1、REPORT No. 808A METHOD FOR THE CALCULATION OF EXTERNAL LIFT, MOMENT, AND PRESSURE DRAGOF SLENDER OPEN-NOSE BODIES OF REVOLUTION AT SUPERSONIC SPEEDSBy CLINTONE, BROWNand HERMOKM. PAEKERSUMMARYAn approximate method is pre8ented for the cultwlafion ofthe ezternal lift, moment, and prewwre drag of 8Lmd
2、er open-no8ebodies of recohdion at superwnic 8peed8. lh? ifi, moment,and pressure drag of a typ”cal rmn-jet body 8hupe are calculatedat Mach numbers of 17, i .60, 1.76, and 3J10; and the lijland moment result$ are compared w“th aiwiiirble experimentaldata. me agreement of the cakulded lift and momen
3、t datam“th the ezpei+mental data is excelent. The pre8wre-dragcomparison uw not pre8ented because of the uncertainty ofthe amount of 8En-fi-ietion drag present in the experimentalre8ult8. It win found that the lift coe$a”ent dq?nitely in-crea8ed unlh incream”ng Mach number, ?.cherea8 the momentcvej%
4、ient taken about the midpoint of the body and the dragcoejln”ent deereused with increasing iJIach number. Themanner in which the method -may be applied to 8knder bodie8of rerohdion w“th annuar air inht8 i8 8houm. T7ie excellentagreement of the calculated lift and moment results with experi-mental da
5、ta indicate that the approximate method may bereiably ued for obtaining the aerodynamic characteristics qf81ender bodies that are required for efieient supersonic jlight.INTRODUCTIONCurrent proposaIs for the design of aircraft capable ofsustained flight at supersonic speeds and utilizing the ram jet
6、as B method of propulsion have established the importanceof knowiug the aerodymunic characteristics of slender open-nose bodies of re-rolution at. speeds greater than the speed ofsound. The lack of theoretical treatments and experimentaldata emphasizes the need for theoretical investigation of thisp
7、roblem to serve as a guide for future work tid as a check onthe reasonabIenew of current and future experimental results.The smalI-perturbation approximation was used in refer-ence 1 to deduce the wave drag and in reference 2 to obtainthe lift and moment of slender pointed-nose bodies of revolu-tion
8、 No fundamental analysis is known to have been made,however, of the characteristics of a slender open-nose bodyshape, such as that required by ram-jet propelled craft. Thepeculiarity of the problem, from general considerations ofsimiIari, is that the flow pattern is two dimensiomd at theIip of the n
9、ose and approaches the three-dimensional patternfarther aIong the body. The present work extends themethod of references 1 and 2 to apply to these slender open-nose bodies of revolution with supersonic flow into the nose.The result is a faidy aimpIe method of numericaI integrationof the differential
10、 equation of the flow. As an illustration,the pressure distribution, wave drag, Iift, and moment arecalculated at Mach numbers of 1.45, 1.60, 1.75, and 3.00 fora typictd ram-jet airplane bocly ahape; arid the lift andmoment results are compared with the experimental data.It should be pointed out tha
11、t- the accuracy of the method,which assumes potentiaI supersonic flow throughout the fieldand also assumes and disturbances, depends on the surfaceFes of the body and the Mach number. The error in-creases mith either increasing Mach number or increasingsurface angles.SYMBOLScylindrical coordinatesdi
12、stance aIong r-axis measured from nose ofbodylength of bodyIadiUS of bodyMach angle (tin-+)perturbation poterdiaIperturbation potential for axial flowperturbation potentiaI for cross flow()tlljaxial veIocity increment x()*radial velocity increment rveIocity in undisturbed streamvelocity of sound in
13、undisturbed streamMach number in undist urbed stream (rla)density in undisturbed streamincremental surface pressure due to angIe ofattackIocal pressurepressure in undisturbed streamratio of specfic heats of air (1.4)angle of attack, radians (except where other-wise noted)angle between surface of bod
14、y and x-axis49Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-50 REPORT NO. 80 %NATIONAL ADVISORYCOTTEE FOR AERONAUTICSliftcOefficien/:v2TR2)dragcOefEcient(Drag/V2m rmdthe sec?ndJh= B cos 6 “j,(x+ cosh U) COSh U du (8)represents the cross-flow potent
15、ial of an arbitrary dist ribu-tion of doublets along the axis of the body starting at thonose of the cone or projectile. The form of cquat ion (8) isthat the cross flow is from t-he direction 19=0, as shown infigure 1.Provided by IHSNot for ResaleNo reproduction or networking permitted without licen
16、se from IHS-,-,-CALCULATEDLIFT, MOMENT, AND DRAG0? SLENDER OPEN-NOSE EODIES OFBy neglecting the smrdI effect of the axial flovi on the lifting pressures,of arbitrary shape the equations:REVOLUTIONAT SUPERSONIC S3?EEDS 51Tsien obtained for the pointed projectile(9),(11)The values Ki in theseequatioM
