1、REPORT 1153ON THE APPLICATION OF TRANSONIC SIMILARITY RULESTO WINGS OF FINITE SPAN By JORNR SPRDrmRSUMMARY transonic aerodynamic charwk%tics of wing8 of jinitespan are discu-s8ed from the point of view of a uki? ddisturbancetheory for sub80ni.c, transonti, and supemonicjhos aboutthin wings. OiticaJe
2、xamina#i.Onh madeof themerii8 of the various statemeni% of the eqwtiow for transonicj?ow thu.! hoe been proposed in ih recent Wrature. It hfound that one of the kS8 widely used of t?we po8838e4 con-skkrable advante9, not only from the point of ti of a priori “theoretical considkrti but : CDOcontribu
3、tion to drag coeflkient due to liftu(t/c)fi (Ad)lift coeiiicient .uJ(t/c)(CJCZ)pitching-moment coefficientzJJc(t/c)(C=/a)preasmrecoefficient(uJc)l/(t/c)/5titieal presaum coef6iient_ce.nter-of-pmswre functionwing chordsection drag coeilicient of symmetrical nonlifting airfoils(Uok)l(qc)wa c% .contrib
4、ution to sectio drag cwflkien$ due to liftI SnpamedES NAOA TN 2TMIentltlod “On tbo Applkatlon of Tmnwnlo SinWarity RuM,” by John R. 8pIdk, 19521055Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-1056c1(CT)%:;dodk.c1wmMMOT8tU*U,vX,y,zxc.p.(z/c)azYht%7
5、P1wJ2. REPORT 1153NATIONAL ADVISORY COMMTIDE FOR AERONAUTICSsection lift coefficientUJ(t/c)fi(cja)dragfunctiondrag function for symmetrkd nonlifting wingsdrag due to lift functionsection drag function for symmetrical nonliftingairfoilssection drag due to lift functioncoefficient of nonIinear term of
6、 d.iflerentialequation for y. (See eqs. (7), (18), (19), (20);(23), and (35).)lift.functionsection lift functionpitchingmornent functionsection pitching-moment function10wJMach numberfree-stream Mach numberpreasum functionstretching factors defined in equation (B8)maxhimm thickness of wingfree-strea
7、m velocityvelociw components parallel and perpendicular,respectively, to the flow direction -ahead of ashockCartesian coordinates -ivhem z extends in thedirection of the free-stream vidocitydistance from wing leading edge to center, ofpressureordinates of wing profiles in fractions of chordangle of
8、attackwhere the slwpe of the wing protlle is given by- z/c= Tf(x/c, y/b) (16)w-here(z/c, y/b) represents the ordinate-distribution functionand is an ordinate-amplitude parameter. Note that, ingeneral, a variation of r represents a simuhlageonschange ofthe thickness ratio, camber, and angle of attack
9、. In thespecial cam of a nonlifting wing having symmetrical sections, is proportional to the thiclmess ratio; for lifting flat-platex of v” thiCkJl r is proportional to the angleProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-1058 REPORT 1153NAHONAL
10、ADVISORY COMMITTEE FOR AERONAUTICS .0of attack. k order to obtain unique and physically im-portant solutions, it is necessary to. sssume the Kutta condi-tion (that the flow-leavea all subsonic trailing edges smoothly).Upon solving the above boundary-value problem for thepotential, one may determine
11、the pressure coeihciaut bymeans of the formula(17)It should be noted that the results obtained by using theforegoing approximate equations might be expected to tendtmvard those of linear theory as the free-strixunMach numberM. depark far from unity. This follows from the fact thatthe product q= beco
12、mes small relative to the linear termsunder this condition. Solutiina of the equations for trawsonic flow found to date have all possessed this property.COhlPAFUSO for instance,it has been suggested that a* be replaced with U. in thoequation for UP. This matter has been discussed at lengthin referen
13、cca 8, 17, and 18. Since no restriction requiringtho Mach number to be near unity is made in the U. analysis,it is informative to examine the relation between the resultsof the a* and the U. analyses. This is done iQAppen- (30)Tho variation of Cp= with M. has been computed by use of”the exact relati
14、on and ewh of the four approximate relations.The rmults arepresented graphically in figure 2. It maybeseen that rLreasonably good approximation for CPa is ob-tained over a wide Mach number range when k is taken as.given in either equation (18) or (23), and that a somewhatgrimtererror is incurred whe
15、n equation (2o) is used. On theother hand, equation (19) leads to a very poor approximationfor Up,.Similar comparisons can be made for local Mach numbemM other than unity by noting that the coefficient lMkw=, -LO -Mo4(y+I)/uo: y:m ,/ .-.s#-/uoz2+(7- I)M?/okAfJ(y+l)/uo/- o , I -.6k ,- Exact km,Qmu? 2
16、.Variation of oritical preswre coefficknt with Mach number.of ff= in equation (7) corresponds, in the present approxima-tion, to 1.kP,thuslW=lMo2k%=l M.2+ CP (31)The corresponding exact relation foi isentrcpic flow iscp=*-l+=a71 32The results % obtained are generally similar to those in-dicated in u
