1、ONREPORT No. 894SIMILARITY RULES FOR TRANSONIC FLOWSBY CARL KAPLANSUMMARYxl method used by Tsien to derire m“milan”tyrules for hyper-sonic $OWS ia utilized to derice J-on K4rmdns similam”iy rulesfor transonic $ow8. A 8ight generalization i8 introduced bythe inclution of y, the ratio of 8pecijic heai
2、%, a8 a parameter.At the lower limit of the tran80nic region of jlow the theoryyied8 a formula for the critical 8tream Mach numbers of a“tenfamdy of aym?netn”cal profie8. It i8 jurth.er 8hown. thatthi8 formula can also b8 obtained by mean8 of the PrandtLQlauert small-perturbation method. ke8tigation
3、 of the be-harior of the similarity parameter in the region where thethickness eoejitient approaches zero and the cm”tical streamMach number approaehe8 unity 8hows that it po8sesse8 alimiting value characteristic of the prescribed family of shape.INTRODUCTIONThe rigorous solution of the subsonic flo
4、w of a compressiblefluid past a prescribed closed body thus far has proved tobe of insurmountabe dif%culty. As a consequence of bdifficulty the emphasis has been pIaced on the estabhhraentof a correspondence between the flow past a given body in anincompressible fluid and the same body in a compress
5、iblefluid. Among the best known results of this mode of attackare the PrandtLGIauert rule d the Von Kdrmthen, according to equation (2) and the awunptions leadiigto equation (8), a velocity potential can be introduced with=”c*2!+(l-M*)p (9)where M*=: and (1M*) q is the disturbance-velocitypotential.
6、Thenancl equation (/3) becomes : . . -: _:_ (7+1)(1M*) * g.+c” po (11)Equation (11) is a nonlinear simplified forni of the fundamen-tal differential equation (1) and has been treated recentlyby Von Kdrmfi in commction with similarity rules in two-dimensional trammnic flow (reference 1). Equation .(1
7、1),when expressed in hodograph variablw, is of the type treatcclby the Italian mathematician F .c.omi somQ years ago(referencti). .-.DERIVATION OF SIMILARITY CONDITIONSRecently, Tsien (reference3) derived similarity rules forhypersonic flows where the fluid velocity is much larger thanthe velocity o
8、f sound, In”thepresent paper the same proce-dure is employed to derive similarity rules for transonic flowswhere the fluid velocity is very nearly that of sound.According to the assumptions leading to equation (11), itis implied that the solid body is thin and possessa no stag-nation points smd that
9、 the velocity of the fluid is everywherein the neighborhood of the local velocity of .scmnd. NTOW,suppose the. profile of the obstacle ti. be symmetrical withrespect to both the x- and y-axes and to possess cusps at boththe leading and the trailing edges. Such profiles with uni-form flow in the dire
10、ction of the long axis-of-symmetry xfulfdl the assumptions leading to equation (11). The flowspast these profiles are said to be similar if the equation ofmotion (11) and the boundary conditions can be expressedin nondimensional variabhs in such a way that only a singleconstant factor is involved. T
11、hus, if 2a is the chord and 2bis the maximum thickness of the body, then the followingnondimemional variables are introduced:(12)where t=: and m and n are exponents yet to be determined.It is clear that the nondhnensiomd quantity invokccl k thethicknepscoefficient t aimx.this quantity dctwmines the
12、mag-nitude of the disturbance mlocitiw. The exponents m and nare to be determined in such a way that the same constantfactor appears in both the equation of motion and theboundary conditions.The appropriate nondhncnsiona.1 form for the velocitypotentifd P is=ac(t,) (13)By substitution from equations
13、 (12) and (13), k cquationof motion (1I) becomes(.y+l)”+(l-ilf”) tfi )A comparison of the differential equation (14) and the bound-ary conditions, equations (15), shows that a single parnrnctmis invoIved if2n =(n+l) or 1“n=- 32m+l=-7n or m=-; - -“that is,(7+ I)m(l ilf*)t-fi=2KProvided by IHSNot for
14、ResaleNo reproduction or networking permitted without license from IHS-,-,-ON SIMILARITY RUZES FOR TR.4W301WCFLOWS 85The undisturbed+tream llach number Mm=! can be.introduced in the folIowing way:The general reIation between M* and Mm isor.1 (1.M*)2- (13f*)1/2M.=1+% (1M”)2-(1 M”Then, if powers of 1M
15、” higher than the first are neglected,1AI*= + (1M.+ . . .Therefore(1MJ(7+l)t-=K (17)The results obtained thus far are such that by means of thesubstitution equationsy=a(7+l)t-u3qt(18)P= ac*(;:;mremains constant, then the flow pat t ems aresimilar in the sense that the same function f (,$,q) describe
16、sthe flows.RESULTS DERIVED FROM THE SIMILARITY RULEPRESSURECOEFFICIENTIn the case of a uniform flow past rLfixecl boundary, thepressure coe%kient is defined as-. where pm and pmare, respectively, the pressure and densityin the undisturbed stream and the static pressure p in thefluid is given byp=pm
17、+=(%+lfi 21)._71Then -+.2(%913 22)cpjM.=7m (23)where ( K) depends on the form of the solution f (, q)for the particular family of profiles treated.LIFT COEFFICIENT .The lift 1 of the body is given by$1= (p)q.(dzBy a similar procedure, as in the derivation of equation (23),the lift coefficient is giv
