Field of flow about a jet and effect of jets on stability of jet-propelled airplanes

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Material Information

Title:
Field of flow about a jet and effect of jets on stability of jet-propelled airplanes
Alternate Title:
NACA wartime reports
Physical Description:
44, 10 p. : ; 28 cm.
Language:
English
Creator:
Ribner, H. S ( Herbert S. ), 1913-
Langley Aeronautical Laboratory
United States -- National Advisory Committee for Aeronautics
Publisher:
Langley Memorial Aeronautical Laboratory
Place of Publication:
Langley Field, VA
Publication Date:

Subjects

Subjects / Keywords:
Jet planes -- Flight testing   ( lcsh )
Jet planes, Military   ( lcsh )
Aeronautics -- Research   ( lcsh )
Downwash (Aerodynamics)   ( lcsh )
Genre:
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Summary:
Summary: The flow inclination induced outside cold and hot propulsive jets by the turbulent spreading has been derived. Certain simplifying assumptions were employed and the region near the orifice was not treated. The effect of jet temperature on the flow inclination was found to be small when the thrust coefficient is used as the criterion for similitude. The deflection of a jet due to angle of attack has been derived and found to be appreciable but small for normal flight conditions with small normal accelerations. The average jet-induced downwash over a tail plane has been obtained in terms of the geometry of the jet-tail configuration. These results have been applied to the estimation of the effect of the jets on the static longitudinal stability and trim of jet-propelled airplanes.
Bibliography:
Includes bibliographic references (p. 41).
Statement of Responsibility:
Herbert S. Ribner.
General Note:
"Report no. L-213."
General Note:
"Originally issued April 1946 as Advance Restricted Report L6C13."
General Note:
"Report date April 1946."
General Note:
"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 previously held under a security status but are now unclassified. Some of these reports were not technically edited. All have been reproduced without change in order to expedite general distribution."

Record Information

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University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003594028
oclc - 70902610
System ID:
AA00009358:00001


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ANkAL-I




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 JET-PROPELLED 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' JET-PR tEiLLED AiEFLAiHES

Ey Herbert S. Ribner


SUT!M. lR7


T-e 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 trer-ted. 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 nor--mal sc eleraticns. The avir;&..e jc-t-induced
downwash ever a tail plane has been obtained in terns of
the geometr:r cf the jet-tail configurit ion. These results
have been applied t the titi the -tn he effect c-f uhe
jets on the static lnn.itudinal stability and trim of
jet-propelled 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 an-i the external flnv is caused tr
incline t-ward the jtt axji. If the jet passes near the
tail surfaces :if et-prop~elie i :irlran~r, the jet-induced
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 analys-s 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 err-rs in the opposite
direction. In the :rezer.t revised trea:-tment, most of
the advan-:ta6s of Simoij.fica-in 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 t-e rep-esentati- n i-f t.ie jet-induced
strewn $.eviaticr:- byl a :-ir-ie 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
n-t tre-t t:, h'-.ot et

T he firzt .-,art r2 the prcs.nt ?. ;r is lonoerned
.;iti t.ie an-naL-s3 I. of fl-v; 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
stabii-ty 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.e-t is neeessar--.





( F r Ji r:grir, rr s c re-resent7r ti- n "l' 3t', s.F 0" the s,-n:t-,ls
rcler-r 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-" -trCa-m 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 r-hture o.er stream tempera-
ture at r-.int x cn jet axis

7 jt;t-t-n,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 h-lds at sub-
stnr.tial distances irr:m ,crifice -:uld h.'ve
zor.) cr'E.ss section

r radial di t *a-r.. 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. jet-st pre Ain n r p r.s:. eter tIken as 0.f,')

r je t-s, r .e rint r-.amet t r I Lr ske :- s r

constar. t .t k n a '..1)

S s t r-e ~ fc t i :-n


C )iNFI DP:iTI AL








4 CONrIDEY:TIAiL IACA ACR N'. L6C'1


c jet-induced 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 ijet-inc.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 nr-ent)


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 -i--t ; pcwer cn

unless n s t:.scri-pi.te !

I :-L-r.ce h rv!.z nt-l 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 coefficie-nt

____ 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 leading-edge 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 jet-induced flw inclination

1 due to single jet

2 due cc two jets

nac due to nacelle normal f'mr-c,-

0 measured at zero thrust; define as power-off
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'.ur-bulerc f reference 2, p 165-6 ). The
flow studied is incompressible '-ut the results are con-
sidered closely apFlicable to all subso-nic 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 I-r 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: i-s 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,.. ,:-' .;cr-z35
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'r-c 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 L-aS,,







. '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 -. v-t.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 urn-tralnsfer theory vhen: the tempa.raturL di feren.ces
are so s:,aall that density charges and Feat transfer by
radi-atin 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 cin-plifying
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 jet-induced 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:: locall-y fro.-.. :ihe strean, r.ou.iGntUnc.: considerations
she":. th.:-t ~ thle ciusL equ,.,ls t-.e ilass flo;- per second
tnrou.7h ,ny el-le..cnt rmultiplid by the excess .' tie
jet velocity ov,:r the streak veocilty at the ele-ment
integrated ov-r 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 air-breathing jet motors. For rockets the
a.ded mass per second equals the thrust divided by the
jet-nozzle velocity. Aspirator-type 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. 163-165) 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 *r-igin, where the
jet a-ditirnal velcity U is large in ccmperison with
the streanm velocir y V, rj is sm-ll in comparison with
unity and equatti(n ,4) is appr.oximateiy




R = kx


CC'IJ I DTI AL


(5)







ITACA ACR No. LJC13


Thus, neu-r 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-
tl-ird 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- .quit-ion (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 or-in 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 o-cci;: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 in-duced by the jet in the
external flow are ignored. The anlse at whi-h 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 2-R 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). ~ asy-mptotic 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 represe-nted 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 -n-l 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 flow-inciination 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
r-e.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 er-cer.t within twice the jet radius at distances
"re-rter th:nr. 3 criflce dimrieters d,-v.nstreoa, of the orifice.
Thi: ecc.ir&c:,, shn-jld 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 iC-R ':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 one-eighth 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 tal-e-'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
coeffic-ent 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 j-ts .re ordinarily
supersonic n aor the nozzlele and the equations are not
strictly applicable.


Hot Jet para-iel 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 b-twveen 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. (3-e fig. 1(b) for rotationn)
-y the serfect-as 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 th-e 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 b-y 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 'r-m


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.at-te:nperature crefficient T may be determined
from the f-llowing 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 applicati-r, 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 ri-dical ions ide-ably 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 v-riation 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 s-mall values of u induced by the jet in the
e:ternral flo,' are ignored. The jet-induced stream
deviation is then, for small ansnes,



Vr C- x


1-2 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
anr-1 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 it-L. this 'hen.r-


--x T T 1I '' 1 5di /V)
- 2 '2 + diT dT-h I+ '
d 1; dV d ]d d1


(15)


where /'c has b-en 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)


t-1 -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 ultimaste-ly .inct ons of r, and r alone;
the I's arid dir'd,. are g-v[.- 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 ), r-espectively.

If r. is pressedsd in terus of c and T by
means of equ,.-tion (15), the flow-inclin'-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 jEt-temperature 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 measur-d in degrees, alone, with the curve
for the cold jet (T C.).


Similitude of Hot ,-nd Crld Jets with

Applications to 'ind-Tunnel 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 jet-inlduced i'o-;' i-linr.tion can be
seen to be small, prviided the ccn. ariscn is made at the
same tf-rust coef1'icicet r'. Tie thrust coefficient is
th'u a suitable criterion for the sliilitude Cf the flow
fields about hot and ccld jr-ts 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 d-ensity 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


CONFTID-ETI 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 dens--ty aid. orifice area is
the sa-;c for both jets. In wind-tunnel teats at the
.\r,es .\eron.-utical Laborato:,. of the :Iie'A (unpublished)
the scale-size orific of the cold-j: r.i odel was restr--csd
to an annulus b- rIeea,-n of a failed rlu;g.

If some of the f-i.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 i-m.a&s flo' i-,nto -cho
inlet is less t.i:ni th1 .:ass lofr' ro: the e:.it, and the
fore-oing relations do, r.t pi;. In th3 ass- s .mul&-
tion of t-rhe -:.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 jet-induced
flow incliniati.n at a ivei- thI-iut ,1 ? t: o sr.al for
the reasons exolN'.ine1 in tL-e 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 :nor-al flo." :n.t be co-nsc .'er... in
estim-iiLons of the ic-ini. 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 the-re 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'c-c 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 pro-gress. _ve devi~ cion, 'howev.i:r, f r. t.he in tial
direction tow i-d 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 dista-nce between the jE -i.' t:- hcrIontal toil,
and therefore t he je -in.'1.iced .o.lovwn.;ash.

Determination of' j3c defl-ct ion.- Let a be the
local inclinaticn of t-he- jet a:- t t-o e general flow,
and let ac be the.' inclination of the th:ru-t axis. C0 t
the basis of metio ntuf-r cons ide:.-ations, the following
approximTate relation for the fractioal an-ulari deviation
of the jet is derived in a.,ppendix C:
C 0: FIDE' T I ,TL


"OI I DETL-L .',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
fi-ure 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 g-jt the curvature .3' trie 11l-t 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 oti-er


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 devi-tion 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

Avera-e 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 jet-inauzed 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.. LC-5


or


-d +. b d +
r ar-1 2 -1 (20)
btrtan r + ta(20)
r r


Lifting-line 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 t-is .nrditin. Thre curves f- f .JLre 3, accordingly,
have beer. p--irared from equations (2-) vith -.9'bt sub-
stituted for b.. The curves give tne variation of
Z/e ; th r, bt ard 2b/b-t ,_:er e is now the effec-
tive ean jet-irnd.;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 pl-rjne, 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

Gene-al c si-ideratins. 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 pit-hIng rerment and
lift coefficient are initially :btair..d as functions )f
the po,.er-off (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 th-e 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 mor-en.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. .1-7 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 power-on lift coefficient CL. In
order to obtain T.' as a function of the power-off
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-
and-try" procedure may be used and a curve of Cr
against CLo can be obtained at the same time.

Pitching moment contributed by jet-induced 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 c-iven in figure o as a
function of the geometry cf th: ijet-til configuration.

The pitel ing;-moment coeffic. ent 3orntributed oer jet
by tne jet-inrcuced 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 I-iAC A T.:. 1;". L -1



(dC^ dC.. C(2
AC free = it d (22)C


If tihe orifice is !ne-r the horiz,.ntal tail, as when
the jet issues from the rear' end of the fuselage, the
horizontal tail will be in i reg-i. 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
downfln--ting tendency vill add a stab:.lizzin or negative
a;n -.unt to the value of the stick-free pitching-moment
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 hinge-moment charac-
teristics :,night be modified by an effect of the jet on
the boundary layer of the e-evator.

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 tching-Mcm.ent incre-
ment wvil be ae;'.rjximately valid :rov.Lidd E is evalu-
ated at the three-zuorter-ch.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 th-rust axis) in
becomin.a aii.ncd vith the jet Sa:s. This turning of the
air gives r-se to a. certrif-;gal force e acting upward at
iie in Let. The i-crce, 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 :'-entr-ir 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.t-tesd from .l i--ure 5 of reference 5.
This upv.wash is large only, when /,'c in equation (L) > .s
small -nd ts neglecL therefore intr-odLces simc ll err-.r
in the m'iomr nt.

Pitchnine t:o::ent contribt :ted t bc'un.-.r-r-ia' -r
removel.- The auction and c'ti.r effects ofc: tcre jvt nay
tend to rem-ove some of the bcund)ry l ":'er ?n c.-djacert
surfaces. rhe Or'essur-e. J stribuc','n ?uld be somewhat
altered. In some instar.ces fl ; s ,-r-aration mTily be In.~tb-
ited, 'whih wvuld result in r-atner lar-'e cl'2r,oes in
pressure d Lstributio'n. In S,.. Lov,. separate _c.n cn t-lhe
Vn ngZ ..s suip'ress.ed,' ani inc ei:assd oc.n,..-,sh ..;ill occur -at
the t-iLl '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:
bcundary-iaver re-ioval with t;7.,L -,n T-. pEc3st lv,, hLe -li:-
fere-I, frcrl Eir, 8cn a cha-nge vw th t .al .-f' because of t-L-
Lnt-rf1e 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- l-F.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 c-hoUid b- o'- i-' 1 .- e the ,o:',r-off
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. TI-s ..eriv..,tives are evalu-
ated ..t n-Y convenient elevator s-..tt-n, for tne stiI-:-
fi'xel condition and at any ccnven,--Int elevator tab
zetti.,g. for the stick-free -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 power-off lift coefficient, and CL is the
p-wer-nn lj't coefficient. Since CL CLn is small,

however, either of the two equations s1 a good first
approximation. The exact neutral-paint shift is slightly
dependet.t on the position of the power-cff neutral point.


Numerica!l Example .rnd Discussion

Specifications for a hypoti.et:cal airplanee propelled
by tvin :. ing-r.-unted jet r.:Lt:rs are s;ven in table I.
Details-: conirputations of the effect of the jets on
longitudinal stabilit-y and trim are viven in tables II
and III. .ny moment resulting :ror boundary-layer
removal that may be caused by jet action is not considered.
The computations c-nver a range of lift coefficients and
both cold arnd hot jets. Th innre important factors cal-
culated are the mean jet-induced 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 stick-free 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 jet-induced 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:t-ninduced downwash
is found to be 7 percent. The jet deflectI.on does not
exceed 15, -ercent of the dict-nce itr-on 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 methi-d has b=ean used to estin-mte the
stick-fixed i.-itcnin -.:;,o0ient inc-re,?r, ts due to jet c-era-
tion for a tvin-jr;t flgnter-ty,.e c.- clr..lne tc.it has been
tested in tne Langley .ull-scale t L..1., The uinpubl lIsed
experimental values are comp' red Lt. the estitr.ated valuess
in figure 9. The flaps-neutral i'rv-es (li I(a)) sr.c
a discrepancy in trimrr, but g- d a.-er.r-2nt in slope. The
flaps-deflected 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
pitching-mo:.ent _ncrE;ients .du1- to jet o, e rat ion and toe
experiLmental inc-rer-ents 3 ,e--rs 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 .c-f te f _eId of fl"
about : jet t fft of j' s ,mn the stabC.lit'y ..d
trim o.f jet-propelled .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 jet-induced 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 jet-induced
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 one-eighth
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 one-third oower of tno axial distance very far
fr".. Lhe orifice.

