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SNATI IONAL ADVISORY COMMITTEE FOR AERONAUTICS WA TIME RE PORT ORIGINALLY ISSUED April 1946 as Advance Confidential Report L6C13 FIELD OF FLOW ABOUT A JET AND EFFECT OF JETS ON STABILITY OF JETPROPELLED AIRPLANES By Herbert S. Ribner Langley Memorial Aeronautical Langley Field, Va. Laboratory WASHINGTON NACA WARTIME REPORTS are reprints of papers originally issued to provide rapid distribution of advance research results to an authorized group requiring them for the war effort. They were pre viously held under a security status but are now unclassified. Some of these reports were not tech nically edited. All have been reproduced without change in order to expedite general distribution. DOCUMENTS DEPARTMENT L Bll. Digilized by Ile Iniernel Archive in 2011 Wilh lundlng Irom University ol Florida, George A. Smalhers Libraries wilh support from LYRASIS and the Sloan Foundallon lillp: www.arclive.org details fieldolllowaboulOOlang HACA A? IF. T, .Cl; NATIONAL ADVISORY C :,':ITTEE FOR AE.Oi:AUTICS FIELD .L FLO'.' ABOUT A T T A:D FECT COF JETS CI! ST.EILITY F' JETPR tEiLLED AiEFLAiHES Ey Herbert S. Ribner SUT!M. lR7 Te flo, t nirn in.liduced outside cold and hot propulsive jets by the turbulent spreac.inz has bcen derived. Certain simplifying assu.!pticns were employed and the regicn near the orifice v;as not trerted. The effect cf jet te:;iperature on the f'lcw inclination was found to be small 1,hen the thrut': coefficient is used as the criterion fPcr s lmrili tude. The deflection of a jet due to angle of attack has been derived and found tto be appreciable but small fcr normal flight condJltinns with rmali normal sc eleraticns. The avir;&..e jctinduced downwash ever a tail plane has been obtained in terns of the geometr:r cf the jettail configurit ion. These results have been applied t the titi the tn he effect cf uhe jets on the static lnn.itudinal stability and trim of jetpropelled airplanes. I 'i'8: r :r': c v A jet, as it spreads b: turbulent mixing, is knovwn to entrain outside air in th= :i.lirq, zor.e. Air is thus drawn into the jet ani the external flnv is caused tr incline tward the jtt axji. If the jet passes near the tail surfaces :if etprop~elie i :irlran~r, the jetinduced flow devition will affe't tne Stni. it.. abnd trim. This flo j deviationr and its effects on static longitudinal stability are herein in':estigatea t;ihe':retically fcr both cold and h.:t jets. Ths present investigation was well advanced w'hen a British report by Squire arnd Trc.,ncer m on the ccld jet (reference 1) beca:e available in this country. The considerable rigor of the British analyss was found to 2 ICNFIDENTIAL NA0. ACI. rIo. / C15 bh impaired b the use cf an idealized cosine velocity Jlstrijbution ir. tije jet, v'hich :rou'uces errors as great s 11 percentnt ,lo, the original verzicn of the present a..lysisN was found to be oversi;rplifiei in one respect, J:rAcn r s;slt'u in c.mi.'arrble errrs in the opposite direction. In the :rezer.t revised trea:tment, most of the advan:ta6s of Simoij.ficain re reteind, but the 0ssic analysis of reference 1 is ,sed to establish the vnlu D ' a cl.,nst.nt. The a :rcx. .te treat.1 e:. t jlven herein .er:nits te repesentati n if t.ie jetinduced strewn $.eviaticr: byl a :irie curve. A comparison of the pre ?it 9.nal i3 for' tiLe cDl' jt '..ita that cf Squire .nd Tr'ur. cer is '.rn in aro Reference 1 dees nt tret t:, h'.ot et T he firzt .,art r2 the prcs.nt ?. ;r is lonoerned .;iti t.ie annaLs3 I. of flv; in: inatiln nd.uced outside : i.a l it ts ;. '.. the j ot defli e t ?or. d.ie to angle of at taTcE The las tp' t is unce"rl i. t. w th acl.icaticns to t'.e :!~ 1ut at irn t:.e eff cts &. th J t on longitudinal stabiity 5r . trn:. ThE c r ..tst .ona procedure is out line i irn decL il . the r.u.. ',e.ca exa." le taless I to III) .co enaL li le r:' ren e to tr.e t.et is neeessar. ( F r Ji r:grir, rr s c reresent7r ti n "l' 3t', s.F 0" the s,n:t,ls rclerr r to .'ets, seF ? fig. 1.) T. deII.ra. t 7 'bs; i.te t.:trex te.n.. r t'.u :, r1 .p s : stre." enFitvy C r':c *f 1 C et dr:. it: f. treata .: &en.it : ': ' ?' "' '* c "i :r " Sin r.' .' i i't vel .cit./ r. v r tre ... ve 'Ycity at :, ., ; >y ,r ) I; inr.e:. l. ir.t rf jet ,j el:.l'icy rve" trCam velrcity at ,,o .nt x on. jEt acis '::.CA.. : :T ". L lC, CCNFIDE;ITIAL 5 t incr'nment of jet temperature ";.r tream temipera ture at point (x,r), degrees tm ircrere.r,t .f jet te:np rhture o.er stream tempera ture at r.int x cn jet axis 7 jt;ttn,Fer.lture ccefflicient (t x a.xial oist..n e frr:1 pr int. at ',which je. in ac',d ance /ith 1 sa cof spr.e asing th..t hlds at sub stnr.tial distances irr:m ,crifice :uld h.'ve zor.) cr'E.ss section r radial di t *ar.. e l'rom .n ,j t 'X s vF ST ' 2 /2' / / l S x T  0, V F CT c' tthruist coeff ii,nt r 3 v;ing area I1, 12 c.nsta.its c.f :eluc it,: .'.. il.:; d:i 'ir.ed in text i rd s. frrt! fur.ti. ns f T nd n; ;el' e i. text I. jetst pre Ain n r p r.s:. eter tIken as 0.f,') r je ts, r .e rint r.amet t r I Lr ske : s r constar. t .t k n a '..1) S s t re ~ fc t i :n C )iNFI DP:iTI AL 4 CONrIDEY:TIAiL IACA ACR N'. L6C'1 c jetinduced inclir.ation of flow' toward jet axis: Iw.th s4bscript w, w'in; uoi.inwsh averaged between jet orifice and horizontal tail e "me n ijetinc.uced dowrnrasi v:r :..:c. i baJl local j.n"lin;atiL n of jet axis to general flow a ar.rL? cf strEcI: nf thrust axis a sn n of attack of thv.ust ax;s relative to average flow ,bet.'een let arld tail a C N A area of jrt oril"ice bh span *.f horizontal tail ,i later.l itan,'e of jet a::1; frrc' center of hori .* tai L tail S ,:ist a.ice .f tnr,l.t axis belo i,. center of gravity ' 'irpiune pitchin:,mcment coeffi,ient (Fitc. h ir nrent) Sv"wing r'L.ori S .iszt.c re of ,nacelle inlet Saeac of ce;t.er of ra;r it', rir.easurJed parallel to t!.rust axis C1 aircleize lift coeffic ient it ; pcwer cn unless n s t:.scripi.te ! I :Lr.ce h rv!.z ntl tail, Jeg.rees 5_ el '..it. r : le, de r:t s ; p.. s ti'e Jo:nwari a. e levatoUr L'lrrne.,or:ent coefficient ____ Hin_.e ..ent____________ \ E / H D.: e r t (.ei n t) .4c .' El, :,tc.r7 soan x (Elevator chord) Cl'?!!FID D ;iTIAL NACA ACR 10o. L',C1 CCNFIDEl[TIAL 5 dCh Cha da dCh Ch5 d dbe np distance of neutral point behind leadingedge mean aerodynamic chcrj as fraction of mean aerc dynamic chord Anp shift cf neutral point due to pov er; positive in forward direction Subscripts: j measured at jet orifice T due to thrust fo'rce E due to jetinduced flw inclination 1 due to single jet 2 due cc two jets nac due to nacelle normal f'mrc, 0 measured at zero thrust; define as poweroff cnndi t inn fixed stick fixed free stick free ASSUME. PT IO1S The basic assumptions for the .old jet are the same as those for Frsndtl's saprcxinmate treatment of the spread of t'.urbulerc f reference 2, p 1656 ). The flow studied is incompressible 'ut the results are con sidered closely apFlicable to all subsonic jets and approximately applicable to supersonic jets. The starting pint for the present paper is a corollary of :0? I DEF TIAL . .CA AC.. dto. L Cl5 the a:_sir..:ions i:f ri'cerence. 2, r?.'erveci in appendix: 9 In sl. r e '.in of a jrt c. .'ii : 'i In rf .. nce t :.r a.e . :. ti.e et cbc nr.ed r: a ri .o:".s Ir c er" a l .. fr:, r'rst r...:c. .1.3 It cs :. n .'.n a.' :,e.i.:: t'.at B' uitalI c. ''o ce .f a ccins r.t tc''? v ie., of th.e t':.. e:.;'r: is ca .r. be. 3.a.C to a.re l very c:l c s l 'The co:nt.. t; as Dbe:', so c:..:s:n i. t:. .',r zent an..a ysi. rin .c.:e ba x.. o2 c;"r,'r..lerit.i C.R.To ( r,.. ,:' .;crz35 a I : ::4 11 i .::lu.: .vc u .' crifice ": .on, t:.c vO i c '..l: .s '.*.'.:.. to I..v e bL.e sr..e '' ',. '. .. I  c..2: a s';" c. 3r:3 or t. e .j'.e. TI'i.c n cial con "ie::'it .. r i .' ' ": , .a pr i'.i ,' ely : v..'i _ce ,_'.. c s I. l P:" l In ,1...':, tr'... t_ cL .c c.:'. i"ro t!e . S , e:.".r.. ., c jVt ., '. ce . C. Y ,C , t.1 r, .I , .,rc.r ?.. ,._ r .: ,:11" :.: o;..*.: tv.. b, '., t ,]j ? .o fo _e ^' *.i.' l ^ r ^ ? _: 'tL'2 L ,'ie, ' .; .'.' h '.L : r" :..e .. o:r.i r _n _, t Clo .".' ;,ia s1'" ... 1..1  * i" .'i .11 j' '.. ..' ... _. : d o t ,. 1 nr .. r 1 s i L'."_F l c''c1 g .: t..L cO : bii ns u .r LI .c r'... c;:'.e .: *.' e ioci t .rc ile n 5 ._1,? rc 2CC' % .i.L HZ', b ic J' .... S :: i: .L .I ; t ,e ar.ol':.s. '?.is m ...s .'n fz'rc a cc z:.d.: :blc sa i, li f 'ati n ii ..at it :..r; 1': _ *.:. :t ti of t e field .' s is '': j .r '. ::;: nce i ;' et o t.e in.duced b i.. C. ,.i P L': "r',ion ::C j s .,t. in' .lid at L. (. t ic z "r:'. . :t iL: L. T' C'c '.. 1 i '.' O c .."c _.c j? et ca.'. :". r. ..,.e. .s clocw.. ;,' e,ui va.le.t to . t ..r'. "'. a t '. cJ.n .s a i, .. f j;::t ,is , fLc LaS,, . 'tr .'.. L. .' c Er.c o .S.... i." .... p, .'i c.' at .z ?*.i 'Ji [' :. Ui ... ," '"rie r. t...'ts C. ....". it ' tn.i s ; rPoA ..t10a t7?' f 14. 2 i3j ': i;t in ;li : 1lity t' th.e .. al n.ci;. c jC u. '. " ic" 5 ci're .: ,: clSjul, def a... 3suEh e.. a i .' ,i: tV e 'rt.: it ia,.or. r:, t. };:. t j,_ t r t ,s:.'. t.':, " ,Ini' [:.ra ss bZ lx .ty 13 'aba .co 'nic ''. r '21 '.. :. tfle 3..t ',"C1 0 re ai;odC for 3'l:" .ib. t e .e c oc 1F'.: is aC led, I ... .. t.. to: .e e e . vt.JI r aIr L. ,oint i a" t_. jet .rtu,.,.d to b C r't:., to t.'c '. f ,'t.ce bet.oen the local Irt vel :: it : the trr ca' 'e oi t. Stcli a C :'. IDEI TI ,L COIT'IDE. TI AL WACA ACR Io. LiC15 COITFIDE:TIAL 7 temperature distribution is knovn to follow froni the mor.ent urntralnsfer theory vhen: the tempa.raturL di feren.ces are so s:,aall that density charges and Feat transfer by radiatin r:ay be neglected. This principle '1ill be applied herein v'i Lhout restriction to small t. I.;perature differences ad "' itLhout regac'i for the divergence frov'. e;peri:.ient. (See fig. 2.' Because of these cinplifying assur.ptions the analysis of the not jet can hardly be valid quantitatively. The analysis should be valid qualitatively to the extent o establishing 'wether the effect of te:.,peratu.re on tL e jetinduced flo.t inlciina tuon is larg. or s.iall. MU I r ,S iS Cold Jet parallci tL Streaw_ Velocity in jet. Ir all the fluid of the jet is tal:e:: locally fro... :ihe strean, r.ou.iGntUnc.: considerations she":. th.:t ~ thle ciusL equ,.,ls t.e ilass flo; per second tnrou.7h ,ny elle..cnt rmultiplid by the excess .' tie jet velocity ov,:r the streak veocilty at the element integrated ovr the crozs section cf tie jet; that is (see fi. 1(a) for .notac..on)), F = p (V + u)u 2rr dr = 2TrrpF.P VI + '2) or 2 (1) L'_ I 3 =k rp V I2F; where S u r dr U R R cl0 COIFTIDEiTI AL 8 CONFIDENTIAL iACA AC.' No. L6C13 122 r dr 2 ) R R If any of tne fluid of the jet is not taken from the stream, the thrust F in equation (1) must be replaced by (F Flight velocity x Added mass per second). The added mass per second contributed by the fuel is negli gible for airbreathing jet motors. For rockets the a.ded mass per second equals the thrust divided by the jetnozzle velocity. Aspiratortype jets lie between the two categories. Equation (1) may be solved for the ratio of the peak jit additional velocity U to the stream velocity V in the former S u ( 1 (2) V 212 where /no2 1 /212 S2 /' \/ ST Ic \I 3T,' and is a nundimensienal parameter. Spreading of jet. By extension of Prandtl's qualitative reasoning (see reference 2, pp. 163165) it is shoan/ in appendix B that dR k d (B2) dx V 1 +f U where k and f are constants that are determined in appendixes A and B, respectively. By use of equation (2), equation (B2) may be written CONFIDENTIAL IT.CA ACR .I,. L6C"1 dR ax 1 CONFIDENTIAL + 2 + + ) When the new variable =  \ STc ' is introduced d,= dR. d: dx 1 + I+ 2 2 + n Iz j /iT; and upon integration '*( i ' + +1)(2 11k (4) Equation ( ) provides the la.: f spreading for the jet since R j and x ~ ; the thrust F is con tained in both 7 and ea.r the *rigin, where the jet aditirnal velcity U is large in ccmperison with the streanm velocir y V, rj is smll in comparison with unity and equatti(n ,4) is appr.oximateiy R = kx CC'IJ I DTI AL (5) ITACA ACR No. LJC13 Thus, neur the origin of the jet, the spreading is approximately linrar aitih c ie axial distance x. Far froi.;: h c rising, where the jet additional velocity is srnall in comparison *:,it the ctraar.; velocity, 1r is large in conjpari_3on vi7h unity and equation (i) is approximnote l 2f 12 2T = kI 9I1 Ci' R = ConstanD x x rhat is, far from ti:e origin the jet spreads as the one tlird power of the axial distance Y.. Some further con r.onts on the spr?."/iri of a jet are made in appendix B. For the velocity prof.Lle (fir. 2), experimentally found for a jet L1 a still fluid, I1 = 0.0O91 and 1 = 0.095. For greater generality kc will be left undeterr.lined for the present. iih} these values of Il and I .quition (I1.) has been used to prepare figure 5, wi',ich sho,:s th.e v.riacior o' R/ iTc with ILx/ Equation (4) has also been used w'ith ecluation (2) to provide the v;.riat:.ci of U/V wi'Lh c::/lTc' shown in figure 1:.. The 1oiLit orin iof tne idealized jet io" thie present treatment, *',i, i is the ori, ,in of the coordinate x, is locat:1 a dit .nce x. upscreal cfl t.. orifice ofi the actual 1:t. (See fi;. 1.) The value of xi varies ath Tc but an average value is 2.5 .rifice diameters. ilore occi;:e val.iec can be obtained froi: figure 3 '."ith R lWLerlo 'ed as the orifice radiu,d.s Flow inclinaticn. The condition of continuity may be expressed by foriiinj the strsa.: function CC '.IFIDTIETTIAL CONFIDENT I AL NAC.'. ACR 0o. L6C13 C'NFID2~TIAL 11 fnr S= (u + V)r dr Jo Outside the jet this expression is appro:imately 2 Vr2 S=' URII + if the small values of u induced by the jet in the external flow are ignored. The anlse at whih the external flow inclines toward tihe jet axis is then, for small angles, E rV 6x i 2 ldR 2 d(UV) t) r dx 2R d l R The use of r] and in place of R and x, respectively, (vith ratios of the farm x/". per.mitted, however) serves to eliminate the thrust as a separate parameter. when this change is made in equAtion (6) xl 2 dn + d(U/V) . ii + r: dL n da if x/ is written for its equal R./'). Then by the use of equations 12) to (r ) there results finally k2 x(2 + 1 2 r  Ei (7) 212 rF 2 2 1+ ++ + 1 in radians, where r is related to tlhe independent variable by equation (L). ~ asymptotic approxima tion, accurate to U'ithin 1 percent for r; 0 o.18, is COCNF'IDINTI AL I .TAC ACR Io. L6dC5 kI2 x 1 2; S (79) 2 12 r2 1 I12 1 + 2f r If rj is expressed in terms rf r the flow inrlinaticn relations (7) is of the fcrmr ST., ' C = Constart x x _'unction o' r 2 x withinn the limits rn .picticr. o'f eq.iatln (7) the flow inclin.'ticon rut.tidj the jct t:,us is inversely pro Iprtlional t th.e ra3dal distance r fri'cn the jet axis. Equaticrn ['}) can be convenr1iently represented by the r T ' ,a ri9tion of '.vIth ST. he values of the con x2 stSnts :', f, Il, anj 12 therein are determined in spowndi'ps A nl B as 0.21, 5 0 5,.0991l, and 0.04,75, respectively, or tI e velity profile if figure 2. For p ST1 those values the variation of E with  is given x 2 in figure 5. Tis single curve provides all the neces sa ry infrrziatio ,,n the fle inclin, tion. typical flw pattern 'is row1i in figure 6. The flowinciination relation (7) end figure 5, w'hi,'h is ccmputed from i,, are limited in arpllcati,)n to r.ints reasonr.bl7T near the jet but well away from the orifice. The first limitation results frrrc. the neglect in the computation of the stren function of values of axral v. lc'. it:T induced by: the j t in the external flow. The seo.3 i limitatolr results from?; tle neglectt of the transition region between the .::ifi'e cf the jet and the re.irn of similar veicocity pr'files. The charts of reference 1, in which these u'nissions w,re not made, show that the.e sriation of question (7) holds, in general, r to ercer.t within twice the jet radius at distances "rerter th:nr. 3 criflce dimrieters d,v.nstreoa, of the orifice. Thi: ecc.ir&c:,, shnjld be suffic ent ror the usual relative pI.s..i..,3 cf' ;e jet eni the h'jrizntal cail for wing m nte i i t :inotc, s. Cc'i,:.'DE;,T1 I L CO1F I D 71TI A.L ,:ACA iCR ':I. L':15 The foregoing remarks may b interpreted from another point of view. The diameter of the jet orifice does not appear in the equations of the flnw analysis, but it has been ascertained that these equations are applicable, in general, for distances greater than 8 orifice diameters downstream of the orifice. The dov.nwash induced at the horizontal tail by v.ing jets at a given thrust may there fore be concluded to be almost independent of the size of the jet crifice up to a diameter about oneeighth the distance to the horizontal tail. For very high ratios of the jet velocity to the stream velocity > 50 rj is very small, and equa tions (.7) and (7a) become approximate 1. kll2 x k \ic tan  (b) I1 r_ r I, J/ where the assumption that C is s'r!i: is dropped. Such conditions may occur with rockets. at tale'ff and at lw speeds. For rock'cts the mass flow fr ., the nnzzle is not taken fror the stream and, as iLas been stated, the coefficent Tc' rust be m.lt iplied b one minus the ratio Of the streanim velocity" to the jet exit velocity for use in the f:rmulas. Rocket jts .re ordinarily supersonic n aor the nozzlele and the equations are not strictly applicable. Hot Jet paraiel to Stream Velocity in jet. The 1i o. air .enrsity in the hot jet will be some v'riable fracctirn a of the density in the free stream. For the present purpose the temperature elevation at any point in the jet will be assumed to oe prcpportional to the difference btwveen the local jet velocity and the ztrearn vel '.ocat (cee s.ection of present paper entitled "Assuanptions') tli..t is, t T u1 T V COONFI DTETIAL CONFIDENTIAL CCrNFIDY'TIAL wneere r i3 a cmnstent. (3e fig. 1(b) for rotationn) y the serfectas Is. then S T T + T 1 t +1 + IT S(3) 1 + T  IV ;iLn t.'. th.: equatir:; incr'r rrtacn r f t'ie denziity factor C, frr the z;a ld jet ;:.11 '3e .:clified to apply et. The nn.en.ntu e .utiuln .:ill take the S 2 2 +  = TTP12 I, r (9) where tJ. u r dr * R h u U 1 + T IT V L; [' 1 t T  1 t r T ' ,.ir ari .r. I2 .3 I I v~it th.e cnrrespunding .r.titie s? f the ccl. ~~et i r1 d 12 (euation (1)), .E:.t.: the f:1" i. r. .rpr'AimT.ti.rs: ,1 FI .7.::.TIAL  n I..,C.C ACR TO. LC615 CONFIDENTIAL 15 12 12 Ti 11 S1 2 1 12 I'   1 +^ I2 '2 u 1 + KT  where K is a constant to be determined by substituting values computed by the exact e ..ati.ns in the second of equations (1C ). An. average value cv'r the range of < 1 < greatest interest, 0 T 1 ..'2, is K 0.51 for the 'I experimental velocity trof~le of figure 2. Equation k9) can now be expressed in the solublie 'rm 2 + +(1 2 + from which U 11/212 Ill1 T2 KT + +2 T2 where is the function of R and Tc' defined under equation (2). The j.atte:nperature crefficient T may be determined from the fllowing considereticn if the temperature at the jet irifacen is known. Equation I';) as applied to crnditionr.3 t the jet r ifi7?e aiesi:inated by. subscript j), across which. the '.locity will .. ass'u:ined ,uniform, take the fern ( Tj U2 + tj T Il V ) / 2A CONFIDE NTI AL CONFI DYNTIAL ,v nence / ( t 91T 'S +: T A2 By applicatir, o its definiuin 1 a the orifice, the temperature coe'ffii2ent is t ,' T T = ,_ ST / V (12) Spread .:. f . It is s':..AI. in ap.erendix B that d A (32) 1 + i .utscttuti:n of equation (11) in equation (32) gives I I / 1 +  (2 KT + vI4 + 2 2IT)2 + T = k L 1 The m.s iorn cf rc' in the ridical ions ideably simpli fLcs the integration 'Jid yields littl. error for r2 << 1. ',t:; tl'.s missionn the inrte,:ral is . + f + a +asi ka (13) CC NFI T I AL C C F I')FErP XT AC' AC., !, LT l :I 1IACA ACF; Jo. LSCl0 where a = \.' 2KT Equation (13) provides the approximate law of spreading for the hot jet, since R r, and x E. The variation of R/)f/T' vitn kx//STe for a typlc al hot jet (T = 0.15) is shown with the curve for the cold jet (T = 0) in figure 5. The vriation of U/V with kx/~ TP' for T = 0.15, obtained by use of equa tion (15) vitn equation (11), is given in figure h along with the curve fnr the cold jet (T = 0). Flow Inlinati.n. The str2.n function for the hot jet is T'J o(u + V)r Jdr Outside the jet the expression is approximately S= V ( Ii' 2') + \1 T +  w' r e r 1 j  r lr  ) 1 r r r= F R 2 u U 0 + T  G V CO !FIDIEUTI AL (14) CONFIDENTIAL TJACA A;2 No. L6C15 if the small values of u induced by the jet in the e:ternral flo,' are ignored. The jetinduced stream deviation is then, for small ansnes, Vr C x 12 r l' U dIlT' dl' d(U/V) r dx L V dR? d dR The introduction of in and 7 in place of R anr1 x, re pecttively, Iwith ratios f.i' the form x/_ per.i tted, i' ';ever) e i.irini Ltes tnc tiru st as a separate nar'uai, c Lc V itL. this 'hen.r x T T 1I '' 1 5di /V)  2 '2 + diT dTh I+ ' d 1; dV d ]d d1 (15) where /'c has ben substituted 'c.r its equal R/r. Accordiri, to the original asulumption that the shape of tihe velocity prcf le i the sv,,? for' all sections, the r,tio u, U ep,d.: rnl' on r/!l and is independent of R cr T. ThE r'er 're 1 r dr' _ L'r :'r  U r d" U drT flu (16) t1 L r r "\ " .. \= I (i 'V 1 + "' L J C'CFID DE TIAL v:.I e r CONFIDENTIAL HACA ACR io. L'Cl'3 u L :) r dr I i = Ur ' I d (+ T r1 Al I T 0 Jo u r it' U RR (1+ 7T \ U V Ailso, by differentiation of equation (11), d (U/V) I 2 12 L" 31 2 r(2 :r) r2]T 2 t The incorpor,:tion of eqj. stions (it1) in eqoution (15) then fieldss the following frnal x ress on for the angle at which tie f'lo," in.lines toward the axis of tie hot jet: .x 2 dT), I2, I d( U, V) r5 r d r, d. (18) in radians. .11 of the vsr'Labl:.s in the equation except x and r are ultimastely .inct ons of r, and r alone; the I's arid dir'd,. are gv[. in term of U/V and T in equations (9), (41 (o1 ), an: ('B2), and U/V, d( U/V)/:ir, arid are giv.n in term's cf r and T in equations (11), (i7), and .1 ), respectively. If r. is pressedsd in terus of c and T by means of equ,.tion (15), the flowinclin'timn relation (18) is of the form *Ot:F ID:;'TIAL (17) C CNF I DE NT I AL ".'CA ..C R IT. L'GCl5 3ST \ E = Constant x Function of C; T As is the ca e for the cold jet, the flow inclination outside the jet is thus inversely proportional Lo the radial distance r. The effect of the jet ter.perature is dter;nin:d by the jEttemperature ccefficient T. Equation (18) f:r the flc.v deviation abcut the hot jet has been evaluated icr the single value T = 0.15. STcm The curve of rE aair.st  is .hc.n in figure 5, x x2 where E is measurd in degrees, alone, with the curve for the cold jet (T C.). Similitude of Hot ,nd Crld Jets with Applications to 'indTunnel Tests A typical value nf the te.nerature coefficient in a rropulsive jet is 7 = .i1 t :.ax..rrunTm f1iht T,'. From the curves of figure 5, therefore, the effect of temperature on the jetinlduced i'o;' ilinr.tion can be seen to be small, prviided the ccn. ariscn is made at the same tfrust coef1'icicet r'. Tie thrust coefficient is th'u a suitable criterion for the sliilitude Cf the flow fields about hot and ccld jrts of the ty:pe 'or v.hich all the flow fr.om the exit is sl;clied frnor the inlet. (Fcr a constant throttle sr.ttirnr t.he coefficient T increases as Tc' decreases, b.it this ,,arletion does not invali date the conclusion.) P.ecause of the reduced density the hot jet from a typical therrmal jet nmotor vill h,.v ofi the order of tvice the exit velocity of a cild jet that develops the same trust from the rccne 'ize .:rif ice, if 1ll the flow from the exit is supplied fro.ri the Inlet. The mass flow of the hot jet, hc'evtr, j.ili be of Lhe order of one half that of the cold jet. Fcr r.iodel testing with a cold jet the mass flow into the nicelle inlet tiat would occur vith a hot Jet should be simulated in order to simulate; the proper flowv about the nacelle. The mass CC" !Fr IDENT T AL CONFTIDETI AL IJACA ACR No. L6C153 flow in the cold js.t can 0. made equal to that in the hot jet bV' reducinri the o.fiiice of ihe cold jet to such a size tlihct the, productt :f a.Lr densty aid. orifice area is the sa;c for both jets. In windtunnel teats at the .\r,es .\eron.utical Laborato:,. of the :Iie'A (unpublished) the scalesize orific of the coldj: r.i odel was restrcsd to an annulus b rIeea,n of a failed rlu;g. If some of the fi.ild of the ccld jet is supplied fro.a a source other than the iI.i'lt of th. i nacile, as in the case of an aspirator j :, the im.a&s flo' i,nto cho inlet is less t.i:ni th1 .:ass lofr' ro: the e:.it, and the foreoing relations do, r.t pi;. In th3 ass s .mul& tion of trhe :.ro r ..ss flo'. int' t' e in c i .'c LsiLbie v.'ithon.t reduction of thle 4 t ?:.L ir'ro. the scal? value. ,i.th an asrirator jc, i vw,/r, th jetinduced flow incliniati.n at a ivei thIiut ,1 ? t: o sr.al for the reasons exolN'.ine1 in tLe an.ly.' i. cf' tle cold jet. (See section entitled 'Cold Jet Parallei. to Stiean.") Ef L' ct of Inclinctc. r: of Jet .':;is General rei.iark.. The effect ,f :,.ir.clin.ti n of the jet axis to the :noral flo." :n.t be consc .'er... in estimiiLons of the icini. ucec. d.:_.'...,c:. it E e. tail plan. Tf the let haved lik3 7. riid bo.y t.E iicli nation wI.ould ,i.ve rise to L in .erf.rence simiilar to thai between the fusela _.nad tihe horizontal tail. Vertically above the jo:t there wouldd be a slii,,t do.mn wash, and on r:h r side, a lithjt ip'."'_sn. .Lv'eraged wasei ,, across tihe tail, tile r.nt eiL'cc would br :nicgig_ble. The j t ac cual.: ap:.:ir :ii:.ites a ri id b d. in tiSEt. it tends to maint i.1 ts :I a .e .nd c ..ec ,icn n spj.te of any inclina.tion to the :aa.in fi'.. TI ..re i3 an aci:r'e ciable progress. _ve devi~ cion, 'howev.i:r, f r. t.he in tial direction tow id the .cr l l ii.ction t, i; can be obtained frc.ii .:': ntui cons ideraci n TIi.i d.Lefi,:c tiol alters the distance between the jE i.' t: hcrIontal toil, and therefore t he je in.'1.iced .o.lovwn.;ash. Determination of' j3c deflct ion. Let a be the local inclinaticn of the jet a: t to e general flow, and let ac be the.' inclination of the th:rut axis. C0 t the basis of metio ntufr cons ide:.ations, the following approximTate relation for the fractioal anulari deviation of the jet is derived in a.,ppendix C: C 0: FIDE' T I ,TL "OI I DETLL .',L 22 COTFI';ETIAL :IAC, A'? 1o. LC 13 2 + II + 2; T12 1 (C ) S+ (211 + 6KT + 12 2 The va.i.tion cf 1 with k:<1 T for the cold jet (' = C') ar.d the hot je (7 = 0.15) is given in fiure 7. 'i, ef''.ct of jet tC,.r..:,er tu''. is se n to be eligiblel. The chance due to jet deflection in the radial distance r firom. the jet axis to t.e horizontal tail is .ven by l = x 1 v) (19) where x x is the distance from the orifice to the horizontal tall end is the average value of aasv 1 bet.: en the jet orifiue and tre hinge line of a0 th. hori c'ntal t:.l :.inas the v l1.'le at the jet orifice. In this ar :licr.t in the eneril fi.%. in the r g.on of the Lt is affcct,d by the .vinw dov'nw'.i'h so th:t, in st rai_ nt fl L. t, ae = Q EW in .der.es, J.l.ere a is the irnclin.tion of the thrust a.i,: to the Cree stream, 6nd ., is the aownnwash due to th. v.'ing< v er'..ed ,vcer tlhe ier:..th x x In accel eoat.d fli gjt the curvature .3' trie 11lt pat., contributes an add .t ona l incrr'tent to e. Th, jet deflection Lr is evaluated in table III of ther numiWrl,al exam.pl, alonp ,:ith various otier CO0 IDE:'TIAL :J'A.L. .ACR 1 1ClC CCI':FIDE'1TIAL 25 quantities, and is shown to be no more than 15 percent of r. Cn the basis of these computations the jet deflection appears to be small for straight flight and for flight with small normal accelerations. On the other nan.d, the average anqular devition of the jet is an appreciable fraction of the ainle of attack. The fractional angular deviation (1 ) is 0.24 or a, & greeter for the several conditions of t:he numerical example. (See tables I to III.) E?2ECT C'F TETS DI: LO:,GITUDI!:L STAEILIrY AiD TRIM Averae Dcownwash over Tail Plane Consider a general point 7 along the span of the horizontal tail, .it'' = 0 directly above the jet. (See fig. e.) Let tie angle subtended at the center of the jet by the le.nth y be T. The jetinauzed flow inclination has beer shovn to b: inversely proportional to the radiril distance from the jet axij.s; therefore, if the inclination at :j = 0 is C, the inclination at y is c ccs 3. The downwash at T i,3 the comoonent of this normal tc the tail planee ccos The unweighted mean downwash ang le over tne La.l plane is therefore bt ' d + I rbl b t , d+ Ol. c r bt J ^ bt C ,NT7I DE'TI ;UL CONFIDE."TIAL [.'T. .' CR. 1r.. LC5 or d +. b d + r ar1 2 1 (20) btrtan r + ta(20) r r Liftingline theory suggests that an average weighted according to the chord .ould providee the most accurate values of tail lift. n. unr'eighted average over, s.y., .c f t.he tail sa.n would appear to approxi nate tis .nrditin. Thre curves f f .JLre 3, accordingly, have beer. pirared from equations (2) vith .9'bt sub stituted for b.. The curves give tne variation of Z/e ; th r, bt ard 2b/bt ,_:er e is now the effec tive ean jetirnd.;ce i:o.'nwash across ti.e tail plane, c is tte fec; inclinatIon at a r!'a us r from the jet, a:wid r/tt ard i2 t lieate th: jet axis relative to the tail plrjne, aC shcvn in fj.egre ?. The curves apply t a single 'et, and the dw:r'nwash s adcitlve for several .1 t S. Pitchzng'.'oment Incremen.ts Due t) Jet Operation Geneal c siideratins. hGc giv.er anle of attack, ncer:ation nf tre jet nrtrs v.ii_, in .er.erai, change both :>e p:_t..ir.g rr.cenrt :ar. the l:_t coefficient. confusion :.11 te avoided if t..e cha'es ..n pithIng rerment and lift coefficient are initially :btair..d as functions )f the po,.eroff (zero thrust) lift oefficient rr, which is a i:nc..n funci:,n 1f an.ie of =tte;:. The several pitc':rin:.c.ent increne:ets due to jet operation are dis cussed in the followiing paragraphs. Each increment is to be re:;:ec as a function zf Cr The increments are g;ven for a sin:le jet and are to be rimu1ltiplied by the nuirber :f .ets. PFtcn:ng moren.t centrituted by% direct thrust. If the tLrus.t xi.s rf t..r je t passes a distance z below the center of rr'vity the th.rust .ill contribute an incremental :itl.ir.g .....nt, ;ich is in coefficient form, . r ?: DT TT AL C 1. .17 1 _r7X T\ 1 IT, C.,CA .::"'. I' L',CT5 CO !FIDE. TIAL 25 Amm Tc The thrust coefficient Tc' ordinarily will be known as a function of the poweron lift coefficient CL. In order to obtain T.' as a function of the poweroff lift coefficient CLr, use can be :ade of the known relation between CL. and c together with the relation Cr Cr = aTp' where CL and CL are measured at the same angle of attack a and a is taken in radian measure. A "cut andtry" procedure may be used and a curve of Cr against CLo can be obtained at the same time. Pitching moment contributed by jetinduced downwash. It has been shown that a jet induces outside itself an axiall, symmetric flow field. The inclination c Imeas ured in degrees) relative to the thrust axis at the point (x,r) (see figs. 1 and c) for a given thrust coefficient T' can be deternmned from figure 5. A small deflection Ar experienced by the jet e.hen inclined to the geneo'al stream can be deteri.:ined from equation (1)) and figure 7 and used tn correct r :nd then c. The ratio of the value of average dow'nJas:i over the horizontal tail T to the value of E Is civen in figure o as a function of the geometry cf th: ijettil configuration. The pitel ing;moment coeffic. ent 3orntributed oer jet by tne jetinrcuced don'Ariash is th'n, for the sticl: fixed, dCT ACmf i 1 (21) fixed ilt If the stick is free and if the jet unit is mounted under the wing so that the horizontal tail is well away from the orifice, expression (21) becomes CCN.'IDE TI AL 26 C N : DT!TI .L IiAC A T.:. 1;". L 1 (dC^ dC.. C(2 AC free = it d (22)C If tihe orifice is !ner the horiz,.ntal tail, as when the jet issues from the rear' end of the fuselage, the horizontal tail will be in i regi. of curved fic,,. If the value of C.h is negative, the elevator will tend to float downward to conforr.i to the cu',v&ture. This downflnting tendency vill add a stab:.lizzin or negative a;n .unt to the value of the stickfree pitchingmoment increment .*iven by equation (22). The change could be substantial for s closely balanced elevator iCh near zero); the magnitude of the chan:r "...11 defend on the type of bslEance. In addition, the hingemoment charac teristics :,night be modified by an effect of the jet on the boundary layer of the eevator. The charts of the present ;::er (figs. 5, 5, and 7) rie not valid within a distance cf at:roximately o nrifice d.ar.eters do'.:nstrear of rh orifice, and ref erence 1 should be 7on3ulted for t. flow in this region. Equation (2i) for the stpi..fIxed pi tchingMcm.ent incre ment wvil be ae;'.rjximately valid :rov.Lidd E is evalu ated at the threezuorterch.rd i'1e of tr1 horizontal tall. Pitching moment contribiutc.d b niacelle normal force. The air taken in at t.e :.acelle inlt is turned through an a nil (ttne anie of attcl: cf' tni thrust axis) in becomin.a aii.ncd vith the jet Sa:s. This turning of the air gives rse to a. certrif;gal force e acting upward at iie in Let. The icrce, v%,h.Lcr: s ne:ligij ble compared with the wing lift, equals tne mt;ss f'lrew .er second through the nacelle multiplied by the streak' velocity and the sine of the local angle of attack,. Tne contribution to the airplane_ piitchin,moment icefficient is ((Mass/sec) L sin (a c) AC r, (= ( ^pJSc Pt C I'l7IDENTI AL iCA ..C. :''. L l, C';IFIDE;iTIAL 27 where is the lever arm fr'rm tiue inlet of the nacelle t tne :'entrir of grsvity of the irol.,ne and c' is the upwa sh induced by the 'in;, at the nlccelie inlet. The upw&sh F can be estim.ttesd from .l iure 5 of reference 5. This upv.wash is large only, when /,'c in equation (L) > .s small nd ts neglecL therefore introdLces simc ll err.r in the m'iomr nt. Pitchnine t:o::ent contribt :ted t bc'un..rria' r removel. The auction and c'ti.r effects ofc: tcre jvt nay tend to remove some of the bcund)ry l ":'er ?n c.djacert surfaces. rhe Or'essure. J stribuc','n ?uld be somewhat altered. In some instar.ces fl ; s ,raration mTily be In.~tb ited, 'whih wvuld result in ratner lar'e cl'2r,oes in pressure d Lstributio'n. In S,.. Lov,. separate _c.n cn tlhe Vn ngZ ..s suip'ress.ed,' ani inc ei:assd oc.n,..,sh ..;ill occur at the tiLl 'with a crnsequ.,nt i'~cit v: itchino:more t Incremu nt. The determirctior. of the :ronmEnt cn~tngez' due t.o. t';ze several ef rects mu, t t I t t. ex':eriment. riy c'.hanr' e in the 1'usel c:a: 1t .I.n rc .u;t d..e t: bcundaryiaver reioval with t;7.,L ,n T. pEc3st lv,, hLe li: fereI, frcrl Eir, 8cn a change vw th t .al .f' because of tL Lntrf1e renc bet:.;een the h.:lrizc.ital tail an:i t;he fuse iage. For this reason th.e c. r.. ris.n 'f tests 7 ..del 'with ta:.i on n. In th tail i ',. fr t r.esess .ril y' eld the psirt o, the r,v. r n '.ai tc i.?,r :n t chn e th t car. be atT rib. ted t: the jet r '''ct! : : .. Tiieutr lF.i t Shi. .t Due Fo;cr T'he o .e r'n curv if C ;.i.:' st f'r various ele'' or s3.ttn! s choUid b o' i' 1 . e the ,o:',roff curves. Th': shift in neutral cr : t 1 to rover is ;.e : "? . r e r c f "n jCL (c',er ,',, r orf in units of the :.ir.g chord. TIs ..eriv..,tives are evalu ated ..t nY convenient elevator s..ttn, for tne stiI: fi'xel condition and at any ccnven,Int elevator tab zetti.,g. for the stickfree na.tion. Cij::' DDEZTIAL CONFIDENTIAL .1. A .CR :.'o. L''cli From the earlier discussion it follows that expres sions ef the form An dACm An_ =  P dCr or dACM An  P d"r are not quite correct, where C,,,i is the sum of the several in:re.ental moment coefficients of the preceding paragraphs multiplied by the number of jet units, CLo is the poweroff lift coefficient, and CL is the pwernn lj't coefficient. Since CL CLn is small, however, either of the two equations s1 a good first approximation. The exact neutralpaint shift is slightly dependet.t on the position of the powercff neutral point. Numerica!l Example .rnd Discussion Specifications for a hypoti.et:cal airplanee propelled by tvin :. ingr.unted jet r.:Lt:rs are s;ven in table I. Details: conirputations of the effect of the jets on longitudinal stability and trim are viven in tables II and III. .ny moment resulting :ror boundarylayer removal that may be caused by jet action is not considered. The computations cnver a range of lift coefficients and both cold arnd hot jets. Th innre important factors cal culated are the mean jetinduced do'.nwash angle over the horizontal tail; the changes in the pitching moment with the stici fixed and with the stict, free due to this down wash, tn tlhe direct thrust moment, and to the nacelle normal fnrce; and the corresponding shifts in the stick fixed and stickfree neutral points. Table II is a suggested short r,,:thod of computation. The ;r.etnod is approximate in that the effect of jet deflection due to angle of attack is neglected, the variable distance xj is taken as .6R3, and the effect CONPIDENTI AL TAC.i. AC1 :' '. ,1,': C ?IFID,'NTIAI, 29 of temperature is neglected except in specifyin. the mass flo'.v per second through the nacelle. Table III gives the detailed computation without these approxi mations. The :nma:imum Influence of the variation in xi on the jetinduced flow inclination is found to ce 1 percent. The maximumr influence of both xj and in  nation of the jet axis on the mean j:tninduced downwash is found to be 7 percent. The jet deflectI.on does not exceed 15, ercent of the dictnce itron the jet axis to the horizontEl taii. The close ajree.mernt between tables 1T and III suggests thli.t the detailed corrom.utation of table III may be dispensed wth in many cases. Coimp rison v t.'t Ex.Jeri.nent The present methid has b=ean used to estinmte the stickfixed i.itcnin .:;,o0ient incre,?r, ts due to jet cera tion for a tvinjr;t flgnterty,.e c. clr..lne tc.it has been tested in tne Langley .ullscale t L..1., The uinpubl lIsed experimental values are comp' red Lt. the estitr.ated valuess in figure 9. The flapsneutral i'rves (li I(a)) sr.c a discrepancy in trimrr, but g d a.er.r2nt in slope. The flapsdeflected curves (fij. 9(b)) s,. gcod sreer.ent in both slnp,e and trim u,, to a :i t Cc eflcien t .' .3.6, but above Cr = C0'. the ez.eri:'ental. c..urve d iverges mar.ke vly from the rather straight es c !:".te,. c.irve. This diver gence is probably associ =ted vit 1 some: suppression by jet act o. cf seP,.r. tin st tht :'ee lle inlets ti'hat was indic ceds b: tuift studies 2r.rri. "ut during tr.e tects. n t ol, the agre ^en:t 0: t:.. l tie estir,.atd pitchingmo:.ent _ncrE;ients .du1 to jet o, e rat ion and toe experiLmental increrents 3 ,ers tc bt e sufficient for design purposes A number .f urtler ccmparisons v;Ith e. :er in:ri nt will h,,e t to be ':;ie before the accura,?. of' the tr.th:' of est im tion can e; sstatlli:hej. Anr anil,sl s ihas ben 'a&de .cf te f _eId of fl" about : jet t fft of j' s ,mn the stabC.lit'y ..d trim o.f jetpropelled .r 1 . 1Th.e if.l ,:o rn cCr.clu sicns include an uillowance : r :n limivra;tations of c:.e simplif;in, assunpt ions epl ': d COnF'1D:!TI.AL NiCA ACR No. L6C13 1. The jetinduced flow inclination varies very nearly inversely as the radial distance from the jet axis within tihe region between the jet boundary and twice the radius of the jet boundary at distances greater than 8 orifice diameters downstre.mu of the orifice. 2. The effect of jet temperature on the jetinduced flow inclination is small when the thrust coefficient is used as the criterion for similitude. 5. The deflection of the jet due to angle of attack is small for straight flight and flight with small normal acceleration. The angular deviation of the jet, however, is an appreciable fraction of tha angle of attack. 4. The downwash induced at th.. horizontal tail by win j.ts at a given thrust is .lr.ost independent of the size cf the jet orifice up to a Jia :ieer about oneeighth the distance to the horizontal tail. 5. The radius of a jet v:rizs almost linr.arlry with axia.l distance near the orifice and varies approximately as the onethird oower of tno axial distance very far fr".. Lhe orifice. 6. 1The equations for jetinduced flow inclination may be appliCd approximately to rocket jtts if the thrust coefficient is multiplied b one minus thel ratio of stre?,r vlocity to jotnozz1.e v'locit7y 7. The influence of viing .], ts on logcitludinal sta bility .:, tri:n may be c3tirm..tld w'.th sufficient accuracy for" dsir. purposes b,; rai pn'r'o; jetinluced boundarylayer re:.oval, Perd :iost of the effects cf jet tempereture. Lagley :'.:eni'lia. Aeronautical Labtv. tor.o ~,sit':_n Advisory Commit:ce for Aeronauttcs Langley Field, Va. CONFIDE.IT IAL CONFIDENTIAL iIAC'I ACR i to. LC1C5 APPENDIX A COMPARISON r'ITH THE A.ILYSIS OF SQrjIT irC D TROUICUNCR The flowinclination charts of Squire and Trouncer (reference 1) differ from figure 5 of the present paser by amounts from 0 to 11 percent when the flov, is .meas ured at the jet boundary c or more orifice diameters from the orifice. Figure 5 is believed to be more nearly correct within its region of application because of the use of an experimental rather than an idealized velocity distribution in the jet, although the treatment is less rigorous otherwise. A'detailed comparison of tne analyses follows. Squire and Trouncer present a relatively rigorous treatment by the momentu~ntranisLer tnec.ry of the develop ment of a round jet in a general streak moving parallel to the jet axis. Full consideration is given to the region, approximately ; orifice diameters in length, in which transition occurs from the uniformn velocity at the jet orifice to the characteristic velocit; distribution of the fully developed turbulent jet. The present analysis ignores the transition region entirely. Use is made of Squire and Trouncer's analysis to correct the value of a constant in an apprrximste equation for the spreading of the jet. (Sec a.,p"nen,.x B.) The equation is derived from the qualitative considerations cf refer:nce 2. In the analysis of reference 1 the values of axial velocity induced by the jet in the external flow are first neglected in determining the strearm function, as has been done in the present analysis. Squire and Trouncer, nowcever, use the result to determine a system of sinks along the jet axis from w'.hch the stream func tion (or, more accurately, its x,Aerivative) 2is reevalu sted. This procedure effectively restores the missing axialvelocity increments. Examination of the computed flowinclination charts of reference 1 in conjunction with the values of 1 C in tables II to IV therein c2aU1 dx shows that this refinement is unnecessary within twice the jet radius at points 8 or more ori:fce diameters downstream of the orifice. This range should cover the CONFIDENTIAL CONFIDE ITIAL 52 CONFIDENTIAL T.. C. :2 To. L .1 usual relative positions of the jtt an.. the horizontal tail for vingr'unted jet motors. Determination of JetSpreading Farameter k The only questirneble point in the analysis of Squire and Trcuncer is the use of a cosinevelocity dis tributicn for reasons of mathematical simplicity, rather cha: the experimental velocity distribution trat was used in the present analysis. The general development of the jet (from considerations of mass flew) is affected only slightly by a moderate change in the velocity pro f:le. (See reference 1.) The determination of the angular spreading of the boundary of the jet by means of the ex: erim.ental data of reference 1, however, is quite sensitive t;, the shape of the profile. The determination may be i..ade as follows. A jet issuing from a small ori fice in still air is known to spread cynically. Aczordirg to reference 1 the cone on which the vlocity,. is equal to onehalf the velocity on the jet axis at the same section has a se.iangle of' 50. ,ith Squire and Trouncer's cosinevelocity profile therefore 0.5R = x tan 5c R = O.175x or k = 0.175 (Al) With the experimental velocity i:rofile of reference used herein (fig. 2), 5.3S5R = x tan 5e R = C.24J0x k = 0.240 (A2) This value is 57 percent more than the value for the cosine profile. CO 'FIDENTI AL IACi.. AC t L.Cl CONFI Ia:'.T I AL Effect f Velocity Profile on Flow Inclination The fl.o inclination about the jet is in turn dependent on the spreading of the jet. If TJ is expressd in terms of , equation (7) is of the form r = 1 Function of c X2 1(2 1 ) I  1 ;iT (A5) where k aRn f are pe'ar.ieta.rs for the spre%',lng ff t:.. .l't, a:i I and 2 re ntegr.is involving the velocity/ prof ile. Vith. Squire and Trouncer's cosine Drofile 2 2 I. C (C. 175) (. 1 , )C (". 1  ,.6,6( 1 fI12 ( ) .t :" If I ';ith the experimental v.elcocit: rf'ili (fig. 2) (.2 ) 2 S" i k'Ii .2 ) .012  0.311j5 COiNFIDE'VTIAL 34 CONF:IDI' NTI IAL :7IA. AC .. 7/ "l: fI2 (.)(0.04895) T1 0.0991 = 1.652 The difference in k2I /12 is 32 percent of the value for the experimental profile. This difference is large enough to reduce the ordinates of figure by from 0 to 11 percent; the reduction is almosL linear with STe /x2 up to a value of 7 percent at S"c .. ..ith this x2 reduction, figure 5 is in substarnt.al arree:.:ent, within its ranae of apFlicability, with; the c:.rts of reference 1. The use of a cosinevelocityv dissrib;ut;.,n instead of the more sharcl: peaked experimental dis cr.'ution thus appears to introduce errors up tc 11 .erc;znt in the charts of reference 1. It is rather striking that the pronounced difference between the cosine .rof'ile an.. che experimental velocity profile results in very litt.lc diffTertjnLce in the parameter fI2/11. Thus the only .Lrportant uncertainty in tie calculations for the cold Jet is the evaluation of the zprepdingprofile parameter kR 12/12. This uncertainty is njt great, since $2 percent error in k2 12/I2 leads to errors of frco;, : to 11 prcent in the flow inclination. These results imply that the calculated rate of chcange of .rass flow in the jet v.,th axul distance is not cr_ cally dependent on tne velc.ity profile chosen. Presumably Scuire and lrouncer .ad this interpretation in mind when they stated (reference 1) that the general develup.tent ofi the Jet is little af~'ected b:' a moderate change in velccity profile. C NI DENT I AL fIACA AC.: io. r. 15 CO FIDNTTIAL 55 APPEiTDIX B APPr XII'.ATE DIFFERPITIAL RZLATICON, FOR SPREADING OF ROUND JTT Cf '.'A IN. Li''VIi..' FLUID ljL ESTABLISHMENT OF TW:T CCH'STAJT r FROMC EtUATICNS '(114) .il;D 15) CF SQUIRE .:C T''.'UICER Ba sc ;Aaliysis Consider a cross section of a round jet or wake for which the velocity at the center is U. The particles of fluid in the section move doArnstrera1 vCith an average velocity + V. Accordingr to Prar:dtl's s orox.mate treatment rf the spread or turbulence (reference 2, pp. 163 to 165) the time rate of increase of tne Je radius is proportional to the velocit difference Ut betr een the center of the j't and the edge. The section may thu.is be visualzed as expondiLn radiall aith velocity prioportlon&l to IllI arn, movin~t downstream with a velocity + V. The slo.e 2 the bundarv of this round jet or w :al is thv.rei'lre k  (. ) .x Ui + 2'; + , 2 wniere k is a constant that is detcrnmined in appendix  from exp'eriliental data. Equatio. ,1) .1 lso ap lii cable to .; t';:ondimensional >'"t o ws,.,i' if i is inter preted as the semiwidtrn. Equation (iB) lelsl to the k:o'.'n linear expansion of thei jet radius vith a.xial d:isttar.ce i'r a round jet in still air anr. to the kno%.n ontlihr'd pov.r la .'or the wake of' a body, of revolution. Th: ;roofs, w nich are simple, art; omit.td. It is ..f interest to note that a highspefe jet in mcvin ai.r s:?ou:!d ; cv. an approxi matal" lin.~cr sprcs.inrg ner th3 .:ri:.ce, v.:er0 te. .ztrean C'7CI DE TI a. 56 CONFIDE:TIAL i ,.:A .C: L6C15 velocity V is small in comparison v.ith the jet addi tional velocity U, and far back where U is small in comparison with V the expansion should follow the one third power law for the spreading of the wake of a body of revolution. The foregoing analysis contains an arbitrary element in the specification of U + V as the effective average 2 velocity in the jet. A more generalized average velocity would be + V where f is a ccnstant that depends f on the share of the velocity profile. Thus equation (B1) ctrn be generalized to dR IUI 1<= k (B2) dx U + fV It will b? s'tn/n that the equations of reference 1, derived on a more rigorous bssis, prcv'.de an expression for dR/dx that appraxim;ites equ ation (52) very closely for a suitablrF value of f, and tlihs establish the cor rect value for f. Determination of JctSpreading prarneter f .qu&ations (14) and (13?) of ref.rence 1 may be written, in the notation of the :resent paper, as UR + Iu) b, (B5) S(bV + bU) + R dfbV( + u bU + b = 0 (B4) dx b1 dxU ( U respectively, where CQNFITDE TRIAL ..LCAi ACR "r. LCOC1l ur dr = 0.1486 U R b = 2(JI b = 2(J 2 2  ) = 0.0578 16/  1) = 1 r u r dr = z. U P R b = J = 0.?091:.  1 2 2 2 r d~ J2 : = 0.0695 J R R b = 2J2 J1 = .I095 c2 b ,e The nur~mlical values ar.pl, to the cosinevelocity dis tr;bu.tCion a.cpted, by Squ.r anc T.ounce r. (The s:~tol c in the e..u.r ion !'cr b is us:di by SQui.re anr T:ouncer and is distinct from the 'in.I r ord c of the present r''port.) Eliminstion of d1./'ix between equations (B5) and (B3) lives dR b5U(liV + 212) dx ( 1iv +eI2UL'(b V+ bU) + IlV+ 2 1)(blV + b2U (B5) If this equation is put into the. f'er'. of eqction (B2), the constants therein are TTC2 k = b b2 COFI DTIT AL I1 = II J pl 2 J = O.R R 8, 0 CONFiIDENT IAL COWUD~N~iL ;: P2 *.l, .i6ct b b r = rf = 2 1 + V bi b 2' l I (b + b (b b I1 + 212 For the values of the constants that apply to the cosine velocity profile of Squire and Trouncer (given under equation (B4)), an average value for f is 2.6. With this value the approximate equation (B2) agrees with the more exact eouation (5B) within 1 percent over the range U U from 1 to = V V For the experimental velocity profile that was used herein (fig. 2) the constants are 12 = 0.J 495 J = o0.07:0 J = 0.0L359 bi = 0.0151h b2 = 0.01764 b = 0.0701 bL = 0.0527 Insertion of these values in equation (56) gives an average value of 5.3 for f. Viith this value the approxi mate equation (B2) agrees with the more exact equa tion (B5) within 2 percent over the range from U = 1 V to Z = m. The value f = 5.5 has b.en used in the V computations of the present :.,>*?r. CONFIDENTIAL (B6) CONFIDENT IAL NACA AC2 i. L/C15 APPENDIX C DEFLECTIONI OF IDEAL TJT IUCLIIIND TO STREAM Let a, be the inclination of the thrust axis to the general flow, and let 9 be the inclination of the jet center line at a distance x from the fictitious point origin of the jet. It is required to determine Qe jet. The momentum relations for the components of the thrust parallel to and perpendicular to the stream are, for small values of ae, R T = pR P o(V + u)u 2nr dr =2nrrF,2 2 1 )2 I . Q\T V + T + I/y 1 2 (Cl) PR aeT = P I Jo o V + u) 2 2 7,r or + P The first integral of aeT is th, crosswind momentum of the mass flow in the jet; the second integral is tile crosswind ciopnentumrr of the distuirb,d outside air com puted frnm the additional apparent mass of the jet. The expression reduces to aS T = g 2uR2 pV2 2 I' + 2 II + r. I Solving equations (Cl) and k2) simultaneously gives (C2) CONFI DENT TA. CONFIDENTIAL H.CA .\C'( Ilo. L'C l0 8 1 e ae 2 1' + IJ' 2 I U 1,+ 2 2 V \ + 2 In accordance with the main text put I 1 = 11 I : 2 I1 V T 2 2 u 2 K2 U 1 + lKT  I, V II V 1. .^ (Strictly speaking, the values of K should be different in each expression.) Then a 1 = Ye 2 + + (KT 12 (2+11 + + 12( CONFIDENTIAL (c3) rOTFT DENTAL NACA AC;l No. L6C15 RLEFE:EiNC S 1. Squire, H. B., and Tr.ouncer, J.: Round Jets in a General Stream. R. & IM. No. 1 74 British A.R.C.., 194. 