17、ie 2, although the relative accuracy of the bet-ter approximations changes somewhat with the situation.All the approximations are exact, of course, when CP=O; onthe other hand, none of the approximations are exact, exceptfor isolated cases, when C, is dHemnt from zero, even thoughall the approximati
18、ons agree among themselves when thefmp-stmam Mach number is unity. b order to provide someinformation regarding the errors that are likely to be in-curred when Cpis not very small, figure 3 has been preparedillustrating the variation of local Mach number with pressurecoefficient for a free-sham lMac
19、h number of unity.A second case where the exact and approximate relations .may be compamd is fimnished by considering the velocityProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-. .- .- .- . .-1060 REPORT 1153NATIONAL ADVISORY COMMFITCEE FOR AJ3RONAU
20、ITCS#-%Approximate; -.4 -.2FMO= LO.2 J JLi+ ; ?Cp . .8 + .6.FIGURE$.Variation of local Maoh number with pressurecoefficientfor IWO=Ljump through a shock wave. If the flow abead of the shockwave is uniform and parallel to the z axis, the results maybeconve.nkdy represented by the shock-polar diagram
21、inwhich Z/a* is plotted as a function of 2/a*. The exactrelation is furnished .by equation (8). The correspondingapproximate relations are determined from equation (12) by=ttiug wzlvnl%2and p=l equal to zero, whereby U.= Ul,fMO=M1, and 1P.:= OM.?+: v%P% (33)once the variation of q,with p% is determi
22、ned for a givI consequenly, q= approaches infinityas fMoapproaches unity and the theory is clearly inapplicable,For wings of finite span, however, the perturbation velocitiesmay be large or small at sonic velocity, depending on theparticular problem m diwuased in detail in reference 26,Speciikdly, f
23、or three-dimensional lifting surfaces of zmoProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-ON THE APPLICATION OF TR,4NSONIC SIa. ccmequently,linear theory is inapplicable within some Mach numberrange surrounding unity.Summarizing, linear theory is a
24、pplicable to lifting surfacesof small or moderate aspect ratio at all transonic speeds,but fails for wings of finite thickness within a range ofMach numbers surrounding unity. The range of inapplica-bility diminishes to zero as the aspect ratio, thickness ro,and angle of attack of the wing tend to z
25、ero.In treating transonic flows for which linear theory isapplicable, it is often advantageous to consider the specialcase of sonic flow (M.= 1) separately. Equation (5) forthe perturbation potential then reduces to a particularlysimple form$%u+$%=oSolutions of this equation, in conjunction with the
26、 boundaryconditions given by equations (13) and (15)are identical tothose of linear theory found by solving equation (5) andsubsequently setting MO= 1, but can be obtained with muchless effort. Since, in addition, the results of this simpletheory, now generally known as slender-wing theory, arealso
27、applicable to low-aspect-ratio lifting surfaces throughouttlm entire Mach number range, a considemble number ofsolutions of slender-wing theory have been presented in thelast fmv yews (e. g., rtifs. 27, 28, and 29). These resultstwo, of course, applicable h flows at ill.= 1 to, exactly thesamo exten
28、t as the results of linear theory.SIMILARITYBULIX3 -In reference 6, von K4rm4n derived similarity rules fortho pressure distribution, lift, drag, and pitching moment ofairfoils in transonic flow using equations (24) through (28).The same equations were used in reference 20 to determinethe transonic
29、similarity rules for wings of finite span. Thecorresponding similarity rules of linearized subsonic andsupcMonic wihg theory were also derived- and comparedwith the transonic similarity rules in the latter reference.It was shown that the similarity rules.of linear theory con-tain an arbitrary parame
30、ter and can be expressed in manyforms, one of which coincides with the similarity rules oftransonic flow-.A derivation of the transonic shnilmi rules, based onthe UOequations with unspecified k, is provided in AppendixB. TIIis derivation possesses the advantage of being basedon a single statement of
31、 the problem of wing theory thatis uniformly valid at subsonic, transonic, and supersonicspeeds. It follows from the preceding discussion that theresults so found me identical to those of references 6 and20 if k is equated to (7+ l)/UO. The similarity rules for(7P,Cz, Om,and CD are given in Appendix
32、 B as follows:where the geometry of related wings is given by equation (16):(z/c) =(z/c, lJ/b)Equations (36) through (39) are functional equations. Forexample, equation (36) is to be intmpmted as stating thatthe pressure coefficient CPis equal ti ?fi/IUJ)lfi times somefunction T of a number of speci
33、fied parametem. Theforegoing equations have been written for flows whereMOWings of infinite aspect ratio.-lor wings(or airfoils), equation (62), representing(62)of infl.nitespanthe functionalProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-. - .z. _ .
34、1064 REPORT 1153NATIONAL ADVISORY COMMPITED FOR AERONAIJTTCSrelation of linear theory for the drag coellicient, reduces tothe following:($):= t/c)x Const., (j) ,=%,= (63).where the value of the constant depends on the shape of the(co)a+,*;(C.)M 1 $%:.(67)The corresponding exact relation can be deter
35、mined simi-larly by use of equation (52) for the slope of the pressurecurve. It is. (%)M.-l=$*-*(o)O-ll%d(:) + (c%)Me., (68)Siice calculations have been made of the drag in transouicflow of simple symmetrical sections at zero angle of attack,it is not necessary to speculate further regarding the var
36、i-dion of o with $0. At present, complete theoretical infor-mation exists for the drag of symmetrical double-wedgeairfoils throughout the tramonic range. Solutions for thissection have been obtained by transforming the nonlineardifferential equation of the small disturbance transonictheory into hodo
37、graph variabka and taking advantago ofsimplification of the normally diiiicult problems relating tothe boundary conditions by restricting attention to polyg-onal proiiles. The results for flows having subsonic, sonic,and supersonic free-stream velocities have bee givm,respectively, by Trilling (ref.
38、 12), Guderley and Yoshibam(ref. 7), and Vmcenti and Wagoner (ref. 8). The linem-theory solution for pure supemonic flows has been givcm byAckeret (ref. 30). All of these results me combined on osingle graph in figure 5. It maybe seen that the preceding2 crFIGURE 10.VarktioI+ with of reduced drag co
39、effloIent of triangularwhiny.If hypothesis (d) is accepted, CLvaries linearly with a forat least small angles of attack. It is advantageous, therefore,to consider the lift ratio C./a rather than a. alone, therebyminimizing the influenw of where Y=If the perturbation nnrdysis is carried out in a comp
40、letelyconsiment manner, the boundary conditions are:at,x= (PZ)O=(JZ?- 1) ) (P.)O=(PZ)O=o (A1o)oat the wing surface(?a(d2)=.0=a* (All)Wd the pressure coefficient is given byc,= $ (o.- (oJo (A12). .The relation between the U. and a* statements can bedetemU =s;uo(138)1(1kfO)=8p i, uok=s Uok or k=$kthe
41、potentirdequations (eqs. l) and (B2) become(o Linear (B9)and the shock relation becomes.321 OOGG3I38Nonlinear (B11)The flow in the primed system is similar to that in theoriginal system if q saties the same differential equationsand boundary conditions as p. Consequently, for similarityto exist, it
42、isnecessary fit of all that the potgntial equationsand the shock equations for the two flows be the same.Therefore, the following relations between the stretchingfactors must be satisfied:8#BA/A Linear. 12)1, 8“1, 8= z82% s2/8kNonlinem (B13)where, for linear theory, AX is an arbitrary ccmstant which
43、can be equated to sf12/8if desired. The constant is writtenas a fraction in order to maintain a certain synnhetrythroughout the analysis.An immediate consequence of this transformation is thatthe wing plan forms undergo an a.ilbe transformation suchthat the aapect ratios of wings in similar flow fie
44、lds arerelated, according to both linear and nonlinear theory, bym 14)“=: =;= (1MolOr by4(1MO) A= A (B 5)Since p is proportional to P, the boundary conditions atx= OJare automatically satisfied. The boundary condi-tions at the wings may be given in either of two forms:, G%).J.0=5(%)0=: % 8p8j7J= T T
45、SU8= SV8Z(dx (1fMO) ,P 1M* Linear (B18)1073Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-. -_ -1074 REPORT 1153NAONAL ADVISORY CO E FOR AERONAUTICSor as7=? Linear (1320).m=l M.=(Uikr)n (Uokr)f3 Nonlinear (1321)The relation between the pressure coef
46、ficients at colTe-sponding points on the wing surface is given byc,=+-, (Sg)x,=o0 z-The foregoing relationships may be summarized in “thefollowing statement: The sW rule for the pressurecoefficient.son n family of wings having their geometry givenby(z/c) =7j(z/c, y/b) (B26)The similarifi ruIes for t
47、he lift, pitching-moment, and coefficients given by the linear theory are .The corresponding similarity rules given by nonlinear theoryareCL=29. Lomax, Harvard, and Heaalet, May. A.: Linearized Lifting-SurfaceTheory for Swept-Back Wings with Slender Plan Forms.NACA TN 1992, 1949.30. Ackeret, J.: Luf
48、tlaitfte auf FNigel, die mit gr&erer als SohaIl-geschwindigkeit bewgt werden. Zeitscbriit ftir Flugtechik undMotorluftschiffahrt, Feb. 14 1925, S. 72-74. (Also as ATACATM 317, 1925)31. Puckett, Allen E.: Supersonic Wave Drag of Thin Afrfoils. Jour.Aero. Sci., vol. 13, no. 9, Sept. 1946, pp. 475484.32. Guderley, Gottfried, and Yes_ Hideo: Unsymmetrical FIovrPatterns at Mach Number One. Air Force, Air Materielco remand, Tech. Rep. 6683.33. Stewart, H. J.:
