1、REPORT 1284.THEORY OF WING-BODY DRAG AT SUPERSOMC SPEEDS*By ROBERT T. JONESSUMMARYTLe relation of Whitcomb8 “area rule” to the lirwarformulu-sfor wave drag at sligy supersonic speeds is discussed. Byadopting an approxhnui.ereluiwn between the source strengthand the geometry of a wing-body combinatio
2、n, tlw wuvednzgtlwory is ezprwwd in lam involviqg the areas im!.erceptidbyoblique planes or Mach planes. The remdtingformulas areclucked by comparison with the drag m.euwrenwm%obtained inwind-tunnel mperim and in ezperinwntswithfalling modelsin free air. I%ML?y) a theory for determining mung-body8ha
3、pes of minimum drag ai XTA2=2A1:52If one of the higher coefficients contribute to the base areaor volume, but they invariably contribute to the drag.The rules for obtaining a low wave drag now reduce tothe rule thrA each of the equivalent bodies obtained by theoblique projections should be as smooth
4、 and slender aspoesible, the “smoothness” again being related to an absenceof higher harmonics in the serk expression for S (X). Thusin the case of given length and volume the series shouldcontain only the term Az sin 2+ (see fig. 3). It should benoted thnt in this theory, the equivalent bodies of r
5、evolutiondo not have a physical signiiknce. The concept is simplyrm aid in visualizing the magnitude of the drag of the com-plete system.430875-57+0hS(x) =Asin2 +(%acs - Hoock body)FIGuRE 3.Optimum area distribution for given length and volume.To check the agreement between these theoretical formula
6、sfor the wave drag and experimental values, we have com-pared our calculations with the results of tests made bydropping models from a high altitude. This comparisonwas made by George H. Hoklaway of Ames Laboratory, whosupplied the accompanying illustration (fig. 4). In some ofthese cases it was fou
7、nd necessary to retain more than 20terms of the Fourier series in order to obtain a convergentexpression for the drag.Considering the variety of the shapes represented here, theagreement is certainly as good as we ought to expect fromour linear simplifications. The agreement is naturally betterin th
8、ose interesting cases in which the drag is small. Theory- Experiment:,p, ,+”:N .9 1.0 1.1 1.2/-.17$II/_Ed./ 1.01.1 1.244FIQUEn4.Comparison of theoq with results of Ames Laboratorydrop te3t9.Figure 5 shows an analysis of one of Whitcombs experi-ments. The linear theory, of course, shows the transonic
9、drag rise simply as a step at M= 1.0. We may aTect sucha variation to be approached more closely as the thicknessvanishes. To represent actual values here a nonliieartheory would be needed. For many purposes it will be suili-cient to estimate roughly the width of the transonic zone byconsiderations
10、such as those given in reference 9. In thepresent case it will be noted that agreement with the lineartheory is reached at lMach numbers above about 1.08, andthe linear theory clearly shows the effect of the modification.For further theoretical studies of wing-body drag, shapeshave been selected whi
11、ch are especially simple analytically,namely, the Sears-Haack body and biconves wings of ellipticProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-_.+. . . .760 REPORT 1284NATIONAL ADVISORY COMMITTEE FOR AERONAU.ITCSplan form, having nspect ratios of 2
12、.54 and 0.635. Figure 6shows the effect of wing proportions on the variationof wave drng with Mach number,” both with and without theTJhitcomb modification. In each case the modi6caon hasthe effect of reducing the wave drag to that of the bodyalone at M= 1.0. In the case of the low-aspect-ratio wing
13、this drng reduction remains effective over a considerablemnge of higher Mach numbem. With the higher aspectratio, however, the drag increases sharply at higher speeds,so that at M= 1.6 the modification nearly doubles the wavedrag.The rapid increase of drag in the case of the high-aspectiratio wing i
14、s, of come, the result of the relatively abruptcurvatures introduced into the fuselage lines by the cutout.Such abrupt cutouts are necesmxily associated with wingshaving small fore and aft dimensions, that is, unswept wingsof high aspect ratio.These considerations led to the problem of determining a
15、fuselage shape for such wings that is better adapted to thehigher Mach numbers. The first step in this direction is,obviously, simply to lengthen the region of the cutout-thusn,voiding the rapid increase of drag with Mach number. Theproblem of actually determiningg the best shape for the fuse-lage c
16、utout at any specfied Mach number has been under-MFIGURE5.Comparison of Whitcombs experiments with theory.04, +,. Unmodified/-/ ,A40dified-y. _ _ _-1.4 .MFmmm 6.Effect of Whitcomb modification on calculated wave drag.+.A SF-%,/,Tmces of- Mach plone;1i4i9 rSw= .5Wsinp-,.,.kvs a2J-( )a (A5)3121 +;: fw
