1930, vol. I,P. 418.7. Possio, C.: Lazione Aerodinamica SU1 Profilo Oscil-l.ante in.un Fluido Compressibil,e a Velacita Iposonora.LIAerotecnica, vol. 18, no. 4, April 1938, pp. 441-58.8. Munk, Max M. : General he,
43、ory of Thin Wing Sections. Rep.iiO., 142, NACA, 192269. Birnbaum, W.: Die ,tra,gnde Wirbelflache als Hilfsmittelzur Behandlung des ebenen Problems der Tragflfigel-theorie. Z.f,a.M.Me, vol. 3, no. 4, pp. 290-97.,10. Kiissner, He .G. und Sohwar.z, L.: Der schwinge.nde Fliigelmit aerodynamiscn ausgegli
44、chenem Ruder, Luftfahrt-forschung, vol. 17, no. 11/12, DecO 1.940, pp. 337-54,11. Ki.issner,H. G.: Das zweidimensionale Problem der be-liebig bewegten Tragf15che unter Be.riicksichtigung vonPartialbewegungen der Fliissigkeit. Luftfahrtforschung,vol. 17, no. 11/12, Dec. 1940, pp. 355-61.Provided by I
45、HSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-,-NACA Technical Memorandum No. 979 2712. Schwarz, L.: Berechnu.ng der Funktionen ,Ul(s) und,u (-s) fiir grssetie Werte “von s. Luftfahrtforsch- “- ung, vol. 17, no. 11/12, Dec. 1904, PP* 362-69.+Ii 13. Nielsen, N.:
46、 Handbuch der Theorie der Cylinderfunk-j tionen. Leipzig, 19C14, p. 224.,:,J 14. Cicala, P.: Comparison of Theory with Experiment inf the Phenomenon of Wing Flutter. T.M. No. 88?,)-l!rAcA, 193915. Kleinw3chter, J.: Beitrag zur ebenen Leitwerkstheorie.Luftfahrtforschung , vol. 15, no. 3, March 20, 19
47、38,PP a71 127-30+.,.,LProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-, - _.-F . v-.-wwAFunctionT(ti).s J5(X) w%0,11919-0,155370,185370,210910,233020,252350,269400,284510,297950,309940,3206603302b0,33884-0,3iG520,353390,359530,364S90,369850,374150,37
48、7940,381270,384160,3S66G 0,388790,36059 0,320T 0,39326 0,39419 0,39487 0,39511 0,30132-0,38475Yfl TMW6 o,09135 0,15042 0,23200 0,28519 0,32080 0,34460 0,36015 0,36978 0,37513 0,37735 0,37725 0,37645 0,37239 0,36841 0,36376 0,35864 0,35319 0,34752 0,34172 0,33585 0,32997 0,32411 0,318310,31258 0,30695 0,30142 0,29600 0,29071 0,28554 0,2
