The flow of a compressible fluid past a circular arc profile

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Material Information

Title:
The flow of a compressible fluid past a circular arc profile
Alternate Title:
NACA wartime reports
Physical Description:
69, 20 p. : ; 28 cm.
Language:
English
Creator:
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:
Mach number   ( lcsh )
Compressibility   ( lcsh )
Aerodynamics   ( lcsh )
Genre:
federal government publication   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Summary:
Summary: The Ackeret iteration process is utilized to obtain higher approximations than that of Prandtl and Glauert for the flow of a compressible fluid past a circular arc profile. The procedure is to expand the velocity potential in a power series of the camber coefficient. The first two terms of the development correspond to the Prandtl-Glauert approximation and yield the well-known correction to the circulation about the profile. The second approximation, involving the square of the camber coefficient, improves the velocity and pressure fields but yields no new results with regard tot he circulation, since the circulation about the profile is an odd function of the camber coefficient. The third approximation, involving the cube of the camber coefficient, permits the use of higher values of the camber coefficient and furthermore yields an improvement to the Prandtl-Glauert rule with regard to the effect of compressibility on the circulation of the circular arc profile. Numerical examples with tables and graphs illustrate the results of the analysis.
Statement of Responsibility:
Carl Kaplan.
General Note:
"Report no. L-216."
General Note:
"Originally issued October 1944 as Advance Restricted Report L4G15."
General Note:
"Report date October 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."

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003593774
oclc - 70891407
System ID:
AA00009411:00001


This item is only available as the following downloads:


Full Text
hCA 214t


ARR No. L4G15


NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS






WARTIME IRElPORT
ORIGINALLY ISSUED
October 191k as
Advance Restricted. Report L4G15


THE FLOW OF A COMPRESSIBLE FIUD

PAST A CIRCULAR ARC PROFILE

By Carl Kaplan


Langley Memorial Aeronautical Laboratory
Langley Field, Va.


*htaIS^


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 DEPAM-"--


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P. a"


L 216


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Digitized by Ihe Internei Archive
in 2011 Wilh funding Irom
University of Florida, George A. Smathers Libraries with support Irom LYRASIS and the Sloan Foundation


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-70g qjL0-7
-3 17f

NACA ARR No. L4G15

NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS


ADVANCE RESTRICTED REPORT



THE FLOA OF A COMPRESSIBLE FLUID

PAST A CIRCULAR ARC PROFILE

By Carl Kaplan


SUMMARY


The Ackeret iteration process is utilized to obtain
higher approximations than that of Prandtl and Glauert
for the flow of a compressible fluid past a circular arc
profile. The procedure is to expand the velocity poten-
tial in a power series of the camber coefficient. The
first two terms of the development correspond to the
Prandtl-31auert approximation and yield the well-known
correction to the circulation about the profile. The
second ap;roximaticn, involving the square of the camber
coefficient, improves the velocity and pressure fields
but yields no new results with regard to the circulation,
since the circulation about the profile is an odd func-
tion of the camber coefficient. The third approximation,
involving the cube of the camber coefficient, permits
the use cf higher values of the camber coefficient and
furthermore yields an improvement to the Prandtl-Glauert
rule with regard to the effect of compressibility on the
circulation of the circular arc profile. Numerical
examples with tables and graphs illustrate the results
of the analysis.


INTRODUCTION


The calculation of the two-dimensional steady flow
of a compressible fluid past a prescribed body can be
performed by a method independently discovered by Janzen
(reference 1) and by Rayleic: (reference 2), which con-
sists in developing the velocity potential or the stream
function according to powers of the stream Mach number.
The first approximation is the incompressible case and
the succeeding approximations represent the effect of
compressibility. The method has in recent years been









2 NACA ARR No. L4G15

successively improved by Poggi (reference 5), by Imal and
Aihara (reference 4), and by the present author (refer-
ence 5)- Although the method can be applied to an arbi-
trary profile, it suffers from the practical restriction
to small stream Mach numbers, because approximations
beyond the second or third entail a prohibitive amount of
labor.

For the flow past a profile of small thickness,
camber, and angle of attack, Prandtl (reference 6),
Glauert (reference 7), and Ackeret (reference 8) obtained
by various means an approximation that applies to the
entire subsonic range of velocity. The present author
(reference 9) extended the method of Ackeret by an itera-
tion process that takes into account the effect of thick-
ness and applied the method to a particular family of
symmetrical profiles. In the present paper, the effect
of camber is inves'.igated oy a similar application of
the method of reference 9 to a circular arc profile. In
the application of the method, it is desirable to avoid
stagnation points so tF.et the variation of the local
velocity frc-m thst of the undisturbed stream can be made
sn:all. For this reason the.- direction of the undisturbed
stream is -hosen pgaraiel to the chord of the circular
arc (ideal angle of attack) and the circulation about the
profile is determined in accordance with the Kutta condi-
tion; na.ely, that the flow past the profile leave the
trailing, edge tangentially. The flow is symmetrical fore
and aft and the velocity remains finite at all points.
The circulation in a compressible flow will be seen to
be an odd function of the camber coefficient. In order,
then, to obtain an imorovem.-.nt of the PranJdtl-Glauert
rule, it is necessary to c~r.iry the iteration process
through three approxiritions.


TE- ITT7HATI,:i; PROCESS

The velocity potential (:x, -i) ol the two-
dimensional, steady, irrot tional flow of a compressible
fluid satisfies tnhe follcving differential equation of
the second order:

2_ _2 2 2 02
(c- u) 2u + 2 v2)(c v) 0 (1)
X) 6 xaY C ''








NASA AiR No. L415


whE. re

X, 7 rectrc'oulqr C :rtesian cooririnates In plane of flow

u =--, v = fluid v.locity: cJCone.:s eLang X r-tr. Y
c, 0-/ &axes, re.s'Iectiv.:Ja,

c local e.' lo t, .)' s:,.;nd

The lozal v.-Lcity of sc.ni c i ex-..re.. 2sed in terms of
the fli v-l.,.cicy q L.- .r r-ns of m;-iOull s equation

r .2 -
1 --- = (2)
j-1

the ejaation definr.ng tae veioct '. .

= 1- (3)


and the i ic .zeltlor. bt:e-a i the r.sssure and the
densi ty


F- = ( (p4)


In equatons (2) (3), enId ( ),

p static prts sujpe i fluid

pl static pressure in uLai13.urbd .t c.-ar-i t infinity

p den'.'-ty of: flu'.d
P c'enity .fi unr l, t r'b crc.C rt L -ir:iby

q n.sg.1itude ,.f v loccl :;. :? -...:i

y adiab-.tic ir.dex (Z::r. ,x. 1.4 ..: sir)








:'ACA A:?2 : L, .' L3.

For the aJdiabatic case, equation (5) yields

=Yp (;)

By means of equations () .d () d (5) Pernc.Ulii's ejuatii-,
equation (2), yields -'re follo.-irn r2.t ions:


2 2 1 2 2i02
_- = 1 1 .)





2 1





where

U velocity of undisturbed str-'-i,. inflity

c1 velocity: of sound in Zun:isturb-J 2tr-.: .' it i-f.intty

:,1. ?' ,ch number of undi:.izturbed sr-ca. inIr.,
low, if the profile is hzld fixei in i.h: un.Lfcir:n
stream of velocity U an if a c,'r. actc.r i tIc leIj-th s
is assu-:,d to be the unit of length and t-,e sre;;:n
velocity U is assui,-d to be th- inrit rcf vlnccit.y, the
furndam.iental differential equ-. tion (1i ) .j ti, fir-t cf
equations (6) become

(02 6 2 .-
,,u2 ..; ---+" --.: 2v'- 3 (7)
12 2 J c^ }.r

and
c2 1
1 i2 -1 (o)
c1








NACA APR :Io. L-215


v:Iiere ,, Y, u, v, q, nJ no'" =nnte, respecltively,
the n'ndimrnsional qar.i. Les /-,'s, /, u/U, v/, o ,
&r.d '/L u .

The iter.ztL,:n process cc.-.a sts in 6ev lcn,"r. tr.'r
ve-lo _-il potentri al 3' in po.v.-rs cf a E raacter h, the
c nu.oer cr tile circul ar .:. "..


lP hS h ()



u =-- 1 ,.- h S -... -

.- (o10)

c= -- -. J
h. ~ -


" esn cl. s .-: -: /, -u, i v,
2 ;


to C:ther


".i*. th e.re-s3si.: to' '-/, ; en :- : tion (j ),
are in utrciu.ce i:-L'.: t}. lun ", 1 .: -'erential equa-
tion (7) ..:: thc eri ce- ft : 0o"' the various powers
io h ?re te," t :cr '.rc lowIr!; ":g'ferential
e u, atins "'-' Ai, ".. u it:


-:1


S L,
---


.2 ,
+ -- =--" 0


(11)


(2 ( Y )


+
yy


S S1 !) 1'' + 2 2
c_': 0% .>2 6Y : '


(12)


(i -







i -1.. A i t [ '. ; "


(1 ;i2)


S 12
,"F7 7- L


S\ (Y/ -


+ y +




+ (y+1)
6X


+ 2 i
\6x 6Y


11
6x2


6X


c2l
+ 1) +
6X-


6x2
~2
6X


.1
Y ~ J ) -


+ (r+1)


62
+ (y



+ (y-1)--
6 Y


+


62.
+ xy
6Y 6x6y


6Y


(1 ?,)


Thr-se differential eui-,'ons r,;y be put into more fail'ar
forms b-- the introduction of a new set of indepn.-rdr.t
'R' '-I:'l1e s x and y, where

x =X

y= YJ
and
S: (1 ,.21/2

Zcr M1 < 1, equation (11) then becomes a L-:- 1sce eLqu-
tion and equations (12) and (15) become '-i:':. eu.'-t 1 ns.
Equation (11) replaces the fc ,:a .-tal .l. f-1er.tiai
equation (7) for flows that differ only .11i',ti.;," :'O' the


c- 2 6y2


62-









LiACA AiRR Io. L 'il5


undisturbed steea&, rind its solution yield, the well-
knowvn Prandtl-Glauert resIul.t. T'.e solutions c.f equa-
tions (12) nr.d (15) .rcvil.e successive ir,-ro, ve -:~;ts in
the apprcoxir:ation to he sl i tion of a co'- -,-sible-f.ow
proclenm.

