Comparison of Measured aud Predicted Flows through Conical Supersonic Nozz]es, with Emphasis OU the Transonic Region

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1 AAA JOUHXAL YOL. 3, NO. 9 Cmparisn f Measured aud Predicted Flws thrugh Cnical Supersnic Nzz]es, with Emphasis OU the Transnic Regin L. l. BACK, * P. F..\1Assn;u, t A!,;D H. L. GERt Jet Pr1Julsin Labratry, Callfrnia nstitute f Technlgy, Pa.sadena, Cal?f. ~'" ulldcr"la,"1 hcuc' C'"ilihril.n n"w,; hrugh,;11pc.'"" 11i. 10Z:t.!C8,~-:.!a~ç,J.)!.c"s~lrc,; 1~,"-h.s.«.111~'r;~1 luzzl.,,, with ci~ulur-arc thrnat..!,,,"illg "iftc'cllt rati"s "f hrat 'a,lius ~curvalurc t tlu"ual radills rc/rth, cir'cular-arc nr (:uical cnvcrgcnl ~cctins, und cnical (li\"t~l'gcul ~ccti()lls. 'rhcse Hcasur~_~lc~.,;,~,, ~,d(~\"i.!:..l~iil.2~~j)alill ~Jnpc!a_lu"c!'i "f 5: u.ll11500h ali"...a sla":latill pl'cs~urc rallgc f"01l1,t _psia. Thc nw lu'ugh tlh~ L.all~()nic.cgin ~:--a~-r;-;~l(1 t dcp;-u.1 ( s~cntjally 011 lcal (~-;:-(igu"'r~n, i.c., u li-e rali rei r,h; t,.-di JlCl~i()lla 1 iscllllpic fl()\~ prc(1 ict íns al!rc(~d wi th lhe da ta ili tlds rcgill ru' lhe "",; wilh re/r'h = 2.0, hllt C'Cillll"e'lualc f",' thc lzzle willr re/r.h = By pari:.;nn, lhe ~ilnph~ une-dilllcn~innal i!'õelllrpic fl\," H'(~dicti()11 was as rlluch as l5'l hig:h in lhe thral rcgil1 (0--;'une ~u~zh~; i;; lhe cnical sectin:-:.. dc\"ialil1s f a slnallcr Jua,::nitutl-;; ~'e f,,"d. Tr" efrcel:~ culillg a,"1 varialill in lhe "H,,"1a"y-layc,' hickllc,;,; a lhe Jlzzlc inlct,,'crc in\"cstigatetl.. as,,'cre diftt~ CJlCl :-Õ ill pr(~:-<~ll'(~ r(~acling::" \,"ith taps f \"~lj'i()us sizes. SO.1C,;cpal'alin ""'SS''''.laa,,,.,, presellled 10 "llow lhe en"'d f wall,~ling. Olhc,' fiw fealures lrat indical. 11", exlcll f clc,-iatins frlll llc-di.l1ellsillal nw include fiw cu"flieiclls, t h, ",,1 ra ius, a 11.1lca ""'';,,nllxe~ t is hpetl hal 11'sc 1'0111pari,;n" he C" ll,'asu"clllcnb alie! p. ~e!i.lills will he u,;cful 111"tue!yill!; 10Z~.lCnws wilh lhe ae!tlilillal eujl.,xily f ehcll1ieal"eact iolls. l' tncncla t li t c a = ~pced f sund a* = spced f sund a the snie endilin A = lcal nzzle crs~-sect ínal,11'1':1 Ath = nzzle-thrat an'a CJ = nw ce!\ieienl d = statie-pressure lap diameter ]) = nzzle-inlet di:unet er F = a xial t hmsl = lzzle appr:1ch sect ln leng;th 7i! = mass flw rate.l[ = :\laeh nnmber p = w,11sla ie pressure 7'0 = ambienl pressure 7'. = separat in pressure 1" ~agnal in pressure r = nzzle radius rth = 10zzle-thrat radius re = nuzzle-t hrat radius f eurvat ure rj = nzzle-inlet radius f curv,1t ure R = nzzk~inlet radius T = stag:nal in lemperalure Tv = walllemperature axial dislance frm nzzle inlet 11 = vdeity empnent in z direc'l in ]' = lw veleity at wall 'Y =,;pecifie-heat mli ~ = veleity bundary-\ay('r thicknl's,; at n07.zle inlcl ~. = di~plaeen1('nt thickness 'e = nzzle eolltractin-area rat i 'E = lzzle expansin-area v = kinematic viscsily mli ~ p = distance = densil \. defined in Eq. (ti) T = wall shear sl ress Sub. cripts e = enditin at freestream edge f bundary ln)'er i = enditin at nzz]e inlet teeeíved September 8, 1964; revi,;in reccived 1lay i, 196;). This papel' pre:;ents the results f ne pha~e f researeh carried ut at the Jet. Prpulsin Lnbratry, Califrnia nstitute f Tcehnlgy, under Cntraet N. NAS i-oo, spnsred by NAS.'\.. Scnir R~earch Engineer. t lescarch Grllp Supervisr" r-el1lber AAA. i lesearch Engineer; nw graduate stutlent a. the Universit)' f Clrad. 8 = cndit in a flw separat in t slag:nat ín ('ndit in 1 = ne-din1pnsinal flw value. nlrductin FLO\YS thru;!:h suppr.-nic nzzles are af intprest in design and develapmenl and in basic researeh.!t~'d _fi('~l~ nazzles a..!.:." u=:ed in rcb~t engine~ amc in measurin~ fiw 111t('~ wherea.- in research they are used J1acceleratin de\'ices and in tlll' study f fiw phennu'na sueh as nnrquilibrium ceet.-. This experimental i\l\'rstigatin was initíatcd t gain a bcttcr understandin;!: f t'quilihrium fio\\"" thrugh nzzles which app('ars t be basic in studying the addilinal dects asseiated with ehemieal real'lins. Data are prescnled thrllgh tlh' subsnie, transnie, anel supersnic fiw r('gins f a number f cnical nzzlcs. Emphasis is placeel un tl)(' thrat regin. whcre the fiw is transnie" rcvius i\l\"pstil'alins f g3.sfiws thrugh ('nical nzzles h[i\~~h2.~)~ elc\"iatins frq!!! ºn.2,:dimensinal i~ntr.ql)ie fiq.\\~ as indicatcel b\' mea,;urcd wall static pr('ssures. Th('sc de\"íatins result frm radial \'Pleity ciúpneills ('aused by llp taper and eurvature f thc nzzlc.' f;imilar de\'iatins havc hepn bscn'('d \\"hcre m(,3.surements \n're madl' in the elivrrgl'nt re~in f enieal nzzlcs2'-5 and where a few measurements wcrc made in hl' enieal (,tl\"('rgent reg;in.6 Lcal \'cleity ml'a:,uremenh; in the thrat rc~in h[1\'c il~eatl'd Ln this papcr, the tw-dimcnsinality mca:,ured wall,;tatie!. tllc)l\\"j prcssul'(,'; are presenteei fr enieal nzzles f varius dimensins t ahw in eletail thse regins where the fiw is tw-dimensinal, and thus wh('re the sim pie ne-elimcnsinal fiw prcdietin fails.1 Thc cnieal nzzles invcstigated have 30 and 45 half-ang!es"crgenee, 15 half-angles f di\'ergcnec, eircular-arc cntranee and thrat seetins, and expansin-area mtis \lp t 6.6. The mtis f thrat radius f Cun"aturc t thrat radi\ls wcre 2.0 and The eftect f inlet cnfiguratin was invcstigatcd in ther nzzlcs with thc col\'erg;cnt scctins fnncd by circular ares. Operating cnditins spanned stagnatií1 prcssures frm 45 t~ 250 psia and stagnatin tcmperaturcs frm 530 t 20000R, \\'ith data reprtcd at 5~0 and 1500 R. The ~J~bi~~t p~~ \\~~ ~)11spheric. ~t the l\\'cr stagnatín.. 1'r\

2 ~ -....,._~----~---- SEPTE~[BEH 1965 l\leasuhed AND PHEDCTED FLO\VS THHOUGH CONCAL SUPEHf:OXC NOZZLES Hi07 temperature f 5300R, colnprc..."seelair was useel, anel lhe nzzles were uncled, 'fhe higher stagnatin temperatures were btained by heating cmprcs.sed air by the embustin f methanl. 'fhe prducts f cmbustin \\'ere then llixed 1õübtam Ulllfrmity bcfre entering lhe nzzles, and at these higher temperatures, the nzzle walls \\'ere cled. :\ t the highcst stagnatin teml2q[aturc,_thcjqtal..hcat~~r frm ~s tolhe nzzle \\'aljraiiles than 10/0.f the ttal cnergy f the gas a. the lzzle in]e.t s that the fi\\' \\'as learly adiabatic illall ca:~cs. -The prelu<:ts f ('mbustin euld bc' treate;1 apprxinately às air, sinccthn1iass fl\\'-raie rab f methãnõl t air \\'as small, 'e.~., a H, the mle<:ttlar weight anel speeific-heat mti were 28.6 b/mle anc~ respectively, fr the prelucts f cmhustlon cmpared with 29.0 b/mle anel 1.35 fr air. :\b, <::dcu!ated temperatures f the~rducts f ('mbu~tin, with til<' assumptin f cmplete euple<:hcmical \1C:lsuremenb reactins, eles<.:ribcelin were withinse<:. the 11,J a~ura<:y uf lwrm Bunelary-ayer thicknesses at the nzzle inlet were varied rq!n_:ilidut 5 t 45Mn11Cm1Ctl'ã(1ilb t inl'estigate bun(c: ary-!at'r J.\i;jp1lCeml'nt effects. Thl' e1tect f \\'all staticpressure lap size n the readings \\'as als il1\'e~tigateel. At the 100'er stagnatin prcssures, lhe nzzlcs \\'ere \'erexpandl'el, and the as50ciated separatin pressures are pn'sc'nteel bth fr ht-fl\\' pcratin with elcd walls anri fr eldfi\\' pc'ratin. Other fcatun'~ i the flw are induclecl in term~ uf de\'iatins frm n('-elimen~inal is('ntrpie flu\'. Fr \'cr-all nzzle perfnnan<:e, flw <:dfieic'nts and thmst rati,.. are sh\\'1, Lcal del'iatiuns in the l\1as:; fiux at the edge f the bundary layer <:alt-ulat<:d frl\1 the \1casured \\'all static pressures are inducled, a,,,;i~ the snic line lcatin. [n the transnie fiw reg,in, 'arius t\\'-dimensinal isentrpie flw preclirtins are cmpareci \\ ith the elata t indi('ate the aelcquaey f thl'se theries. divergent regins anei mst f tlh' cnvergent reg;in is less than~ Only in the nzzle-inlet regin at the lwest stagnatlon pressures, \\'here the elifference between t he static anel stagnatil1 pressure is small, is the estimateel erl2.!:. 2!.. 5% significant; this errr wuld be less at higher stagnatlon pre&,urcsj fnsielerably!argcr clifterenccs than the previusly mentinecl % thru~h l\1st f the 110zzle are fund \\'itllciifjerentsizccl taps, as cliscll.~d in.ce. V. 11"r ht-flw pemtin \\'ith clecl walls, wall temperatures \~C dctermineel frm thenncuupips embeclflecl in the wall f the nzzle anel frm calrimetric wall lh'at fiux measurements \\'ith the nuzzl<] T. St a t ic- Prcssl r'c Dis t r-ihll t iol";; :\[casured static-t-stagnatin-pre~sme mti~ are SO\\"1l in Fig. 1 fr the luzzlc','c'r a range f ~ta~natin pre~surcs fram 45 t ).10 psia at a,'ta~laliul temperat.ure 01' 1500!t, \\"ith <:led \\'alb. TH' pper limit uf LiO psia \\'as dictated by manmeter limitatills at thc' time the tests \\"ere maclc. Frm bundar~'-a,\'('r t ra ver~es, the ra ti f inlet bunclar '-la\'c'r thi<:knes,; t lozzle-inlet radm \\"as estimatcd at abul 5 'R..."0.2,1. The pre";';li'(' rati..; are ncarly in\'llriant \\'ith sta~latin pre:"sui'(' ('x('ept in the fi\\" separatin regill, \\'here lhe rise in ~tati(' prc'ssme i~ <:allsed by \'ercxpalldccl nuzzlc!)('ratin' n the thraat rcgin, there i;; sme data ~pread, bul it ~('s nt appear t vary systc'ma tically \\'i th stagna tin prc..;sure. Cnsiclerablc de\'iatins fum the predi<:tin fr ne-dimcn~inal (1 d) i:"entrupic fi\\' ('Y = 1.40), :::h\\"1[1:-; <:ln'f~a, an~ apparent in the transllic rc'gin; in particular. t hc' measured pn's:"urcs are as mu<:h as abllt 30% lid\\' the pi'('dietin jllst d\\"nstream f the thruat. DC'viatins f a smaller magnitude are bsel'\'able in the irtlet and <:nic'al euj\'ergent sc<:tin, 11. lnst r'lll1cn ta Ín Xcar,dJere the lan~eney data are f sjightl~, the circular-are abve the l d thruat fi\\' and predidin1 ('nicar',.,.. clivergent sedin, lhe measurcd pres:::mc ratis <:hangc slpe ~lp fl\\' and lstrumentatlon dlagr:\l!~ f the sys~elll t abruptly and <:rss ver the l-ri fi\\' predidin further rl\\'n-,~'\lch the nzzlcs \\'ere atta('h~d :' 5h\\'n Hl'f. 1..Stagna- ;;tream. ~ 1-d isentrpic fi\\" predictin \\'ith \'ariable spet\on prl's~ure,l"as lleasur~d lst upstre:11~1 he a ) )r~~<:h <:ifieheat \\"as ab made ancl fund t be at mst 2% ah\'e the ~, t lc engtl: f wlmh ('uld )e adjusted t speclhed predictin sh\\'n fr 'Y = This small elifferen<:e indivalues. ghanges m lel:~th \\'crc, ~n:lde t \'llr\' the nzzle- cated the predictin \\'ith 'Y = AO t be adec uate fr COTinlet bo,undal'\'-lan'r tll!~kncss. 1hese turbulent bund,alt- parisn purp~es if the cntire nzzle fi\\' regin! is cn~irlcrectj ~a\'er 1111<:kn~~sE'~ \\'ere estllnatccl. frum measurements de~e~'lbed Fr sub~equent <:umpari~ns in the thrat regin, 'Y rliffer~ m Ref... lhe appra<:h S~,( tlon \\'as cle~1 :11, the lughcr little frm thc stagnatin <:nditin v:tlue userl. stagnatlon t.cmpl'rature f ~voh. Dt~gnatlOn temper. ature Anther -d fi;,' vrcdictin referred t in the liter:lture is l'as dc'tpl'l1uned by a\' the n':\(hng~ f \1,'0 sll!~'lded that fr <:nical suurce fi\\' in the divergent regin. 'fhe therlllcuplps p!a<:ed 0,20 m. up,..;trc~ln f the nzzle 1:let1 prcdicted \\'all slati<:-pn's:"ure distributin fr isentrpi<: fi\\' le5(' t,\" thermcuplc's, 10<:ated 1 m, frm t he C~!1-~tj!!.1~ ('Y = <:n..;t)is./ _ \\'cre spaced 180 apart cir<:lljnf('l'pntially and genl'r:dly rcad " " ~ \\'ithil ~(/Q f C~.C~1ther. The air m...;s \" "tl' \\'a~ meas- l!... = { _ 'Y--=-! -1 2 [-.ig~/s)4 _ J} ~/h-l) ured wa~ u~ccl \\'th t aliwasure nfic(" aliei the fr masst.he fi\\' lot-tlo\\' ral<' f<'st~, ll1cthanul. a rtmeter The / Pb 2. b (P/pb)2h. () aecuraey f the' ttal ma5s fi\\' rate is estill1atpd t be 1.%8: The sub~('l'ipt b denutes a pint aln!!; lhe cni('al \\'all \\'here at stagnatin prcssures aoõve aõüt 100 -psi:l. At l\\'cr the indicated variables are kn\\'n, anel s is thc radial disstagnatln prcssurcs, the rcadinj!;s are le5s a('curale. tance frm the suree. lf the experimental prcs~lre and Fr the fllwing nzzles [the Fi". ), thse <:rrcspnding.:\1a('h null1ber at the circular-arc-thrat cnical shwn in F!ll~ and the 45 _] 5 (fig..jd]' t le chall1eters tangen<:~' pint are useel, Eq. () is sh\\'n in Fig, by curve f the \\'all static-pressure taps \\'ere O. 20, and b t prediet substantially l\\'cr static pressures in the <:nical in., respcctively1 and the ratis 01 hle depth-t-dianthter sectin than t.hse btained experimentally. Althugh better \\'ere abut 8, 4, and essentially infinity, respectively. The agreement <:ulc! be btaincd if the predictin l'ere initiatcd hles \\'ere as sharp-eeigec! as thpy culd be made by drilling dwnstreall1 f the tangency, the cmparisn nevertheless and thcn smthing the burrs \\'ith emery elth. The axial indicates that the aclual fiw differs frm cnical sollrce fiw lcatin f cach tap was knwn t in., and thetaps \\'cre in the first part f the enieal 'spacec! <:ircumferentially ãiidãxially alng eaeh nzzle ldata were als btained bth at a lwer stagnatin temwall. 'fhe static prcssures were lleasurcd either \\'ith mer- p~ature f 5300R uncled walls and with n appraeh eur)' manmeters r, at the hwher )ressures \\'itll ~ length such that at the nzz mjet 51 R ~ These data, gãges, which had O,25-psia markec! increm~. The ac- t ug 1 nt shwn, indi<:ated that the effect f \\'all cling,. cumey f the readiilgs S chctatedliytllcdífference )et\\'een ~dtered the measured pressure rat.i negligib!,l excel~t in th~ the statie anel stagnatin prcssures and thus depends n lca- fiw-separatin regin, where, at the same st.agnatin pres- Lin in the nzzle and n the stagnatin pressure, 'fhe esti- sure, the separatlon pint mved dwnstream with wall mated errr in static pressul~thrughut the thrat and clingj 'fhe dependence f separatin pressure n \Vali

3 ... : : (f) O NOZ ;:) (f) a: a.. 1,0 ~ ẇ..j 1.5 O a: w O , O O 3.0 BACK, l\asser, AND GER + AAA JOURNAL Ec=7.90 EE= 2.66 'é =1.800 in. -~ GAS FLO TANGENCY z= 0.71 in. _----L --L-- ---ct li =15000R' ' Tw COOLED ALLS'jf=OA- 0.64; ô l.l R ~ 0.25, O = 3.60 TEST P' psia t:: <J O V' \ O ,6 TANGENCY TANGENCY in. z = _ r-l-..+-t=--~ --~ z = in. Z = in. LJ Z~~R5~;Tin.l!.1 _ ", a-one-omensonal SENTROPC FLO - (Y= 1.40) b-sentropc CONCAL SOURCE FLO (Y=1.40), EO. () c-hall d-sauer (REF 10) e- OSATTSCH ANO ROTHSTEN (REF (REF. 9), EO. (2)'TO-OMENSONAL 13), EO.(3) SENTROPC FLO (y= 1.35) b AXAL DSTANCE z, in. Fig. 1 Static-t-staguali()l1-pl'(,~~ll'c rati:-i all1~ lhe nzzl{. cling is discusscd in Sec n. [i'he effcet f inlet, bundarylarer thickness n the measurec!'!->ressure ratlo \\"as nt (liscerlll ) e. Thus these' iuvestigatins revea e a negligil)[c liõüiídarv- aver displacement effect, which was als agreemeut with preclictins f the clisplacement thickness ó* frm the turbulent bunclary-layer analysis.ij ln cnueetin with the preclietiu, it shulcl be mentinecl that ó* became negative upstream f the thrat as a result f the embined e!tect,.: f wall cling and fi\\" acceleratin. Preclicted values f ó are sh\vn in Fig. 12 f Ref. 1J [j' iuvestigate the effect f inlct cnfi uratin measnred pressure ratis are shwn r' 2 fr nzzles f different cntractin-area ratis but havmg the same thrat raclius thrat radius f curvature, and half-angle f divergenc3 The cnvergent sectins were fnned by circular arcs f equi11 radii f curvature. Fr cmparisn purpses,c he cld-fi\\' data shwn are average values fr eaeh nzzle ver the range sh\yn) f stagnatin are dlscussed 'pressure~ l!l indicated1 Sec v:' ehrugh S~pamted the fiw thrat data (nt and divergent regins, the measured, Rre.-ss.l!!:.e _'::!!:ll.0~~~ tiallv inde )cnelent f inlct cnfi"uratiun anel de cnel nh' n the lcal nzzle cntur. The ma~nitllelcs f the deviatins rm - lsentrplc fi\\' curve a) are similar t thse funel \\"ith the nzzle (Fi~. 1), which has the same rati f thrat raelius f curvature t thrat raeliw; (rjrtl. = 2.0) an half-angle f elivergencc as the llzz1cs sh\\'n in Fig. 2. The cffcct f thrat cnfiguratin is sh\\"n by cmparing Fig. 3 t Fig.. n Fi". 3 measureel pressure ratis fr the nzzle are sh;\\'n' fr a stagnatin-pressure range frm 45 t 250 psia and a stagnatin tempcrature f 1'500 R. lbecause f the smallcr thrat radius f curvature t thrat ra~ls f 0.625, as cmpareel t 2.0 fr the llzzle, there are larger deviatins frm the pressure rati fr' -d isentrpic fiw m the transnic regin. Thcse amunt t as m\lch as 45% just el\\'nstream f the thrat' ln the cnical cnver~ent sectin, there are als lárger <re'viatins frm l-d fi 11' than with the nzzle because f the larger 45 half-angle f cnvergence. ln additin, at the higher expansin-area mtis f the 45 _15 nzzle, the data nce mre crss ver and becme less than the -d fiw pre-

4 SEPTE;\1BEH fl6.~ :'EASURED AND PHEDCTED FLOS THHOUGH CONCAL f'uperso:\'c NOZZLES li) 1.0 ::;) 15 <! cr...j N z L2 Ls 'i = 1.55 in. (CENTER AT NOZZLE EXT te' : r. /h ' n. NLET), fé ' -- 1,h= 'c' 4'02- ç, ~ 0.7 C;.- - Q 0.6 <! cr w cr ::;) li) li) 0.4 w cr Cl r- r, ~520R. UNCOOLED ALLS NOZZLE p/.psia NUMBER (; RANGE OF TESTS.l... c AVERAGED O 1.64 O t--- r -- -] j -\ L 1-0- ONE -DMENSONAL SENTROPC FLO (r' 140) c - HALL (REF. 9), EQ (2). TO-DMENSONAL SENTROPC FLON ( r' 1.35) ,- -r - ---< -j tj. 402 ~ J L AXAL DSTANCE FROM THROAT, in. Fig.:! Static-t-~tagnatin-H'c~sllrc. ali~ "ilh \'~u'iull':';'"t (~()nligunlti()us. dictin. ith the nzzle, nly ht-fl\\' eledwall d:üa \\'ere btained, and a larger ufjstream length (l/d = 8.4) \\'as u~cd such that /R = 0. i5. -AltilOUgh n (Tã!:1 were btaincd with a thinner inlet. bundary-layer thickness, bth experimental bservalins with the nzzle and predictins indicate that the mcasurcd pressure ratis wuld be alt.ered negligibly. V. Tw-Dimcnsinal Flw PrcdictiH; Exact slutins f the tw-dimensinal (2-d) isentr )icflw cqllatlons are '.i~~y~existen_t Jr_ 0\\' rc!!:ime~ thrughollt a SllperSOllC nzzle. :\pprximate sllltin~ cnsist Clther af calculatmg the fiw field by numerical 01' graphical techniqucs 01' f btaining analytical slutins in particular fiw re~ins. Since, as cxpccted, geviatins frm l-d flw are larp,est near the thrat...1h.e transnic regin is f pmnary cncem. Als, a slutin in t:his rep,iõii -S needed ~ ~~ ~()lutlon by the methd-f-charact~ristie.:~_jl.llie ~persnic regin. n thc transnic re in, Hall9 btained a slutin f r he velcity field fr isent.'oplc. irr tatinal fiw (' - COl~ Ly a series expansin f the vclcity cmpnents lu mverse pwers f r./r'h. The first three terms fr the velcity empnents in the series slutin were calculated and appear in thc reference. Frm these, the wall stlltic-t-stagnatinpressure rati can be calculated: p/p, = - [(' - 1)/(1' + 1)]ir:}'Y/h'-) (2) where V is thc lcal velcity V at the wall, nndimensinalized with respect t the speed f sund at the snic cnditin a*. ~'he predictin frm Eq. (2) is shwn in...!'2!l.!...bycurve f, ll1chagrcci' well with the mcasured pressure distributins in the thrat \'icinit~1 n thc regins that extend circular-arc-thrat colllcal tangencics, the requirement t the Lhat thc 'lelcity at the wall be parallel t Lhe wall is nt exact1y satisfied as a (;n~eqllencc f the slutin methd; this undubtedly leads t the inferir predictin near these tangencles. l!he crrespndellce f the predictin frm Eq. (2) with the data is cqually gd fr the nzzles with \'arius inlet cnfig\tl'atin~. as shwn in Fi/!. 2 by cur\'c c. t shuld be nted tha t lall's predictin depends nly n thrat cn figuijtin tinj 1'he thrugh mcasuredthe rati pressure rcir'hratis and nt in n the inlet thrat cnfigura rcgill shwn in Fig. 2 display this same trend. 11'01' lhe.t5-15 nzzle (r./r'h = 0.625), the pf,niptiu...ísllt applimble since the series slutin divenres fr r.jr'h <. lnstead, Fil!:. 3 shws as cun'e d lhe Sauer prealctiõi1'which, as Hall has pinted ut, is the first-term apprximatin in his series slutin. This uredictin is cnsiderably belw the data L1Jis predictin fr the nzzlg) n Fig. 1 (curve d), indicates the imprvement upstream f the thrat aftrdcd by the lall slutin, curve c, in which three tenns are uscd. Hwever, at the thrat and dwnstream f it, there is little difterence between the first and third apprximatins. t wuld be f interest t cmpare the data with predictins frm the irrtatinal methd f charaetcristics in the supersnic regin; hovever, predictins by Darwell and Badhaml1 and Migdal and Landis12 fr cnical nzzles with circular-arc lhrats reveal fiw cnditins near lhe nz~je

5 1610 BACK, ~[ASSEH, AND GER AAA JOUHNAL c: () ::> O ~ ! N 1.0 O Z 'í = in. Z=~: LT':GENCY Z = in. ri TA~GENCY. TLGé -~! :JZ n. CONTOUR TANGENCY c': 9.76 E : 6.63!Er:" - rhroat z = in. r'h = in. : z : in. <::>..~ º ~ 0.6 :::;: r () () r Tw r; : Ô e R '" : 8.4 TEST O 343 O 342 V V 348 O 351 Pt. psic O O c - ONE -DMENSONAL SENTROPC FLO (y = 1.40) ~~~ V O e-osattsch AND ROTHSTEN FLO (y = 1.35) d - SAUER (REF. 13), (REF. EQ 10) (3) ) TO- DMENSONAL SENTROPlé. O g~~ ~J, ~ 10 O i=t 6! 1 16T AXAL DSTANCE z, in. Fig.3 Static-tn-1-õtagnatiun-pre:-.surc 'ati~ alu#! lhe 'l5-5 J1zzlc. axis that \\'uld ]cad t shek fnnatin, :ld ensequently inyalidate the irrtatinal a~sumptin. Predieted shek frmatin ecurs where ~ach lines l'i~inatin~ just d\\'nstrcam f the crcular are and cnical t:lgeney appraeh the nzz!e axls. Dar\\'ell anel Dadham dlscuss the predietin in elet:ul \\'ith respect t the aceuraey f the numerieal slutin and its initiatin, using either Hall's9 r Sauer'::;l transnic s!utin. As indieated in Fig. 1, fr the nzzle with r.lrtl. = 2.0, either f these slutins shuld pryide a gd apprximatin t the actua! fiw in the regin f interest. t shuld be nted that bundary-ayel' displaeement effects, nt aeeunted fr in these predictins, d infiuenee the nzzle freestream fiw bundary t sme extent; this ean aher the predicted fi\\' field. hether r nt shek frma..tilln.. actually ceurs ª axls ean decided by expenmental bscry-atill. Rather than \\'all static-pressure measurements, either ttal pressure-prhe traverses alng the axis f enieal nzzles r sme means f yisual bservatin in a transparent nzzle with a sufficiently lng divergence sectin wuld be required. Anther apprximate slutin valid thrughut the nzzle, prviding that wall curvature effeets are nt large, is the predictin by O~watitseh anel Hthstein.13 \l"hcre L p, = [ 1 _ ~ ') _ ( Jl -- O)2(r)2J'r/(Y-) a, 1/1 The subseript 1 dentes averagc quantities fr 1-d iscntrpie fiw, in \vhich "/ = cnst Fr caleulatin purpses, it is cnvenicnt t use the relatin (/ul)(duddz) = (2/r) X [(drldz)/(m,2-1)]. ln the predictin that applies fr a cnstant,,/, the veleity distributin is cmputed frm the lcal enfiguratin f the wall. The requirement that the fiuid veleity at the \\'all be parallel t the wall is nt éxactly satisfied because f the way in whieh the slutin \\'as btained. The predictin frm Eq. (3) is sh\\'n in Fig. 1 as curve e t be in else agreement with the Hall predictin and,

6 :-;EPTE:'BEH!)(i.~ :'EASlíHED A:\D PHED[CTED FLO\\'S THHOt:GH COXCAL SUPEHSO:\C XOZZLES 1611 thu~, thl' data in the thrat vieinit~ f the nzzle. At the cireular-are-thrat cnieal tangeneies, the predietin is disrntinuus bccause f d'lr/dz2; these disentinuities are indirau-d by vertical dashed lines. ' Near the tangencies, the slutin becmes mre apprmâmate, since the restrictins n the magniiude f ihe nzzle radius and its derivativl's implied in the analysis are nt satisfird. Thr nzzle cntur deviatrs even further frm the rcstrietil1s impsed by this anal~'sis; h\\'evpr, fr cmparisn, this predictin is sh\\'n in Fig, :3 as C\1l'VCc in a narr\\' part f tl1l' thrat rcgin. The ratll('r sharp decrease in static pl'essure is prl'dicted, but tl(' pl'edietin is bel\\' the data. V. Static-Pn~,,;slll'e Tal' Sizc {j'11p pl'l'sencl' f a sta til'-pressurc hule ('auses sme fi\\' dt:::turbah'p, lhi('h altl'rs thl' ml'as\1l'l'd statie pressure fram the tnlp ndlll'. Fr shal'p-edj.!:l'd hles, dl'l'per ihan abut 2 diam, lleasllrt d statie pressurl's in f()''; \\'ith negli/!'ible prcssul'(',!:radil'nts han 1)('1'11 fund t incl'case \\'ith hle sizc; it is bdic\'l'd hat the slllall hles rl'ad 1l'arl'l' thr true static P'C';SU'l.. Othpr 'Te,'ts, sllch as sli),(ht hurrs and the prl's'1<'1'f fo'( ij.(npal'licll's, hal'c' 1)('1'11fOlllld t alter the rcadinj.(s a,; \\'1'11. Fr nzzle /Ju\\', in additill t the prcs";\'e J.(l'adil'l! indul'cd in the flll' h~' thl' Pl'p'C](" f the hle, the '-"tl'mal pn,s,;un' J.(radient is superilllp'ed in lhe fll\' directin, The fi\\' elisturbanc(' is thus expl'cted t incl'l'ase \\'ith hle sizl' )('causc a larger prcss\ll'e drp exist,; a(''oss til(' hle than "!1<'nthp fr('('stl'pam fi\\' is nt acccleralin!!j T innstigate the 'ftpct f hle sizl' n thl' stalic-pl'essul'' 'l'adings, the 2,51-1 nzzle sh\\'n in Fi!!, 2 \'as il1strumentl'd \'ith 0,010- anel O,040-in,-diam ta) Jair,.;. As llh'ntin{'d ljefrp, t Je 10 'S \\'ere as sharp-cd)!:ed as they culd be madl' hy 'nh\uthin!-( with cmer~' ('10th any sllrfacl' burl's lha! \'l're prdu(,l'd hy drilling:, The rati f hle dcpth t eliametl'r \\'as abut :3, ~tati('-pressul'e elistributin~ wl're btaitinf 0\', 1' a rang,e 01' :'tagnalin pre"s\lj'es frm 4;) t 170 psia, \'ith air :l a stag,natin tl'll\pcraturl' 01' 520 1\ (unell'd \':lls), '1'\\'0 01' thl'~1' clistributins arp sho\\'1 in Fi}!;, 4. T ali\\' a dirpet poll\parisn f t]1(' diftcrences in the re;;'f. illgs. thr 10\\'1'1' par f Fi)!:..t :lis ~hws the percentage dilterence be!\\'el'n the 0,010- and O,040-in,-diam tal' reading,.; at lcatins \\'hl''(' t]wse t\l'o taps rerl' axially \\'ithin in, 01'e:1<'hthpl'. This is the limit t \'hieh the axiallcatin 01' t}w taps is kno\'1. :\s S('l'n in Fi)!:.,, thl' pressure elifferenel' bet\l'l'l'n the smallc:'t and lar;;est tal' '('aelings varies sy~tell\atie:dly thrug,h thl' nzzle; ':nelll\ differences (pllls r minus) thai \'lll,[ be assciatcd with reading ace\ll'aey are nt pviclent. The smallest tap has the 10\\'1'1' rl'adin)!:, as lul:ó been Obsc'l'\'ed in fi\l'" with neg,ligib}e acerler:ltin. fthe pereentag:(' (lifference in the tap reading,s is a ma-"ill\ull\ m t l' transol1lc reglon. \\' lere l' pressurc grarlient S largest. '1 he clifterence S 1i:i'r(]101lscernible in 111e nzz}e-inlet regin \\'here the freestreall\ \'Clci ty is l\\' anel in the fi\\'-separatin regin fr t!1<' l\\'-st.agnatinprcssure tcst \\'hl're the reversc fl\l' velcity is re!atively 10\~ The O,OO3-in. uncert:.líni\' in the axial elistancc bet\\'een hé ~lnd-.õ=i'o-in,--jiã~n- tap pairs \\'llld alie' the re': centage difterences sh\\'n 1l l' 19. :tõf'o, 1 at m:ót. - The trends f ihe difl'ci:ei'icp."sh\\'n fr the t\\' tests were typieal f thse funel at intennediate stagnatin pressures f 75, 100, aud 125 psia, and dllplieate tesis at sme f the pressures inelicated the data t be reprdueible. ln a system in \\'hich pressure gradients ean exceed thsc alng the nzzle, JaiviuH folln differences f the sl1me magnitude as thse shwn 1l Fig. 4 fr tap diameters ranging frm 0,0016 t iu. by measuring the pressure distributins alng a fiat plate u which a liquid jet impinged. 1n that investigatin, a limiting value f the tap size was fund fr \\'hieh n further ehange in the measured pressure distributin \\'as bserved;~ O,004-in,-diam tap read the same as the 0.OO6-in,-diam tap.j \\ '\ O. /Q/6 w ~ t:, 0.8 ~ 0,7 f= 0.6 <J: a:: 0,5 w wg: ~ 0, '~ 1'1 ~ TEST 103 J [;1 0.8 r,=534 R,.~ i'i l- ~- 13bB TAP d DA ~j n 0,7 11=170,3 pso ~ 0.4 r '03 i '" 0019, 02' 01 ~ 0,----,.. 8r 'Q TEST 100 ~ Ti =550R,_L_-, PO 45.1 psi i j! f) r; Fig. l Statie-t-~tagllatic)-)l es~\l t~ ratis alllg' ti " 2..~ ",,;11. di'!'l',',,"1 siz,~d J"'"'''''''' lap", 25 ~~ 1 j J t}46j. - -!'J r; ti <'>} 1 THROAT º 61 TEST p. psi J X ,1 8 "'" 2 - /j j., ~Q.. Q ~Q ~ "O v_, ' Q.: AXAL DSTANCE z, in, T imli('ate the mag:nitwll' ()f the diftprcnces bet\l'een the tap rl'ading;s in antllc'r \ ay, rderpncl' is made t an estimatr 01' tlw stat ie-pl'( sslll'(' tal' 1'1'1'01', H,\' elimensinal allalysis fr li 10\\'-spel'(1, 'ssl'nti:dl,\' eollstant-prperty tmbulent bundar~' layer \\'ith nl'.l.dig;ible fi\\' aeeekrati0ll, Li\"es('y ( t al.15 and thers give the cltr,),p fr dl'<'p hles: - D.p.1p. 'T = - (r/*) T \'here d* = r(t/p)/2/., ri is the tap diameter, anel T is the \\'all shear slress, _\eerding t experimental measmemenh by jjvcsey et a. and llll'rs. D.p/T inpreasl's mnlnically \\'ilh d* t'abut 3 at d* ~ DOO,lhe limit at \\'hich mea:;mements have been ma,jp. Hy cmparisn, in the thrat rcgin fr the highl'st slagnatin-pres:óure test, \':dul's f (p, PO),'T are a" large as 40: tl1<' errt'spnding nllue 01'r* based n the O,040-in,-diam lal' is abut ü200, Fr hl' stagnatin-pressnre "st, lhe pn'clicted \'alue in the thrat rl'gin i,,,:lcs:;, (PO - PO) r ~ 10, as i.s t he crrespnding \'alue d* ~ 1900, ln these estimates, the \l'all.shear stress \\'as predictl'd frm the analysis f Hef. 8. Cnfrtunalely,.since the true statie \U'PSSUl'eis nl kn\l'n, ne can nly enclude fl'l11the data sho\\'1 in Fig..t tha~ static-pressul'e distributin.s sh\l'n in Figs. 1-3 arepr "'G71bly;lightly-hig;her-than the tru~ -- \'1. Separalill P"essul'cs 1}"r verexpanded uzzle p<,ratin, the ralis f separatin re rth = 2,. 'fhere \\'1'1'1'100 few pressure taps in the divergent t ambient~ressul'e regin f the.t5 -\5 are sh\\'n nzzle in fr 1~'lJr a l11eaningful nzzlesrepre.senta tin. The data are sh\\'n by thc barred verticallines. hc that have ' ujp_er bar is the tap reading upstream f the sepanítior pmt; lhe l\\'er b:j.j'\\'uld be ihat at the succeeding tap if separatin haa nt ccurred and was determined frm higher stag,natin-pressure tests. Thus, the actual value lies between the values shwn, Ehe?vaeh number at separatin was ealculat d fr isentrpic fiw (y = 1.4) based n the average ambient separatin jjressure frpressure1 th!:_hõ\.fi\l' The lests mtis with f separatin-t clq.e ~Us [shaded symbls) are generally abut 5 t 10%-')cl~ cld-fiw values (pen symbls). These l\\'er values at the same stagnati?n pressure crrespnd t a releatill f the

7 161~ BACK, :\Af'f'ER, AXD GrEH A1AA JOURKAL <>..~ º 065 ~ 060!;:; a: UJ 050 º a.. '"UJ ZUJ iê :J 045 ::<... Z UJ a:... cr NOZZLE. 30-SO FG 30-SO O 30-SO O 2 S- 1 ) O 164 _ FG 2 1;,R SOO} COOlED ALLS < &. < OS8 520 UNCOOLED S20 S30} ALLSTi "t + L 7} 36 O a R ~02S ~005 ~010 Fr the nzzle, the pl'edictin frm Eq. (,O is ,_ in fair agreernent with the exp<,rimental values. This crrespnence is expceted frm thr elsc agrcement f prediete and measured statie prcssurps in the thl'at regin. H\\'ever, fr the nzzlc, the experimental values exeeed the predietin frm El(. (4), in \\'hi('h nl~' the first term in lhe braekels is rctained. As menline befre, the lall slutin di"erge;,; fr r,/ru. < 1. f nl., the first te~ in the brackets is inc!udeel, tlll' pn'dietin is iuenti('af \'ith that f either Saucr10 01' Os\\'atit;:ch and Hth;:tein.13 Thus, tjhe l\\'er!)redic!l'd values ~f.tll(' ~h\' tlll' ll1aelequae~' 01 PXlst~ J(J\. cpfli('ient P'('(ll('tln5 fr furtla:r nzz]ps \\'ith r" ru. -:.) \'111. Thl'lst Batiu,.; SEPARATlON MACH NUMBER M, St"H..atil1 tu alu"ielll-pl"'~~urt nll'ill~ lzzlt,,,,,,. n.ti~ f,' ~epamtin pint d\\"nstream 'ith wall elin).!:, as ms mentined in Sel'.][1. This indieatl'~ t!!at wall (;!!g..l'1G- imprtant in the interaetiõi11)rt\\"cen the shck \\"ave and bunelary layer ~dlll'h ):-esulh in fi\\" separatiçm frm tlh' -nzzle \\"al1. :\hlberg l't :11.'6 ab fuund the same tn'nd. Fr ur data, it is nt c!car \\"hetlll'r tlh'l\\"er srparatiun pressures are due nly t l\"flll cling, sinl'e, \"ith hut fiw and cled walls, the bund:u'~'-la~'l'r-thiekness Reynlds numbl'rs ai the separatin pint estimated frm the analysis f Hef. S wcre abut ne-half f th"e fr l']d fi\\" at thr same stagnatin pressure and bundary-byer thickne"" at the nzzh' 'inlet. n this regar.!. the data shu\l'n in Fig. 5 fr the nzzle 'ilh ht fi11" and COOll'd walls d nnt n'veal :ly delinite trcnd bet\\"l)en thc results with relati\"cly thin bundar}' a~'ers (ó/ R ~ 0.05) at thl' nzzle inlel :wd thosl' \\"ith thiekl'r layers (ó/r ~ 0.25). Hwcvcr, it is diffil'ult t determine lhether there is an~' depenenl'e n lhe bunelary-laycrthickncss Hl'}'nlds numbcr, wing t the sl'attcr f thc rcsults at the hig:hest separatin :\al'h nlllnbers (la:'t fur :,hadeel pinb), \\"hich l'rrc"pnd t sl'paratin near the nzzle-exit plane. The data in Fig. 5 cxtpnel n'r rathpr 101\" rati;; f :'tagnatin-t-ambient pressurl' ]JPcause f thp relatin'ly small expansin-arca ratis. "'1l'n these data arl' cnmpared \\"ith thers bt:rineel with un('h'd \\"a1b. s\l(,h a., tl1l' :l('('umulated results f Arens