1、REPORT 1306BODIES OF REVOLUTION HAVING MINIMUM DRAG AT HIGH SUPERSOIYtCBy A. J. EGGERS,JE., MHIIR M. RESNIKOFF,and DAVIDH. DmNISSUMMARYApproxiti shapes of dijtingpremure foredrag at high supersonicboditx hating minimumair8ped.s are cdwh!ed.With ilk-aid oj Newton8 luw oj res-istmce, the invedi4m isCa
2、rri-dOuifor variOu8CO?nbinution$Oj the conditiOm3of givenbody lenh, base diameter, swjaa area, and volume. In gen-ewl, II is jownd that when body length h jikd, the body b ablunt nose; whenxw, when tlw lenh is notw, the body luMa8hU?p71.08e. % lldiitkd eJt Oj CILTWIJhLTeOjtheJ?OWOVeTthe surjase i8 i
3、nw8t however, it does not become unacceptablypoor except for values of K below 1 (e. g., the pressurecoefficients War by from O to 35 percent for a K of . Itis therefore concluded that for valuea of K greaterthan 1,equation (1) may be used with acceptable accuracy for thepurposes of this paper to pr
4、edict the pressure distributionsand thus pressure drags on bodies. For this remon, andbecause of its simplicity, it is tvnployed throughout thesubsequent analysis.If the manner in which the pressure coefficient varies overthe surface is known, it is Q simple matter, of course, toevaluate the pressur
5、e drag of a body. Neglecting the bsae-drag contribution, we have then(3)where y denotes the derivative dy/ok. This equation maybe expressed in a form more convenient for use hereD1.=s c, i thus the variation of y with z is readily deter-mined with the relations of equation (25) for a given 1 and d(c
6、orresponding to a given% and y,) of a body. These relwtions for a body of given fineness ratio cm be shown to beequivalent to those originally developed by Newton (meref. 6).Given length and vohune,-For these given conditions,the terminal conditions (eqs. (19) and (22) require theslopes at the nose
7、and at the base to be, respectively, VI= 1and y=O. The iirst integral to the Euler expression (eq.SlmilarlY,it cm beshownthatthareamnocornersbetwoa (O,VI)and (ZI,YY)onm y of tho to k trfateilhem.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-BODIES
8、OF REVOLUTION HAVTNG MINIMUM(17) then leads to the following parametric representationof the minimizin g curve:(26)I?rom the relations of equation (26) it is clear, again, that theminimizing curve cannot pass through (o,o)j the conditionyl=l determiningg a value y,O. These relations, togethwwith the
9、 volume condition (eq. (7) and the given lengthcondition, serve to determine yl and x and thus, of course, theshape of the entire body. As the length approaches O, x be-comes infinitely negative; while, as the length becomes in-finitely large, x approaches O. (In the latter case the bodyBhape approa
10、ches the minimum-drag shape for the givenlmgth and diameter condition, Z/d+ .) Intermediate nega-tive values of x correspond to intermediate values of lengthfor a given volume.Given length and surface area,-li this caae a fit integralto the Euler equation is given by equation (18), and theparametric
11、 representation of the mhimking curve may bewritten immediately in the formconst. (1+#*)2=4#-A(l+y2)3fl -z= Jr,?/(27)Upon examination of this equation and equations (2o) and(22), it becomes apparent that, again, the mhbizing curvecannot go through the point (0,0). The latter equationsdetermine uniqu
12、ely, however, the values of y, (y2), equation (29) has two solutionsin yz. One solution yields values of y; greater than a result which violates the Le.gen namely, whenthe length is given (fixed) the bodies assume blunt noses,whereas, when the length is not given (i. e., is free), the bodiesassume s
13、harp noses. The former characteristic may betraced to the fact that with the length restricted, the netdrag is reduced by accepting higher pressures on a relativelysmall area of large slope near the nose, thus achievinglower pressures on a relatively large mea of small slope nearthe base. On the oth
14、er hand, when the length is not re-stricted it is evident that a sharp rather than a blunt nosewill obtain for minimum drag, since the drag of any blunt-nosed body can be reduced by simply reldng the requirement on length, thereby allowing the body to be made sharpnosed and generally more slender.In
15、 order to permit a quantitative comparison of theshapes of the calculated minimum-drag bodies, typicalmeridian curves for these bodies are shown in figure 2.For simplicity the bodies are compared on the basis of theProvided by IHSNot for ResaleNo reproduction or networking permitted without license