For the present l.obler, th> procedure to be fol-
lor.. d :in solving scoua:.ior,3 (11) to (15) is fir-t to
obtain, th? vclo .t, p-r' nc .1- i for the irn;-:.)'v.,ressible
ccse in ti,: fc~i; of a -owr' -, .s in the cari.ce' coeffi-
cient h of the cir'cu.r *Lc ::'-,file. The solution for
the fi.rt .:.rr.>m-'r--i:n .-, of the compressible flow is
then cbt.ineda b:;- rao0;' fro, Lrir.e form of the coefficient
of h of tLe i"nci.p':..csi ...:city potential. Th.
scluticns of e-.u'r.i n (1?) and 15) for the second and
third a.prc;.:ii .ti' rrs '-2 a' -d ,. follow b:, a straight-
fcrwar-d proceiureo. Ti t U: Li: .conditions that t,',.
flo1 v b:. ta.:int 'i 1 t: Li.- .': i .-:- ',.:l that the distui.r',-
ance t' thj rain stre.-c-'.ir vrih L infinity are satisfied
to the same ,o'e.- cf 'tc c'.m.ber coefficient h that is
ii, volv- d in the :ppr.:;.eti.crn J'7o r the velocity poten-
tisl ~, The caiLcuIt.eC-or- 3 1- 1 borious when more than
two ste Ls in th-i it.-''L a. n -c,,ss are involved but the
third st'o is n -cs:.r'y to .:bt-'pi results th&t extend
press .nt-da;, l.nndlr.- jjic ..o c.i' .,he details of calculation
are given in Lppenoi.<:-.- in ode-. not to obscure the
present tioL cf h tr.- .LL. / n ci'.


r SUL'.S '" ,--: :...NALYSIS


E::pr s cicr. f.r- t.-.-:, v'? ,.1-,1t.' pot nt~ The choice
of the clr-.ular -: asc ti :.l. L,.-.:nj -.- was n.i.-l for
.wo rea-.on.3: (1) the :.oluticn f the -r.:r.n-ressible flow
c'.1i be e'sil.l,r e.xpr-ss--d in ,lo-ed form, an' (2) when
the ci:cul.l: ,.rc : f'co in r .- iform stream flov':,.n; in
a :ir.ection rarealel to the ,c i.' end ivhen the K.itta
cor.dit cin !:] : the l:' \..- :he tr-il i e.-ize tCne n-
tially is r:: ii h1-' vlcities at the nose and the
t-il a f inj.te :e .: t :'-....' zero. PI stZ 0..tion
points ocC& u..', ti-. .-i-.. c T ':he boundary or in the field
of flow. and a jre*,t.r .*-,:.r .- accuracyy in the iteration
prcce 3 is a:L s r-. -. .... i .. contains s the calculation
of the income .res-.ble ;l.-..- ,-.t the circular are profile
and appendixu.s ., C, :.' D c'-r.tin the detailed calcula-
tions for ), and '7, 1 -'L2ctively. The final










exrre-s-on for the velocity potential
following form:

= cooh Cos h h2 h3/3

w'h,-re, fro.im qcuaftion (2D),

f- ( 1_) -2,
Pl = -2sin 21 r)

from eq-.ation (017),


Sta&-es the


('5)


= 2 (y+1) D++41 e-+2De- +2 +-D+D y +) "+1)- i1 e- e .,s n

(y+ +1) 5-e + {12+12D-D y+ 1)D+1I e-= cos 5']

and from equation (D18),

G= G1(() sin 211 + G2( ) sin n

/I sin 2T ) <
+ G(s)i ---e sin r- -: sir.
S \2 cosh 2 cos 2ri


F2G-(0) + 2)

In these equations

D -


'-i(1), G2(,), and


G( E)


fu.,-tions of t 'iven by equa-
tions ('l'), (-.-), ;.,nd
(19), r.::_-'ctt"ely


elliptic coordinates z l'..tedc to t.rcul-r
2 -rtesian c : .-.. .-: tes _, i' ; e.L 3ti.:'s of
transformat" on:


r.ACA 'J, 'io. LT 1G15








NASA ARR To. L',ul5 9


x = = c osh cs Ti

y = |'Y = .:irt r_.. "n n


T'e c i cul. ticn co. reaction i'r m" .- .. t '- ( C,
rerr.reT'nts uPLCe sa '..t9t i-. r-.- e & i.: :- t i *,:- c. I -
c'0 ustic Fn (1) thf-it S t t 1fie tL.e hbo n :]s-': ncit .', c
che surf.CrOc of the ci i'c l :', t-rc -,.cfil, c. i:-n ir. t
inscfa' ." 6s. ft.t tei', ini., l i1"e ;" t'..e tl-j'-.i ,.-,. .;'' ,f I:t ,e
c.-rnbar cc.e f iie it r-? :re r.er c. -con of 0c.e
expre i.c.ns I, -9 1 obt. aired in i sc d
ronr ard are f i .n.c fi i lu.s ~~s ,- tl-: n tfes .- i
n, uiber !"1 r-ir.. c c o u t K'.t t i .' '1.1 g u. .; T.re
KutLa ronditd c.., F:1. i : t Ic l t I n
un~iz .- bu Z t -t. ... :.': .
t ailir,. ed i' FE .e J ,i c... rf3ol i:." .
c c' la ttion r' r (s ..-- e ri :n L ') ):



p-0 ] i 1 ,-
+ I 7)- --( + + 1 j\ .;.



1 2 22 ..
+ -L j 1) + F ) (S )



where 'r. bit Pi ar i-h : -ct r .:.;c :lz-- c "- l :-:s
in the ccr'rT'h:'S iJi'- ndJ L: coi'' ..1 ].- 3s. Th 1:.:: -,:-
prE sij e circu latli i (- i '- 1 t 'L:- I s
po.v. r of. h sc t'L t the c -o -. "1 c .-cui- -c .c ,
C
i-3 'ii r-.JA u :t o ..f h T. j
is tb-,err:icre i ent ','"r1 ..t'. :";Z -*.r t ".', "o.,: r tC'.t c;-1
no epar'tu.re e f P." L 'Lt J--..r v ->-t P 1 i -..i
u'cil t.. third p '..Lr ..' h r iL -. P r It
e6 p-ain.3 v.hy .l. .. l r-n. L J .- '. -t rn'1 ..C, '
effc& t cA con'ress'.bili:. .-n e ciil;.:' ..c C. lift ':f
an cJrfo l- .: thi? I. tr'" "-: .. -l.. -"C .

For compr-':i S:! e .'c rul .. E L :.LI. t ..t :'..,. (1-.,
ba' bee L.bt 7 ined bL .''' n l-, ',, ;.'.,.. .n-': ie..
viccity corC r"'cti ro f rc.,'u t-o th: ci' 'cuJl.r r :'.. r:."i11 .







10 : ..' .... ; '' L ".. I

From reference 10

1 p.l
1 1 -i

where
qc velocity of crripveszible fl'id

qi velocity of ii.cc..':..i-si ble fl2id



1 + (1 2)


By an elementary in. --ati-.', -rou.'.d t.: c.r.,1 corre-
s3 n:..;l--c i co /tormall t h .. t ,r .r c-.i r ar:, t:- foilc.."in-
relation is then o1' L ?i.,


S- ___
1 -I 2 I/2 Q 4'"
ri 4pl/2 sin26/ + r,.r "s 1


I + lJi/2 cos .,
-J 2" -(17 )

L1 + 2p/2 1 + sin26. + u -.j

where the angle 6 (see fig. g is :-el -t-c ti the cFmb.'nb
coefficient h by means of thU e u tin
t an 6 = 2: -

Table I gives values of the ratio /o F for va-rius
values of the stream Kach nuin:r. -.. tre c,.".ber coeffi-
cient h, calculated by means cf c qutionr (1) and'-l
(17). Figure 2 shows the .,r :-.- c," F'i. as functicns
of T'l for various v-lue-: c' h. The cirvs b.seJ on
the von Karman-Tsien velocity cc.rrct-icn crm.il lie.
between the Prandtl-Glauert cuv.' cd ti-: curve basLd















.AC A AR i:0. Ll4G1 1i


C')











GLI
0 L

















.0 "--
, ;.- *.






4)















C)C






-- C
l --- ,














,I-
r., H, ') -.,
S-- L.

.3










o .O--
rlj .,- -







,I)
0 .
H I..
Il- *.- L







1. ^
F-i C.i

oS C-' ":
















r -- I L_
''A "
-^ '=1


i *Ic










C ,:.)1



p.o
I
(1.* U)


C.) r' -.

n- I (
,. -, -
0 7 **;

L.}*"







ci (L-4




001 4-.-1


I'
S
nj





F---
C', -
"V

C-



ri
,~~~









(-I





ri -
-I'-
'.1





'4'-








"I




'4'



I K


-I-
A
4-
I,


'-I


'.1

K

4-
+ I-I.-


~ L ___ I


+
4)

-4

'K,

C F-



-, *1
ii~1 *Ui


+
C-i
.-"'-'- "I




-4- -9-
r j

-: _
"--4
--I -

42. CI,


rH


N
1-2


--4




+


*1 -4



T

"-

ii
--I





II

'II..


K) *'~~"


+




ri


4-i







4

II


N '.N -A-






I.

'-4



.-JK-le



-i


-I

--A


+








+



I-'



riI N",







1.



L.j.
4' ]

~1~

-ii





-r



4'
Li

'-'







-4-


-d
'--I










I,- -


- -. I




--


, i




t.-
























.1'i


.4.
* -I
* 4 -*
















L '.1 .'"
-% i*




.-I




I





~.1


I-.


+










+




L


I.-
II I













:'f.CA AV.R :-*-.


e--J





bO

$-l
r1



4-3


0

















(O
-4 -













O C\
C)4



0
o t


-r









O II













0


-:0
.4 1


I.
















--K
0


+ I

I" *


i -

*- I

r'

,- I





0 +




+ i



,-j






*' j


"l -
r-


L;;'rl5


a-"
c*)


..







04
*1-
fCo
















r-
Io








II


0o i


J






(* 0











C?
o -.-.













.4-3







C.
C 0
,-, -.











F-
..- T"





















i-I d






ci, CJ


0




U-'
+

+













r,,,
--








.1
I I
,"" j !' '^













s I
i + 1









1-- ----- ,


jJ









II
I
r -l




I
1I' JI T'
C"I -r


c- I"


II








,--I
ci





c- 1
I.-

*-I


=.-1






I."









1
n.
.

!









NAC A ARR ,.o. Ll015 15



shows thc graphs of 1 a. fuinct' in- of ,I

for the three cppro;nr.mations for various vcl'u-- of ti.
camber coefficientL h.

The critical veloci,;y q.r., defined c s t:' '91... -
for which the velocity of the fluil equa'-. tie l:.al
velocity of sourd, is oburined fro n the fPir'z. olf q.Jr-
tions (6) by putting, q = c = Tr: 1hus


/( + t 1 ;'.r l' N
----- \
S + ( 5)



The velu s of cr Fre g'. en in tbl'e V in Lc.. colu:-'n
for which the local L.:'fh numberer' is ..iity. 1-,o r'ti-.
q p/qci iz easc .' c.,r-ul t o r t' : ve.-'.us &-'pro.<., -
tions. Thie gr-: hs of cr,- the third r-por.x ir :C n j'
qcr
-l1 Pre in':lui .,: in C".- -. '.. Table 'I 11 Lts tLe
q4
first, second, and ti.'.u.d .Fpr ':;:.'ete \&el. .' the
critical store a,. is:n. r.LmUbe. ,d :-:. .
the corresponding gra:phs a: f'.ll-tin n. :7 c t cr .,-ber
coefficient h.

The grPrhs of th ti. t '.L r, 'rcx L "itio-. fC t -I '. 'LL'
and minimriu values c'' q,, ,a;: -ineid fr c.. tb'e:.- II1
and IV, are shov r. in ri urc 6 f. tnttior,, -f tl-h s.rc..M
Sach niinmoer r The cc.-os t n-t oc. l c i cC!i U 02:' 11.-i i :s
shc.vnn in in .gre 6 'rrc cbtes red r.cm .. tu e 'i l j) b
introducing th'' lc iVe h :.u .bu r in ce t-.
local velo..ity if so.nd 2. T1:J:

1/


ote tt ec (2 + '

Note th-. t ec'ueti::, (2.'.) be': r, s C -:...-tl'r. tf-,_ ; '.:..on









'ACA Afl' "o. L1,C5l


!I = 1. Table V contains values of q for various
values of I and ii7.

t\ ccnmos ison of the re'-ults of reference 9 on the
comp'rezsibi".ity f'fect cf thLckrness .-inrl the results of
tie o"c.'ent paper' cn uhe ccrripressilbility effect of
camber is f interest. Fo' this purpose, a s,..-...ietrical
shape of r:fer-etnce 9 wra compared with a circular arc
profile with th.; snm? inc :.,npresslble .neximu- i speed at
t.e su'-f S e. Re-nults of this comijri.-or. for several
ccrr-,-3s',cn.Jdin tn'clne': Ean] camber coeff.,i. 1nts are given
in tsble VII. lT.e d.'. 'cd curves in fi gure 6 are asso-
cipteal with the v.'ri:*n. sy.-inctri;al shapes. For moderate
values orf i*b..er en. t.ic -.s Lthe difference mnoay be secn
to be negliibl_ cve"r The en.-tire subsmnic range. -his
observation i r ic.te Li c t, at I:asc to a v-r y -ood
approxLimtion, tLe effect o cc.ucresiiliiit: in the
susoric r'a'e can be ecnsidIered to aepEcr.5 oxpliLi'tly
only on tic ri nc:.1~-r. _.- fluidJ velocity and th*. stream
JMach numw.ber Lnd :o be ride-npdent of the shape of tl-e
profi-le. This re-ult therel'-ore substantiates the use of
velccit'; cor'c ctt.:n fo.ules suc. as the F-'andtl-Glauert,
the von 'C:.ar. .-Tn L n, L.e T.- le-_'ar,..:.od, ana the Garrick-
Kaplan (refrecncc 11) J.:.rmulrs, v.ii ch deprn; only on the
income s Lble fluid ,'Ilo.lty F.-._- on t ie st,,ez.,- ;-.ach
number.

In geneirel, the ve'.ocity q at the surface 3f the
circular arc prtfi.e mie,' bD "ritc:ii s3 olio.s:

2 )
q = 1 + alh sin 4 + h-(-. + a3 30o c72)


+ h (aa, sin + ar sia -' + ... (25)


whrre fr..-i ,e q.:atlon (12),




(-.
=. = -2 + i- + (-, 1 -
r'i-F










NASA AR2 :ro. LLL:15 15








= + (0) + -(0)


a- = (' -i- + 2+(2-D 5) + (A' j(


Values '"f &~, a a ri .. '-.l'E
of the str. ,-'. n ,l ..,r- :. e '.Ii .
As an excV-I l. c,' tie 1 v. vl .-.- : : t. L ci: r.. b -
ticr. v:r a c'2r 'r a 'f il t c. r, D ..:-ch
n'ur.,ter s cv -i J, the c. e : t. ,1 = ,
0 7 ., and' 0.'.7 is .? ICL4TatL: 2 .:, ,- .., I'd ; tt', the i .im-
:,press ible cai e 'Ti.: c .c, ," ,al.ue o1 tr. 1.:.:"'i ty
trh l p 120. a IJ sP :.s &i'f c t. 1t
a; th ur'p-:r a& id Icv.:--.. s, rf. 'f S .. t"; cirJulr a :.-
profil an '."', for t.- : .T orG '7
giver -in tebte I" : ra : i'. -I .. 1 ":it -
diL tribciti.n ct.r-v : .2 ; e' ." -

ThE pr -asu.r: c r- th- c t "'":' 9.i '.' r"
fi w p.. t a ciysd L.- ..An. a.... '" e.- co.: I'Ll rt i.:


')-
--
dei -, I L


Fi:n the thi- -:f ic-c tl -r.a i c c1 t.c .' "-.- I; L tL



"" P i -P 1I P i ( q- -
S',', .


For the inc O"rIe? I""- -.' *- I ;' 1 I


CpJ = ]. q









16 NACA /W? 'IJo. LC- GIL

For the sonic case, q = q ,

or j L+
? 2 + (y 1):
(C -r '1- 1 +


For the limiting cas-e c' absolute -':.cuur., = w

a- *1id 1/2
and q =

\2
(cp,"2- -


Table X gives asstocited. VEIES Lo' th,? vIoc-ty and the
pr-essurt cceffi .i nt ,C, 'r. vc.ri.u's %ai.-es of' the
stream 'ach rn1um;berh r' d f'i ti e :' 3 shc,, s the corre-
spor-ding c r-:ph -y' r..:*ns tbLt X a--',; figure 3, the
velccit:y readiins firm .'i -Mres I6 ?idn 7 c',n b: replaced
by the corr.-sronliniig :.'rur.? ccsef.icieints.