6. 1The equations for jet-induced flow inclination
may be appliCd approximately to rocket jt-ts if the
thrust coefficient is multiplied b one minus thel ratio
of stre-?,r vlocity to jot-nozz1.e v'locit7y

7. The influence of viing .], ts on logcitludinal sta-
bility .-:, tri:n may be c-3tirm..tld w'.th sufficient accuracy
for" dsir--. purposes b,; rai pn'r'o; the ,sfficts 3f jac deflection, siz. of the jet orifice,
jet-inluced boundary-layer re:-.oval, Perd :iost of the
effects cf jet tempereture.

La-gley :'.: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 flow-inclination 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~n-tranisLer 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 strea-rm 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
axial-velocity increments. Examination of the computed
flow-inclination 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 ving-r'-unted jet motors.


Determination of Jet-Spreading Farameter k

The only questirneble point in the analysis of
Squire and Trcuncer is the use of a cosine-velocity 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 one-half the velocity on the jet axis at the same
section has a se.iangle of' 50. ,ith Squire and Trouncer's
cosine-velocity 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 express-d
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 cosine-velocityv dissrib;ut-;.,n instead of the
more sharcl-: peaked experimental dis cr.'ution thus appears
to introduce errors up tc 11 .er-c;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 zprepding-profile parameter kR 12/12. This
uncertainty is njt great, since $2 percent error in
k2 12/I2 leads to errors of frco;, : to 11 p-rcent in the
flow inclination.

These results imply that the calculated rate of
chc-ange of .-rass flow in the jet v.,th ax-ul distance is
not cr_- cally dependent on tne velc.ity profile chosen.
Presumably Scuire and- lrouncer .-ad this interpretation
in mind wh-en they stated (reference 1) that the general
develup.t-ent 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 visual-zed as expondi-Ln radiall- aith
velocity- pri-oportlon&l to IllI arn, movin~t downstream
with a velocity + V. The slo.e 2 the b-undarv 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';:-o-ndimensional >'"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 on-tlihr'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
high-spefe 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't-n/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) v-ery closely
for a suitablrF value of f, and tlihs establish the cor-
rect value for f.


Determination of Jct-Spreading 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 : = 0.06-95
J R R


b = 2J2 J1 = -.I095


c2
b ,e


The nur~mlical values ar.pl, to the cosine-velocity 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.r-e anr T:ouncer
and is distinct fr-om 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


CONFi-IDENT 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,- cross-wind momentum
of the mass flow in the jet; the sec-ond integral is tile
cross-wind 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 k-2) 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 K-2 U
1 + -l-KT -



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. 162-175.

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 Pro-eller 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 Jet-induced 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 Jet-induced downwash angle over
horizontal tall for two Jets
(2 x step 5 x step 8)
10 AC, .0231 .0455 .0455 .0777 PItohing-moment Increment due to jet-
Ef5xed2 Induced downwean; stick fixed
a step 9

11 &C, .0175 .0526 .0526 .0583 Pitching-moment increment due to jet-
free2 induced downwash; stlck free
[(dC, dC, Ch, stp
dit dO, Cha x) 9
12 AC .0160 .02020 .0 060 Pitching-moment Increment due to
T2 thrust-axis 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 Pitchlng-moment increment due to
-eac2 nacelle normal force, with wing
upuash neglected
(41 step 15 x sin step 14)
"16 an- .078 .073 .075 .068 Stick-fixed neutral-point shift due to
power [slope of curve of
(step 10 step 12 + step 15)
against CL]
17 Anf .068 .064 .064 .061 Stick-free neutral-point 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 Jet-induced 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 jet-Induaed 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


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P1
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T

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t2
z z
I-rn
8


t


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-O
IU








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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).


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NACA ACR No. L6C13


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UNIVERSITY OF FLORIDA
DOCUMENTS DEPARTMENT
120 MARSTOf SCIENCE UBRARY
PO. BOX 117011
GAINESVILLE, FL 32611-7011 USA




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