16、from IHS-,-,-560 REPORT 180 thus, the data presented do notinclude the forces acting on the bases of the test bodies.Reynolds numbers based on the maximum diameter of thetest bodies were:;:lm;$ whereas when the length is not iixed thebody has a sharp nose.Several bodies of revolution of iinenem rati
17、os 3 and 5,including the calculated minimum-drag bodies for givenlength and base. diameter and for given base diameter andsurface area, were tested at Mach numbers from 2.73 to6.28 in the Ames 10-by 14-inch supersonic wind tunnel. Acomparison of the relative theoretical and experimental fore-drag co
18、dicients indicated that the calculated minimum-dragbodies were reasonable approximations to the correct shapes.It was verified, for example, that the minimum-drag bodyfor a given length and base diameter has as much as 20 per-cent less fored-rag thap a cone of the same fineness ratio.The cone is, ho
19、wever, the calculated minimum-drag bodyfor a given base diameter and surface area.The comparison between theory and experiment alsoindicated that the centrifugal forces in the flow about bodiescurved in the stream direction may influence their drag.The relative extent of this influence was found to
20、be pre-dictable, particularly at the higher Mach numbers, with asimple motivation to the impact theory of Newton. Itwas therefore suggested that improved approximations tominimum foredrag shapes at high supemonic airspeeds (forwhich the hypemonic similarity parameter has a valueappreciably greater t
21、han 1) may be calculated with the aidof the motied impact theory. Such a calculation wascarried out for bodies with the same given conditions asthose calculated with the Newtonism theory. In general,the resulting shapes were found to be somewhat blunter inthe region of the nose, to have more c-h%atu
22、re in the regiondownstream of the nose, and to have slightly lower drag thanthe corresponding shapes obtained using the simple impacttheory.Amm AERONAUTICALLABORATORYNATIONAL ADVISORY Commrr m FOR AERONAUTICSMOFFETT I?IDLD, CALm., Dec. 14,1965Provided by IHSNot for ResaleNo reproduction or networkin
23、g permitted without license from IHS-,-,-564 REPORT 1300-NATIONAL ADvrsoIIY coMMITITl E FOR qsoA3gGIVEN BABE DIAMETBB AND SURFACE AREAiY=: Jl+ti3B 1.=j;=:p()!2(B6)with yl= O andA= 121.6(/s)3The minimkkg curve given by equations (B6) is compnredin figure 9 with that determined earlier (the cone) by c
24、on-sidering impact pessures only.REFERENCES1. von Kdnmin, Th.: The Problem of Resistance in Comprmible 9. Forsyth, A. IL: Cahxdus of Variations. The Cambridgo Univ.Fluids. GALCIT Pub. No. 75, 1936. (From R. Aoad. DItalia, Prms (England), 1927, pp. 320-324.vol. , Ronw 1936) 10. Busemann, A.: I?hm4gke
25、its-nnd Gnrbewegung. Hnndwortorbuoh2. Harmk, W.: C%chossformen kleinaten Wellen-widerstandes. der Naturwim enschaften. Gustav Fisoher, Zweite Auflage, Jonn,Lilienthal-Gw41 sohaft filr Luftfahrtforschung, Berioht 139, 1933.Teil 1, Ootober 9-10, 1941, pp. 14-28. 11. Ro.ssow, Vernon J.: Applicability o
26、f Hypersonic Similarity Rulo to3. Ferrari, Carlo: The Determination of the Projectile of Minimum Pres_wre Distributions Which Inolude the Effeots of RotationWave RAatance. Reale Aoademia dells Science de Torino for Bodies of Revolution at Zero Angle of Attaok. NACA TNAtti, 1939. (Imued as British, M
27、.A.P. RTP Tr. 1180) 2399, 1951.4. Lighthill, . J.: &WISOniO FIOW pest Bodia of hVOhltiOI1. 12. Bolz Oskar: Leotures on the Calculus of Variations. G. E.& M. No. 2003, British ARC., 194S. Stechert and Co., New York, 1946 (1904 edition), pp. 22, 27,5. Seam, iVii R.: On Projeotilea of Minimum Wage Drag
28、. Quart. 38, 40, and 47.App. Math., vol. IV, no. Jam 1947, pp. 361-366. 13. Couran R., and Hilbert D.: Methoden der Mathematisoho6. Newton, Iwac: Prinoipia-Mottes Translation Revised. Univ. Physik. J. Springer (Berlin), vol. 1, 1920, pp. 17f)-180.of Cdif. Pres, 1946, pp. 333, 657-661. 14. Eggem, A.
29、J., Jr., and Nothwang, George J.: The Ames 10- by7. Sanger, Eugene: Raketen-flugtechnik. R. Oldenbourg (Berlin),141nch Supersonic Wind Tunnel. NACA TN 3096, 1964.1933, pp. 120-121.15. Ivey, H. _ Klunker, E. Bernard, and Bowen, Edward N.: AMethod for Deter8. Epstein, P. S.: On the Air Resistance of Projectiles. Proceedingsmining the Aerodynamic Charaoteristios ofTwo- and Three-Dimensional Shapea at Hypersordo Speeds.of ATationalAoademy of Sciences, 1931, vol. 17. pp. 532-547. NACA TN 1613, 1948.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-