Laniley 'morirl Ael'cnstuticai LC'ab-'c. ,
National Advis r'y Ce,,rn.ittee fr:,' Ae.'onsutlics
Lan-ley Pleld, v..,









liAJ'. IRR !Io. L.<;u 5 17


APP:LI[ 1 < A


DETER.'iiL'".TT. O TW-' CL ..PL.T : PCTEPTIAL C'L :: C-7 w

The Inrcom.,,'.r--"-c FloI.o tast a Circular Arc Profile


c.nsi.der t'.e IrL.,,ping of a circle C' in the Z'-plane
into Y ci:'-c2r -c C1 in L!h '.-plane. (See fig. 1.)
If the cntr is a t m1," .) r, tiL.- Y'-exis and the circle
pass s rt-.hrou:r t, ;:.'.-! ("," ) '.:. i (-a,0) on the
X' -a..:iS, t'i L t!. J:. '.u. r" rj iisformation

.a2
:= + -- (Al)


maps thie cic'l:- 2' id th-. '-jllne into a circular
are C in be '-c:1 -nI. ''.,- equation of the circular
arc is

.2 :"-/2 + a2(2
+ y -----/ :- (. 2)
m /

The parts of tn-: :.c.:i C' lying above and below the
.X'-axis corre .r on..,, r-e. t" .- '"-, to the upper and lower
surfaces :,.f t.-- r O.. a -. C. The end points I and
Bc o.f the circ L-.r, a. .:-.:: t' :e points



a n d thle nr, a : .,mu, 'i -.I r," '-,. :-- 1.
'. ~ ..&n 6
*1 ar



The c:r'b er' ccefi7. e, f* ffined as the ratio of
the rmLnX.ic.'I or nt : to t,-e c. rd, or



= n 6 ( )
: :,t~n 6 (.@)













The c".,rplex potential of the flow pt a cic.ul&ra
cy.ine:e of radius R fixed in a unif'orr, fi:'.. ?
velocity U :t zero anifle of attack an'. vwit;, a c cI -
lation I is gi',.,rn by


+ + o -- (1'* )




Z" = Z' is cn 5

For the purpose of the preser.t pa. r t.e ..lclation r
must be so chosen that tlie st -i; -.. cr. :.ir.ts .r t.e
circle C' lie at the point ,' = i :t '--fSr. D r. to
the le;dJJ- ana t r.il r.: -dg :2 t : c rc,!-- sr.c C;
that is,

r = .-

= .T (A5)

T'lth this value of t;-e circul-ti-.i irns3.'t: '," ir. eLqa-

tion (.t) and '..ith '" .-:-laced by Re t. cc..v -x
velocity at the surface of the circul- ar rc C Lb'-coimes


dw -iG sin + e in C e i -
-2iUe- (e- 1 z Cin *5)
dz 1 2ie sein 6 e2i9

The :..a.-itua of the; velocity is

dw r1,1/2
q =- -
(dZ d2/


= (. 1 + 2 sin 0 sin 6 + si'- ) (/.6)


It is recalled U..-t the -yp;er surface of t .2 circul'
arc is traver -.a in a clor-.'v. ise s ;,je e 5 --es 1'"f..r.
-6 to Tr + 6 a,,d :be lJwer siface, ,3 gpc.s '.-or:
-(IT 6) to -6. The velocity at tl.-e .-e or tall is
then given by










I;.,..'c JP'R '.o. L -.'] 5




n!s0 a qt=a. "; ,:cos-

Tr
".he mc":irr.'Ln anj "m niTu. e -.'it'..s c, ''r' at -- a ....

at = --, re'spct '.',] : .--J re -*. y


q O 'I 1 + '"n Jl

c. i i ] L' r. 6 i


(A7)


Lc.u.a" n f 'i -. ., e. series in h

'}-6 eQ t t..- : "., ,: T.h'. .i' obtained d from
e t ":.. n (. c, t'c r. -


/ = L...; r K^ -,1' (p.2)


wher-e r- r= -,-- -i iL : .1'.


Sr circ Le. .x'.& i-


s on of t. r. J L.. 1 i.i, c ... ti.i, ,.) c dirg to
of ,/i- y .. .l s


----
.4 1
':"- 2;7


-I
1 C *


E.y us- c = --



r


Then ecuatir (, ) '..


S= l .a- -j --- -" +
y, 9 .
-7'


I-


- + .


( -1 ')


'1 --.+ i + ...
, a /


, = .-n


rowers


(19)








w~c,; ;x~.


X X I.
io":, put = cos 4 and replace and by X
2aa 2
and Y, icspzcctivaly. r.iEution (AlZ ) t'..:. 'ozome:
Y=2h sin2 +"2.:3 sin?24 + .75 sin22% cos 21--+ ... (.1i)

and
dY -hh cos 4-l1h3 cos 4 cos 24-16h5 cs (1 + s a ,)- ... (..12)
dX


-..L'ation of w as a .c. :ri :2 e in r

Consider c i-atonr (.1.L) with a = In",- aor
R2= a2G( + jh2). Then

S=-U z + a2 2--) 1- ,-. 1 (_.15)

Now

Z" = Z' ia tai.

= Z 2iah

Then by expanding the right-hnl. side of c.,utlon (;.i7)
accor di.' to p .'.,:rs of' h and re. lac:r. ':

z + Z2 a obtained fr..; the Jcuk.:'..k 1. nsfcr-
2
nation (A';), it follows that

S,1/222 2 ,,.
w=-UZ+2i & il-- -..logL Z ...

If w/2aU and Z/2a are writt., resr ct'-1., w and
Z, then

w=-Z+ih 1- Z- 2- -2 l Z+:2-! + ... (A14)
LJ









hNACA -F.R o10. LIcl]5 21


From equ-tion 41(, ) for' a.l a .orres.-.dii'.g e:juation
fo" the cotm.plex conjugste the noir. :~rnsional velocity
potential 1 becomes


-- -J2 12
/ -2

z-. .. +A~-- ._ + .. (A15)
+ --)








22 IA.CA A.R: i.:. L.;115

APPENDIX B

DEI'Ei,J~!L"ATICN OF ITEi FIR3T APPRCXI ::ATICI 1

By means of transformation (14), equation (11) f',r-
fl becomes

a2/, 12/
+ -= 0 (31)

A comparison of the expressions for 7 1iven by equa-
tions (9) -tnd (A15) sT,;Cests the assumption


S= k a 2z2 [ i 7 2
-

z + ( 1 2
+ 2 log (P2)
.+ (+ 1)1/29

where z = x + iy, z = x iy, anr. k is an arbitrary
constant. Since this expression for l is the sun of
a function of z only and a fu.iction of z only, it
satisfies LbrFlace's equation (51). The arbitrc,-y con-
stant k is to be determined from the bounua-ry condition
p dY 60
6X dX bY
or

L d 2 6/
6x d.': 6y
The expression for /, insofar as t'o first p'..er in h
is concerned', is
$ = -x -














a.nd, to the f i2r-.t p ."r in h, fron- c .1.ton (. 12),

C:



The bounda-ry cnditio, e.ati.on (13), thor bec.ar es



,Ph COS =-.'.




\z z/




first z /
t 1' '' 1 1 I 'i" 1 -.1








r .s t- e ..e i.- I,,


z 't + 1 iL.'


I = cr C -


- I: n i si' ',


Z- : + siih -.- -


Z- 1 -


(2 -" =



(5 <2/-: =
I' j


.= -" 2.-'C


.i In 1 (. .,i' 222 : )


-1 1;- '- [ '-^ \: P .: *


1 -:
i 3 l


LI: .


S-.1)


Then


I-il.:A AF.R :Jo. L',.il5









24 1 ACA AF F T.?. L4-15


or

k = --

The exi ress.ion for the first appreciation ef is then


z + 1
+ 2 c.- --
S+ (2 1 i


(B5)


This expression for j can Ie .i.-:'pfied ccrnsidercbly
by introducirg elliptic coord'In-ts .n ad Thus,
let

z = cocv ( .I


where


= .4- 1-


Then


x + iy = cosh (A + ")


= cosh F cc,3 + i .rh i cin


so that

x = cosh r :c, r, j

y = sinh ( siji T

Equation (B5) can then be wrftt:en

= oh + cosh ) e- + 2 lo. e

or
c = cosh cos 1)


( 7)


(d3)


- (-72


- 2










I'ACA APR !:o. Li-,u15 25


Fro-" a cci.ir 3scrn :1 ec i-aions (A15) tnd (B5) n1 -te
that, if T'-i and 1, de-n te the circulation in the
incomipr-.-ible ca:e a.r the c '.essble cas', then





: 1 0)

1- M
(i )

Equation F.1') is t' .- 1 n..wn Prendtl-Glauert rule
connectii,- tn: cir-:'-AJ I 'r- (or lifts) in the incom-
pre sclb e -and .'. c:.r e .i':.i..: cc: es.

In orp.j-e t i".l z: ., .:ion ( -) for the celcule-
tions, the e u..tion o1 f r- 1-'ns formationn (B7) must be
inr. '. rted. T'nu ,


S_ I


(B11)


CO T, .-

Fron equation: Ell,


Sr = + o +

2 inr b +y2)1/2

where e

b 1 2)

3y rrfear, :.., tr'- rzi. .r '- 'ti n "( i ),


2 si -
2 sin = b *- 1b ^P2-'J









rIACA /,iR '1: L:..15


where


b = 1 (X2 +2,


In terms of the complex variables
velocity compcnenrts in the direction of
axes nre


1i 1 r2 1
sinh t t sin h .



s i -


( cnd !, the
the coordinate


(B15)


2~1 *'~

~Y)


Let be given by equation (5-); t.crn,


u = e sin


v = 4-e cos nr


(Pi!)


Now, to the first power in h, at the bow'-d".ry,
T = 0

Hence, if q and qi denote the mr.initu"-.-s of the
velocity at the surface of the circular ar.:. Profile for
the coinpressible and the incen.p :!:.ble c._-: :, reSji:c-
tively, then

q = 1 + sin 4
1 (315)


qi = 1 + 4h sin 4

or, when h sin is eliminate'-2,


%i 1 ( qi


(Bi6)


U
v= =



6Y
bY









NACA ARR No. LJ4Gl5 27

where

= 2

Equation ( B16 ) re rezc nts the veloci ty-cor1rect ion formula
for the Pr-.natl-Glauert approximation. Equations (B15)
can also be wriLten as fol ows:

qc---- (B17)
qi -

Since the Prandtl-Glaucrt approximation is strictly true
for infinitesinmal Jtsturbances to the uniform scream,
equation (bl6) may be replaced by tne differential coef-
ficient (frcm reference 11i

/(c 88
-- = (B18)
q I1










!ACA 1ARR No. L4015


APPENDIX C


DLSTRJ.IAT:I:E C' THE SECOND APPROXIMATION 0'2


Dy means of tra.isfor..n ion (14),
relations,


6 _6 +
Sx 6+z 6+
Ox bz


6x2 6z2
-= +
Sx2 6 z-







2 ~ z2
8 z


O n
-2 '2
A i --.Z


2---+
c z


o 6





+2 62
0o 0Z


62


62
, z-2


the symbolic


(ci)


ana the equation oD transformation (36)

z = :-oh

or


. = .:c sh h


diffrer,-nt i.1 eu quat ion (12) for ~';
term of the ccmrilex vacriables


.2n1 b, expressed in
and t as follows:


-2)














T T A *, '.o. L


El
c.-i
















- I 0
C"-













-- f




r .I
+




































C ,



,7,









r-l

cri
'I-










,-41.i -"









-J-





r'l


' I

i..j., i |.,..,
ir4-\
6.3.

...i



.O I .,--
I i -.






^ii





41.:










Cl C
-I







C', r"
&*. ,.

C. I C-'.


I ',c"






I, -
, -,
-' I .

















*Lw
i












4.-.-)
_, .2



-







^. "'-







i .


-N
7 N

.1



Ij,~


-J -4
--N

-I-







7,~fl
rH I -,

'a


.-




-j
..-
'- L .-i





















*' 1 (*- )
+ --1
*2
"' "--

,-L 0


I .

i -5


(If)

S-'l J i
-i
,i (+kl



-) o

-i 0
I- L
*j 4
-I -





.-. 0)

-i .r-1
-t --I -

.01



-1 0


.... r.
.--{
*-I 4t)

.-1 I
a1 H

o (0
*-i













p. e -(c e sinh j + e s ir.h

Finally, by pucting = + ii and = ,


S 2 )

+(y+ ) 1 p2) e co ) Ai (C)
/
The right-hand side of equation (C3) s.g .- :ts a solution
of the form

12 = PF() cos n + F(?) coC: 5 (CL)
By substituting this expression for 92 into equa-
tion (C3) and by equating the coefficie-nts :-f cos r End
cos 3r- to zero, the followvinr, differential equations
for Fl(,) and F3(P ) are obt.:ine .:1

2 F
- = 1 2 (y+1)-(y-5) e -+L2e (C)
,--1 = ) kY+l)- (Y4)p' -- c5)



d2F 1 3l r
9Fq = 4(+1) e (C6)


The solutions of these equ L.ions 1 r

F=2 Y+ )- (Y-)P2 e-e ++ 20 '.- (c7)
V4 ^ J/


IA;A A. ::o. Le415








NACA ARR No. LAG15


F = ( + )(e + e-35) (C8)

where AI and -z are arbitrary constants to be deter-
min.d by the bounderry condition at the surface of the
profile. The other tvo rbi-htrary constants are taken
equal to zero since F, -,-a r F must vanish at infinity.

In terms of the variables ( and r, the boundary
condition (B3) takes the form


L 6 =/ p y(c, /',\ ds Pn .
sinh cos n-e cosh sin -- = 9 Icosh sin


+ sinh cos r- (C9)

where tini velocj' ty potential V has the :orm

= cosh ( cos n -- sin 2r, 2 n)


h ,(F cos n + F, cc3 7 + T) (C10)

and v.bere F2 is3 an rbltrar; circulation to be deter-
mined by the Kutta cun:ition aL the trailing edge of the
circular arc profile.

In order to makl- us' *'L. th boundary equation (C9),
the various function.3 of a and rT e ppearing in equa-
tion (C13) must be expressed as functions of 4 evalu-
ated at the boundary. From equ-.tioons (All) and (Al2),
the boundary and it. slope arm no- J:iven by

=- 2 sh sirn2$ P+ Fh n 2in2- cos24 + ...

Xd -_ = h cos f ...
dx








32 NACA ARR No. L4G15

ft the boundary then, with x = cos 4, when powers of h
above the second &re neglected,


b = 1 (x2 + y2)

= sin2- 2h2 sin4

Then, fror" equations (?12)

sin2i = sin2 (1 + Up2h2 cos2)
s In (i sin i2h


cos- = cos24 (l &2 s2


sinr = sin. (i + 2W2h2 cos2 )


c.)s ') = cs z 1 2v h2 sin2)

h2 2
sinh2 = 4I2- s in-4

1o- D I + -. -
cosha f = 1 + jb t:'.1 *


s trLh = L[h sin 4


cosn = 1 + 2 2h2 sinv4

e"- = 1 20h sin 4 + 2P2h2 sin2-4

S= 2ph sin 4


Vh'ben these expressions, v.ith equations (C7), (CS), and
(C10), are' utilized in the bon.lIory equation (C9), the
following results crre obtained:









HACA ARR No. L.4G15


1 A-


nL4


7 + 2(y + )2


The value of the arbitrary constant I2 is deter-
mined in the following w&y. The magnitude of the velocity,
when terms containing pownr-m .of h higher then the second
are neglected, is given by


= + + h' p 1' 1
q = 1 + h + h p- -
6xL T)* ^7


-I-I
rtyi.~I
+ ~t~j+


or, in the variables _, and -n,


q= + 2h nh cos h -cosh sin _]i N)
cosh 2 co 2n
2h2 /
+ 2hnh c os -- o'osh c sin I -2)
cosh C=-cos 2- q ] /c

+ 2.2 1. 2sh b2 sin T: 1+si.h i cos )4-2 ... (012)
(cosh 25- cos 2r).K c ,


From e4uetions kBq) and (ClJ),

/ = (e sin 2r 2
-


and


(ci5)


2 = FI cos 11 + F5 cos 53r + F2r


(CII)













34 'AICA ATRt I:o. LG1-5




Wr-I




-4 -. "0 ) 1



+_ 42 ) .
00 'a a.A ..





S* 0l 4 0 >-' .o


(* N- -,-




mW I ~ -. )- >r :e|
0 -. .Z U -i I r-- II
S- ") 0 0

O ,.- 4 r *- I C I El

'-.- I ,. 4 -
> ,C E *- r i|


LDI 0 0 0 NS


04 % Li) (, .+ 3
'-I e C, -I







;0 I | 4-1 C 4- ,..
+ o ,n j_ -"; ..- 4
o c r(- 4 C. ) T -~


.Jr Z) A I I W


- + Q .. U -
,r ,i p u -i o eL. o ;i0




C4 + Co1' *-
H \- L- .,-1 O c *








S- ( U- D r O
0 L-.L1 I U co H )








;ACA ARR Ho. LIGI05


For the position of m.txinmun velocity, 4 =: 5-,


1+^h 2-
1 + + h I -
L


1 1
+ ) -..,+1 + Sy --
\r ) ^


(C 17)


i(1 + 2h)


For the position of minimum velocity, 4 = -


1 -+ h
+


q
1 i


+ -+1

2( 2
(1i 2h )


/IvV


(CiS)













36 NACA ARR H;o. L1.G15








N
0i-


01 "- '- M
C r 1

+
p\ 4-3 N


c ,--r 0 c




1 -1 m .,) ,--- ,"
-" -- 01) .
1)N









0 < .j CI N"
It1 +


)_, I r-k + I

S -- n ;--' -. N j

., r )5 -J e J C)
CM CEll 0r


N N 0
W4
S- ,Ns









-.2 + I



%r
C- -I
N-4 I3 CN I



a + N I1
-4 L4 N c
Z: C r-r- +




c-I U' + +-a



4X.4


2: -1 ?
+ ,- .1-I












1= --2
L


1l/2
z + (z- 1
+ 2 log --


Introduce ne -:ormpl. x ,variables X and -, where


S= z + (z2 -

A = + ( -


1 /
i i .