18、en by(24)whereL(K)=J:lP(E,o;K)4and where for an extremely thin straight-line profile thethickness coefficient t has been replaced by the mgle ofattack a.DRAGCOEFFICIENTThe pressure drag d of the body is givm by the followingexpression:=2J:.w14YxProvided by IHSNot for ResaleNo reproduction or network
19、ing permitted without license from IHS-,-,-.=-. -86 . REPORTNO. 89 4-NATIONAL ADVISORYCO_.TTEE FO.RAERONAUTICS Hence the drag coefficient is given by(25)whereUm =2J”1 !I(:)W, 0; m ADDITIONAL CONSIDERATIONSThe results derived in the present paper apply to two-dimensional near-sonic flows past thin sh
20、apes. .Such flovshave been calculated for. a family of symmetrical shapeswith cusped leading and traiLingedgm (rerence 4) and fora family of elliptic cylinders (referenm 5). These calcula-tions are valid at least up to the critical stream Mach rmm-ber h!.,. The critical Mach number may be considered
21、from two points of view. First, it may be considered. todenote the lower limit of a mixed subsonic-supersonic flow,that is, where. theimbedded supersoriicreon is simply thepoint of muinmm fluid velocity at the surfwe of the solid.From this point of view, according to equation (17), acritical value o
22、f the similarity parameter K can be defied.Thus(26)This equation can also be written in a form that yields thecritical stream Mach number for a given family of shapes;that is,Me,= lK, (7+ l)ta (27)Second, the critical-stream hfa.ch number may be consid-ered to denoti the upper limit of the purely su
23、bsonic rangeof speeds. This point of view suggestsa derivation of equa-tion (27) by means of the Prandtl-Glauert smalI-perturbationmethod (reference 6). Th procedure is as follows:The relation between the local and the undisturbed-streamMach number, within the approximation of the small-disturbance
24、theory, is given by=-1+2$(1+=2)128)where u is tho disturbance velocity and U is the undisturbcd-stream velocity,By cle6iition, M= 1 for M. =M, and, $nce 0, = -2$-approximately, equation (28) becomwThe relation between CP, and the pressure coefficient CP,of the incompressible fluid is given by the Pr
25、andtl-Glauertrulec, ,c”=dllwc:” “ x.-Hence equation (29) becomes(1M,?)”2 =R=rC”o (30)As seen from equation (29), 1M,? is of the first order in thesmall perturbation u/ U and, accordingly, equation (30includes”terms .of higher order. This fact can be seen byrewriting equation (30) in the following fo
26、rm:(1mf,?)i = c,7+1 _T1 1(1M,;) (1M.)Then, to within the lowest order in the small pwturbation,( +1Cp,o 2/8Jf*r2= 2 )or, approximately,(31)It is qui easy to show the connection between equation (31)and equation (27). Thus, for an incompressible fluidl tiepressure coefficient is given by0,0=1$ - (32)
27、where Qis the magnitude of thefluid velocity at any point inthe field of flow.In thecase of the family of shapes of refcrcncc 4, the maxi-mum v?locity at the surface is given by+=l+; t+ . . .Hence -.-c,o=3t+ . . . -ant equation (31) becomes .()ill.,= 1 (7+ wlgm (34)An emwninationof equations (33) an
28、d (34) stiggest.sthatK., possesses a limiting value; that is,(35)Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-ON SIHJARITY RULES FOR TRANSOC FLOWS 87-lt is noteworthy that the value of (Km) ,k depends on thefamily of shapes, although in the limit
29、t the profile inevery case ia a straight-line segment. The numerical VdUHof (ZGJiahown in equations (33) and (34) repremut, how-ever, only the effect of the PrandtI-G1auert or first-orderterm in the power-series development in t of the maximumvelocity at the solid surface. It is rather surprising th
30、at thehigher-order terms imrohing the higher powers of t alsocontribute to the value of (Km) f. The procedure is simplyto replace the maximum velocity by the critical apeed; that is,(3)=(5).=(-.:2 crorwhere al, az, aa, . . . depend on the given family of shapes andinvolve only 7 and 31,.Then as sugg
31、ested by equation (26), vvhen 1Mmzis re-placed by 2 :6.S12.Km.Ina. 6i60:;l.ma.704:%:%!.647O.iw.?36.782.72s.716.m%!.awvalues of t different from zero -were obtained from equa-tion (26) with the required values of Mmdetermined by meansof the theoretical results of references 4 and 5. The limitingvalue
32、s of thesecond and third columns show, respectively, the effech ofthe second-order and third-order terms in the thicknesscoefficient t. The successive vahes of Km for the variousvalues of the thickness coefficient tindicate good convergence.LANGLEY MEMORML AERONAUTICAL LABORATORY,lVATIONAL ADVCISORY
33、COMMITTEE FOR AERONAUTICS,LANGLEY FIELD, VA., December 9, 1947.REFERENCES1. Von K6rm6n, Theodore: Supersonic Aerodynamics-Principles andApplications. Jour. Aero. Sci., vol. 14, no. 7, JuIy 1947, p. 396.2. Tricorr1947, pp. 142-148.TABLE IIVALUES OF K., FOR THE FAMILY OF ELLIPTIC CYLINDERS -OF REFEREN
34、CE 5Frr air, _L4 for Frcix- pl.13Sl=Atro L L lo 1,m O.ml? 0.627 O.m.(02 .flss .986 .494 .bm .650.Q74 .972 :%?:E .420 .M4.Om .95s .4s7:% :%:% .020 .b27.4s1.Oio .fml .s94.616 .s26.470 .Mi5:%? :g .4!3 .516.W .s73 .M3.Om .s4s .E37 .s22.m7.4W .mo.m .s?7 .S13 .Sw .4*7 .4s4 .404Ra.m-12I.og . $i6.002.94.0(8.Sm$%1:&o.974.969:%!.s72.S47.Sml.07a.9Ea:E%.S71.845.s22Uxc.407.40s.491.4s4.474.4&3:%0.656.6s2:U%.613.Eo6.4m, 1,.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-