and :-;piegll'j' (Rl'f. 17. Fig. 2), the ht-flo\' data \\"ith a l'led wall aj"('fund t lic bl'o\' thnse J"('sult;;. V. Fl',- CefTicicn t~ Ld tz;1 Fi~. O.Lemparisns bet\\'cen mcasu'ed ma;:s fi\\" ratl' at a sta~natin temperatul'e f 1500 H (cled walls) and eolhputed values fr l-ri iscntrpil' fi\\' (' = 1.35) are sh\\'n ill terms f the fi\\' l'eflil'ipnt Cd = li1/,i,. At the 101\"('1' sta!; natin pressures, there is apprccl:1) e Sl'atter in the value::;,)=: Sme f the scatter is undubtedh' due t errl's in the mass fl\\'-rate measurem!:nts. ~nhe ';lp;her stagnatin p~, the scatter is]ess,-and fr bth nzzles, the fi\\' ee(fjcient i::; apprximately bel\\'cen 0.98 and LQ. Sinl'e the fi\\' thrugb. lhe transnic l~or1 detellnines ti1c mass f_~'j'ate th[ugh the nzzle, Fig. O als cntains the Hall predictin! - - T elparly illu~tratr the d""iatius, lh~ nzzle.~ ar!:. a;;"1ll)).!:..!.l t dischar~ int~ ae ('.!!.!.UH) and tlh' thrust rati,; are sh\\'n fr hypthetil':1 expansin-area rati~ frm a ":llh' nf t that f thc slatie-pn'ssure tap lcatin neare~t the nzzle pxil. C'H' terns in the thm,' ('XH pssin.l:c.llj'~j thl' ' n tlll' nzzje-inl('( arca 01' dtel'tin' l'hamber ellll wall :1d tlll' integratl u \\'all pl'e:'sure distributin resulting in tl\' axial fl'<'p n the nzzle sidc \\'all. The "hape 01' the thrustrati ('un'e is depp)h!ellt n dl\'th'r the 'al pn'"sure is le,.;s 01'!-!:reatcr than the -d fl\\' ":lup :\d n the ma~nitude f thr dittej"('''.as shll'n in Fig". 1 and :~. :-;h\\'n fr l'mparisn is the ften C/uteu ('urre('till faclr ~( + cso) ú.ll noll:lxial exi t fw fr i'qni!:a! L1Qzz!':J Thmst ra tios tr clei flo\' \\'õü1(l ije thp sanh' a" th"e shulm in Fig. 7. hased n thp data nh'ntined in Sec \:. Snie Line t'able cnt.ain:< a l'mparisn at a stagnatin telllperatul'c f 15000R (cled \\'alls) f pr~lil'tcd and ex~~'imqlt:ll 1.calín f thc intersectin f the.'unic lille \\'ith the edge f the bundalt la' pr. The cxppllnh'nt:11 "aluc::; ('õrrc;;pnd t the pilrt at-;;i~h the :-ach num!jpr btained frm the mcasured ;;tatíc-pressure distributins fr isl'ntrpie fi\\' (' = 1.35) ;;: :( 0ge Li ~. zuj 096 UJ 102 LL u '20 LL.. O'J.~,-,"... ;=...J ge T, "5000R, COOLED ALLS NOZZLE ~ 30- S' ~ ' ~045 E11 f~ r_nozzle 4S-15' FLO SEPARA TON N NOZZLE CJ = 1111 '~l= _ (' + )(~') r, T 90 _ 81' ('t') r, '% ] "( (r'hy r, -... ] (4)~ Fig. 6 STAGNATlON PRESSURE P,. psi Flw ccllicicnts. '( r,;, é )',TERM EO(4) ONLY / e

8 SEPTE~mER 1965 ~EASURED AND PREDlCTED FLOS THROUGH CONCAL SUPEHSOi\"C XOZZLES? 16~ Tahlc 1 Cnlpllrisn f prcdicted nnd cxpcrinlcntlll. nic linc il11cr"cctil1 with edgc f holl.ulary layt~r at T, = { (cled wall. ) Nzzle ') 4;' _lfi (tlr'h )xv. 0.lrrO O.:H Ulr'h), ms equal t unit~'j Une predietin fr iscntrpic flw S frm l'ither Saucr10 01' O,;watitsch and Rthstcin :13 ~/r'h = b + 1):2]'/~(r'>.'r,)1/~ () x LL ::;; < (f) '<. a: :::" (f)::'" <1-06. ~ '<. 0.9 i= 98 05U ~!(~---- ~. r'~d~~ t 6~ "--"" b --.:-~V-&~ 01 '--p-..., ,"-- li '1500R, -' 10! D COOLED ALLS ' NOZZLE 1 CONVERGENT-'-DVERGE';;:;:-Ti ' --r NOZZLE AREA RATOA/A,~ Fig. 11 Lcal t llc-tlilll('l"illal 11a",", fllx "ati,", alllg lhe tlzzlcs (vahll:s \\'c.'e dctc.",ui,u c fnuu a.vel'ugetl 1'/1'1."ati,.. sltwn in Fig~. alui :3). ~ \\'l('n' ~ is the axial distan\;.l:u :-:trcam f til!' "l'mctric thr at Thl' t H'r Sl'nhpic flll' 1)1'C(ctln :; rm!-a s analysis.9 Fr the 300-lli nzzh', thl' predidin frm Eq. () -!;i\'c,; a \"alue lar;.:!'!" than thal mea,;lrcd fr thc \lpstrl':un distance \O the snic li/h', \'ith hettl'r agrl'clllenl aftrdcd by /[all's prediclin. lall's a!lal~'si:; is!lt applicahll' t the 45-1.,) nzzll', "incp r,ir', < 1: b\' usin~ Eq. (), thc predi<.:teu elistan('l' i:-: 1110n' than tll"ic{,-tll!' ('xperll11ent:lll~:- deducecl \'alu!'. -'1 1l' ('lcl-t!j\' ('xp(,1"nh7õiãi 'alue:; \"(Jl1d 1)(' tfi(',;aml' as hsp indiea ted in Tahl(', haspd 0(1 the data mcntloned in S('e. 1li,.'as,; Flu,'\: Hatis Fr th(' eall'ulatin f hul\(lar~'-la~'l'r fi\\" and hl':\t ransfpr 10 nzzlc \"alls, tlh' l(wal ma~s fiux ~p!'), at thc l'd~p f the hund:l'~' aye~' i~ lh'pdl'(1. rny assul11inl-!; isl'nt rpit' flu\\' (1' = ].,1), tlus leal ma,;s '7rux \\"a:-: eall'ulalt d frm thp mcasun'd stalic-pr('~"ure distrihutins and is sh\\"n nnrlinh'nsinalized hy the ]-d isplltrpic fi\\" 'aluc in Fi~, 8. Thru~h mst f lhe nzzh', lhe leal mass tlux is c:;s than H' lurhulenl -d \'alu~\\"hieh hundary-ayerimplips flo's, lwer ) \\"all a: heat (p \"),',5. flux% 'fll!' sinccmas:; fr flux dp\'iatin amoll11t~ t ah\lt 20/c in tl!' inlet rel-\in and just dwnstrcam f tht thrat fr the _1.') '15 nzz!(': \\"ilh the :30-15 nzzh', til(' dl'\"ialins an' ks:;. The maximum \'alue~ f lhe mas:; flux (p \'), ceu~usl up,;tream f the thrat at hciiltl'r;('i'!ion ôf the -;Qjlic!ine ~'ith til!' edg;e 01 f]lltoull(lar~' ":iycr. l1 is in thi,; reginth;ll ilcat-tran,;fcr measlii'('ml'nls indieat(' til(' maximum hl'at fiux t lhe \\"all,l.6.1~ COJlclusin;; "'all static-pj"l,,:,;ure nll'a:-;uremenb hal"l' heen presenteei fr air flwing hru;.:h cnical nzzh's \\"ith circular-ar<' ' ~ 0.98 "- Q >- < a: >- V ::> a: :x: > ~ ' T, '1500'R, COOLEDALLS NOZZLE 30-15' ~ ~ ' ~ 0.45 thrat.-. ('ll1parisolls ha\"l' 1)('1'11 l1:ldl' with prl'di(,tins. Tl!' rl'sults indieate til!' fll\'inl-!;. ) 111 th(' thrat J"l'gin, w}wrl' lhe flw is ransllie, t\l'odill1l'l:-:illal isclltrpie flml' predidills an' ill elsl' agrl'l'l1ellt with lhc.laia fr tlh' 10ZZ!