= -

S= EZ


1/2
(z2 1)

(- 1 1/2


The r*:ltins b ,c..een the complex variables and
a;rnd the complex v.riabLes .e and (, respectively,


. = e

S= e2e


X=

= e2i1
e


(D5)


are


(D4)


Then


)


S 2 Log -) (D5


Similarly, the .*:.;-r-ession for 2, obtained from equa-
tions (Ch), (C7), (C8), and (Cl), is


=,-.=2(D-E) log kX+D +2 +(-5C +D -5) +-+


.2K2 x323


)


(D2)


- -2 1 2








fIA: A No. L- .15


2
0 =- -I -
3

D-
2


E=-
4


1 p2
p2


1 1 .. p2V
12


r 2
p2


and


C 2 D E = 1
3- -S =


From equations (D5) and (D6) with the use of equa-
tion (D5), the follow." ing relations are obtained:

2 1

lzz -
oiiz 42


2 =4 D + EX
\ x


1 -1 X)-2D
K -.1 X i


+ 2(C -D+5-) 1 +2E
\2 -_1 2R2(X2 1


\ + 2
,cXe( 2 i)

1/ .


where







;ACA AMFi To. L4015


log ~. _3 2x3(. + X) 2 -
-(X2 1)3 K( 2
xA- _k 1)?


2zz =8(D -


D -8 8(50 3D + ,E)
?,


.7 + 202 1
2(2 2 (2 )


-Ez -i -dC
2zz \\ /(x2 -iQ 1 1 )


(x+ )( -\)2
.(:. l 2 -2 )


ana expressions for the corre-.p-'rinRdlng conju:'.cate complex
quac.rt cities.
lien th, forrgoin'. express ions are introduced into
equation (DI), and .h-en equat ions (DL) -re used to express
the various qiuantitles in ter-,is f" tL.: v-riables
and r, the foilo' in. d'iffcrentini eqi.tion for 9z is
obtained:


















-4




I
U3





7 O






i 0
r -4








cu
-44





+ am

+ +


















+ +-


(-j
-:t r
_.j V
in







+ p
9.0 0 0
+






































r l If


0


U,

2

F-,



-t
a,


9.'e
C
U.'i





1-fl
I




.1
-I




U)
I

('I









+
1-'




Ij.









C.,
C"'






C.J
4


14.)
LO
L




C.
C-
t-J.




1^
+







C"

















It),
4i

I-



















wl
N-



ii-

vCu
c'j



+




1..




U-


i-

iji









C!
fJl
0








L.
c___


!*.0.. A^ ', n. I0 GL)



Fr-
-- cu

"-

o t3
91-1


U,)


+ +


.9 .4


II 1
IJ i I-


Io +


C)


90
.:
+ CI


*0 <0
o I I

I U)J CJ
Cu 4- I


4"+ +
I i'l N









(di I I
,,cL. Q

L'.'.









+
I ,L D,
"-,C.G /~*

02 I Id
^J1















+t) C kfl
0 '-0



I + I




._J1l 1 i
'< + r+

krj 0 -.3


*I IJ'

j C I




co
-u "u I
I ~.%


OJ J. +
--,J
+ 0 Sd
,.u*'I @ CJ]

a ,
L__C1 c.[


+

1,-.

4-
co



CM







r-r
4.

















I
(M




0






C'0
Cu]
('.

0,.
I.
U)








FACA ARER 110. LG!C-15

22
A. = 80pD(D + 1) + 8rD2 (y + 1)(6D + 1) + PD4(y + 1)2

A62 = -1920D2 2pD' (y + 1) + 12pD4(y + 1)2

A2= -6OpD-' (,y + 1) 1PD4 (y + 1)2

A102 = 96pD2 + 4hD5 (- + 1) + 6p.D14 (y + 1)2

A2 = '5 (Y + I) (15D + 8) + 4F'D2 (y + 1)(7D + 9)

A = 96 16 D5 (y + 1) 10 ~. 1(y + 1)2

A64 = _UD- (y, + 1) -.D4(y + 1)2

A6 = -224pD2 16p5.3 (y + 1) + 22pD4 (y + 1)2

A10 = -84PD3 (Y + 1) 214 (y + 1)2

A12 = 123pD2 + 64PD5 (y + 1) + 8pDl (y + 1)2

B2 -8pD2k + 1) D + 4]2

B 2 = -12D (y + 1) (Y + 1) D +

^2 = 16[Z2j( +1) D + 2

B42 = 12pD5 (Y + 1) L- + 1)D +

3, = -16pD2 [( + 1) D + 4]2

B64 = -20PD5 (,- + 1) (y + 1)D + 4]

B3S = 2! pD2[(y + 1)D + 1]2









42 :IACA ARR cI:. LLG15

Ci = v2 (y + 1) D + 2] y + 1) D +

C2 = 8PD2B7 + 1)D +

Dl = L'pD2 L( + 1) D + 2J

D2 = P3 (y + 1) (Y + 1)D + 2

D = D2

D, = 2EP'5 (Y + 1)

'ote that

B62 = -2B22 2 B' ,12 1 2


B2 = -B24 3 4= A L A.

The ri'ht-hand side of equation ('7) suggests a
solution of the form

Gl(G() sin 2r+G2( ) sin :r,+j Gn(~) sin 2nrj (DS)
n=
i..zan this expression for /. is inserted in the left-
hand side of equation (D7) and the coefficients of
sin 2r,, sin .r,, and sin 2nn are equated to zero, the
folloA'ricn differential equations for GI(*), C2(),
: Gn() result:

d2G
= (k +d+ .+ (A2 +. ) B)e-

2 Ae-" A 1 (De9)












l A A AF'R LL CIY


Id
o


-t
a'

+
-4


+







I
'1)

-t



-4-









'0


'C
+




-,
s-4

+ kfi












-1 -'
"a- C)l

S-4
N) 'CJ
4

itj 0

-4
+ "I



II

riJ

cuI-,
1-1

0dj^


+ Cj
,2 -- ,
I ',,. 0@




IJ n
CJ + -

cu .-. I' ".L-
c' ^ f"


k, c'j Q.', q
L) ~ N'l -4
Cd ,. Q, -^ +~
+ C) +


j -j o
++.--+
0_ r-. -.$ I C.]* Ci .j, ri-.4
"J I




+ ^ -I+ ....
L' t ,'-.n ^O / l' ^ I ,+- ,"


J ,N
'-, 'C -, I r'-




,' -_ -- '-' +J >

c -- ,
I I +_ C.
-i- ,N C' + -.. c cd




i---+ + + + + o
+ --
l w' C +



+ C + 0 + 0
I- ,-t _-, o *-'


--.-l + +J + + + oS
Clil

0
II 0
+ d,--4











,---
o..I 0


+- i
I
c' ".J
-. _
La1S+l C _










41F A "A.9 I. LL.013










- -4++






R a
Q)) '0






90






mI I i a
S + +


'.0



++ + + 9-
r-'4


lJ J) c -


I I I VRI I c i
s ) C,,) r,






I OI r- r4 +
i + +
RrI H "eI







UAC A ARR o10. Li5 1 45






R lo0) .-
r1 9 r



t-:4
-kfl + -,
iI H + 1



+ + c



0 + +- +

-+ .-
r-4> C. -
+ +


I ,--- 1 6A + +J fl

+ +








+ T 00C N
N- + + ,-4 ?D
-- l j -- > -- -o + +
+ +


--+ +._
I \ 0 -I10 _. ,_
H + ^ ? -J- .-t





-+ + + + H
+ H +



++ N + H
+ H L + 0t





12 -- + 0A
/-. ~ ~ r-4 c ~ ^l~l H Q
a S-^ + ++
C-r &D
r~103








D1 -D5 = PD3 (y+1)

D2 -D4 = D (Y+)2

Dl+D2+D + D = D (y + )2 + 8pD (Y+ 1)+16D2 = +C2


Di +D2 -D5 D4 = D (Y + 1)2 + pD3 (y+ 1)= C C2)


The arbitrary constants kl, k2, and kn(n 5) are
to be determined by the boundary condition at the sur-
face (the boundary condition at infinity is taken care
of by f-utting equal to zero the other set of arbitary
constants that normally appear in the solutions of linear
second-order differential equations with constant coef-
ficients). It is now anticipated that the arbitrary
constants kn are independent of n -nd equal to k,
say. Then


Gn sin 2nr = G
n=3 L cosh 2, ccs 2-j


e2z sin 2T] e~4 sin (15)


where


N/.CA AfRR No. L4G15













; AC A A'.A T' LL&_.

II


S
--j ,, 1' r-- ... -"
rl _.Ji +
'i i "' 1 ,










-- j
ri+ i r-- i.-di


+ >- -





-i
i4 -_ .2i .



