1、NACA Tit 1051AN ANALYSIS OF BASE PRESSURE AT SUPERSONICVELOCITIES AND COMPARISON WITH EXPERIMENTNational Advisory Committee for AeronauticsWashington, D.C.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NOTICETHIS DOCUMENT HAS BEEN REPRODUCED FROMTHE
2、 BEST COPY FURNISHED US BY THE SPONSORINGAGENCY. ALTHOUGH IT IS RECOGNIZED THAT CER-TAIN PORTIONS ARE ILLEGIBLE, IT IS BEING RE-LEASED IN THE INTEREST OF MAKING AVAILABLEAS MUCH INFORMATION AS POSSIBLE.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-
3、L,_c_ a-f tc_5-tREPORT 1051AN ANALYSIS OF BASE PRESSURE AT SUPERSONIC VELOCITIES AND COMPARISONWITH EXPERIMENTBy DEAN R. C_APMANSUMMARYi,In the first part of the ince._qigation aa analysis is made ofbase pressure is, an inci.wid fluid, both for two-dlmensional andaxially symmetric flow. It is shown
4、that .for two_timen,ional:low, “and also for the -flow over a body of recolutioa with aoflindrical sting attached to the base, there are an infinitenumber of possible solutions sati._fyi_g all necessa_j boundarycot_ditions at any gicen free-stream Mach number. For theparti4u, lar case of a body havi
5、ng no sting attached only onesolution is possible i_ a_ invi._cid -flow, but it corresponds to. zero base drag. Accordingly, it is concluded tl, at a strictlyineiacid-fluid theory cannot be _.ati.gactory.for practical ap-#icaaon,.An approximate semi-empirical analysis .for base pressure-in c, vi4wuo
6、us fluid is deeeloped in a second part oJ the inoestiga-_. The semi-empirical analysis is based partly on _inci_cld-“ fl_w calculations. In this theory an attempt is made to allowfor the effects of 3Inch number, Reynolds number, profil shape,and type of boundary-layer flow. Some measurements of base
7、pr_sure in two-dimensional and axially symmetric flow arepresented for purposes of comparison. Experimental resultsalso are presented concerning the support interference effecto-f a cylindrical sting, and the inter-ference effect of a reflectedbow wave on measurements of base pressure in a supersoni
8、cwind tunnel.INTRODUCTIONThe present investigation is concerned with the pressureacting on the base of an object moving at a supersonicveracity. This problem is of considerable practical impor-tance since in certain cases the base drag can amount to asmuch as two-thirds of the total drag of a body o
9、f revolution,and over three-fourths of the total drag of an airfoil. In thepast, numerous measurements of base pressure on bodies ofrevolution have been made both in supersonic wind tunnelsand in free flight, but these experimental investigations havehad no adequate theory to guide them. As a result
10、, thepresent-day knowledge of base pressure is undesirablylimited and some inconsistencies appear in the existingexperimental data.Various hypotheses as to the fundamental mechanismwhich determines the base pressure on bodies of revoh|tionwere advanced years ago by Lorenz, Gabeaud, and yonKtirm/in.
11、(,See references 1, 2, and 3, respectively.) Theset Euper_._des NACA TN 2137, “Ale Analysis of Ba_ Pre,.sure at Supertonl Velocities andrel, ren,e t_ mine expcrimen_ net dlscu,_qi therein, end inCOrPOrates s mort detaih,d ana|ysl$. _. _,m_-_t_._-_llhyl)ottmscs, which neglect the influei|cc of the bo
12、undarylayer, do not, appear to be adequate for lWcdi(ting tlm basepressure or for correlating experiments.A semi-empirical theory of base pressure for bodies ofrevolution has been advanced by Cope in reference 4.Copes analysis and the semi-empirical analysis of thepresent report, were developed inde
13、pendently and are similarin one significant respect; both consider the influence of theboundalT layer on base pressure. The basic concepts aitdthe details of the two amdyses, though, are entirely differ:m1.Copes equations are developed only for axially symmetricllow, and (lo not provide for the effe
14、cts of variations inprofile shape on base pressure, tie evah|ates the basepressure by equating the pressure in tile wake, as calculatedfrom the boundary-layer tlow, to the corresponding pressureas calculated fiom the exterior flow. In calculating thepressure from the boundary-layer flow, however, se
15、veral,approximations and assumptions are necessarily made which,according to Cop% result in no more than a first approxima-tion.TIle primary purpose of the investigat, ion described in thepresent report is to fornmlate a method which is of valuefor quantitative calculations of base pressure both on
16、air-foils and bodies. The analysis is divided into two parts.Part I consists of a detailed study of the base pressure intwo-dimensional and axially symmetric inviscid flow. Thepurpose of part I is to develop an understanding of the prob-lem in its simplest form, and to study the effecis on basepress
17、ure of variations in profile shape. In part II a semi-empirical theory is formulated since the results of part Iindicate that an inviscid-flow theory cannot possibly besatisfactory for engineering estimates of base pressure.A co,nparison of the semi-empirical analysis with experi-mental results is a
18、lso presented in part II of the report.Much of the present material was developed as part of athesis, submitted to the California Institute of Technologyin 1948. Acknowledgment is made to H. W. Liepmann ofthe California Institute of Technology for his helpful dis-cussions regarding the theoretical c
19、onsiderations, and toA. C. Chart_rs of the Ballistic Research l_boratories formaking available numerous unpublished spark photugral)hswhich were taken in the free tlight, experiments of refe|cm:e 5.Comparison with Experiment,“ by Deta R. Chtpmta, 1950. The present report includesof the ellecta of va
20、riations in profile sb*pe on b_te presstJre in h_vh_cid _ow. 3“Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-REPORT 1051-NATIONAL ADVISORY COMMITTEE FOR AERONAUTICSNOTATIONC constantd rod or SUl)l)ort diameterh base thickness (base diameter for axi
21、ally symmetricflow, trailing-edge thickness for two-dimensionalflow)L length upstream of base (body length for axially sym-metric flow, airfoil chord for two-dimensional flow).tt Mach numberp pressureP pressure coefticient referred to free-stream conditions“1- 2_p.U.Pd base pressure coefficient refe
22、rred to conditions on aPa_ base pressure coefficient for maximum drag in inviscidflow over a semi-infinite profileP_* value of P6 obtained by extrapolating to zero boundary-5layer thickness the curve of P_ versus _q dynamic pressure (2, U )R gas constantRe Reynolds number based on the length Lr radi
23、al distance from axis of symmetry to point in theflOWif temperaturet thickness of wake near the trailing shock waveU velocityangle of boattailing at base,), ratio of specific heats (1.4 for air)boundary-layer thicknessp densitySUPERSCRIPT conditions on hypothetical extended afterprofile aver-aged ov
24、er a region occupying the same positionrelative to the base as the dead-air regionSUBSCRIPTS conditions in the free streamb conditions at baseo stagnation conditionsI. BASE PRESSURE IN AN INVISCID FLUIDThroughout this part of the report the effects of viscosityare completely ignored and the flow fie
25、ld determined for aninviscid fluid wherein both the existence of a boundary layerand the mixing of dead air with the air outside a free stream-line are excluded from consideration. It is assumed through-out that a dead-air region of constant pressure exists justbehind tht. base and is terminated by
26、a single trailing shockwave. As will be seen later, the assumption of zero viscosityoversimplifies the actual conditions; the rcsuhs obtained withthi-_ assumption agree qualitatively with a nunlber of ex-perimental results, but provide quantitative informationonly on the effects of profile slmpe on
27、base pressure.TWO-DIMENSIONAL INVlSClD rI,OW OVII A SZ,II-INeI._ITE PliOFILEIn order to achieve the grcatcst possible simplicity at theoutset, the case of a semi-infinite profile will be consideredlirst. By this is meant a profile of constant thickness whichextends from the base to an inlinite dista
28、nce upstrcam(fig. 1). The problem at hand is to determine the ltow pat-tern in the neighborhood of the base. Since the effects ofviscosity are at present ignored and only steady symmetricalflows are considered, the problem is simply that of determin-ing the flow over a two-dimensional, flat, horizon
29、tal surfacewhich liars a step in it (fig. 2).-.,. “-.Dead ate“Ft(_calt l,-_.,mtdnfln|te profile._/llll/I/I/I/lll_/lllllll/lllllllllFlGtrlglt 2.-Example of inviseid fl0w over s two.dimensional profile.It is easy to construct a possible flow pattern which satisfiesall necessary boundary conditions inc
30、luding the requirementof constant pressure in the dead-air region. For example,suppose the free-stream Mach number is 1.50 and someparticular value of the base pressure coefficient, sayPb=-0.30 (Trip.=0.53), is arbitrarily chosen. Since thebase pressure is prescribed, the initial angle of turningthr
31、ough the Prandtl-Meyer expansion (fig. 2) is uniquelydetermined, and in this particular case is equal to 12.4 at 13. The pressure, and hcncp lilt. w,lo,ity and .M.ch 1511155-ber, must be constant _lh)ng the free stlcaunline BC. Forthe example under consideration, the Math number .longthe free stream
32、line is calculated from the Prandtl-.h,ycrequations to be 1.92. For u uniform two-dimensiomd llowover a convex corner, tlw pressure dcpcnds only on the angh,of inclination of a streamline, hem.e it follows that BC isa straight line. The triangh, BCE therefore bounds a regionof uniform flow haviug th
33、e same pressure as the dead-airProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-AN ANALYSIS OF BASE PRESSURE AT SUPERSONICregion. As the trailing shock wave (fig. 2) extends outwardfrom F to infinity, interference fiom the expansion wavesgradually dec