2. Prandtl, L.: The Mechanics of Viscnus Fluids. Spread of Turbulence. Vol. III of Aero'ynrAmic Theory, div. G, sec. 25, v. P. Durand, ed., Julius Springer (Berlin), 1955, pp. 162175. 3. Fluid Motion Parel of che Aeronautical Research Coc;,littee aid Others: i:odern Develoanments in Fluid Dynauics. Vol. II, ch. X.II, sec. 255, S. Goldstein, ed., The Clarendon Press (Oxford), 1953, 7. 596, fri. 256. L. Cor'sin, Stanley: Investigtion of Flow in an Axially Syimmetrical Hleated Jet of Air. iTACA ACR 'C. 5L25, 5. Ribner, Herbert S.: iNotes o01 :he Proeller anc Slip stream in Relation to St?bility. NACA ARR No. L112a, 1914. CONF IDX'T IAT CONFIDE.7TIAL Ci.\.A ..CDI 'o. L6 C1 TABLE I SPECIFICATIONS FOR NLUMEICAL EYXAMPL Twin wing jets S, square feet ...... . 27 R foot . . 0. r, feet . . x x (to hinge line of horizontal tail), feet 3 d, feet .............. .. 5 bt, feet ...... . 12 t/c . . . 0.5 z/c . . 0.1 dCm/dit . . 0.0 0 dCm/d e . . 0.015 Cha/Chj . . 0.5 Te' per jet .. . 0.160L Jet tmnperature minus stream temperature t oF 1350 Stream temperature T, CF abs . 550 NATIONAL, ADVISORY COMMITTEE FOR AERONAUTICS CONFIDENTIAL CONFIDENTItAL NACA ACR No. L6C13 43 CONFIDENTIAL TABLE II SHORT APPROXIMATE COMPUTATIONS POR NUMERICAL EXAMPLE [Jet deflection neglected and xj taken as 4.6RJ; Jet temperature neglected except in step 13] Jet (assumed) Cold Cold Cold Cold Remarks S lap deflec Step ion, deg 0 0 15 45 Given Prame ter 1 CLO 0.5 1.0 1.0 2.0 Given 2 Tc' .08 .16 16 .52 Given 5 ST,'/!2 .227 .155 .455 .909 a step 2 4 If .222 .420 .120 .750 Prom fig. 5, by use of step 5 (curve for T = 0) 5 g, dog .75 1.8 1.8 2.46 Jetinduced downuash angle at section of horizontal tell vertically above jet (step 4 6 r/bt .25 .25 .25 .25 r and bt given in table I 7 2d/bt .5 .5 .5 .5 d given in table I 8 r/I .26 .526 .526 .526 Prom fig. 8 by use of steps 6 and 7 9 i2, deg .77 1.45 1.45 2.59 Mean Jetinduced downwash angle over horizontal tall for two Jets (2 x step 5 x step 8) 10 AC, .0231 .0455 .0455 .0777 PItohingmoment Increment due to jet Ef5xed2 Induced downwean; stick fixed a step 9 11 &C, .0175 .0526 .0526 .0583 Pitchingmoment increment due to jet free2 induced downwash; stlck free [(dC, dC, Ch, stp dit dO, Cha x) 9 12 AC .0160 .02020 .0 060 Pitchingmoment Increment due to T2 thrustaxis offset (2 x z step 2; 1 from table I) 13 Nas/sec .00470 .00654 .00654 .00914 ass flow through nacelle at aea level; ovs hat Jet; In coefficient form (given) 14 a. deg 5.7 10.5 .5 15.0 Given 15 a. .0006 .0024 .0001 .00142 Pitchlngmoment increment due to eac2 nacelle normal force, with wing upuash neglected (41 step 15 x sin step 14) "16 an .078 .073 .075 .068 Stickfixed neutralpoint shift due to power [slope of curve of (step 10 step 12 + step 15) against CL] 17 Anf .068 .064 .064 .061 Stickfree neutralpoint shift due to power [slope of curve of (step 11 + step 12 + stop 15) against CLO] CONFIDENTIAL IHATIONAL ADVISORY COMMITTEE FOR AERONAUTICS 44 NACA ACR No. L6C13 CONFIDENTIAL T ABLE HIII DETAILED COMPUTATIONS POR NI RICAL XAMPDLE Jet Cold Cold Cold Cold Hot Remarks (aiven) lap lap doflec stop, deg 0 0 45 45 45 Given Parameter 1 CLO 0.5 1.0 1.0 2.0 2.0 Olven 2 To. .08 .16 .16 .52 .52 Given ) t/ 0 0 0 0 2.70 Olven I4 Uj/V 4.15 6.13 6.13 8.87 17.5 Ratio of outlet velocity ainus strom velocity to stram veloloty (from equation (12)) 5 r 0 0 0 0 .159 Ratio of absolute temperature to veloolty stop 6/ 6 Rj/A .085 .060 .060 .OI) .0)4 Rj and S given nl table I T' given n stop 2 7 kxo/A .096 .066 .066 .047 .04 Prom fig. 3 with step 6 used as absolass 8 ft 1.88 1.8 .85 1.8l 1.68 DIstance upstrem from orlfloe or point origin of equivalent Ideal Jet 9 f, ft 9.88 9.68 9.83 9.84 9.68 Axial distance from origin of equivalent ideal Jet to point under consideration; in this case, the binge line of horizontal tall 1 o10 Sto/2 .225 .455 .55 .g09 .959 STc'/(*tep 9)2 11 I e .220 .20 .420 .750 .722 Prom fig. 5, Dy use or astps 5 and 10 12 k/Si6' .506 .372 .372 .252 .248 (o.24l0/,1S ) step 9 15 1 ) .54 .51 .1 .21 A.vrage of curve of between values of kx/ T/.f given by steps 7 and 12. respec ivelyl, minus value or 1 B for step 7 IL a, deg 5.7 10.3 .5 15.0 13.0 Given 15 w, dog 2.5 5.1 10.0 15.1 15.1 WIng downwuh, estimated 16 a,, dag 1.2 5.2 10.5 2.1 2.1 Average inclination of flow relative to the Initial direction of the Jet aia (atep 14 step 15) 17 Ar, ft .06 ..25 .45 .07 .07 Jet deflection at horitontal tall due to inalina clon to the stram [(a xj) a stop 15 Stap 161 57.5 i 16 r, ft 2.94 2.77 3.45 3.07 5.07 Dimension r (fig. 7) corrected for Jet deflec tion ().00 4 step 17) 19 (, dog .74 1.49 1.20 2.40 2.27 Jetinduced flow inclination at point of horl ontal tall vertically above Jet f step 9 ep step 18 / 20 r/bt .245 .231 .288 .256 .256 Step 18/bt 21 2d/bt .5 .5 .5 .5 .5 d and bt given in table I 22 7/V .522 .502 .570 .553 .5 5 Prom fig. 8 with the use of steps 20 and 21 23 72, dog .77 1.55 1.57 2.56 2.42 Mean jetInduaed downwash over horizontal tail, for two Jets (2 a step 19 a astp 22) .77 1.l45 L.45 2.59  Approximate value from table IT for comparison Prom this point the procedure of table II is followed. CONFIDENTIAL NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS 41 NACA ACR No. L6C13 4 Fig. 1 \L ioi P1 u0l '.. T Zu t2 z z Irn 8 t I s o 1 I 03 .E L E 0 0 O IU L O CL L E N 0 .c L "I, r. .I C 01 o 4 3 L. 03 . o 4 d 0 L 4r NACA ACR No. L6C13 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS U O or .6 tm 2 0 .2 .4 .6 .6 A? Figure 2. Velocity and temperature profiles for a round jet in still air. (a) Experimental velocity profile adopted for the present report. Replotted from reference 3 with r/R taken as the value therein divided by 2.74. (b) Experimental velocity profile of figure 20 of refer u ence 4 fitted to curve (a) at V 0.5. (c) Theoretical cosine velocity profile of reference 1. (d) Experimental temperature profile of figure 20 of reference 4 to same r/R scale as curve (b). /.O0 Fig. 2 NACA ACR No. L6C13 Fig. 3 E4 *2 _; vl r 0 0 .r4 ., A N 1I) Fig. 4 NACA ACR No. L6C13   E .. .. 0, O 1 0I S= /O O o\ ^  ^ s ,r 1 1, C SO o  i. __ __ _ /' / 0  7 ^  __ _^ / __ _ .' y * ___ __ __ __ __ __ ___ __ _^ * /IYIIIIIE %o \ N 0 NACA ACR No. L6C13 Fig. 5 'A 8 t UU z I0 ' _ ____  X 11 I u'< a \' a __ __ __ __ __ y __ __ P, \P \[ a ,1;  ^  & ^^    1,; ^    \  v NACA ACR No. L6C13  LIILUiIIVU:ULILI 111r l11 tIn Vr I 1 I I  U _ ". jf\, i'4 . i ! i . .. 7 1~~n~ IC : C :1 t  : :~ I I h ~Hull M' i 4I l l i i l i TI l "i I'11 1 1 1 1i i 1. 1 1 1 1 1. Fig. 6 fttttCttttii IN NACA ACR No. L6C13 Fig. 7 uA 0 04 2g z 8 \Q ^0 \ \ C'I P3 I 0 ed 3 0 o I ba 0 44 a) Li 0 0 4 o (U ta .4 I ' a) ' N ro IQ NACA ACR No. L6C13 0 44 3 8 I 4i8 om I 0 d .0 *J, to ,I u. C GeJ E 4.I a ' *r I u aia * k * O *J 1 Uc U V *^ Fig. 8 I(44(4 NACA ACR No. L6C13 Fig. 9a I ma) 1o U ) C ) .0 t \ \ \ o Tz \) + 2 > 4 * Z a 2 o .E w 0 w 0.) U oW r Z I4 ___ i) it N \ I >), > 0) s r... 0I 6 L4 M ) d \10) 0. La O 1 H\ A I    ___ iQ t < e o n i e \ V c) >. ____ ___ ___ ____ ____ ___ ___ 0>0)t *'"a a) i d() a \io I 3 *' Cd II NACA ACR No. L6C13 \ \V _1 _ i \ ^ rx~^ r ^ \J  ^ = l  t ^1 o4 w> 5w i 8 0 J^a a '*^ (VI ' Fig. 9b UNIVERSITY OF FLORIDA DOCUMENTS DEPARTMENT 120 MARSTOf SCIENCE UBRARY PO. BOX 117011 GAINESVILLE, FL 326117011 USA 
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