17、l+vwhere S is the plan area of the wing.By mdcing use of the reverd theorem for drag we maycompute the wave drag of any body from the fictitiouspressure field obtained by superimposing the perturbationvelocities for forward and reversed motion (refs. 12 and 13).This process leads to some interesting
18、 relations for the shapesselected. Thus in the case of the Seam-Haack body it maybe shown that the combined pressure distribution consistsof a uniform gradient of pressure over the whole interior Rof its “characteristic envelope” defined by the Mach conefrom the nose together with the reversed Mach
19、cone fromthe tail. (See fig. 11.)By thinking of the characteristic region R as a region ofuniform horizontal buoyancy, and of the body b in terms of acertain volume, v,we see that the drag is simply the product(A6)The existence of a constant pressure gradient makes t,hccomputation of interference dr
20、ag particularly simple for suchshapes, provided the interfering body lies mtely withinthe characteristic region R. Thus the additional drag of anairfoil a placed within the double cone of the fuselage will begiven byDab=vaVb(A7)Now, by the mutual drag theorem (ref. 13) we haveDab=DW (As)or, “the dra
21、g of the fuselage caused by the presence of thewing is equal to the drag of the wing caused by the presonccof the fuselage.” In this way we obtain the general formulaD(a+ b)=Dt,t,+ 2Db.+D. (A9).(b)(a) Body of revolution.(b) Elliptio wing.FIQUBE1l.Characteriatio envelopes.7(32Provided by IHSNot for R
22、esaleNo reproduction or networking permitted without license from IHS-,-,-TREORY OF WRTG-BODY DRAG A? SUPERSONIC SPEEDS 763and for the special shapes selected:“a+b=D”(l+23+D(AIo)The effect of an indentation or cutout in the fuselage mayI)o calculated by introducing a second “body,” c, shorterthan th
23、e fuselage, and haicing a negative volume equal to thevolume subtracted by the indentation. In order to simplifythe situation as much as possible it will be assumed that thewing lies entirely within the characteristic region of theindentation, and furthermore that the latter may be repre-sented by a
24、 “negative” Sears-Haack body with volume equalto that of the wing.seors Hood body;Negotwe volume Wingc-.,.,D(O+C)=DaO+2 Negative votume; UC=-%D(o+c)= DOa-DwSeers-Hoock bOdY,/ - ,.-b ,”A+/ ,., / / DtO+b+C)=Da.+Dbb-Dcc / /FIrJUEE12.Simplified caloufation of interference drag.The calculation of drag in
25、 this case is illustrated in figure12, l?or the airfoil and cutout we havell(a+c)=ll=+211u+lla )but, SiUC13,Da,= D. ./D(a+c)=DmD. J(All)hTow, the combination (a+c) maybe placed inside the charac-teristic region of the body b without interference, sinceva+v=O. Hence,D(a+b+c)=Du+DbbD& (Al)This formula
26、 yields the minimum drag for the shapes selectedunder the assumption that VNCis fixed. In this case thedrag saving is equal to the drag of the indentation alone.The negative S-Haack body is not the optimum shapeof the indentation c for the elliptic wing, as shown by theresult of Heaslet and Lomnx qu
27、oted earlier (ref. 10). Again,how-ever, m the case of the optimum shape for c, our previousequation holds. However, the calculation of D., is morecomplex in this case and its value is somewhat greater.REFERENCES1. WMtcomb, Richard T.: A study of the Zero-Lift Drag Rfse Char-acteristicsof lVing-Body
28、OtonbinationsNear the Speedof Sound.hTACA RM L52H08, 1952.2. Hayes, iV. D.: Linearized Supersonic Flow. North AmerioanAviation, Inc., Rep. No. AL-222, June 1947, pp. 94-95.3. Ward, G. N.: Supemcmic l?low Past Slender Pointed Bodies.Quart. Jour. Meoh. and Appl. Math., vol. II, pt. 1, 1949.4. Graham,
29、E. TV.: Pressure and Drag on Smooth Slender Bodies inLinearized Flow. Douglas Airoraft Co., Rep. SM-13417, 1949.5. Heaslet, Max. A., I-omax, Harvard, and Spreiter,John R.: Linear-ized CompressibleF1OWTheory for Sonic Flight Speeds. N7ACAReP. 956, 1950.6. de IUrnutn, TL: The Problem of Resistance iu
30、CompressibleFluids. Estratto dagli Atil del V Convegno dells FondazfoneAlemandro Volt% 1935, Rome, Reale ccadeia dItalia, 1936.7. Seam, IV. R.: On Projectiles of Minimwn lVave Drag. Quart.Appl. Math., VOLIV, no. 4, Jan. 1947.8. Jones, Robert T.: Theoretical Determination of the MinimumDrag of Airfoi
31、ls at SupersonicSpeeds. Jour. Aero. Sci., vol. 19,no. 12, Dee. 1952.9. Busemann, A.: Application of Transonic Similarity. NACA TN26S7, 1952.10. Loma.x, Harvard, and Heaslet, Max. A: A Special Method forFinding Body Distortions That Reduce the wave Drag of Wigand Body Combinations at Supereonio Speed
32、s. hTAC.kRep.1282, 1056.11. Fern, Antonio: Application of the Method of CharacteristicstoSupersonicRotational Flow. NACA Rep. S41, 1946.12. Munk, Max M.: The Reversal Theorem of Linearized SupersonicAirfoil Theory. Jour. AppL Phy., vol. 21, no. 2, Feb. 1950,PP. 169-161.13. Jones, Robert T.: The Minimum Drag of Thin Vlings in Fnction-less F1ow. Jour. Aero. Sci., VOL18, no. 2, Feb. 1951,pp. 75-81.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-
