1、REPORT 1374THE SIIVIILAIUTYRULES FOR SECOND-ORDER SUlk30NIC AND SUPERSONICBy MILTON D. VAN Di-mmow 1SUMMARYTh.a similarity ruikefor lirwarizd comprewdle * theory(th+?lf$Tide Wld ii%SUpt78kC COU?ltWj?Wt) (We td+?.dd tO8econd order. t is showm Lhui any seed-order subsontifiwcan be relatd to “nearly in
2、compreeea”li.djlow paxt the samebody, which can be cakulated by the Janzen-Rayleigh mdhod.INTRODUCTIONThe linearized small-disturbance theory of steady com-pressible flow, based on the Prandtl-Glauert equation, yieldsa first approximation for thin objects moving at either sub-sonic or supersonic spe
3、eds. More precisely, it provides thefirst term in an asymptotic expansion of the solution forsmall disturbances, provided that the f3ight Mach numberis not too close either to unity (transonic flow) or to infinity(hypersonic flow).The similarity rule that governs linearized subsonic flow-past genera
4、l three-dimensional objec was first given cor-rectly by Gdthert (ref. 1). It has an obvious counterpart insupersonic flow, and the rules have rendered great service inboth theoretical and experimental hence the linearized solutioncontains terms of order A?n T and ?, and the second-orderincrement the
5、n consists of terms of order AA, r41nr, and r4.Nothing is known of the convergence of these series; theyare perhaps only asymptotic expansions for small thickness.Second-order theory, like linearized theory, breaks down inthe transonic and hypersonic ranges, though it may pene-trate somewhat farther
6、 into their fringes. A similarity rule for second+rder theory has recentlybeen given in the special case of supersonic flow past thinflat wings by Fenain and Germain, who demonstrate itsusefulnesa in theoretical studies (ref. 5). However, as inlinearized theory, the rulw for flat wings are only spec
7、ialcases of those for general three-dimensional shapes. Thepresent paper is devoted to deducing the general rules forsubsonic and supersonic flows, and examinb g their implica-tions. In particular, it is shown how the rule for subsonicflow relates the second-order solution for any object to nearlyin
8、compressible flow past the same body, which can be calcu-lated by the Janzen-Rayleigh method.The author is indebted to Wallace D. Hayes for suggestingseveral improvements that have been incorporated in thepresent version of this paper. In particular, the procedurefor recovering the second+mder solut
9、ion from the Janzen-Rayleigh solution (p. 930) is simpler and more logicxd thanthat originally given iD NACA TN 3875.DERIVATION OF RULES FOR BODDN3 OF REVOLUTIONA body of revolution is the simplest shape that is not aspecial case, but displays the fdl generali of the existingsimilarity rules for sub
10、sonic, supersonic, transonic, and hy-personic flows. The same can be shown to be true of thesecond+rder rules to be discussed here. Hence for clarityof osition, the se AZ, y, 7). This notationindicates that for each family of bodies (associated with agiven function R(x), the flow field is regarded a
11、s dependinot only upon the two independent variables x and r butalso upon the three parameters following the semicolon:M free-stream Mach numberr adiabatic exponent of gas zT characteristic slope of bodyThe aim of a similarity analysis is to transform the prob-lem so as to reduce the number of param
12、eter appearing igit. If that can be accomplished, flows having diiferentvalues of the original parameter are related provided onlythat the reduced parameters are equal. The transforma-tion to be used here consists in separating the dependemtvariable into several components, and then stretchingeach c
13、omponent and the independent variablea by factorathat depend upon the original parametem. It is convenient,and involves no loss of generality, to leave stmamwisecoordinates unchanged, so that r is to be stretched but not Z.Perturbation potentials are first introduced by setting1%=+ . . “ (1)where 4
14、is the potential of linearized theory, and q thesecond-order increment.EIJf.ESFOR LINEARIZED THEORYThe linearized problem isa75 = (lMW=+A,+=O4+ fM,T)=p ( ,Br;BT) (5)SECOND-ORDER lZULE9The sendrder problem is found to be (ref. 3)at infinity)(6),= Tr#rzR(z) at r=rR(z)Note that the first equation conta
15、ins not only qundmticterms on the right-hand side, but also the triple proclucLr#II#Jfiwhose contribution is of second order in some cases,The parameter y appears only linearly in the combination(Y+ 1) d can accordingly be separated out. Tlms thoappropriate transformation is found to beThen equating
16、 like powem of iLf2yields the following set ofthee problems for fl,z in which the parameters M,y,rappear again only in the form of the single similarity prwnm-eter /37:Aj,=Of,+) at infinity (sfL)flp=f%)+ (%W%m1 .fl(v#z; BT)+JX )+ (7+0 $ M ) (loa.PDi.flerentiation yields the corresponding rules for v
17、elocitycomponents (those for w having the same form az for o):;=1+Z7(Z, ,pz;T) +m1 Z,/,Bz;B)+1 tw%pz;t?d+mb( )+(?+0 $%( ) (mF(The functionz appearing here are actually related to deriva-tive of the functions in equation (lOa), but the connectionis of little interest.) To second order the pressure co
18、efficientis given byc,=2= (+q %Oz-2(#,%+hw) + (M1)4.+fw(w+) +*%, 19z;l%)+;zi( )+z( )+(7+1)*?3()-j(lOe)c2=T3(z,B,pz; BT)+T z, ( )+M?2( )+(7+1) $33( )(lOf)Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-. . . .-_ .-A_ . .928 REPORT 1374NATIONAL ADVISOR
19、Y COMN1lTIE.El FOR AERONAUTICSIn addition, the fit two second-order terms can be manip-ulated, using the connection between M and 19,to yieldadditional alternative forms such as the following, whichcorrespond to the three forms above:(log)C,=;RW%JJ%)+Zi()+$32()+(?+1) +$13()(lOh)cp=Tfi(z,Bu,Pz; BT)+T
20、PL( )+12( )+(7+1)$53(). (lOi)FORCE COEFFICIENTSThe rules for pressure imply rules for the lift and dragcoefficients. For example, equation (lOe) leads toTLow-$pfk)+mb%)+(7+1) $ M%-)c!(fM,7,)= 1.- (lOj)+ Upl(l%)+mwd+(+l)$%mCD(M,7,T)=p(1Ok)if the cdlicients are referred to some plan-fomn area. Ifsome
21、cross-sectional area is used, each term is reduced by onepower of . Various alternative forms are again useful Inthe case of lift coefficient, one fl ordinarily choose toidentify r with the angle of attack.RULES FOR QUASI-CYLINDRICAL BODIESA special class of objects must be distinguished, which will
22、be termed quasi-cylindrical bbdies. These are shapes thatlie everhere so close tQ some cylinder (not necessarilycircular) parallel to the free stream that to a first approxi-mation the condition of tangent flow can be imposed at thecylinder rather than on the actual body mrface. Likewise,in second+r
23、der theory the tangency condition can be tmns-ferred to the cylinder by Taylor series expansion. The sim-plest example is an airfoil whose thickness, camber, and angleof attack are so small that the tangency cition can betransferred from the airfoil surface to a mean plane parallelto the stream (g.
24、3). Another example is an open-nosedcG= - -:J/ / / /777n+7-FIWJEE 3.Esamples of quasi-oylindricd bodiesbody of revolution whose radius varies only slightly. Otlmrsare biplanes, cruciform wings, any of these in an open orclosed wind tunnel, in combination with one another, etc.A quasi+ylindrical body
25、 can be regarded as consisting ofa skeleton upon which is superimposed a small slope distri-bution. The skeleton is simply the projection of the bodyonto the basic cylinder. For example, the skeleton of thoquasi-cylindrical body of revolution is the circular tubeshown dashed in figure 3. The speciil
26、 place of quasi-cylindrical bodies in similaritytheory arises from the fact that the skeleton and the slopedistribution can be varied independently. This extrafreedom is important. For example, it leads to a usefultronic similarity rule for quasi-cylindrical bodies whereasnone exists for general sha
27、pes. It is convenient always toleave strearntvisedimensions unaltered. Hence, we consichmfamilies of quasi-cyhlrical bodies that are derived fromone another by a lateral compression or exprmsion of theskeleton, and a quite independent magnification or reductionof aU surface s10pe9. Two members of su
28、ch a farniIy areshown in figure 4._ _ _ . _i J _ _ _ ; . ._ _ . FIURE4.Exampleof tworelatedquasi-oylindrioalbodiu,Distortions of the skeleton will be measured by somecharacteristic “aspect ratio” A. It is importrmt to notethat the term “aspect ratio” is used here in a very gemmdsense to mean any typ
29、ical ratio of gross cross stream tostreamwise dimensions. For example, in the last shapein figure 3, the ratio of wind-tunnel height to airfoil chord kan appropriate characteristic aspect ratio. Changes ofslope are measured, as before, by some chamct eristic slope T,The preceding similarity rules ca
30、n be sirnped for quasi-cylindrical bodies by using the facts that iirst-orcler pw-turbation quantities are directly proportional to r, undsecond-order terms to /. The simplithtion can be carriedmt by first imagining the quasi-cylindrical body to bewstricted to be a general body, which means that bot
31、h3A and M notd previously, tba ilrat-order prmuro036iOiElltOIi86MOOtbskndor lted bOdy Of IOVOhltkIII W n 7 foc 81TI r.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-THE SIMLURITY RUTJ3S FOR SDCOND-ORDEIS SURSONIC AND SUP13RSOMC FLOW 929# outside the
32、 second-order terms. For the pressure co-efficient, this form is that of equation (lOe):u,=; F(wwz; /wo+$ M )+wz( )+(Y+Q 4$%3( )So far the functions , have been supposed to dependparametrically upon both IM,7,T,A)=;P(X,PY,BZ;M) +; 2A% M)+MY,( )+(r+l) + in subsonic flow P and PI aremore complimted (r
33、ef. 8). This rule implies a corresponding,but more complicated, rule for surface velocity (ref. 8).In addition to the restriction to single bodies and planeflow, these rules are not similarity rules in the sense of thepreceding results, because they apply only at the surfacerather thrm throughout th
34、e field.EXAMPLESThe rules will be illustrated by two simple examplw, aY(5M-1) ln;+:w+(y+l) 1$ (13)This has the form of equation (lOi) with2 2 135=3 in li75 ln#z 3=1WAVY WALL IN CLOSED SCBSONIC WIND TConsider the sinusoidal w-ally= sin z at a distance h froma flat wall (or a distance 2h from its mirr
35、or image) as indi-cated in figure 5. Subsonic flow between the walls at a meanYAn/ / / / f / / / / / / / / /II :-y= rsn-lxIo / xFmmm 5.TVavy wall inwind turmeLMach number fM can be readily calculated to second orderby separation of variables. The resulting pressure coeffi-cientt on the surface of th
36、e wa wall is .The relevant aspect ratio is the height h (which is really amultiple of the height-chord ratio, because of the choice ofscale for the wavy wall). Thus the result is seen to have thesimilitude of equation (Ha). As the tunnel height increasesindefinitely, the last two terms disappear, an
37、d the remainderfollows the similitude of equation (12) for the surface of asinM,7,T) =o(z,y,z; 7)+M%(z,v,z; )+.(r+l)il!f%( )+M3( )+ . . .by theon thein w.(15a)The two terms in lM* are ordinarily considered together,but for present purposes it is wsential to separate thembecause only %2is required. T
38、his is fortunate because Zcan be calculated almost as easily as %, whereas the deter-mination of 3is much more ditiicult.The small-disturbance and Janzen-Rayleigh series repre-sent two diflerent asymptotic expansions of the actual solu-tion. They are bdieved to complacent each other, so thatan expan
39、sion of the Janzen-Rayleigh solution for smallthickn- must be identical with the expansion of the small-disturbance solution in powers of M, as has been veritiedin all worked examples. This fact permits the small-dis-turbance solution to be recovered from the Janzen-Rayleighseries. The converse is n
40、ot true, however, except for bodieswithout stagnation points, because the small-disturbanceexpansion is not uniformly valid near a stagnation point.PROCEDUREFOBRECOVERINGSECOND-ORDERSOLUTIONAnother alternative form for the second-order veloci po-tential of equation (lOa), which is useful here, is+-
41、(z,y,z:M,7,T)= z+#F(zJ,/9z;/37)+.+ A()+$%()+(7+1) $%()Jlol)ere F and f, $, f, are not the same functions as in equa-tion (lOa) but related ones; in the notation of equations(lOh) and (lOi) they are actually fi and , ja, t,.)For presentpurposes itisuWecessaryto distinguishbetweenthe fit-order term F
42、and the secondarder increment fl;combining them as F1=F+fl givesThe Janzen-Rayleigh solution is now to be manipulatedinto this same form. The first three terms as given in equa-tion (15a) are equivalent to; (z,y,z; M,i,T)=o(z,q,z;T)+$%lo+.()(7+1)$%( )+0 $! (16b)which may be rewritten aswithox*.P(16C
43、)(15d)Finally, with the aid ofJ-V=PY l+;=w+$+0 ($) 00and corresponding expansions for z and t-, this may bore-exprmed as=$%(ww%sd+$%d)+!#ti()+/32 1 lidazo+g$%ro+(7+1)jji$%() (150)which is the desired form. Here, for example, pOY(Z,W,;Pr)means /by) w(z,y,z; r) evaluated at Z=Z, v=13y, z=flz,and r=r.T
44、he secondarder solution is thus recovered from theJanzen-Rayleigh approximation simply by calculating inturn the expressions in equations (15d) and (15e). Theprocedure can actually be expressed by a single equationas follows. From the Janzen-Rayleigh approximation in thoform of equation (15a), the s
45、econd-order small-disturbancesolution is recbvered according to%(z,y, z;7)z M2 % y %-z+7+”+=;4).+(9)J+NV172+27-Wwhere . means a(z,v,z; T) throughout, and subscriptsindicate differentiation.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-THE SMDARITY
46、RULES FOR SFICOND-ORDIIR SUBSONIC AND SUPERSONIC FLOW 931APPLICATION TO PARABOLAAs an a-ample, consider plane subsonic flow at zero angleof attack past the parabola described by y= thecomplexity arisiig in a proliferation of functions multipliedby powers of (7+1) -il&n&zP. Likewise, the smaU-disturb
47、-ance solution to any order can be recovered from the nearlyincompressible solution provided by an appropriate numberof terms of the Janzen-Rayleigh solution.AarEs AERONAUTICALLABORATORYIVATIONALADVISORYCOMMITTEE FOR A.23R0NAuTIcBM!OFFETTFIELD, CALIF., Oct. 18, IWREFERENCES1. Gi5thert, B. H.: Plane
48、and Three-Dimensional Flow at High Sub-sonio Speeds (Estension of- the Prandtl Rule). NACA TM1105, 1946.2. Broderiok, J. B.: Supersonic Flow Round Pointed Bodies of Rev-olution. Quart. Jour. Meoh. and Appl. Math., vol. 2, pt. 1,Mar. 1949, pp. 98-120.3. Van Dyke, Milton D.: A Study of %oond-Order Supersonk FlowTheory. NACA Rep. 1081, 1962.4. F* J., and Leslie, D. C. M.: Seoond-Order Methods in InvisoldSupemonio Theory. Quart. Jour. Meoh. and Appl. Math.j vol.8, pt. 3, Sept. 1955, pp. 257265.5. Fenain, Maurice, and Germain, Paul:
