On the flow of a compressible fluid by the hodograph method

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Title:
On the flow of a compressible fluid by the hodograph method
Alternate Title:
NACA wartime reports
Physical Description:
41, 8 p. : ; 28 cm.
Language:
English
Creator:
Garrick, I. E
Kaplan, Carl
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:
Hodograph equations   ( lcsh )
Compressibility   ( lcsh )
Aerodynamics, Supersonic   ( lcsh )
Genre:
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Summary:
Summary: Elementary basic solutions of the equations of motion of a compressible fluid in the hodograph variables are developed and used to provide a basis for comparison, in the form of velocity correction formulas, of corresponding compressible and incompressible flows. The known approximate results of Chaplygin, von Kármán and Tsien, Temple and Yarwood, and Prandtl and Glauert are unified by means of the analysis of the present paper. Two new types of approximations, obtained from basic solutions, are introduced; they possess certain desirable features of the other approximations and appear preferable as a basis for extrapolation into the range of high stream Mach numbers and large disturbances to the main stream. Tables and figures giving velocity and pressure-coefficient correction factors are included in order to facilitate the practical application of the results.
Bibliography:
Includes bibliographic references (p. 36).
Statement of Responsibility:
by I.E. Garrick and Carl Kaplan.
General Note:
"Report no. L-127."
General Note:
"Originally issued March 1944 as Advance Confidential Report L4C24."
General Note:
"Report date March 1944."
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."

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University of Florida
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All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003613067
oclc - 71200631
System ID:
AA00009399:00001


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Full Text
(AAc 4


ACR No. L4C24


NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS





WAIFTIME REPORT
ORIGINALLY ISSUED
March 1944 as
Advance Confidential Report L4C24

ON TIE FIOW OF A COMPRESSIBLE FLUID
BY TE EODOGBOAP METOD
I UNIFICATION AND EXTENSION OF PRESENT-DAT RESULTS
By I. Z. Garrick and Carl Kaplan

Langley Memorial Aeronautical Laboratory
Langley Field, Va.







1N* ACA -A
..ACA. .


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.


L 127


DOCUMENTS DEPARTMENT


L-/ 27


I









































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hllp: www.archi ve.or details onllowolcompressOOlang




2. 0:I3o7 I
.. i 3 (1 7



NATIOTHAL ADVISORY COM'.TTTTEE FOR A'.i>,IVyTCS

ADVA'"CE COTFIDE'T'TI.L RPORT '0. L4C"4

ON TTT FLOW OF A Ci'.,FPRES-IBLE FLUTD

BY rE7i !IOr'lODiAPH ":ETHOD

I UNIFICATION !'D X.T;.rOn'! OF PEE3T:T-DAY RECLTS

By I. E. E .rriick and ,arl K.aplan


SU:T AF'P


Elementary basic solutions of the e-rqations of
motion of a compressible fluid in the hocr-.ra-ph variables
are developed and used to pr.:viie a basis for co'-:-:.ris?,n,
in the form of velocity correction formulas, of corre-
sp.ndinri com.irressible and inccm'n'-essible flows. The
known aprjximate results of Chaoni-.:.., von Iairr-a:n and
Tsien, Temple and Yarwocd, ar.L Pran-dtl and Glauert are
unified; by means of the ani."--is of the present paper,
T'o new types of :lp:.rx: nations, obtained fro'"; the basic
solutions, are introduced; they possess certain desirable
features of the other apFr;.iz.~tions and ap.par prefera-
ble as a basis for extrapolation into the range of high
stream :'ech numbers and larg. disturbances to the main
stream. Tables and firures giving velocity and pressure-
coefficient correction factors are included in order to
facilitate the practical application of the results.


T F ,- I CTI 0.::


The present aperr is coiererned with a theoretical
study of the hydrocl-yna.ilcal equations of a perfect com-
pressible fluid in two diiensi.ns, in which the so-called
hodograpr variables are used as the iniep6ndent variables.
Tt is hoped to achieve herein a unification of the
present-day results obtained in this fleld and also to
provide a working basig for fiirther develop;meits. The
earliest contributors to the hcd,,graph r eth-d for treating
compressible fluids were iolenbiroe!-: (reference 1) and
Chaplygin (reference 2). The. remarkable w'or of
Chaplygin on gas jets appeared in Russian in 1904 but re-
mained relatively unnoticed. In recent years contribu-
tions to the hodograph method have been nade chiefly by









?;ACA ACR 'o. 1,4CZ4


Demrt. er-n::o referencee ), von :Arr-an referencee ) 'ien
(ref.-.rr ince ), ingplj l: (reference S and .'- rple asrl
Yar'ijod (re'l'erence 7 ,

Ther c.ief reason, and perh?; s ti-e ornl rec-orn, f'r
pr:ferr.n' t',e iiod 7.c: :ph 'voiab ,. s to the ;. h':?icril pllrne
coordinates is that the relations of motion in the
hocdopraph variables are linear. T'i. r i.:nl' fiMc.' tn is
schlIe'reri, h"'v ,'e 2; at the cost .f j*-,,rF- li' ou l] t our'darT
conditions and nt a l].?rs ?7 rr.ny cal ir."i: .t. 1'The reat
simplificatir-n in th- r:athe: ratis '1e to li cai r,- rnever-
theler brakes it 'desirr.ble to ;n.sue Li-s lMne u.f attack
as lo.rin as it a': ears I r3.' t ab to r: so.

The miqt'.e:..It.cs for her.dli.. : o" -c :.ti 3c re-
ceived a sibstant.r al ir et's. :-,y L'e .:or': er. rr, d
Celbart (reference 8), v".:) Cev3;l.;: I l. ..cti.c theory
aralo: a us Eo or inuary tnl2:ric a^i tfor. t1-lnr^-v. Thl
present paper utilize? the .: -,;:.. :,f thi" rev f ncti..n
tlheor- to .Ic VFlop cert'#.in ',.('. ti)..r. s csrent .1 t tl e
rc. r'C3ssid]C-fl:-.) prnible. It is D,." I. ist ",ic.l fl.tC1PSt
that Ideas si;:.i:tl t. t.:D ;oe of 'r..s a i 'I.elW ; t .r re (::-
plcred by the r:r ov:r-n at'. rr-atilcin a:" l:, rt (rr.-fcrr.nc< 9)
ir. the earl-- g: rt oCf t"is 3 nt rjtj but do not on:,'P.ar to
have b-: En fu1rthe- r c irl'-, c 's at I:E ti -.

The mrLteri l to be t'en.ted i.s )r.- nieri:,.r1 1 srp. rated
Into tvwo. parts I I, the present 'ai-r -, l--sc
particular Pcluti.onrs of th'- hodc:?raph flow r :.;1ti.cn are
d-veloped djpd eFnploy.)I in u'm ni f.. r a .' e":: i, t C l1 e re-
sults obtained by Chaplygrin, von I- 'rm-an, '' T':ri-le FaL.r
Yarw.ood. T-:e r,'sults :.lbtit rned in r art I aI 'f I t e!.t te
prctlcSal applications Fr.ni -ir c- rien in th-- f'-r:n -' t.lfe.]-,
anrd r;re, hs of v-lacit r ar. pre ss:' re-r'-effic i nt c'lrrrc'cticn
factor's. Tn nart II, r'hic. will aoner r Intrr, g'cre'-Fr
partictlsar solutions ofI tn hd-chraphl. Plc c-: .' t.r:.s are
develojjped a:'r. discusCsd. The -.lat.ria- in .art 71', .
Is hoo) will l ad to a crtl.od for n.:-idlin'- t !-,n actun,.
b'.undary, problem of thr flo- of a co.- ?res? i3e ." flid pest
a prescribcc boi".




Flo-" Equations of an Incoirpresil.'le 2 .ull

It is well known that t i r'ela rtior.,s bett..een the
velocity potential 10 and tr.e st;.. functIon for

C C:IDE'.In IrL


C OTTT'EETTITAL








JIACA ACE Ho. L4C4 CCO iFIDEIjIAL 3


the steady irrotational two-dimensional motion of a
perfect incompressible fluid are



O x y I
(1)




These equations are the Cauchy-Riemann equations and
therefore + i' is an analytic function f(z) of the
complex variable z = x + iy.

The coriale:. velo-rity or reflected velocity vector
u iv is obtainr-d from the cmple.:x potential f(z) by
differEntiation. Tiis,


u iv -
dz

= qe-i


= -i(++ilogs) (q)


where q is the rapnitude of the velocity vector and 6
is the angle the vector .akes with the positive direction
of the x-axis.

The variables 8 and q are sornetime. referred to
as "the hodograph variables." The flow equations in the
variables 0 and q can be readily derived by intro-
ducing 0 + i lo3 q as the independe.it complex v.rialble
in place of x + iy. Then, in analoo7. with equation (1),




> (3)

6 log q 6l
='


C OiFIDENTIAL








!:ACA ACC:, Io.. I.4C24


or


Sq
39 qTq
1> (4)



'TIzSe eq:mtlniir ar., kn:r'n ss the ii,-,ceraph equations for
ti.n floyj of an incomirpraibhle fluid.


Flov' 7T;uatlnns of a. Ccrnpress.t.blp Fluid

T~'h' r-tintion.ns orrc,: ?-r.di to equation (1) ar-e, for
a c o res: r. le "1lul1.,

-7,, F 1, C
2AA iP
(5)




where p 1 t dF.Asity of tiFe fluI.T& at any n'Ijrt (,y)
and p i a contain er.- it?, which for con'cr.iencr is
referred to ar -tage i -In roint.

A s'.ort way- to r-lerive the hciJo-reph ecuaticns -'fr a
copres-ihle :luid, ttr.'uterl to 'l"snhrock, is as fol-
lo!s :

According to equation (5), vith u = and v=-
c o7

do + I -- = (u dx + v dy) + i(-v ,ix + i dy)

(u Iv)(d- + i dy)

= qe"3 dz

or

dz = a e(d + i di (6)

CO!' IDENTIAL


CONr- I DE'; TI r.









NACA ACR !o. L4C24 CC,'EDE"TTAL 5


It follows fro'r equation (.3), by considering and q
as independent variables, that

_z 1 18 Po 64;)
-- 1 = --e + 1 Po
LZ e o6Q + P) 6e
6e q Tp Te

and

z 1 iee6$ Po 6"J
q q q P 6q

Then, by assuminr that p is a function of only q
(equivalent to a-s-rLming that the preSsure is a function
of only, the density),

62z i 1 6 r (,./Pq) 2X1 i 62 Po 6 \
.. = + + + i 8 t
e + --- +- e
6q 6 [q2 66 Cq \q c9 P 6q 66,

and

62 i Lf Po 1 ei 62Po 2
S- e + 1 + e1 + 1 e -'
6,e q q P q/ q e 6q P 693 ,q

Since, by continuity, thee two expressions are identical,
it follows tl.at

i e + % + e ) ei -- +
q \q P 6 r 69 .jq 69

Hence, by equating rep.l and irna.irary parts,

p0q
69 o cq
>(7)
.5q _)
6 q dq J
-


CO:IF I DET IAL









m CO!TID,.TIAL 'CL, .-CR -:D. L4C'4


'".here are trhe hodi rraph eq'upi ti nns, first bt,- ne 1 by
".-leint:'cek, for thc f.ir; of r cm'.x ess..ible r ic' al... '. E
rdecenr.en t nf tir fr... of -the p-t SsLll-den': ty re] ttlon.
Tt is o1'se.rC tiat, \iwhni. = po Conirtant, eqlua-
tions r.') r-ur.;ce tc e iu ti:nl (4). Equat i-.,r-r in
contra w it i e L- q lol r. (5) sra lin A? ir- th-.r C ';. .Ei nt
variab.lr s.


Ernroulli's rq'a ticn ri :.r Lqu&ti n ?,f ate

In th- ;n-esent Fre.t .c. *. "-e l :. I F lits tcr a col11 etion
of form.: las and dc' Iniinr ton' .r ne 2 -';1 i. tl anr al'sis

Bc -roulli s c'quatl r f: ." ca r. re' .iblr fluidc is

p




v'-whicre

p static re 's1.ure I'n fluid

pn static pre!v.r:re at sti~narstior. pA in.t (q = 0)

p density of fluid

q rnr.nitucde 3~ velncity of fluid

Th- adinbati' relit.icn '-et"-e: tne tLcr i-- re ar;: t "e
densit-y "s


.- (9)


where

Y adiabat'.c irndx (appro-x. 1.4 fo- c-ir)

PO dersit- of fluid at stcr.nation point (q = 0)

The local velocity of sound a is oatsired from


dp
CG:FIDENTII.L









TACA ACE IHo. L4024 CC :TTLE!TTAL 7


For thb adiabntic care,

a2 = y (10)
P

Fr'-i Bernoulli's eqi.astion (8) and from equations (9)
and (10), the following relations may be obtained:


2 .2 1 2
a = a j (y l)q



1 1

L .

P : Y-1-


(11)


Fro:1i equ.ticns ( .I), for > 1, a r''::i-':ur v' i'city
q = qm is obh tr.ed for' th lin'ti;; c nilt n
p = p a = '. Th


qr ".'- ao


P a.)-


/-)


wI. ere


1
C -


.he flndai :er.tal r.on-lir e! crioir, l -pc-ed
genr-.al, is q/a', bit It IT find .i-eful
to employ a nondi.r-.e.niicn-tl sp>' d vsri~Fble

T = q2z /

For 1 > 1, t'.i- rar'.a cf t!-L -r/a.ia'bl T


v;r -i-bl iiT
in che anDl.iysis
T r-dfine:: as


iS 0 T < 1.


The v lsue T = D has u d'.al remain T -- C in the case


C. TTD: 9rL~L


wh- Ir c ,q
(q _- D) .


s '1,F :JF:1CCjt'~ rJ~ S~:~'J1~ .~.t; Sl;?cll~.;:jCI~ ;1711:t








8 COi Dr TT...L -;. A AC1R 'T. I. 1'


of a compress ible fluf.' corres!onds t") a sp;rnat'on point
(q = 0), or T -= n imai re:n the limiting case of Fn in-
cor..pressible fluid (Rao = .

.V.th the defini t'.os of T and 3, e.qutio ns (11)
become

1/2
a &,(l T)

p = p(l1 T) (14)

P = P)(1 T)3+ I


'.P loc9l T1-!, numl.er 'I= q,/a mn- b:e exr,: r ted in
terms of th-t rceed1 vai-iale T In the :ollovir.g way:


q2 m2 '-
0 2 2 2


2 T
1 T

or, by sclviinr fr T ..n ters of '1

,.2
T =: (16)
2 + 2


The value of T 'for vich the local velocity of
the fluid equals the local veocity of round (; = 1) is
given by


T = (17)
1 + 1


In the -c. e of uniform flo..,r past a fixed boundary,
the pressure coefficient is defined as

P PI
Cp,1!1
1 1
C PlD l
C O'I' IDE-'TTI1.L









I'ACA ACR To. L4C24


wh1re the subscript 1 refers to tihe undisturbed stream.
The pressure coefficient for the incompressible case
(TT = ) is


Cp,o = 2 (18a)


The pressure coefficient for the cc-:pressible case is


pM1 -1 + 1+ (y -1)2 :1
1 -Mi 7 1 2




(or q = q) (+sonic(),
r Y1
S2 +2 ( ,1)'2 +-
(CpM, + -- J (18c)


For q = qm (vc.C'.r.) ,

CP' 1. = Y- (l:d)



Ba.lc Soluti ~ns o .TodoYraFph Equatir'on

Consider the inconrpresible car? rer--Fr,-ted by
equ.tions (.3) or (4). It is clear tl.at e = 1 id
, = l3fg q Fatiffy thefce eq-uati'os. 7n fact, an, c.n-
verr'ent power series i4. w = 9 + i l~'? q rIe.PF resent? fan
ar.alytic fur:ction of v.'hich the real ans! imaj.inar': prts
satisfy equations (.) or Ii). 'e cla f s lytic
functions in w (and' the concept of ail'tie conrtinr'al-
tion) then yields all t-he particular olu-lionr.s Dr these
equations.

The particular solution w = 9 + 1 loq q can be
obtained by: reans of an inte-raticn that is instru cti-.v
in the generalization to tie compresible Case. It IF
well known that


CO: IDE;T I AL


CO:'0IDEITIAL









10 CO'"h'TDE P'TIL InC,' ,C? 1:o. L4C24


'(w) = f(w) dw

ccn ",e repre senFEd as the .-..m of t';.'o line irte ~ rel

+ +
F(w) = ,(P 0 d 1:- q) + i (2 dl + P *ll -i;)

r U re

f(w) = F + i4

Thus, -iven a pair r uin' .'nc Q t:!lia s.ati:fy
equac' -cri, 13) or (4), i vioe..- T ldr ri nai
of solutions. narr :ly, e r' ?.1 n:.6 '..e n ra r'" rarts
oJ' (w) Fbir e:xa :,-'. P = 1 arnd = C,

F(w) = w = + 1.- q (19)

Arair, if P = 0 an.l = 1,

?(fa) = lu' = -lor- q + i0 (20)

The phyf ican intrri.'ret, tlo.. of crqutr il r, (10) andi (''),
coriI--dered s f"lo oattternc;, is o' so!:e itr..; 1 r 1wt In corn-
nection wit h later-' d ?veldp( rts It i. cFr that
equations (19) and (,C0) repres-nt a v.:'rte:: :.:.d a o,'lrnc
located at the orsirin, respectively.

rhe ;}rcnralizatl3r to the coi nreos'l a e:1'as. f the
fore- irn. cfEnlment.arv rc-sults vas acco-_rlisr. lied b:. r.er?
and "Gelbart refer ; ence 8) by mea.s of si.nlrF ; t fertile
ideas. Tere and Celbart trcat equations .' th'e for:!



le -
(21)

cq ,- c .

and sh:ov, as is readily verified that, if P anr Q are
a pair of solutions, the real san imarairary arts of the
followinr.f sun of line rinteirals


COITF DEiNT I.L









HACA ACR No. L4C24 COUFIDL'ETIAL 11



P dO \2(q) Q dq] + 1 d9 + 1 P dq (22)


are also solutions of equations (21).

In particular, corres-ponOin, to the pnir of solu-
tions P = 1 and = O, there is obtained

w= e +.l/ d (3)


and, for P = 0 and Q = 1,

~rq )
S= i + I/X2(q) dq (24)


By repeated application of the process of intei 'ution,
indicated by expression (22), a renerral set of partic.llar
solutions of equations (21) ma7 be obtained. These
particular solutions are discussed in yart II; in the
present paper, only the solutions given by equations (23)
and (24) are needed.

The general hodorraph equations (7) are of the form
of equations (21) with

poq
P)

and

Sd(p,/pji)
>2(q) = -q
d,

For the rest of this paper, thi adiabatic pressure-density
relation (9) is used. 3:- reans of equations (9) and (14)
and the relation

dp_ P -2
dq q


CO"FIDZ.rTIAL









14 CO: FIDE!'TT L : CnC CCR i'o. L4C%4


and

1 l 1 GT dT
I(T) = 2 1 -
t' |_(1 T)' '


1 1 1
= +
(1 T)5 (1 T)3/2

S1 + (1 11/
+ 1- 1 (29)
(1 T)1/2 lop


Table 1 contains values of f(T) and -'(T, and fig-
ure l(a) shov's these fui.ctions p.lotted flga-inst T. Ob-
serve that f(T) and F(T) are vell-behavcd functions
in the range 0 T < 1. In fir-ure 1(b), tli-se functions
are clotted against the local Marh n'mnber '. in the
practical spel;ed range.

Other interesting cases for which the functionss f(T)
and g(T) can be expressed in closed icrms. are y = m
y = 2, Y = 5/2, and y = -1. For y = c ( C = C,
a = z, inconprescible case),

f(T) = C(T) = 0


For Y = 2 (W = 1),

1
f(T') = T

1 1
F(T)'- -T lo (1 T)


JFhr Y (= = 2:
1 2
f(T) = -T + -- T
4

1 1 1 + T 1
C(T) = (1 -T)2 (1 T)


C IDE!; T I n L









FACA ACR iT.. L4C24 CJ. FTDEjTIAL 13


and it is observed that the functions f(T) and g(T)
vanish for T = 0.

Equations (23) and (24) can be written in the form

W = 9 + iL

and

i" = i(9 + if)

It is important to note that, in the rin.c:r-uressible case,
W and i, reduce to w and iw, sl:i-u-e L and L re-
duce to log q. Thus, there are -n the compressible
case two basic functi :..' L .ani L co2ri F pcr.dlin to the
one function log q in th'- incinpressible case. It is
of interest to rent ion that the functions a:' d 1'.'
considf-red as flow pat':erns in a .'c!-pressible fluid, can
arain be interpreted as a vortex and a source.


Evaluation of 'unctions f(T) and ,(T)

for Varic.is V',luo3s of

In -eY.er l, i.Ye intefra's in quati-.n (' .) iard ('7)
rrpresentini- th- f'uncr: ions f(T) an.d f(r) are r'.r'*-
sible -i... i nri itrlt : 'eries. FP r tc impio tant eca- r f air,
or'ever, vwitlh the adia'-batic irde': y p ut eqiw-: to 1.4
instead of thie usual vaIlr: 1.40', these fu~nctilC.ns c.n ieC
obtained in close.i f r1. Thus, .ith = .",


f(T) r= .1 rT) -' 1


S15/2 1 1
= ( T 1 + (] )"'


S(1 T) 1 3)


C G,:FIDENTI .L









14 CO:FIDE'iTTd.L 1&Cj4. .C.CR 'o. L4C'4


and

1 1 T dT
(T) = (I-- T)


1 1 1
7777 2(l737
(1 T)5/2 (1 T)3/
(1 1 + T
(1 -


Table 1 contains values of f(T) and p(T), and fig-
ure l(a) shows these functions plotted asrainst T. Ob-
serve that f(T) and g(T) are \.,ell-tehavcd functions
in the ran-rc 0 T < 1. In figure Ib), these functions
are plotted against the local MIach nwmiber : in the
practical spe-ed rarine.

theirr interesting cases for v.hic'r the functions" f(T)
and C(T) can be expressed in closed former are y- = c,
y = 2, y = 5/2, and y = -1. For y = C ( 5 = C,
a = .-, inconpres-ible case),

f(T) = r,(T) = 0


For y = 2 (p = 1),

1
f(T) =-T


(T)' 1 T lo (1 T)

3
For y = : (3 = 2'
1 2
f(T) = -T + 4 T
4

1 1 1+ T 1
r(T) = 2 (1 T)2 1 log (1 T)


CO.,. IDE TTI..L









LJACA ACR No. L4C24


For y = -1 ( = ,

1 + (1 T)1/2
f(T) = g(T) = log 2


For the isothermal case Y=l (p = o), the velocity
of sound a = ao = Constant and the fuLctions f and g
are obtained as infinite series in the ratio q/ao. Thus,
in the limit p--->,


q/ao


(q/a! \ Lim 1I
f(q/ao) = --


1o2f
-u 7-
7 er


n-
n=l


2 o2


9 -

q/ao


S d(q/a)
Sq/ao


n (q2/a n
2n+l n ni


t'q/ac

J-


g(q/ao)


Lir. 1
l-r>- 2


1
t2
<-


'a 1-


1 a 2


2
= 1 e


+z
+nl
n=1


(q2/a2 )n
n+l n n !


COTTFIDE',TTTAL


anrl


- 1


d (q/a)
q/Ao


- 1J


q /a.


C 1'i I DEiTTIAL










16 CO:.'IDE '' .L ,;7A C' '. L4C 4


'ol- arbitrary: values -f Y tor ) th- e-: ,rl Soios
for '(T) andd ), ittaihed vf t .e aid c'o t.hE c1-
nr7.ial exp'inrion, are


f'T) = 2 2___ (.-)' ( .)-r
11



= T- '- )T ...
n=i






11 v,





ane,
1 .- .
r(T) = -_ ,

n -- -



4 T


Sc I q-r










4 2



anK does not involve xplititl; t"e aditatic index
This cirmcumstr.cne rirderll e th e po'e-:ent-c'ay LpnroxJmrate
ncthod;: for obtainir.n velcrcity and prezr.ure-"cefficient
correction factirea; in the 4fo]lolvirr sectit.o1, tlis point
i? brought out more clearly.

CCOTPTDF:T T..L









NACA ACR I, ,. L4C24


Application of Paric Functions L and L

In this section, the basic functions L and L are
employed to set up relations between velocities in
"correspondinrg, cor.pressible and Incomrpressible flows.
These relations are o2 the nature of "stretching factors"
or velocity correction formulas and contain the results
of Chaljlyin, von Karman, Temple and Yarwood, and Glauert
and Praindtl. It is important to recognize at the outset
that no single velocitW correction formula can represent
in an exact way the correspondence of flow patterns past
a prescribed body in a comp-essible and an I incompressible
fluid. A single velocity correction formula is actually
feasible in only two cases: (1) The stream ;ach number
is small (even tho',.c-h the disturbance to lthi main stream
due to the presence of the body may be lar.~).c) so that the
compressible-flow pattern differs only slightly front the
incomnprssible-flow ,pattern or (2) the disturbance to the
main stream is varis'hi-.rly small (even tho.ir the stream
Mach number i:ay be .i'-h) so that the effect of the shape
of the solid boun..l;nr is small. The various velocity
correction formrular discussed in the present paper differ
essentially only in the degree to v-hich the requirements
of these two cases are satLified. Despite their limi-
tations, single velocity correction formulas are extra-
polated, in view of the lack of more rigorous solutions,
into the ran-e of large disturbances to the main stream
and h'igh .Tach numbers. This extrapolation can be
justified by further theoretical investIratlins and by
comparison with experimental results.

Consider again the corresponding pairs of functions

w= 9 + log q
.(51)
; = + iL |

and

iw = i(a + i lor q)

i. = 1(9 + iL) J


It has previously been note that the pairs of functions
in equations (31) and (32) denote respectively a vortex
and a source in an incorpressible and a compressible


COUJFIDE1TTIAL


COIF IDEI'T AL









LI.CA ACR 'To. LC124


fluid. Ealch pair cf function- can be nemplo:ed to define
a correcrpendcnce of fl.):w pat-terns in vhichL ccrespor.diniF
nnints are identI..jed bl the same values (4,1). Thus,
in the case of the vortex (equations (31)),


'/ = '4 = 9


ri = \fe -= log qi = L


where the 3s .bscripts i and c refer to the incorprc-n-
sible and to the cor'r.:'e-sibll case, respectuLvely. It
follows that

T
qi eL


(T)
= qe-


(33)


Similai'ly, in the ca-,. of the sjrcFe (Equationn (32)),





1 = = e
1' L


and


q. = eL


= e(T)
p0


(34)


tt the end of the prenedin'- recti.'n it vas pointed out
that, to a fir-:t ar-proximaticn, the functions f(T) and
g(T) are equal. Thih fact implies thai:, to a first
appro:-l-atior, i F'-in 'le velocit.r correction formula is
feasible. The assu.ioption is nor:.' ade that either equa-
tion (33) or equation (34) can be ad-pted to provide a
correspondence of flo-w psttyrns in the case of uniform
flow past a bcdv in an incnompressible and a compressible


CO'TIDE'NTIAL


CuF: .IDEFTIAL








;'ACA ACR 7o. L4C24


fluid. VWith the undisturbed streams as convenient refer-
e'nce., the following nondlimricnsional forms of equations (33)
and (34) can be written:

f (T)
i (35)
(J). = JL EI 55

an.


e= ((T) (36)


where the subscript 1 refers to the iu-ndistur'ed stream.
The use of the 'rd;st'rbed stream as reference in the
nondimensional form of the velocity correction formula
vias introduced by I'sien in rfere:,ce 5, where also the
details of che von l:r..n apiroriEmation are developed.
It is shown in the following section that either of
e.-uation.: (35) or (.6) containii the results of Chaplyrin,
von K.ar.:prn, and Terrple ,'iLd 'ar.,ooo. As -has been previ-
ously pc.inted out, t}-e concept of a slinrle velocity cor-
rection for.niuln is feasible in only two cases, namely,
small stream l-'ach numbers anl[ vani-ri.arn *: y small disturb-
anoes to the r-ain stream. It is desirable their, to reek
a single velocity correction formula thut combines the
features of th-ese v.v'o cases. From this point of v:".e.,
equation (35) or equation (36) is not the best choice.
A better choice cf a sinrrle velocity ,jDrrection formula
apriears to be the fr.llownr. coil,ti nation -,f equation (35)
and (356), bazed. on the ar-ithn1eti. :-een :i f.T) and g(T):

-fl (T)+F(T)]
2 -

ql/i 1/c f(TI) 1e


In a later sectio'-, still es a.il.er conrbination refer-
red to as "th.e geo'metric-mean type of approximation" is
introduced; in the section djalin,- with the G]auert-Prandtl
approximation, certain 'eatiires of the forero lnr arithrmetic-
mean trype of Eppro;:ir',atio'n oan Of the reometric-mean type
are discussed.


fc (~TWTLIlPPTAL


COiFILEI'TIAL








20 CcITIDLE;T. ..ACA ,C. ,'o. 1402


4t this point it 's desirable to discius- the crac'icRl
application of equation (.27). According to equation (16),

I..%
T = --
213 + :2


f1
1 2 + M11

and

Ic 1 )

./1/2
/ O'i3 + :II
=- ---(33)
1 3 + :',

Equatior (37) then. yield, for a givenn set of vsl-.c-e of
the stream i.ach numberr I: and the local M.ach number I,
a value for the ratio (q/'l)i of the local velocity q
and the stream velo"l' qi in an incor:pressible fluid.
Table 2 shows correspondin', ,alues of (q/q ) and
(q/ql)! for various values of the stream Mach nmriber i11
with e = 1.4 (. = 2..$). ?lis tabulation is performed,
for the rpe f c', f the urpe f c: te three cases repre-
sented by equations (3.), (~6), and (37). Values of

q/qi, (n )0q1; c and 0q \) obtained from equa-
(ql01, "l)ip
tions (37) and (38), are plotted against t'-ie local i'ach
nu Fiber iM in figure 2 for various values of the stream
.ach ni-.tbcr Ml. Table 2 alco shows values of the
pressure coefficier.t Cp,o and C,., calculated by
equations (13a) and (18b) for these corresponding values
Df (q/q1 snd (q/q) c Figure 3 ivshows the curves
of pressure cnefficients rnrresr-anding t t he curves of
velocities a.' figure 2. Use<:l aross plots of the
curves inr figure 3 are shc-'n .n fi-ure 4, in which Cp,M!
ij plotted ag!ir st "1 for vkrrious values of C-p,. Tr

CO:.PTDE.I IIAL








IFACA ACR" I4. L4C2'i


addition, curves are shown in f "ire 4 for (C, ) and
and (Cp,,I ,T calculated by equations (18c) and (1~1),
respectively, The curve for (Cp, :)s corresc:.i. to
the sonic value 1! = 1 or T = Ts = 1/6 and in effect
divides the region of flow into a sut-.nic and a super-
scr-ic part. The curve of (C, I.- corresponds to the
ra:xi;urw! v.lue ;! = o or T = 1 and represents the outer
limit of the s aer. nic rr-:ionI (or a perfect vacuum).
In order to exht'bit the main diifferences between the
various correction formulas (.5), ( rr), Can. ('7), the
ratios of the sonic values Cp ,l\ n t corr-rnn
in- incompres-sible values C ,o are plott,:. a"-,irist the
streak Mach nlun~er T in f:IJu'e 5o

Observe in fi~'urr. 2 that the (q/q curvesrs have
raxrinu- points. Ti's fact means that the value of
(q/q c associated with a value of /qi is not
unique. Anal:rically, the cr-'.terion for l!-i maximum
point is equivalent to


I (.:3)
dT

or, frc:o velocity correcticn formula (7),


(1 T)I' (Cj + 1)T + 1 = ,


For 2 = 2.5 this equation h>s cnly cne pcsltive roct,
T= 5/24 or = 1.15. It Is irnt'restfrr tD note that
velocity correction fcrrmula (36) yields as the crrter'ion
for the maxirm? pgint

1 (29 + 1)T = 0

1
The root of this eq'ntion is T = Ts = 2 1 ar., fcr
S= 2.5, is T = / or = 1. Velocity .-rrection
formula (5I) yields no n3;i"nui value of T cr :'


CO.r TDE!TTIL


C.'TFT DE,'TIAL







2? CONFIDENTIAL NAC. aCJ- :Jo. L+U2LL


'..eaning cai be given to t'h value T = 1/6 (N = 1)
in I1- ca.7s of equationn (3M) :tl. rcfcrei to the origi-
nal ntL-r.-v:-tation o' the flow -attern ;s tl- ;-t of a
source. It can be jilovr. thpt t'_e acceleration
(iq along a stretsline is inrfinitc at all points

for which the loccil Nuich number i. unity ('r = 1/6) cnd
:h,:-t f_ flo--.- dis.ontir Litv e-::ists th;r: In the case
of the. ,,-ortx flow:: c.rter., (tquati- (qu)t), no f! low
jisc-ntin-uity occurs for" M < co. The velnoity ccrrec-
tibn fcrm.ui: -. ( ') su ,e-sts "l-.niting" v'aluc ;.. 1.15
fcr a Lpi'.al. 1l''., .,,i .: equati...n (-'1) is anR 1lo cus
to ci ,onijtit:n i' inf. :nit- .c -: lel .tin. T.iu the
:xistcnc.; of' a .i:: e : "bs'-r2: .i' s.iup rscn':' r9 .i:fn of
lo.' ..ith ii :.-onL'nuitl,.- 1.: in i tL linc- the
ocrrr:ne of t','s llviiti:i7 vl!e If ." 1 .3 a cornequenco
cOL' t .si. orm -urd for -ic v-locit'y corrc tion
1,crm'31., -o und cl I.IAifi:."T 3 h L-c1 t. .i~ ac'vi'. tj any
partleuJ 2:, 1.'1 i-, tlhe ; r -'*. it -ire-.


r:.- Chsaprly-in Ap,"o.vi,.=. ion

From tl'e o-'nt of vie,: f tre p,'Oent ,-cOcr,
Chla--..in' s approzination 21o- ',soni, seeds as e ate
sil.'r aid luci. forimn. C!-iaplyrin inz:'oduces In lac3
cf q v nVw i..: e .1'nt .3c-.ri '- r'i.=.,le i equivalent
to the qua .tit" -.e. m n ct.hc -i.i.Lt-h :id .ile ..o'f eqa--
ti)n- ,53.), rar..c..,


The hcdor. ph fl ni.n.: (7' th r EuI t- frja
The hidor. ph flm: equali: n.: (7 t h n aSur.. thc f orml


= 0 i
> (.0)




rhe e

F(T) = 1 (2 + 1)T
(1 7)20?+1

2 2
= 1 P;2 + 1T p(2 + 1) (2 + 2)T ..


CONFIDENTIAL


11-I









ITACA ACR No. L4C24 CO!FIDE'TIAL 23


Values of the function '(T), for several vilies of
(or p), are given In table 3 a-.d are plotted In figure G
against the nocal I'ach number !. Chaply-gin noted that,
in the case of air (3 = 2.5), F(T) differs but little
from unity over about one-half the subsonic range
0 T / 1/6. His appro::xiati.nr in the renre 2f low sub-
sonic speeds consists in replecting- power-s of T hIgher
than the first or in replacing F(T) by unity. Equa-
tions (40) can then be written in ti.e Cauchy-.iemann
form


b8 6 log -


c log Ti >19

and 9 + i/ therci'eore. is .n analytic f'un-ction of the com-
plex variable r + 1 logr. Char.llygin's approximation
thus leads to LIt 'rFlocl.t, co)rrectiikn formula


1- T
4c (41)



where powers of T higher than the first are neglected
throughout. The -i3e of equation (.14) instead of equa-
tion (Z.3) also leads to this result to the same order of
approximation.

i I
The von !arnran Approximation

'Vo. Ernan's anpproxir ation corresponds to the case
S= -1 or 4 = -=). It follows at once frcm the
inteFral exp res-ions for f(T) and g(T) piven by equa-
tions (25) and (27), respectively, that for this case


1 + (1 T)
f(T) = f(T) = log
2


CO I FIDE7,1TIAL








TIACA ACR :Io. L4C24


or, with the use of equation (16),

f() = () = lo 1 + .2 /2


This function, plotted against rI, is included in fig-
ure (b). Corresponding to equations (25) and (ZG),
there Is a single eqation


(--
Vq41


' + T- 1 1/2
S+ ( T )1/2


Replacing T by
to equation (16)


T (ie1
ielid-


:41-
and T1 by -, according
M1 1


1+ ( 1 )


1/ e
k(q^


ql


- r2)11/2


Then, by solving~ for (/qll c
the stream TTach nniber MI,



l ) \ ,l
=i~ (t)1


1 (42)
+ 1 MI + 4 2/2
L ,3


in terms of' (q/qli and




I ____ (43)


S ./ a


where


+1=-
CG+ I 1 ) _JTA
COIFIDZ:TIAL


COFFIDEITITAL


= ~-/)


1 I 1
91A








;IACA ACR Io. L4C24


The pre-cure cceffiicien-t Cp ;l, e.pre .sed In terns
of the inconipressile I se nfessre cocffticlent Cp P, is
easily obtained fror. the general formula (18b) by, putting
{ = -1 and nkrl infl, us of eq'i.ations (43) and (18a).
TIIus,


C.. 1 C yo
C,L;l = C (44)
(1 21/2 + '1
1 +/1



,b.serve that for this case ct:. fufncttrn F(T) intro-
duced hy Cnaplyrrin and given in e,'us tion (4'") is exactly
equal to unity. .Pr'o:, the noint o' view of the present
paper t.hn, "on ;rra-ani's ap;lroxi-at.ion ap;:eLrs to be
equivalent to that of C"haplyin, who si, r.'-iirates F(T)
by unrtv. Tt follo'::s 1-1.t the rr.gre nf validity cf
von iu arrman' approxinration and tirit of Chplyrin, in a
strict sense, coincide. ?urther-nore, it i? pointed out
that the von, l.Ar':I' appi'oxinrltion does net pernit a
super-oric rrr-ion. 7on Kgryr.5n~t choice of Y = -1 has
the advanta.re, h.jw-evr-r, of :rieldinr srlr le e::plicit ex-
orefssions for (qq in terms of (qi'ql) and for
C in terms of Cn, Several values of Cn,~rl
calculated by equation (44) are includ'.d in figure 4.
For the. purpose of comparison with tin- other approximations,
there i. plotted in figure 5 the ratio of (Cp,;.11)s to
Cp o against the stream "ach nunLmber '1 in the case of
von Karran's ap.,ro::inat-on. The values of Cpo are
obtained wit' the use of velocity correction formula (42)
for the local Mlach nmiLber I = 1, but the values of
(Cp s are :alculatpd ,lith y = 1.4.


The 'enple-Yarwood Approximation

The functions and r' related by the first-order
simultaneous equations (21) separately satisfy the second-
order equations


CO:T IDE.'i rT L


C FIT IDEI'T IAL








T .ACr ACR iNo. LC02L,


62-.
--+ (q
'2 0


1 ='1
1
:.2( )


2. ( 1.
N2 +-.( -- Xfflj


^ =0
- I
J.


In t r .i of the nondi 3 'L i ec' v .i-aj'. e -r and with
the val'L:S c \ '-i() Ia (> ) c :&I adiab2 tic Icas
given I ticri (2 ), t-' l: .: 1 tt.:I the for


+--
- T ,



1 1 (2b + 1)T V
T(1 T ) l -2


.- T)P+1 -
- L + ~ T
C TT7TT


STI


+ -
CTLI 7?


(46

= 0
I
)>(


For.r..l I tli ns of' th:-' eqm i-:c:zi .ie:- given by Cha-ply-in
in th-e frc f nf tw infr.iite 3S' z


m--1
v-22


Ir "l (T) sin(:.iP + c )



Et 1,(7) co3(n0 + EM)


where tli- fun tin.-nz '.(T ) .n. ,.(T )
h,-c rr.C:et:ic series ad B, -,' and
c" Ist s. I t


> (47)

I
i


are obtained from
.1 are arbitrary


A c'isa:1.vant-A-- cf the ii'... l s'lutin.n, as r:nl;arked
by Teh.I: X and Yarwooc', is tha t I: .ns-uitable for
". -- --- r -,
UlC.; I.' ..;n;.L>


(15)


COFTI',Iq II..L







TTACA ACE No. L4C24


n'l, cric2l ccmputvtion Iecatisev theF hvperreonrltrie functions
involved airc comrplicated and ae r. tt tabulated. Tc- ;.le
[iC .orv'od th-ere force looi:ed for approxi-mai.7on tVlat -re
cf 1 ct tic.l valve In calculati.rs of cicnuress blc .fLor s.
.,-- .rL'n ? o*" a sk:illf1ul ania-ly-'i they fI'nd s; ch a -,ro,::i-
T ti: .'11o ad -i '...'p d that th:e s l:-'es3 fornm for 'V lri
i a;, ,r the t'"re


,(T) [= r,(T)"m

.(T) = (jm ')

-(T) = 10 4 (T)



where (T) sad T(T nd e:-..- c t of t.- in,.rx :r, are

,. -. $\9
: -'- -(':')

Sinf. ica-ntly, .rom 'the o..int of vie'.r of the an~llysis
of the pre-r-nt :;aeper, the ."uncrtior:s r, and v, n.pp-roxi-
mated by 1 are no:-i otir thatr. the functiono,.
defin-ed on the r'ght-';anrd .siCEs of equEtilns (U.) alnd
(24). 4 "r approi')'ia' icn of ?er.i le arn Yarv.wooJ then
leals to thcl- saic velocriity: correctin/ ielatiocn a: v.as
obtain-dc. b.- r.eapn of CL.apli'nn's sapr.xina*ion (equa-
tion (41)).

The velocit;: and piress.ure-ccfficient correction
formulas obtained by TIer:eple and Y'a.rwood arFe more involved
than the e.-rlicit ex.nrssior.s (47.) and (44) obtnr.ed bl.
von KIrr.n. ReplaclnfC in equation (41) by
Tia t) thus yields


1 T1
\ 1c 1 1 ^ (50)
4i 9 -- TI


CCOF TD TITAL


CO:IFI DF:!TIAL

















f+ 'C.
28 CCFPT:Di':TTAT MlACA nCR :t?. L4CL4







1

The r'luti.n o,? thii.; r.ti.- equi tion for ."1l i

1











and 0 e r Ctc trv'F ts' e1 C, r '
,--





urnde $ is re-.a"h" ci th cr equ's







ti .n ( ,i1 t.- lc It. c'Irrr-tio tom'~ '_.. (E .r.ds a
liiir alue 1 a c




resent er. it s of iter: t 1t cL r.rr
Ci 3/










f.'..thet .e:t stelcatec io L L.d I a, thto the tu-n.Aircl
Fpart-icu.: solut io:- e 'l-.is E 'rt ncion, w jc .li:e .
2and L reduces to lo1 q fcr T = t. i di 0! E.;.,, by,


,%


It is rerar.'-:e: that H(T) i3 el,.:]': relatL t aa
function Ks(T) em.lo;;e by I'n cpl. Eti ";,r".; (s-,"er-

enc_ 7) ir theF detr.'; rr.it.i. n .:!' tn-. ir i"-q',r.-oxr-. t ion.
In the iie.:t sectior., it :' il be ;-;-en thlt the, function


C ,O.IT'DtiDriTIAL








"ACA ACR i!:. L4C24


H(T) -lays an important role in connection with the
Fi2n.ltl-Glauert apj roximation.

Fr.o- equations (26) and (27),


dL = +- dq = (1 T)i q


a.r.d


d= 2dq = 1 (2 + 1)T d
(1 T) P+ q


Then,
1/ = \1/2
(dL d)1/2 =( dq =


and, from equation (52),


H(T) = 1o': q + h(T)

v.hc re


S 2 T J
S (2 + _! i,/
h (T) r-


1 (2p + 1)Trl/2 dq
L 1 T J q


The fu:ction h(T) can be ,',ain _d in a
any value of Y (or ,3) and 5 s


hT) = -lo
h(7) = -10m


closed form for

1

- (Ts-T1


l-T)12- 17 \, -')
1

2 (1 'T, 's


CC:;FID:7-.T: L


(55)


(54)


- C:
- -


(55a)


COTFIDEUT IAL








30 CITIIDEi'TIAL ILACA ACn 1o. L4C24

1
where Ts and where tii.s expression is valid
in the s.ubsinic rai.,e 0 T T .ith T replaced
by and 0 I 1, the exr'ession fo.r h(T)
2, + i
b comes

S+ 1/2 l-1 2 -r)1/2
h(T) = -1 --- og -,-
1-2 2.YT 1 Ts

S+ s 1 + \T-(1l i 2 1'/2
+ -- -- (5b)
21"I 1 + ,A


It 1. observed -th at, for the su.L'rsDrs-ic r-;g on
T T 1 cr 1 > 1, H(T) as ciefined b:. equation (52)
beconies a cor.plex function; but, 'fr present purpcsel,
orly the real f'incti3n of the 3lslu-onic ranje is utilized.
The function I(T) mray be utilized to obtain a
velocity correction fcrmula in the sari maranner as the
functions L(T) and ((T). Thur arnalc.ous to equa-
tion (35), (36), or (7),


(- < .I) (50)
q- h(TQ.

It is instructive to romnrare equation (5,S) vith the
approximation p-iver: by equation (37). Equation (37) may
be written as


(dL+d!)

1 cI _-(dL+df)
L 2 -1T=T


CO 0IID'T'ITAL









'ACA ACR IT-. L4C24 CO: i1TIDFITTAL 31

-.r, ceq.-ation (56) rr'a.. be written as


e/(dL dt)1/2

()i eL/(dL df)1/2



Thi.S, the .:'ve t he the rE-r.r, .r.ential is in one case the
cL +
integral of the arithliEtic mean and in the
other case the inte.'-'cl of the geometric mean
(dL dL)/"2. Table 1 shows values of the functions
f(T) + g(T)
f(T an() d h('T) in the case of air (y = 1.4,

p = 2.5, and T. = 1/6) anrl fir.ie- ? (a) and l(b) show
tiFse functions p lotted ar air.-t T and M, re'-!ec-
'Ively. Observe that ~e -- fu;.:tions, and consequently
the velocit:- correction fo.rulas (77) r.nd (r'), differ
only slightly in the subsonic ran;e 0 < 1 < 1. Fic-
u.re e.-ib. t.3 gr ph c.'e.ll cor:;.'r- ; ,f T he ': f, -.t:y
ccrr. ?i n a .' l ila (7. anr ( .) for" = 1. e
liritinrr 'ai _e n ,f ': ide'finr d : -. e qu t:i.-n I 'TI -is
- = 1 in t:, c-.e of r-quaticr (5C' a c :,rt 'red w- th
I : 1.1E in the raise .f -.q'..tion ('7 .


Co'nm prison of Frsisults *-'f' Ir,:' r, t P'- I -

v.'i t'I Prand .i-,l'.mi rt .* .., .1-. .m: -,i-n

The t/eli-k novn Pr'nc -t1-.lC ,., r ,," ::._ t i'n i?
based .n the ass urpti r .n f :. i. '. : .i sturb-
ances to tth main Streen. T' ..- -.; t .'-i :'. t
corre, "c i n -r '.r'_' v'.yr tbe e:. -. -





/ 1 ( .
C I
\ 1 /i

where q q, is vanisr iri.,y s'-. .'The left-hand side

CO FIDE) 1T.'









32 CO'TFDEWTIAL IIAC. ACBH I!o. L4C24


of this equation is actually- t.ei 6.lfferFnta'l coeffi-
d(q/qi)c
clcnt ev/ aluated at the main stream velccity
d(q./ql )i
q = ql (or T = Tj). An e:--act 'or:- -f the' Prrndtl-
Gla'iert approxiriti n then is


c
JT=T


1

1 ,7 )l


( S)


The clfferenriual coefficient in fquat'n (E) i" rowv
evaluate fo:r th- various appr.-'j.xi,rna': ns treated in the
present paper.

Fcr the a:'ith.~.t'cr -'Ce:'.n ar',rc':: t:;at ion f the
present caper ,iven b:;. equation (37) (y c r 3 arbi-
trary),


Fdq/ql)
I cl
d- T=-T1
L_


2






( -l21
2 1 +


1 + (l -


w'l1
1 +
C.I::


= 1 + + 4 16
i, +
I+ .

5 l1 3 + 4" + I
7 _-6 +
q* I-


CO:FI IDE: IAL


(59)








IACA ACE t.l. L4024 CONFIDENTIAL 3

For the Chaplygin or the 7U7-.ple-Yarwood arproxt-
mation given by equation (41) (y = 1.4 or p = 2.5),


d(q/q1) 1 7
i=T 1- -- T1
cd q/q 15

1 2
1 -20 1
11 -- 2


1 + + "1 0 + ,

For the von T:arman ap "rrximation given by equa-
tion (42) Y =-1 or p = -


S1 1/2
(1 T)
1 iJ T=T,

1
7-12 (61)
(1 :I^ )

For the re-oretric-mean approximation of the present
paper given by eluption (56) (y or 3 arbit-a-.),



d A
d 3 T Jrl

= ,-,- (621)

(1 -- )

Equvition (e) i irltep r, :. r)ff .-}e value of the
adiab'tl Jc -rl'ex *. irr 11 n -an a pr-:-
maticon C"'-- s rv ,-. t
yields tl'1 P: di *- li r -. .. '. erer t r
ari thml tic-rE an nL. roxir so.. ,'.: ,: ; F ,:; ,, tl-.I j:rt


cC 7 _,.- 1 17 .T









iLACA ACR :I?. L4C24


result insofar as terns inricrsive of Tl6 &re concerned.
The Chiarlyuin or the TrmplC-Yars'ood aproxir:n.atiori con- o
tains the Frandtl-Glaucrt result only insofar as the fl"-
termr is cDonce.rrnr.


RE '-E :..ID C i'j: CL.;LI R_-. .Tl""


1. "asic CeI .rCe tE ry' sColut :ns .:f' thll : -, ..:t-ra:h equa-
tion. have 1.een FrI.r-lo",rd to paro."ie j:a--ir f r- corpar'issn,
ir. the f'oim, of vclo.ilt: co-re ctinrn t"-..rrult of corre-
spondinri corFress!ibtl and i'.c,;i:.'"e sia'. e fl:ws.

2. The veloci tU corre- tic.n I .-r,.I;&? tal,-eca. a e .by
Chaplyf in, by vorn il: r n, r. 'i 'ilempale cnr Yar.'cccd. have
been .unif i :e b'y :ieani- :-f th}n se '.:, r olultins and shown
tc be esse-ntia11y ej- :, v@le't,

In the pres-nt ;.-cr t' typie' of r.;:inaticns
have t.en introi.'u,:ed c r-" c-: .... the hy ric e.-.entary
so luti nrIs, r Ar 1i:' t:- "an r ..t -. ic-.r'.-an" t. 'e :':-: the
" eeo .C tri --v.ian" type. Tn-: C aI..1rr I.;.-. ti -nP I-c' ude
those cbta ied b:: Cnhaly in, ':."- on !: rr:i, ond t: T-.r.ple
?,d .'Yarwood.

4. TeF. anro?.:ii.at :i ns dis Sr-Ped in th" ;presc-nt raper
have been cor.pared with tihe- v~ ll-1.-1".'n rr-suilts of Praindtl
and Glc..uert. For thi. Isur:se, it ha- been emphas ized
that the Prandtl-C:laliert r;?u]lt is vu ,ll for vanishingly
small dis t',rbances n and, in a strict Srrse, is the sl-)p
term in a Tay, cr Cxp.:n ioni in a qua.i tty Vwhich -,e, surfPT
the dist urtace. It vw-'i fourun.i tist ti'e ar th.etic-mean
type y~!eldj Lhe FPrandtl-;l.eArt res.~Ii; to a hi: Thri- order
of aTp.roc.imr.t on th an thie .-spl:.! in -j -or th--' 'er I:-'lr ar-':.ood
type and that the- reo:,etric-L,.ean t:y .e c.ontain.rs the
Prandtl-Giauert rcsulL ex:stly:. The Fxo t"'Les of :tp:'roxi-
nations introcidu-ed in the j'res .nti a. ..aer the-n -;::-ar.r to be
preferable to t'h others' as basis fr- e;:trap:la lion into
the rar. e of high stream ii".-h n'.u:..er, ani large distarb-
ances to the mainr stream.

5. The results of theve prFnt paper ,.ve bern ob-
tained without co'nsiderat iorn of :.ny particular 'cou'.ndary.
The actual boundary pr,,btlem of de-terrilninac the flow past
a prescribed bod" is of a hi,-h order jf difficulty and
involves in general all theo'articular solutions of the
hodograph equations.


r',.;rT -rTTMT AL


CO! TFIDE'.ITTAL









ITACA ACR No, L4C24 COIFTDEN TIAL 35


6. The particular solutions discussed in the present
paper are well-behaved functions in both the subsonic and
the supersonic regions. The hodoEraph equations give no
reason, in general, to suppose that a discontinuity neces-
sarily occurs in the solution when local sound speed is
attaine'. Rather, it appears that the first breakdown of
the solution is associated with the vanishing of the
Jacobian of the transformation frcm the physical to the
hocograph variables. Indeed, von ?.arman has made an equi-
valent sur-estion in that the appearance of 'nflin!te accel-
erations in the flow solution is a condition for flow dis-
continuities. Interesting speciulations on this matter are
suggested. b the results of the present paper since the
"limitir.g" curves discussed in the present raper are
defined by a condition that is equivalent to the condition
for infinite acceleration, The arithmetic-mean t:'e of
approximation thus :yielOs a limitiur.- value of the local
Mach number M = 1.15, and the geometric-mean t"-pe. of
approximation yields a limitin- value of the local Mach
number 1M = 1. The value ? = 1 appears to be exact for
vanishingly small disturbances; that is, local ,:il'
number M = stream IPach number 11 = 1 (Prandtl-Glauert
approximation). HTowever, for finite disturbances to the
main flow due to the presence of a tody in the fluid,
infinite accelerations ray occur, for stream rIach r.umbers
less than unity, in reicns where the local Nla&c' nrimber
is greater than unity. In this rearr, the aritchetic-
mean type of eppro:imartion, considered as an extension of
the Prandtl-Glauert relation to finite disturbances, indi-
cates the possibility of a mixed subsonic, and supersonic
flow without discontinuities. It i]. inrcrtant, however,
to recognize that in reneral the limiting value of the
local "!ach number P' is a function of shape parameters
and is a result of the blending rf nan-" part.cul'r solu-
tions of tht hod-rprap'h flo' equations aeeordinr to the
boidarry conditi-ons.


Langley femorlal Aeronautical I.ab-ratory,
rational Advisory Comnittee for Ar'-nautics,.
Lan-.ley Field, Va.


CC!TFIDE!TTAL










.ACA AC.A I o. L4C24


F.LT'.E: 'CE.


1. ?:ol rntroe k, P.: P'l-r ein-ue eev lgur-en eines GQ.ser
Irm.t Anneime eine; Cec'.'l; Tinjd.Jkeitroternt ials
lr} iv r. ihLt u. Phyr', (2), vol.. 9, 1l .O, p. 157.

2. Chapl,-':i, ,.,: Orp Ir-7 etr.. (rc. ,t in Rurs ian.)
Sci. ''n., 'Toc). Iprili Univ r tiath.-PhyV S. c.,
vol. 1, 1904, p-. 1-1'1.

3. Dentc-henl.,- T. 'I l ue- p ro'-.! 're.- d'h"-drod- na.mique
bidinr.r ienrlle ids s filirles ,, r7m- er'- -si") es. Pub.
ITo 1, ,, Fub. -'ct. t c ch. dt llnis ti c do 1'air
(Paris), 13" .
P i
4. von .,''-r'Vi.n 'I.: CO '-.r... "; 'i. ] ty '': ] t C in e-.e 'o-
dvr inic', ,ouvr. ..-., ."-o, v'ol, no. 9,
July 19-1, pp. 7"-: 5

5. Tsi-n, Hi-iiE- 'hey : ?'c-Diren. si ; ,l t son'.ic i'.cv' of
Corpre. 11it l Fluids. .'our, ..-c c. u ., vol. '
no, 10, t.'.i". 1'7', pp. Z9.9-4(07?

6. Piarlel- r'.r ".cT: r :.-' te .. T -. er f t'r-er.tial-
rleic n-ij r e -i n r .! .'t L .*7 iif ,t crTr.T .C:
f'- Tj .. t -' 1 ,'-1 95.

7. Tenp-le, G., *:d. -. ,c. J.,: 'he ,.ppr:.:-:r te solution
of. the ;od:.: F -*''..t otrs for C,!pre osble Flew.
:^ !" -, ?.0', :._ .: i, .?.1.A Jtui. 1'9'- .

8. Fers, 1- .:r '. r.- -., .-.:r-- 'i. a 7 's o 3D Di'f-

u rt -. ... l r.. L. .L u" 1943,
pp. IdJ-i?'

9. Filbert, ,7vvid: G .:.:. e, In e,] al '-.,.,1 r n Thieo1'ie
der lincaren Interra L-;.',i.. B. G. ,Iul-ner
(Leipzir and Lerlin), 1':'4: ., 7" .






is


c o~: I' PLT 17 7 AL~I


'"IU'':' T.LIL
~L7iJrl L~L~.









NACA ACR No. L4C24


POO 'CO -- t-00 t 10 0a O 0a! aS; m t0o at 1 to Qa O
8 (0107 M E' toM 0I CO t 0 E s)0 rl 0 08
O 0! (0 rl- CICW.




0 00000r-fO00 000Mb% toV 0 000000 a 0
.- pe 8p oor4 w .-s0 W f-m10 r 34 10q o r4 D04 0n V)b4 O 01to 800
+4 cm o 0 o1 o CSM to oe L- CM r0 T to to 0o8oo
* 9. ... .9 00
r4



MO*.. n.-s10 o o w w P~-4v00 <-- 18 1 o100oo -10 0WoWwo o0
#o o> c f ont co momcom o> o noo C0 0aooo
e eoe a me)U emem ees amoorol ee aoo





5000010 n w 000 V-4 H 0 n 100 00 0
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OEM t t-10 U 010I10 0 100 00





* De4iCoOIe W 4).-4 tOa 0tEo-0 0I0 ss -
.. ...o. a *U | l .

fll




IioI Il I I I I I I
50 q.I!o N10 0toP Nw, 9H0SO4I0 o 1111 miasas



S001 0 0 t-5- 0 ,- 04 v 0 r4 m0 t0 I a sla s n
ba000 00l s 6 >#ON co wm lifMC 0 Io t 0- L- i V L ,









In 8oo 0 0 G 0to000 iR01 0 c0 (n' t1w o 040 "ED
040 1 0 L- I 4 qr 0 t- M r 0 0 0 0 a 10 0 4 0 10
ooQ v r4 0 I~0 0o o~ r V
V08Oo e v e otooneeov OmomeomsL MoOID2
840 OOOOO0 Om- m H0 0. Cc0 000 04*0 01 CIS r M10110 2%1 100 IV t W 0 C0 01 -
S 0* 0 ** *. 6 0 ** *
o-N1 1 11 o1t1 I ll I l llt A ill l l 1. a O
I II


rl cO1 o 010 t- V 10 010 0 0- 1 O C 0 D n 10 ne o0 00- CO 0o
10 rt- H10 0 10 00 1- 00 0 %W 10 4 0 10 1Q V C4 QC 0 1 H 0 ," V410 10 M 010 -V0 -10 *
oolnormoo oo
04011 0010 1 0 t00 0 mIwI Eo-10 10M10 10 10 0I 00oIIw-
8 Q-N0 0 1 1 0 10 CO 0 001 n 0 010 (M to60 s0 O0 0CM CM -0 10 t- P) W 41 10Mn CM 0 & 4
0000O i1I rf -O lo o1 1 0 011 10 1- Sot 040




*o .. ...** .... ... .* o o*.. *o e.
SOla lmla I lil I lal ilI Iaot m -o0
S110


00 C M %0M 010 0M0 a10 r 4- %0 010 0 C 0 W# 01 1Q In 010 10 0010
o4 (bD o1 otI1o Ho 4o 0 H0 0 -c rn to 1 o 10-r- 10 I c 0
o m no0 0v 00Ow- w40M0 00o 0 CM10401 0 0OM1010 00

01 040-4.-10410.10 -O. 100





9.9.9 0. *00 10 0110100 00*0 0 0in 00rl
Oi -0 1o 1cNIOo 0 o i r0404 1 CMo nwom0o n
o 0 H n w r- t- r, r-- M 0 0 1. (m 1- ro 01 0O 0



00000
1001 O10O m00o aveW040000 0000000
. 99 ....... m ....... ...........
0 5- r4 r4 P-4 r -I -4 W m n v Q








NACA ACR No. L4C24


CONFIDENTIAL


TABW 2.- VAWU OP (q/ql)o, (q/tl)1, Cp,o. *WD Cp.M

FOR r 1.4 AIN FOR' V~RO7a VALUS OP 1 N


NATIONAL ADVISORY
CUMMITIEE FOR AERONAUTICS


S 10.9 0.3 0.4 0.5 0.55 0.6 0.7 0.8 0.9 1.0 1.1 1.2
0.00794 0.01768 0.031011 .04762 0.05705 0.06716 0.0892 0.58 0.15942 0.16667 0.19485 0.22"30
yR 0.2
(q/q1) 1.00 1.49262 1.97680 2.44948 2.68106 2.90907 3.86349 3.78124 4.19121 4.58255 4.95484 5.30790
Eq. (38) 1.00 1.47471 1.92120 2.3335 8.52518 3.70706 3,04051 3.33365 3.58793 3.80599 3.99122 4.14788
(4/91)1 l q. (36) 1.00 1.47331 1.91444 2.81288 2.49242 8.86746 2.9370 3.14735 3.27893 3.31937 3.827590 .14780
1Eq. 1371 1.00 1.47401 1.91782 2.32296 2.50876 8.88213 2.98917 3.28912 3.42841 3.65441 3.6169 5.61309

1.00 1.01263 1.03065 1.05446 1.06868 1.08461 1.12188 1.16737 1.22249 1.28926 1.37028 1.46907
c Eq. (37) 0 -1.17271 -2.87003 -4.39614 -6.29583 -6.19382 -7.93614 --9.9190 -10.7400 -11.83385 -12.07502 -18.06446
(aEq. ple)) Sq. (Il _
OpI 0 -1.21250 -2.82393 -4.78600 -2.81893 -..92214 -9.238M9 -11.62214 -14.00071 -10.31387 -18.51393 -20.57000

N1 0.3
r'f11) 0.6MW 1.00 1.32425 1.64107 1.'9621 1.94698 1.26672 2.55331 2.60797 3.07016 3.53196 3. M1
1q. (36) 0.78610 1.00 1.30277 1.59824 1.71133 1.83657 .0617 2.26065 8.4899 2.58067 2.70646 8.81285
(q/ql)1 Iq. (34) 0.87875 1.00 1.29942 1.566971 1.69172 1074 1.40374 JS .1386 2.38 835 2.2804 2.23 M 2.13*M
oq. (37) 0.67843 1.00 1.30110 1.87590 1.70200 1.00 1966 2.02793 8.19758 2.593W 2.41140 2.45317 8.46183
0.98753 1.00 1.01779 1.04131 1.05535 1.07107 1.107?0 1.12880 1.20728 1.97319 1.3316 1.45075

CPBo Eq. (37) 0.83973 0 -0.69286 -1.4858 -1.8980 -8.51113 -3.1120 -3.82009 -4.409 5 4.01485 -5.01804 *6.00863

ef*1 0. 94 0 -0.74111 -1.S694 -o.11686 -2.8281 -3.8928 4.76.21 -*6.830S -6.94746 -7.86968 -8.90671

N 0.4

(q/ql)o O.509 0.73614 1.00 1.23924 1.350 1.47174 1.e86t 1.91500 .L12060 3.316 0 6.10374 8.63 16
lq. (35) 0.58061 0.76759 1.00 1.81481 1.31488 1.40903 1.88860 1.73519 1.86784 1.98106 8.07744 2.1886
(q/qlJ)i q. (6) 0.28836 0.76367 1.00 1.80801 1.30190 1.38811 1.83603 1.64401 1.71116 1.73387 1.7111 1.84488
1.__ (7 0.5163 0.76858 1.00 1.81124 1.30813 1.39683 1.16668 1.68897 1.78376 1.8635 1.86642 31.8650
S0.97026 0.98253 1.00 1.02312 1.03690 1.05235 1.00868 1.12884 1.18614 1.26092 1.32964 .48640

CP90 Eq. (37) 0.7811 0.40930 0 -0.4710 -0.71120 0.98589 1.46m30 -1.8681 2 -2.19369 -2.43494 -2.55481 -2.64982

p.mI 0.7664 0.43714 0 -0.88420 -0.8118 .1.12259 -1.7418 -S.3B -S3.03 -3.6 06 -4.25938 -4.81750
ilk. ( ( 1 ) )0 .i
1 = 0.6
(q/qlle 0.40825 0.60956 0.8069B 1.00 1.09454 1.18783 1.38606 1.54570 1.71106 1.87084 E.02281 2.16694
Eq. 131) 0.42857 0.63202 0.82338 1.00 1.06222 1.16017 1.50307 1.4 1 1.6769 1.83115 1.71061 1.77737
(q/ql 1 Eq. 136) 0.43240 0.63706 0.89781 1.00 1.0777 1.14909 1.97069 1.34092 1.41862 1.43531 1.41861 1.36110
Eq. (37) 0.43049 0.63454 0.82560 1.00 1.01998 1.15462 1.B8679 1.39440 1.47068 1.53012 1.81086 1.68889
(0) .0.94834 0.96032 0.97741 1.00 1.01348 1.028i 9 1.06395 1.10707 1.16935 1.28867 1.28991 1.3318

CP.o Iq. (37) 0.81468 0.673 0.31858 9 -0.16653 -0.53316 -.6663 -0.94435 -1.17822 -1.34127 -1.42307 -1.41924
(m1. lies)) I _[L I
CPI1 0.3771r 0.68386 0.36646 0 -0.19564 0.40000 0.82766 -1.26764 -1.70687 -2.13346 -2.53960 -2.91903
N1 0.65
|q/q1ll 0.37299 0.65672 0.73726 0.91363 1.00 1.08504 1.E8081 1.41036 1.56326 1.70923 1.84810 1.97973
aq. (381 0.39801 0.58399 0.76083 0.92404 1.00 1.07202 1.80408 1.32017 1.42086 1.5072 1.58087 1.44836
(q/q1)1 lq. 156) 0.40122 0.59110 0.76811 0.82788 1.00 1.06621 1.17906 1.262 11.31434 1.33176 1.31436 1.26894
Eq. (57) 0.3981 0.58763 0.76446 0.9295 1.00 1.06911 1.19180 1.29114 1.36657 1.41680 1.44133 1.44019
(q 0.93573 0.84756 0.96441 0.98669 1.00 1.01490 1.04978 1.09234 1.14393 1.20640 1.28222 1.37467

CPO Eq. 137) 0.84111 0.6481 0.41560 0.14282 0 -0.14300 -0.41967 -0.66704 -0.867E1 -1.00132 -1.07743 -1.07418
lk te11))
CP,1 0.91635 0.72685 0.47249 0.16787 0 0.17497 -0.54078 -0.91731 -1.2999 .1.5681a -2.00660 -2.35058
(Eq. 18ib))
,ONFIDENTIAL








NACA ACR No. L4C24


CONFIDENTIAL


TABLE 2.- Continued


39







NAI1UNAL ADVISUHY
COMMITTEE FOR AERONAUTICS


S = 0.4 0.6 0.65 0.7 0.75 0.8 0.85 0.9 1.0 1.1 1.2 1.
0.03101 0.06716 0.07792 0.08925 0.10112 0.11348 0.12626 0.13942 0.16667 0.19485 0.22360 0.25262
MY = 0.6

(q/ql) 0.67947 1.00 1.07707 1.15277 1.22704 1.2998 1.37107 1.44093 1.57527 1.70324 1.82460 1.93939
q. (35) 0.70971 1.00 1.06348 1.12318 1.17914 1.23146 1.28018 1.32559 1.40596 1.47438 1.53201 1.58014
(q/91)1 Eq. (36) 0.72041 1.00 1.05612 1.10583 1.14869 1.18436 1.21245 1.23289 1.24908 1.23272 1.18451 1.10691
Eq. (37) 0.71804 1.00 1.05979 1.11447 1.16382 1.20767 1.2485 1.27841 1.32521 1.34815 1.34708 1.32253

(/ 0.98025 1.00 1.01631 1.03437 1.08432 1.07630 1.10061 1.12713 1.18869 1.26339 1.35449 1.46642

( Eo 1Eq.B (ST) 0.48872 0 -0.1218 -0.24204 -0.3446 -0.45847 -0.55214 -0.63433 -0.75618 11 -0.1 81462 -0.74909
c.P*1 0.8492 0 -0.13786 -0.31929 -0.48310 -0.84774 -0.81258 -0.97599 -.943-1237 .59762 -1.88099 -2.24151

*I = 0.65
(q/ql)o 0.63084 0.92843 1.00 1.07029 1.13924 1.20681 1.27296 1.33765 1.4625B 1.56188 1.69405 1.80060
Bq. (S5) 0.86734 0.94030 1.00 1.05615 1.10878 1.18797 1.20377 1.24630 1.32204 1.38638 1.44067 1.48682
IqL/q)l Eq. (56) 0.60812 0.9468S 1.00 1.04708 1.08766 1.12142 1.14802 1.16723 1.18271 1.16723 1.12156 1.04808
Iq. (571 0.67469 0.94366 1.00 1.05159 1.09816 1.13953 1.17586 1.20612 1.25044 1.27209 1.27110 1.24789
(q./q1)c
T-, 0.93500.9010.98396 1.00 1.01778 1.03741 1.05904 1.0886 1.10906 1.16963 1.24314 1.33274 1.44292
Cp.a Eq. (37) 0.54479 0.10969 0 -0.10584 -0.20596 -0.29853 -0.38192 -0.48473 -0.56360 -0.61821 -0.61870 -0.55723
(lBq. (18)1 ______ ______ _______ _______ _______
p.'1 0.64074 0.14002 0 -0.14333 -0.28862 -0.43476 .0.68086 -0.72571 -1.00862 -1.27767 -1.52913 -1.76024
(1g. (18)) ______ __________________ ____________ ________ ______ _______
MI = 0.7

(q/q1]) 0.58942 0.86747 0.93433 1.00 1.06442 1.12766 1.18936 1.24980 1.36650 1.47752 1.58280 1.68236
Sq. (38) 0.63186 0.89032 0.94683 1.00 1.04983 1.09641 1.13978 1.18003 1.26175 1.31266 1.56398 1.40661
(q/ql)l Eq. (36) 0.66146 0.90429 0.95504 1.00 1.03875 1.07099 1.09640 1.11475 1.12953 1.11476 1.07114 1.00096
_q. (371 0.64159 0.89728 D.95094 1.00 1.04429 1.08362 1.11788 1.14696 1.18909 1.20969 1.20874 1.18666
q/qe 0.91869 0.96678 0.98283 1.00 1.01928 1.04055 1.06394 1.08967 1.14920 1.22140 1.30946 1.41772

CP.0 q. (37) 0.58836 0.19489 0.09571 0 -0.09054 -0.17423 -0.24668 -0.11649 -0.41394 -0.46335 -0.46105 -0.40616
(IB. (1lea)) _
Cp.l1 0.70680 0.26516 0.12904 0 -0.13082 -0.26254 -0.39397 -0.52440 -0.77907 -1.02140 -1.24781 -1.46689
(Eq. (18f)1 _
MI = 0.75
(q/ql)c 0.58374 0.81497 0.8"'78 0.93947 1.00 1.06931 1.11737 1.17416 1.28380 1.38810 1.48701 1.98054
Bq. (351 0.60187 0.8480 0.90191 0.95253 1.00 1.04438 1.08668 1.12404 1.19235 1.25038 1.29926 1.34006
(q/qll Eq. (3s) 0.62715 0.87056 0.91942 0.9668 1.00 1.05105 1.06550 1.'7317 1.08740 1.07317 1.03120 0.96862
3q q. (37) 00.61439 0.8924 0.91062 0.96769 1.00 1.03768 1..0704 1.09831 1.13868 1.15940 1.16749 1.13636
e 0.90128 0.94848 0.96394 0.98108 1.00 1.02084 1.04381 1.06906 1.12745 1.19829 1.28468 1.39088

