1、REPORT 1274SECOND-ORDER SUBSONIC AIRFOIL THEORY INCLUDING EDGE EFFECTS By MILTOND. VAN DrmSUMMARYf3everal recent advanctx in plane 8ub80nic jhw theory arecombined into a uned second-orb theoryfor airfoil 8eciionJ3qf arbitrary dupe. sobuhbn i8 reached in three 8tep8:me inwmpres8ibt?erewdt i8 found by
2、 ini!egraiion,it ti wn-verted into tti conwponding subsonic wmprawibk redt bymeans of the 8ec07uLorderwmprtxwibili-tyrule, and it is ren-dered unrmly redid w -aiion poim%by f however, just as for the ellipse the pressurecoefficients calculated by second-order theory maybe slightlylCESnegative than t
3、he true values near the minimum.Experiments on NACA 0015 airfoil.-Experimental pres-sure distributions in two-dimomiomd flow over the NACA0015 airfoil at high subsonic speeds are reported in reference41, For zero angle of attack, the critical hfach number isapproximately 0.70. The mwwurements at thi
4、s Machnumber me compared in figure 13 with the results of first-and second-order theory and of the two common compressi-bility correction formulas applied to the incompressibleflow values tabulated in reference 38. Unfortunately,the model was imperfectly constructed, and the ordinateswere inaccurate
5、 nem the nose and midchord. Otherwise,the measured pressures are in satisfactory accord witheither secondarder theory or the results of the Kfmnhn-Tsienrule.!I!omotika and Tamadas ai.rfoil,-Using the hodographmethod, Tomotika and Tamada have calculated the flompast n certain family of symmetric airf
6、oils (ref. 42). Asusual in hodograph solutions, the airfoil shape varies some-what with free-stream Mach number. The critical Mach-1.7Cp Formal t%st-ander theory- Formal second-order theory- Prandtl-Glauert rule- K6rm(n-Tsien ruleExperiment (ref. 41)0 Upper surfacea75 LowersurfaceFmction”of chord)FI
7、GURE 13.Comparieon of thearetioal and experimental pressuredietributionrs on NACA 0015 airfoil at .ii=o.70, zero angle ofattack.number is 0.717, and the corresponding shape is shown infigure 14 together with the surface speeds predicted byvarious theories.For mathematical simplicity, Tomotika and Ta
8、mada haveadopted a hypothetical gas, which is fitted at Mach numberszero and unity to a polytropic gas having = 7/5. At anyintermediate Mach number, however, the hypothetical gascorresponds to a polytropic gas whose Y is greater than 7/5,reaching a maximum value of 1.91 at i14=0.78. To seeondorder,
9、any such hypothetical gas is equivalent to a polytropicgas having the veke of -r corresponding to the free-streamflOTV,gh%ll by()=1+ $% -_where p is the density and c the speed of sound in the hypo-thetical gas. For Tomotika and Tamadas gas with -ii=0.717, that value is 1.82. Actually, the second-or
10、der solutiondepends so alightly upon the value of -r that the change fromy=7/5 to Y= 1.82 increases the maximum value of gJUby onlytwo parts in a thousand. However, the nonpolytropicProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-. _ .554 REPORT 1274
11、NATIONAL ADVISORY COMMITTEEFOR AERONAIJIMCS“ Hodograph method, Tomotib and Tornado (ref. 42)1.41.25Lo“%?FIGURE 14.-6peed on Tomotik*Tamada airfoil at M= O.717.nature of the hypothetical gas must be considered in con-verting Tomotika and Tamadas values of surface speed(which are referred to the criti
12、cal speed) to the form gjU.At the free-stream Mach number of 0.717, the ratio offreestremn to critical speeds is 0.769 for the hypotheticalgas, compared with 0.748 for a polytropic gas having -y=7/5(and 0.774 for a polytropic gas having y= 1.82)S Thus, asTomotika and Tamada emphasize, their resndta