17、me um to be constantsfor each interd of the step-by-step process. The momentcoefficient of equation (10) is resumed positive for nosing-up moments, these moments being taken about the nose.OPEWXOSEBODIZSThe flow conditions over an open-nose body ditTer fromthose of pointed bodies in that., for finit
18、e angles of the noselip, the flow is two dimensional at the lip. This problemVWLSnot- considered in references 1, 2, and 4 and the generalsolution should therefore be examined to determine itsapplicability to this special case. Lamb has shown (refer-ence 3) that a sufficient requirement for the exis
19、tence of thegeneral solution to the differential equation of motion(reference 1) is thatj(x-l?r cosh u) be zero for all values ofthe argument less than some arbitrary limiting vahe. Thedetermination of (zl?r cosh u) such that the boundaryconditions at the open-nose body are satislied assures thefulf
20、dbnent of this general requirement. For the usual caseof supersonic flow into the nose, the boundmy condition re-quires the surface of the body to be a continuation of acylindrical stream surface of radius RN in the undisturbedflow ahead of the body as shown in ure 2. The perturba-tion potentiak, eq
21、uations (3) and (8), therefore must be zeroat the cylindrical stream surface ahead of the body. Sub-stituting BRH COS 6 d8R dx (16)whom 1 is the length of the body, 1?. is the nose radius, and the moments are taken about the midpoint of the body.By substituting the expression for b thus,W2()Va Cos 6
22、=& ,= (19)for which the radial velocity ia assumed to be normal to the surface. -A more rigorous bounchwy condition taking inkaccount the slope of the body was given by Ferrari (reference 4). For small surface angles, however, cqu at.ion (19) iswithin the accuracy of the small-perturbation assumptio
23、ns. The exprtwsioniip,() CQS6 J-Rf,(f)(Xg)dg5F,.,=w IJ(xf)2lFR2 -is integrated numerically for constant values of fa (t) =Ki over the ith interval of integration to obt tiin the suma#)*.() IPcoson Fe“iii,nR -Kicosh-l(T,_,) cosh-(T,”) + (T,_,n) (T,.ln) llinl-1(20)(21)Substituting this equation in equ
24、ation (19) givesl=g: cosh- (Ti_ln)COSh -1 (Tin)+ (TJI) I/(TM)2- 1 Tin(TtA)1 (22)i=llFK,With the values of determined, equations (17) and (18) becomeln equations (23) and (24) the pressure used for a giveuintegration interwd is the average of .&e pressures at thebeginning and at the end of the interv
25、aI. This scheme ofusing average lifting prwsure is paptic.ularly necessary inregions where the pressure is rapidly changing. The methoddoes not give the pressure at the beginning of the firstintegration interval, that is, at the point n= O. It can beshown that, as the &t interval approaches zero, th
26、e pressureat the lip (n =0) is obtained by letting the expression inequations (23) and (24)m-lhave the vahm 0.5 when n=l.METHOD OF CALCULATIONIf calculations are to be made for an pen-nose body, thetotal number of integration stations chosen is a compromisebetween the amount of Iabor involved and th
27、e accuracydesired, due consideration being given to the limitation onthe accuracy imposed by the basic assumptions of smaII(24)disturbances and potential flow. In gcncmd, wlwro thopressures are changing most rapidly tho intqyalion stalionsshould be the most dense.The integration stations must first
28、be chosen. (Sue fig. 3.)rva71Body axis zFrauim 3.Dhgr.arnmatic sketch to Illustrate IntmatonproccsaforIMYWcWJlorcalcultilona.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-CALCULA172D, IU034.EN9?,AND DRAG OF SLENDER OPEN-NOSE BODIES OF REVOLUTIONAT
29、SUPERSONIC SPEEDS 53The calculation for the Iift and moment then proceeds asfouows:The boundary condition, equation (22), is appIiecl to t.lwpoint n= 1, the summation reducing to a single term, whereT“=%andq-oTp,a= R7Equation (22) then givea a value of IFKJ2T”a. Next,equation (22) is appIied to poin
30、t 2, n=2, containing now twoterms. Substitution of the expressionsT=zTof=%tiBR,and use of the value obtained for lPh1/2Ta perm”ts thecalculation of the vahe of B2KJ2J”a.By continuing the proc, at each successive station onemore term occurs in equation (22) involving one nevi &which is then determine
31、d. With the values of F-h”J2T”adetermined, equations (23) and (24) are used to evaluatethe lift and moment coefficients. lt will be noticed that,vrhen equations (23) and (24) are evaluated for n= 1, thesspression occurring after the last summation symboI isunobtainable. For reasons previously stated
32、, the mpre=ionmust be given the value 0.5.The procedure for calculating the drag pressures is similarto but somewhat simpIer than that for the lift. and moment.Equations (4) and (5) are evaluated at the point n= 1, thesums reducing to one term. By applying the boundaconditions (equation (6) ) and us
33、ing the known slope of thebody dr/dz, the constantAl occurringin equations (4) and (5)is determined. Substitution of , back into equations (4)and (5) gives the increment velocities at n =1. BernonIlisexpression (equation (7) ) then gives the pressure ratio atn= 1. It is to be noted, since “=la, that