Cpa Eq. (37) 0.62252 0.26171 0.1'0o7 0.08302 0 -0.07678 -0.14691 -0.20628 -0.29659 0.34189 -0.33978!-0.29131
(Eq. (1891) _______ II________
Cp.l, 0.76361 0.35519 0.23700 0.11937 0 -0.12005 -0.23997 -0.35893 -0.89124 -0.81227 -1.01872 .1.20883
(lg. ( 3B)) _____ _____
I1 = 0.8
(q/ql)c 0.52274 0.76934 0.828e4 0.88687 0.94402 1.00 1.06482 1.10842 1.21193 1.31038 1.40375 1.49204
Eq. (35) 0.57831 0.81205 0.86359 0.91207 0.95753 1.00 1.03956 1.07629 1.14171 1.19725 1.244061 1.28312
(/qlq)l Eq. (s3) 0.60827 0.84435 0.89172 0.93371 0.96990 1.00 1.02372 1.04085 1.05467 1.04085 1.00014 0.93460
Eq. (37) 0.59209 0.82805 0.87766 0.92283 0.96370 1.00 1.03162 1.05843 1.09734 1.11633 1.11545 1.09610
{q/*J 0.88287 0.92910 0.94425 0.96103 0.97958 1.00 1.02249 1.0472;3 1.10443 1.17383 1.25846 1.36247

CPo Eq. (37) 0.64943 0.31433 0.22989 0.14838 0.07128 0 -0.06424 -0.12027 -0.20416 -0.24419 -0.24423 -0.19864
(Eq. (18s)) -)- ___I
CP.Ml 0.81520 0.43547 0.32940 0.2206 0 011 0 -0.11065 .0.22036 -0.43464 -0.63857 -0.82904 -1.00411
(Eq. 1881)_ _

CONFIDENTIAL








NACA ACR No. L4C24


CONFIDENTIAL


TABLE .'.- Conclidea


40



NAlllUNAL ADVISURV
CuMMiiEE FUR AERONAUIlIC


M 0.4 0.6 O.i 0.65 0.875 0.9 0.925 0.96 01.0 1.1 1.2 1.3

,r 0.03101 0.06-'16 0.11962 0.12626 0.13 79 0..13942 0.14612 0.156563 '.16667 0.19485 0.22360 0.25262

1 *0.625
(Iq1).e 0.50872 0.748-1 ].00 1.02665 1.05276 1.C6'870 1.10434 1.13972 1.17941 1.2l-24 1.I6610, 1.45203
3q. (35). 0.56494 070 1.00 1.01905 1.0338 1.oo.06 1.07205, 1..94'6 1.1191' 1.1"S63 1.21950 i.25781
oql), Eq. 1361 0.60061 0.83360 1.00 1.01093 1.020;1 1.027W8 1.01380 1.03929 1.04147 1.02784 0.98764 -.92292

Eq. 137) 0.68253 0.81469 1.0 1.01497 1.02874 1.04136 1.052"2 1.06re.5 1.0"962 1.09833 1.09746 1.07744
tq/ql)e
0.87329 0.91901 1.00 1.01139 1.02335 1.03586 1.04902' 1..50 1..>9:44 1.16107 1.24478 1.34767
Cp,o q. (37) 0.6606 10.3562 0 0 -..03016 -0.0531 -0.08443 -0.10924 -0.10.3774- -0.16!l- .2063 -0. 0442 -0.16008
(Ei. (186)) _
Cp,% 0.S39.50 0.47350 -0.056329 -0.106311-0. 1510 -0.21149 1-0.'B409 -..36F71 -C.56i4'-0.'4599 -0.91487

M. .5BS

(q/q.') 0.49558 0.72936 0.97416 1.00 1..:.0, 1 .050i 1.07580 1.110 1.146-94 1.,4?28 1.3.300 1.41451
Eq. (351 0.55438 0.7,114 0.98,13; 1.00 1.173 1.0353. 1.05:2 1.0r430 1.09626 1.15171 1.19672 1.23432

(q/q1)1 iq. (561 0.59418 0.92478 0.98920 1.00 1.00917 1.01673 1.02264 1.02~0 1.030:1 1.006173 0.97697 0.01295
Eq. (87) 0.67394 0.80267 0.91826 1.00 1.C1357' 1.02599 1.03721 1.05091 1.06371 1.08212 1.08128 1.06156
] l0.86347 0.90a6" 0.96873 1.00 1.0118. 1.0.420 1.03721 1.05647 1.06013 1.14801 1.25076 1.33249
CP.0 Eq. (37) 0.67069 0.355 .20.02926 0 -0.02732 -0.05266 -0.07580 -. 10441 -0.13146 -0.17098 -0.16917 -0.12689
(Eq. (16.))
C0.,1 0.B6288 0.5095 0.0l.147 0 -0.06129 -0.10126 -0.1294 -0.22305 -C.3i199 -0.4920.0 -0.(69K, 1-0.83274
(Bt. (1lb))________
1'. O.8r5

(q/4ql) 0.48322 0.71118 0.94988 0.97608 1.00 1.024653 1.04898 1.08259 1.12031 1.21133 1.29"'63 1.37926
Eq. (3s6 0.54468 0.76734 0.96396 0.98232 1.00 1.01703 1.03340 1.05530 .0'6685 1.12134 1.175EL 1.21248

(q/qll Eq. (36) 0.58877'0.0172 10.98020 0.99091 1.00 1.-0048 1.01533 1.01870 1.'.085 l.CO-49 0.96867 'j.30464
_q. (371 0.5662610.79192 0.97207 0.98661 1.00 1.01225 1.02331 1.03684 1.04947 1.06"64 1.066'8 1.04733
(qf "ql), I
0.65335 O.Bg!90510.r'717 0.98831 1.00 1.01223 1.0509 1.044 1.1349 1.21640 1.3693
CP.o Eq. (37) 0.67935 0.3726 0.0.508. 0.02660 C' -0.02465 -0.04716 -0.0704 I-0.10139 -0. 1986 -0.136 -0..09690
(Eq. (186)1
Cp,0 l O.B8B64 0.542"" 0.09960 ?.04971 -0.04943 -0.09846 -0.16642 -..24286 -1.42701 -0.59898 -0.75712
(Mq. (18b)) ,.
MI = 0.9

(q/q]), 0.47160 0.69409 0.925705 0.95164 0.97595 1.00 1.02376 1.056E?7 1.09338 1.18220 1.26644 1.34609

Eq. 135) 0.5554610.75449 0.94 3 0.96687 0.98325 1.00 1.01610 1.03764 1.060'79 1.11240 1.15,6EB 1.1218

(q/qli )l 61 0.58439 0.81121 0.97291 0.98355 0.99257 1.0O 1.00579 1.01113 1.01327 1 .nooc.O 0.96089 0.89792
SEq. 1371 0.55939 .78232 0.96:29 0.97466 0.986788 1.0 1.01091 1.'-j4Z9 1.03676 1.05470 1.05387 1.03463
(Lq I0.84306 .821 3.96539 0.97638 0.9332 1.00 1.012"1 1.03151 1..05461 1.126B9 1.20170 1.30104

P q. (391 0.68908 0.391 0.07"84 0.06004 0.02409 0 -0.02194 -.04917 -C -.11239 -0.11064 -0.0'46
(Mq. :B m))I 001
"p.,I 0.9078? 0.E7490 0.14460 0.09617 0.0499 0 -0.04762 -0.113M5 -0.18788 1-0.36660o1-o.336. -lj.68 1B
(Eq. 1Pb)) _____I __
N1 = 0.925

(q/q1), 0.46066 0.67798 0.90552 0.92955 0.95331 0.97678 1.00 1.0320b 1.0680C' 1.15476 1.23704 1.31486
Eq. (35) 0.52697 0.74253 0.932'9 0.95057 0.96"68 0.90415 1.00 1.il0;119 1.-43j7 1.03476 1.13756 1.17329

(q/q91, q. (361 0.8103 0.80654 0.96'29 0.97787 0.98686 0.99423 1.00 1.0053 l. .?I'45 '..9424 0.95.34 0.39276
Eq. 137) 0.55335 0.7387 0.94991 0.96413 0.97"21 0.98919 1.00 1.01323 1.''..156 1.04332 1.04249 1.0-547
f 1 e 0.83249 0.87609 0.95327 3.96413 0.9"554 0.9874E. 1.00 1..185- 1.04135 1.10681 1.186(2 1.26471

P.o q. (37) 0.69380 0.40113 0.09767 0.07045 0.04506 0.02150 0 -C..2664 -0.,i177 -0.06882 -0.08679 -0.047491
(Eq. (18)) I )10 .02- -
Cp.Mi 0.92963 0.30574 0.18708 0.13991 0.09301 0.04633 0 -0.06425 -C.1364' ,-0.3103B -0.47285 -u.E.2;'22
(Eq. (18b)

CONFIDENTIAL






NACA ACR No. L4C24


TABLE 3.- VALUES OF F(T) FOR SV-ERAL VALT.ILS OF y


F(T) = (2- + 1)T 1 M2
F(2) P -1

2
(1 -T)2P (1- 2-Zj


F

Y = 1.4 Y= 1 7 = 2 y =
r (p = 2.5) (p--,) (p = 1) (p-->0)
Adiabatic Isothermal Hydraulic Limitinr
analogy incompressible
(1) (2)

0 1.00 1.00 1.00 1.00
.2 .99901 .99918 .9979 .9600
.4 .98228 .98575 .97377 .8400
.6 .906106 .91'73 .89113 .6400
.65 .96634 .88113 .84726 .5775
.70 ._13.94 .83248 .79050 .5100
.75 .74558 .7676s .71822 .4375
.80 .65738 .63273 .62728 .3600
.85 .54489 .57155 .51421 .2775
.90 .40258 .42710 .37504 .1900

.95 .22355 .24041 .20534 .0975
1.00 0 0 0 0
1.05 -.27752 -.33870 -.24667 -.1025
1.10 -.62069 -.70423 -.54102 -.2100
1.20 -1.55960 -1.85711 -1.30158 -.4400
1.30 -2.95915 -3.73944 -2.34862 -.6900
1.50 -8.01227 -11.8597 -5.64466 -1.2500
2.00. -56.6884 -163.79 -26.9981 -3.0000


1 y= 1, F = (1


- r )e,2


2 y = F = 1 M2


COF IDENTICAL


CONFIDENTIAL






NACA ACR No. L4C24


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