13、should beregarded as exact for the hypothetical gas; and the presentmethod should be rP,7-fP6e:JJpIIP7TI;( )=( )b(). -( )M( )rn( )x( )Xl( )s( )t( )0( )1( );coefficient in trigonometric polynomial ap-proximation to Tfree-stream speedvelocity perturbation parallel to chord linovelocity perturbation no
14、rmal to chord linoabscissa8/(p/2)ordinate of airfoilordinates of upper and lower surfaces ofairfoil, respectivelyordinatecomplex variableangle of attackmcoefficient in numerical calculation of uadiabatic exponent of gascoefficient in numerical calculation of Ysemivertex angle of sharp edgepolar angl
15、eangle of airfoil surface to chord lineterminal angle of camber line to chord linocoefficient in numerical calculation of Y“pressure coefficient on parabola in subsonicflowradius of round edgeairfoil thicknw ratio4T for Joukowski airfoil3 (l+COS e) (C3)Then the streamwise perturbation velocity on th
16、e airfoil isgiven byu ?)(0 appo 2 ap_ .u-ax W= G To (CM)Now according to Watsons equations (10), (24), and (27),in the absenca of circulation the values of Z)O at the pointsem are given byaP)ZN1% = ,o BFY.+P P,=.$V, p=oo, p=even, not Oi1N(lcose,) p=odd J(C5)Now ShlC13 /ntional Airfoil %ctons. NACA R
17、ep. s.32, 1045. - I 482-492., , ,TABLE I.PIVOTAL POINTS, INCLINED FLAT PLATE SOLUTION, AND INFLUENCE COEFFICIENTS FOR P, aI N-Sn.123456:II1112;15N=16r 1r. r. (for% (2A) Co%)($.).3. M43o. 22420.224203.11H3. . 1-aowa hem E.0273 62.54%9.14646 2 U42 2 U42 13. S.2Q9M 1.4fw3 1.4066 ? OW-rl -11 Awl-lI. 646
18、7 0 ; 1469. 4616:. 2bw o -. w16. 19W o-.1242n -. I -: Ws-6.4423 lIL31250 -y3. 67%5o . 6743-. 2W9 o0 . 1831. Cw37 o-0.62620. 6%2o. 23310. 7346no. 64370. 11090. 23240. mmJ. 737419.24W7. mo. 7053-o.Olrl:. CL 377s-L 02400-. 4edoo. 196soLo1e3ox 60152656sL 22-. 2z Bs9. 63490.mloL3!?371: Es27o.Em-.3934.139
19、2. 23s2.le3a-! -6.25.930. 247401.EllsIi 7610-g. 00?a. W42o. 1%so. 64474.770117.3m-:5193.m1 61030.151S. m.1642. 26.27. 1W3.U42.2516. 97417: %19.919!37. !MB31.73UL 07WI64U-. 301s. US16o :i%. KIMo -: E. 72S3-1: E%: 6E25 1.0125ZL6274 11. w7. 321So M%16.!3!3s n. 41w-0.0134 0.64690 : . 01s2o . 7617. 03170
20、 %-RI. 0e. . Kk37o i 2377. 37400 :%13. Wa4LSIMI 3% Hi16.CKd4 14a704s.4%9oL83711 lmProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-_. 562 REPORT 1274NATIONAL ADVTSORY COHTYI!EE FOR AERONAUTICSTtiLE 111.INFLUENCE COEFFICIENTS FOR SLOPE(a) N=8f. (slope
21、duo to -mber)I% (slope due to thkhmss)2 I s I 4 60.6340-L404626131-h m.4483L 71E3L 6W729. 2202L 4142g: ag-Ii %213m$20.48s3-l=. 44s3IL 3181 26HlL 9134L 4046: %x.X3:%ki%47V35-m. 0423. 2023la 4526h 1269:%:%. 374. 33%3-aW78i=i w1; E%180M7-4%L3364. w.3978aw. 74L 10251. E:%.3.6748Uh 3%.44!33lcl 4112-4 7M2
22、6%91. .5307.ml-o.m-1:%1.41421. wi E: g-l : Pmla 84W -IL XL 46%5 17.047!3-31. 4m41%-4.7a?6 1?%f. (SIOWdm to camber)1.0100-L06W1.lmo1. 1349L 2018-L4C4KI-k E26466-3.3439;: -3Wo-.E. 16Ea14-144.1647L77-m:93. 464L 58 CEJIL 92$ -i 4st 49Q 37: 10.33-lL 93lL W 14.mI IxJ-N ;E z-%w CO47L77 In %144 M 82a21S-:%
23、g-m. 45-% %-%_E#lL 93t %a E 11.371.2 m% 5m 01 2 E337.78 210.19214.aa 844 m54 53 21a192ha7 m. ea;: E :i 387.H 5. g-IL 48 a.m.76L 2447L 3345L 5M9L M71L E3711204382 Z312m2 ml?3.33Ma.wtl446mh 2512am7.3s361! %!16mH%Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,- . . . .Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-