34、relues its strength until it eventually becomes aXIach wave. That part of the shock wave from C to F“mustdeflect the flow throltgh the saine angle as the expansionwaves originally turne(l it (12.4 for the particular exampletinder consi(leration). This deflection certainly is possiblesince the ._,Iac
35、h number in the triangle BCE is 1.92 _hich,according to the well-known shock-wave equations, is capableof undergoing any deflection smaller than 21.5% As thetlow proceeds downsticam from the trailing shock waveC E F, the pressure approaches the Dec-stream static pressure,thus satisfying the boundary
36、 condition at infinity.It is evident that a possible flow pattern has been con-structed which satisfies all the prescribed requirements aswell as the necessary boundary conditions. This flow, how-ever, certainly is not the only possible one for the particular._Iach number (1.50) under consideration,
37、 since any negativevalue of P_ algebraically greater than -0.30 also wouldhave permitted a flow pattern to be constructed and stillsatisfy all boundary conditions. This is not necessarilytrue, though, if valfies of P_ algebraically less than -0.30are chosen, as can be seen by picturing the condition
38、s thatwould result if the base pressure were gradually decreased.The angle of turning through the Prandtl-Mcyer expansionwould increase and point C in figure 2 simultaneously wouldmove toward the base. The base pressure can be decreasedin this manner only until a condition is reached in whichthe sho
39、ck wave at C turns the flow through .the greatestangle possible for the particular local .Mach number existingalong the free streamline The base pressure cannot befprther reduced and still permit steady inviseid flow toexist. The flow pattern corresponding to this condition ofa maximum-deflection sh
40、ock wave can be considered as a“limiting“ flow of all those possible. There are obviouslyan infinite number of possible flows for a given free-streamMach Number, but only one limiting flow.The limiting value of the base pressure coefficient can becalculated as a function of the free-stream Mach numb
41、er byreversing the procedure described above for constructingpossible flow patterns. Thus, for a given Value of the localMach number along the free streamline a limiting flow pat-tern can be constructed by requiring that the angle of turn-ing be equal to the maximum-deflection angle possible fora sh
42、ock wave at that particular local Mach number. Byuse of the Prandtl-Meyer relations the appropriate value ofthe free-stream Mach number is then directly calculatedfrom the angle of turning and the local Mach number alongthe free streamline. This process can be repeated for differ-ent values of the l
43、ocal Mach number along the free stream-line and a curve drawn of the limiting base pressure coefficientas a function of Math number. Such a curve is presentedin figure 3. The shaded area represents all the possiblevalues of the base pressure coefficient for two.-dimcnsionalinvi,_cid flow. The upper
44、boundary of the shaded areacorresponds to the limiting flow condition for various free-stream Mach numbers.There is no reason a priori to say that for a given M.the limiting flow pattern represents that particular onewhich most nearly approximates the flow of a real fluid.VELOCITIES AND COMPARISON W
45、ITH EXPERIMENTMoch r_an_berFIGI_al 3.-Ba_ pressure for two-dlmensiona Inviscld flow.The curve representing these limiting flow patterns can beconsidered simply as being the curve of maximum base drag(and hence maximum entropy increase) possible in an in-viscid flow. This is the only interpretation t
46、hat will begiven to this curve for the time being. Since it is theselimiting solutions which will be singled out later for furtheruse, a special symbol P,j will be used to designate the base pressure coefficient of such flows. It is evident from figure3 that in the Mach number region shown the value
47、s of P_for two-dimensional flow correspond to very high base drags,being almost as high as if a vacuum existed at the base.At Mach numbers greater than or equal to 6.0, the valuesof P_t exactly correspond to a vacuum at the base.AXIALLY SYMMETRIC INVISCID FLOW OVER A SEMI-INFINITE BODYIn principle t
48、he same method of procedure can be usedfor inviscid axially symmetric flow as was used for inviseidtwo-dimensional flow. The axially symmetric flows, how-ever, aresomewhat more involved than the correspondingtwo-dimensional flows. For example, in axially syinmetricflow the expansion wavelets issuing
49、 fronl the flow pat terns as was done iuthe two-dimensional case, the method of (.hara(,t(,risiics re,“axially synunctric flow must be used. The details of thepartieula,“ characteristics method employed are described inreference 6. B)“ employing the characterislics method theinviscid flow field correspon(ling to a given base pressu,e canbe colkstructed step by step re,“ any given value of the Mathn