8.i1 -I *- '-
.-- -3 -N -'
+ + ---'- ,

+ I
r "
+ _--a _- "- + *... A



L J L--J T *-- I-


+ 'a,; I tJ1 -'' -I



1 (2 + I >1 |

.r-i- + +.I -, C I')
l I >I J '



I I i I 11
S, ,+ -' "4 r"





II.j r-" II ':-
+r -I rIi I f

-. -- I + .2
+ -+




iC I -i 'i C,

r *, _! i t' i I *I '^--i
--- --- +- .- 1 '
S+-
S:,,
1-."'I -+









NiCA AHR !lo. LGI15 -


from eq-I,.tions (C3), (C6),


(07), and (C10),


/f2 2 y+1)D+ ("-+2.-+ 2 D+D ( )D+& e
=(2 ( y+ 1')+ + -'5 3 + D[ (y + 1 D 4 e 0 Cos oa

(y+1) D2e'+- 2+12D-D (y*1)D++ -o ) cos 35n (D17)


where


T-


and, from equations (DS),

5=C,() sin 2-n +G () sin 4T+G(+ G(


e"2 sin 2r, e-' sin /+r) + rF+T

wr're GI () and G-(,.) cre -,ven by
and (Dl5), resp,-ctiv-.-, and J q() can


sin 2T)
coalh 2e -cos 2r,


(Dl8)


equations (D12)
ce written


( ) = Je -I + (-J + K -


2 -- 2E
+ K2 Je t. 0, 2 2J e
~ L) \


+ 1- le --
2 -4


(D19)


f. e- +- ~+ e-3 +k
10


w ith


J = pD1 (y + 1)2
K = D2L (y + 1)D + 4

V'JK = D (-y + 1)[(Y + 1)n +









!lACR A RR If:. L 015


The arbitrnr, '. tont, k,, I.:, an.. 1: app 'ring in
the :-x rezsions for 2, nrir G, respectively, are
deteriired L-.b the bound,-.ry c n -iti.r. at th.'e sL..rfac.e D
the c irc'.ul r ar. proPiL e. Tlhe ,J l i' -nf t.e -r- LLra. r'.~r
circulati:,on: F is dlr-teir.nined t'he V:ttc c :'nditin t
the trailin- JEc, that the v l'j-i tvy t-here be frnit e.

In r- .1 tr eve1 liTe thec v .r' us LP-erm:'- (.pp. r.in;
in the b:u .1-rv cn. i ti on, ae.- u ti.-n (C? ;, the oll1v ing
r'el t ions '-r ntc,3 sarz-y F:.or. e.iuL t iz:,n (All) ,nniZ (A12)


= p2h s/ + sir- zcs) +

=-- cos l6 h c,0 -.c c3s 2. ...
CC,

From 'i-q tion 3- .2)


b = + (L4-. c .- + .





0s35L n = 1 + 2F~ n s :+ + .. .


e = 1 + : :. .9 + 11> ba 3.ii-%

S 2, -? sin 4 c + 3 ~~. -2 + ,in -, + ...


= 1 2"h s s L .- + ^, ni sir- .

;*h^ s". i + 1, .':. 4 .' ,. : { + ,in.. : + ..


r2= h .in -r + 2ph-' sin 2 c

i 2 s-i. + cs- + sin1') + ...

sin r. = 3in 4, (1 + cr. .c + ...

'20- = ccs ( 2 2in1' ...,









50 t A.Ci, L- R 4('15 -

hten the e':,resion 7or, ,' giver by -iquation (DI6)
is sulbsttited into the aun.oar' ondi ti.-,n, equation (09),
the coefficient of h' i t.e le 't- ha d since is

pD y+1) D+J -l4p+6P3# cos 4 + .-2,D y+1) D-) +cos 27

+ )(y+1) D+14 +2"- coGs 5 (D2')


and the coefficient of h- on the rLght-h,:,n:. side is

-36[- D+6 + D2 17 2(Y+))-? )2

AD(+1)2Ios F 2 7 D 5 D2



Ir i -k1 + Ck- k
y++) +4 (y+1)2+ y+ 1)2 + ck + -k7 o 3
o 5 +10 P
r~
4+ p+ 4- D+D+D(+l) ++D (y+ I) + r (,+)+1)+ (y+1)2


(-2-2 +. D
+ 0o S 4 (D21)









:ICA ,PR ,J. L'G15 51


It oitc1-1inr z -'21), th f i lowing rela-
t ions re a 1.t11i .r :


= 2 pl 2
- __ M-_ 7


+ ~..,2 ( + 1) + 2 .- (. + i)
> 5


I
02 *iJ)~ +' I -~ eE


~
-j]


2.L-' (y 1 i7: 1U
I


G0) =- -
IC
G ( 0 -4


pD (C+ .- ',- +


14*- ~L< +P


'I.-
-- I
:~-i Q~~J


4)= + 6.,2(y+ )


+ PD3(Y+I )2- 5


-5


;D4 (y + 1)2


+ -- ::.'~'~ (*~-+1)


--I -
-I-


- 4k2


=


: ".'ic ents of cos -, cos -4,
ar-.r f .. e : ) ons' (oL.r) ai nd (C2 L ),
tr.:- 1 i. '.I i .' l or the 'rb '-r'r constants k
I-- -I :- [ I *-a .' .: ,I r : J :


G ( )


. .:,i '


> (D22)


-L^ (y+i )








iN.CA AitR. 3 io. L C- 15


k1 -k = (Y] .D -- +1 2+- D- D2 D(y+1)


+ D (y ) +)2- (-+192+ 5 D4 (+ 1)2 (D25)


-kl+2k2 k = [(y+ 1)D+ j + li D+ D2


29 4 7 2









-D2 (. + -' (y+ )- D (y+ +I) (D25)



I'3te that the sim. c .' ::. i- (r2.) end ('23) yields
e- i:..tion (L25), sf t' et' t rs f r <.-, ks,
and 1- ae no S .;.. :' -:i. linc one 0i' tihe constants,
s .:; j, .s en i'L .,-L r ':;t. ary. Ic .;ill -.e seen in the
foll '. ing dis suss-'in ..I-sL -.rliit ur, disposal. of kL
is necessary in crie. t.i6: r.3 jn'iiiitc v lc-cities oci'ur
on the circular nae pr
The velocity comror.,-;nts a.L.n.- the X E-nd Y axes
are Jiven by

u = -L=X sinh C nos n -T- cosh sini ) YJ L
oX cosh 2Z-cos 2n 6c, (D
0> (026)

v = -_ ,o'h ( sin2 E-:inh cos r1 (
6i cosh 2 cos 2I i







NACA ARR No. L.'G15


Along the chord of ti'e circular are profile, 0, = 0;
equEtinrsr (D26) the-efora teco mie


sin r dr


sin r, d


*1
7

-a)


By means o:,f equator. (D.16) for /J and the
for il, 02, and VA, it fellow. r asivl
tons (D27) tlhct


u=- 1- sin r+h'" <12 os 22 + + 1 )D+ (2


+ -- 1231G) .os +G0"3) L Gr+F
sinr L -


+ -- --(0) -2 cos crs 14 c csrj
s in n L


expressions
fV'oli, equla-


cos Lr -


+ C L) + ..
d 3)in-r; j


1
-j


(u12 )


v=:4h cos r,+4'Q (2D+5) sin 2i-2n h cos (I-


dG \ 1
-( -_ I-- cos 2r0
1-j? sin- i


+ 20(0)(1 + cos 21)j + ... (D2?

At the trailing: ede, r, = rT or 3sin r] = C0. lr',.ce,
acc Prin1 to equation--s ( ..) ana (D2'.), infinite
velocity sce-ns to or':,ur t-r? The Kutta condition at


([27)


+ 2 +
7 a2-







4h "ACA A-?R F:. 1.1l5

the trailing: edre, ?ov'evr, dei&ands that the velocity be
f in.. tc. .-r.om Lequ&tZLis (D2) Ut is seen that (d = 0

so th--t tl-,L velc.ci ty ccqmnonen-t v is finite r.n the
bournd:ary. The velocitty -m, rpcnernt u can be r.-cndered
finlte by showing tlh.t theft co?-,'ficients o'" h in
sin
equ'-ition (T2-) c.:n b, made to equal zero ben ] = 0
or n. Th.is, sirce the- constant k c. cu .".r.ng in equa-
tion ( 1. S i s ;..l- tr' ry,, It can he 2h: sen so that
3) O J. .ain., if the i' -rst cceffi.tjent of
-i- i-n uaticn (D.L) v-.ni-nc for r, = then the
sin fl
ci, r l] &.ti'n cOal 11 -cnt'


rv = -roU (0) 4a Io; (D15')

w bhe'r- -. (u) C:;-. .. 2. ) ...e .L.. .j b;.' eq,.ations (r'22),

rje '-Lbitra- cons tant k is -'.n dete rmined by
L1. e- :a1i.- n r )- = '. Fr'r equrt ,.r [,2) Lherefl',re,

-, = 1 -JL + )2 2.,"( L) + (D )
2 I

arm -i .,UtiL (D2. ) a.. (D "- -* -pa L vely,

= -i( +1)L+ + 1.2 + D2+ (y+1)


+ (7 + i) + 2 'y 1), 4 (Y + 1) (D2)
S+ 0


*- -- (_Y+ -) D+ 5 2D + L2 +2,2 (,+ 1) D (y+ 1)



+ r D (y +i. ) + (. +:.)2 (D55)







EACA AP5. N,. L'.31] 55


et"te tih t, t o tl: i'n.:- "- ibl fl: past a .-icular
arc p .,of[i1 teen .'e t n. nr ac a cli.- tc th' me ,-.': s of
che. pr--' enc p ..-. c -i n ..n.il to t.l.? e .o :i rc'
w-ulJ h1 j then .. vces ..,, 't, L-. L-- .- ult t l.at I = ,
k2 = -4, k = 0, rilLd 7 = 3.