('S wilh a rati f thmal radill_ f (~llrya'"n' t hral radills r, 'rtl, = 2.0, hllt inadl'qllale fr the lzzll' \'ith r,ir'l> = 0.ti25, j)l'\'iatins as :trge a,; 30 and 450/;. frum thl' sim 111' lll,-dimpll-l:d isplltrpi(~ fl\\' pn'( wtlol \\"l'n' follllel ill tlll' lhrat rpgill fr thl' lzzlcwith r, 'r'h = 2.0 :llld O.G2.,), n'sp('cti\"['~', 2)rnlatic-pr~~l1paSlln'('lt,,- ill t 111' 1hn~l' alld dil'crgl'ltilf't'illill" uf lzzll';-' \ ith thl' :-:allh' r, 'Tt, :11r! half-alll-(]' 01' dinrgcnce l'n' f0'd t be es:;l'ltially illd('h'ld('!.!1 ~ari()ls illlpl l'nfi~ura tins, :» :'111:111('1'de\'i:ffills in statil' pn'sslln' frm (H!e-dillll'lsinal fi 11' \"('re folllld ill thp enil"al sel"lins, 111 til!' enyprgl'll"c rpgill, the l1agllitlld(' illl"l"ea.-l's \'ith l"lll'l'rg('('e alll-!-c, \\"hpre:ls ill hc din'rgl'lt'e rl'gin, this efteet was 10t im esti;.:atl'd,;ill('p ali til!' lzzles tl'sted h:ltl a 1,') h:llf-all.!:dl' f di~'gell("p. 4)tFr lllderl'xp:lnded peratill, ~rl's,;urc.jn!:.l~lt:. nll'nt;~werp in';l'lsiti\"e t lhe cltl'("ts f btlll"ll'd :llld 1111 t'll'd mlls alld tu lzz!' illlpl bolllldary-:t~''r thicknl';;~ t 0.45 f tia' nuzz!p-illlet radill.-. Thus. hundalt-layer eftect:; l're follnd t he negligible, 5) 'rps:;un:f(.;itfíng,; depend U tap size, wilh signific:llt dittej"l'lce;, folllld })('tween a!ld O.O-O-in.-diam hles. The smalle:-:t tap rl'ads thc 100n~r prc,;surc, which i,; belic\'ce 6) Fr~yerl'xpallded nzzll' peratin, thc separatiun pint t 1110\"['( hc~arerd\\"n~trc:lln the truc statie wlth prc~sure~ wall cling fr lests at lhe s:lme stagnalin H'es;;ure, s that lhe mlis 01',;eparatin-lalllhient pre:;slrl' fr thc ht-fll' tc,;ts \\"ith cled walls l'ere gcnel:ajly í) S-las;, abut fi\\' 5rate 10 10% and thrust ]wlw.elepenel the cld-fll\" n the\":llues,l tw-(funl'nsillality'ór thp fi\\" hrugh th(~ trallsnic regiul, with boulld:jr' laycr cttpct,; rp!:itin'l\" l1imprlallt fr lhe lzzle:,; im'l',;tig:~tcd that ha~pã~siõll-:ii'ca ratis t 6.G.J S) n the tran;:oc rcgin. lcal ma,_s fiuxes at lhe ed~c f lhe holllldary la~'er dedllced froll thp static-pres:;ure nll'asurcml'nls deyiale less frm nl'-dimcnsillal fi\\" \'alup,; than d the statie pressure,;. n the ('nn~rl-!;l'nce sl'clill, the magnitudes f these rll'\'iatins are abut the same as thc~' are in til(' lransnic and depellel 011 thc ctl\'crgence angl('. Hefcl'ClCCS ~\ -... 'Back, L. 11., ~[assier, 1'. F., :m Cier, 11. L., "Cnvective heat transfer in a en\'ergent-divergent nz7.le," ntern. J.!leat '" ~ ias:; Tran:,fer 7,,,)4D-.'j68 ( D(4). ""! Scheler, K. an 13ierlein, J. A., "Sme experimcnts n fi\\" :,eparatin in rcket n07.7.le:,," A!{S J. 23, 28-3:2, 40 ( 1!);,;n,, R. P., Eisenklam, 1'., and \Vilkie, D., "nvestigatin f super:<nic fiw separat in in n07.zle:,," J. :\[eeh. Eng. Sei., Campbel, C. K and Farley, J. "., "Perfrl1lance f several Fig. 7 Vnriatin NOZZLEEXPANSON-AREARATOA/A, f t1u'18t ru1i with cxpnn8in-urea.. ~267-:27!)(J959). 'nical cnvergent-divergent rcket-type exhau:'l n07.zle:,," rati (curves wcrc b1ail1cd frm avcraged 1'11" ra1is NASA Ti\" D-467 (Septel1lber 1!/(0). ~ shwn in Fig8. and 3). Arens, :\1. and Spiegler, E., "Separate fi\\" 111 verex-

9 ',BACK,.\L\SSEH, AND GEn AAA JOUnXAL panded nzzle~ at l\\' pres~ure mtis," Bul. He~. Cuneil bmel " ;\[igdal, D. alld Lalldi:;, F., "Characteri~ties f cnic!\l superllc,45-;j5(1962). 7' 6 Frtini, A. alld Ehlers, R. C., "Cmpari:;n f experimental '-... ~ '- snic u Oswatit~eh, lzzles," AHS K. alld J. 32, Hthsteill, 18!JS-!JO \V., (1962). "F\\' paltem in a cnt predicted heat tranhfer in a bell-~haped lzzle with upstrcam verging-divcrgillg nzzle," NACA T;\-12\.,) (Mareh 194\1). fiw disturbances," NASA TN D-1743 (August 19(3). 11 Jaivin, G. 1., "EfTect f hle size n prcssure mcasurements 7 Stantin, T. E., "The variatill f velcity in the neighbr- made with a fiat-plate dyllall1ic-head prhe," Jet Prpubin hd f the thrat f a colstrictill in a wind challnel," British Lab. Tt ;~2-617, Pa.. adena, Calif. (June \.\ H(4). Acrnautical Hesearch Cullcil H & :\ (lu30). " Live~ey, J. 1.., Jacbll,./. D., and Suthem, C..., "The _ 8 Ellitt, D. G., Bartz, D. R., and Silve', S., "Calculatin f static: hle errr prblem," :\ircraft Elg. XXXV, (lu62). ~urbulent bundary-layer grwth alld heat transfer in axi~~ymmet ric lzzles," Jet PrjJul:;ill Lab. T{ 3:!-3S7, Pasl\dena, 16 Ahlberg,.1. L, lamiltll, S., ;\igd,d, D., and Xilsn, E. X., i'truncated jjerfe"1 lzzle~ in pt ;llul1lzzle desigll," :\ ns.. Calif. (Fcbruary 15, 106;{). 31, 614-6:!O (1!J(1). v Hall, , "Transnic flw in -dill1en~i(jllalalld axially- 17 Arells, ;\1. alld Spiegler, E., "Sh(H'k-indlleed hund'll'~' symmetric nzzles," Quart. J. ;\ech.. -\ppl. ;\lath. XV, 487- layer ~cparatill ill \,cj'('xpahh'd ('(Jni.,,1 exhau~t nzzle<'.')(js ( U(2). A AA./. 1, :;7S-5,-'; 1 ( 1!G:j). ~ 10 Saucr, n., "(;pncral ('hara, t!'ri~ti( ~ f the tlw thruugh ' Kl()z~i, J../., ".-\11 illvbtig:atillf hpat tr:ll~fpr thrll~h 1111' nzzlcs at ncar eritiml ~pecd~," NAC:\ '1';\[ (./UC 1047). turbulcllt b<hl1ldary layer in ali axiall~' "Yll1etri(', ('Oll\'er~ellt- ') li Darwel, l. ;\1. alld bdhalll, l., "Shck frmatill in diverg:el1t lzzle,",\erdyl1ami(' Lab., ()hi" :-,tate Ul1i\. T.\-S c()l1icall1()zzlp~," A[A:\.., 1!1:~2-1\);!4(ln6:!). (.uly l!);js).

1.3. The Mean Temperature Difference

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