34、 actuaIIy valuesof ilJa, r,/a, and uzlfa are determined. Proceeding to thepoint n=2, one new is irmolved in equations (4) and (5)that is determined by the boundary condition in equation (6).In the same way, the velocities and the pressure ratioare calculated at n= 2. The process is continued at succ
35、ess-ive integration stations over the whoIe body. The calcu-lated values of the pressure ratio allow a pressuredistributioncurve to be drawn and the drag coet3kient to be evaluated inthe usual manner.DISCUSSION OF RESULTSRZSL_LTSOF CALCULATIONSC%Iculations were made in order to obtain the preesuredr
36、ag, Iift-, and moment of a typical open-nose body. sketch showing the dimensions, the integration stations,and intervals is gi-rcn in figure 2. Calculated pressure dis-tributions at zero angle of attack for the llach numbers1.45, 1.60, 1.75, and 3.00 are presented in re 4. Thepressure rise at, the n
37、ose Iip is approximately that whichwouId be obtained over a twodimensional wge of thesame angje. ckerets theory for smaII disturbances (refer-ence 5) gives for the pmsaure rise of a 3 viedge = 1.147 at31= 1.45, whereas the pressure ratio on the Iip of the open-nose body is about 1.140. This agreemen
38、t is considered areasonable check inasmuch as the pressure on the Iip mustbe an extrapoation of the pressure distribution from thefirst point of the integration process, which must be a tiltedistance back of the lip edge. It can be shown that, as thesize of the first intervaI approaches zero, the pr
39、essure at theIip becomes that given by the two-dimensional theory. Theeffect over the nose section of the size of integration inter-vals is iIIustratecI by figure 5. The 17-point method waschosen because a greater number of points resuIted in onIya small increase in accuracy and a large increase in
40、the laborinoI-red. & would be expected, for the case of the straightconical nose, the pressure fds off from the leading edge andapproaches the pressure of a cone of the same surface angIe(reference 1). At. the corne of the body, the pressure falIsapprosimat ely in accordance with the PrancI tl-leyer
41、reIation (reference 6) for supersonic flow around a tvro-dimenaional corner. On the center and t aiI sections there arepositive pressure gradients that for actual flight conditionswould tend to cause separation. The source-distributionfunctions corresponding to the pressure distributions offigure 4
42、have been pIotted in figure 6.The incremental surface pressures giving rise to the liftand moment are shown in figure 7 for the Mach numbers1.60 and 3.00. At the higher Mach numbers the pressuresdecrease less rapidIy over the nose section. The doublet-distribution functions at the four Mach numbe ar
43、e pre-sented in Iigure 8. The curves of figure 9 show the theoreticalvariation of lift-curve slope, moment-curve elope, and dragcoefEeient with Mach number. The rather interesting resultobtained is that the Iift coe.flicient increases -with Machnumber. fiperiment al rewdts presented in reference 7sh
44、ow an increasing lift-curve slope with Mach number for apointed projectile. The, fact that the doubIet- distributionfor the open-nose body is sindar to that of a typical projectile(reference 2) Ieads to the expectation that the Mach numbwcharacteristics of the two shapes viould be simiIar. Thecent e
45、r of pressure at the lower Mach numbers is ahead ofthe nose and moves back with increasing Mach number.The drag coefficient can be. seen to decrease with Mach1number but to a lesser degree than the Cofp-oy Tm =ER1,: 1 aw f tvo-dimen9ionaI wing profiks., . . comparison of the calculated lift and mome
46、nt coefficientswith some experimental data obtained in the Lan#ey 9-iuchsupemonic tunnel is presented in urea 10(a), 10(b), andlo(c). The contribution to the lift of the internal air canbe shown to be:A(7=2ciThis increment vdl appear at the nose and wilI thereforegive rise to a moment:ACa=aProvided
47、by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-54 REPO NO. 80 8NATIONAL ADVISORYCOMMIITEE l?ORAERONAUTICSJFIGURE4.CalauMed preaeure distrIbntIone for 0”FrouRE 6.Etl&ofnumberof lntwation pofnta onpremure dhtribution orer the now ofthe bdy04 8 12 f6 20 24 28fl
48、mFIGUItE6.-Eource4lstrlbution functions for u =OO.FICWEE7.-DfstrfbutIon ofhrcrcmentslsurheo prasurea omrtho bodyrd two Mach numlwraw%FIGURE8.Doub1et4MfbutIon funetlona.FIaUEE 9.-TbeoretfceJ vdethn of Uft, moment, and droc medlelcnta with Mach number.Provided by IHSNot for ResaleNo reproduction or ne
49、tworking permitted without license from IHS-,-,-CALCULATED LIFT,MOMENT, AND DRhG OF SLENDER OPEN-NOSE BODIES OF REVOLUTION AT SUPERSONIC SPEEDS 55-(s) M=L45.(b) M=Lm.(c)i4f-L76.FIGL!EBIo.-coxnparfxm OfCalculwd lLftand moment nmcients withexperfmmtddata.Provided by IHSNot for ResaleNo reproduction or networking permitted with
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