SuL titL..ting fr:.- q.t-lcn IL2T1) for 0 (0) and
G-,(0) into ec j.o' :,.,.n ) "ivs

G .-- -0 c. c, -,- S
T-"-









r.
+ Y +






te;'r2i ,-t'l:. t :i. s : ,-.:,sib e case, obtained


T F- c r c -wi (


T e c-r, iIA Ll'r, P, thle c.rr:ebible OZG, Inclusive of
te rr..z .,n.--nr i t ,: V. & ,,, r h is obtained b-
adJir.g h~. ci.c. it t ..:rn. 'ci.-. .-iuation (l E ) to the
v-l.:. "i' r., r.I'.,en 1 ., :'.. (:' 4) and multiplying
tbc result u,- !TUa. T.us, if D is replaced by ]- .,


2 1 2 2 2
4-h rst- 5+ + ( + 5



+ (Y+1)2 (- ( p )+ h (r, )
12 7J

T'h circuilt.L.r. c --rection formula then becomes


-^ + (T+i) (s+P2)
L + 5 P


+ 7 (y+ 1 + D 3 + r + (p6)







56 FACA A: : .. ",' i-

The first term on the right-hard side is the familiar
Prandtl-Gi.iuert term sc that th-e second term represents
the first jpearrtLui e fro thL? Pi' ndtl-Glauert rule.
The .na.nitude of the velocity: at the s,:rface cf the
circulLr arc profile is caic.lat.. by the us.e of equa-
tions (D26). Thus

,_ 2 ha2 2., 1 .2 ,1 (1
q= 1+h + + +h
-x 2 / -x 6;/ \cyl

+ P2 +1 + D3T
6y Oy 6x

vwl.ere, symbolically,


--= tr sinl cos 7 2--ch sin -
ox cosh 2 cos 2rj r



- =-cosl S I nr r, + 1. ?. co r
.v cosh 2 -cos 2T 0



and the expressions for i/, vnd J are given by
equations (B9), (D17), an. (l '). V.hLn ell bhe fJnctions
of L and ir are e;.;:r'.' c-Q f..inctio.? of 4 at the
SL-'iiace of thT profile h.fL t..'. .nt.-'l"'_i' g p .r-rs of h
higher than the third ar r.;- ,t, t,- ex~:u.zion for q
becomes











I Ala Ar;R i'c,. L-.i 1r


0 1

CL



-....
*;-.


-r-'t N"
L.-


U


t'-'


<'%





'-I







+



*~ I "


lj-i
tj^






Cil
'2








(--





+



-4
*. 1



' +


1-1
+
r-




+


















,7 '
I



-I
.-I
'..*2-.














4--Hi -0
i' i

















a
.--




-i- -



,'-l If J









(I

0

1


I



,II
0


[-
-1-
+
,i.n
--4





-I

+





t-l
+





>-



+
'-U









+
J.%
N- !









+

01
o

--It
+







0


-o
i 0








o4)
4~0
4)




O4-

H 4-

13




0 O

O *H













a )
-P ^

4-p
















So
H .H










0 o0


0 0-
r-4 0





S0 1)
00


po
-pC


r -f



-J
+


+
1-4

-4-
'-'I
+



_____ I

+ .4


1--4
-I


F -
a

01 01
o
01

+ 0

o 0
-H
H -P
a)
+



1.~~'
C.,

I.-'
I.-.

-I-


+





(M

H
+






HH

+









58 ,..A A<..i y N-;-




W 5 C. 3 -

S. ---

*-; '
o o o
4- 0 1


f oo r '
r -4 0 0



!./ 1, 1 0 J
0 ,0 C
4-1 > > i





*H -
0


VII 02 1777 0






S- 0
0 0









o J0 I + ,
0 a) -0a 0 I



Co r-4
co o 0 ,

w0 Vi x/ o
0- _C CC


S, r- r-

0 \ ) ,I
o W:o -





o -4 *4 + H


S <.4 *-4 +


0- -





+ 0 4




1 0?0 4-5 r-4*
i, O0 0 C )I
a 0 F > 1









IACA AR: No. L. 15 '9


1,FP F "c F S


1. Janzen, 0.: 5litrag zii einer. Tleporie der stationiaren
Str6mung ks.nwressibler 'lUssi.:ke Lten. Phys.
Zeitschr., 1 Jah:g., :r. 14, 15.July 1915,
pp. 639-6L.5.

2. Rayleigh, T,crd: On te Plow of Co.-nres -.ble Fluid
nast sa Obstacle. Fhil. ?a.," ser. 6, vol. 52,
no. 137, July 1i96l., pn. 1-o.

5. Foggi, Lorenzo: Canmo d- "eloc.th.f in una cor'rente
p..ana dl fluido compres itie. Parte II.- C sc. dei
orofil" ctt3nuti c.n r:-J.resc.rntazione zonforne dal
c6rchio ed in carticol'cie d.',i profili Joukowski.
L!'. sroeocnica, vo X;V, f;--c. 5, [[ vy 197 ,


1. Imai, Isao, iand Aih&' c, Ta':asi- On tlhe Subsonlic Flow
of a Cf1nmrre.. -ibl F1-lid pasc aUn EiEptIc Cylirnd r.
Rep, No. 194 (vol. XV,, eru. Rps. In.t., Tkyo
ripe ial FUni'., AI. 19s40.

5. IOr-laf, Carl: Cn the Use of Resiue Th-.eo-: for
Treatin: the .juc-sor.ic F"low of r. C2mrressibal
Pluid. A ... .e-.. 'c. 7., 1 2.

6. Frandtl, L.: re..ar :3 cn naper .1; A.. ?.usemann entitled
"Prcfi e:..es3'.en-en ,:ei ', sc.v,",.di,':eiften t,ehe cer
Sch-allgesc-t.indi':eit (:mi H_nblic: auf Luftschrauben),"
Jah-rb. b .G.L., 1- ', pp. r%-.,

7. Glaliert, _,. : T-'e ECXect -f Co .c-rres: bil t,' c n the Lift
of an Acrofol. ,'. S& P. I.. 1135 rritish .F..C.,
1027. ( 3so, F.rc. R_.. Soc. (Lcr.cn), e /.,
vol. 1, nc. 779, rch 1, 192:,, pp. I5-11).

3. Ac.r-et, : .rer LJ'.ftk. fte hol aehr gr:'. sen
'-ch-vird'i.:eiten ir3bescndere .e-i eb.nen .3tr.ir.'ngen.
lv tc P.-.,sica Acts, vol. 1, f7c. 1p l,
pn. 501- '22.

9. Ka.pl.n, Carl: The lcw ,'.f a Co::.pr-szible Fluid past
-- Curved S'ui'2!ace. 1' -. P '"o. 5::2, 1975.









60 Ti. A /R.I if. 14015

10. von :.TrmIn, Th.: Compressibility Effects in Aero-
idnramlcs. Jour. Arco. L!., vol. 8, no. 9,
July p. 537-556.
11. Garric, I. E., and planla, Car!: On the Flow of a
Comi. presible Pluld by the Hodograp)h Ye thod.
I Unification and Extension of Present-Day
Results. NA"A AC R Fo. 1+C2\1, 19-4.









NACA ARR No. L4G15 61




TABLE T

RATIO OF CIRCULATIONS FOR COMPRE3SSIBLE AND 7COMPRRSSIBLE FLOWS





NM 0.10 0.20 0.30 0.40 O.4B 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90
Appro*1-
iation

Prandtl- 1.0050 1.0206 1.0483 1.0911 1.1198 1.1547 1.190? 1.2500 1.3159 1.4003 1.5119 1.6667 1.8983 2.2942
pladert
h = 0.010
Third .B0050 1.0206 1.0483 1.0912 1.1200 1.1550 1.197 1.2508 1.3172 1.4029 1.5170 1.678 -1.9348 2.4603
von Karndn |1.0050 1.0206 1.0483 1.0911 1.1199 1.1548 1.197 1.2503 1.3165 1.4013 1.5135 1.6692 1.9025 2.3030
h = 0.015
Third 1.0081 1.0207 2.0484 1.0913 1.1202 1.15535 1.1984 1.2517 1.3189 1.4059 1.5234 1.6940 1.9804 2.6680
von Hirmn 1.0060 1.0207 1.0484 1.0913 1.1202 1.1553 1.1983 1.2511 1.3175 1.4026 1.5152 1.6720 1.9078 2.3142
h = 0.020
Third 1.0051 1.0207 1.0465 1.0915 1.1205 1.1558 1.1992 1.2530 1.3212 1.4102 1.5323 1.715b 2.0441 2.9588
von tirman 1.0061 1.0207 1.0485 1.0915 1.1801 1.1557 1.1987 1.2520 1.3187 1.4043 1.5179 1.6763 1.9153 2.3301

h w 0.025

Third 1.0051 1.0207 1.0486 1.0918 1.1209 1.1565 1.2002 1.2547 1.3242 1.4158 1.5438 1.7427 2.1262 ------
von K'run 1.0051 1.0207 1.0488 1.0916 1.1206 1.1560 1.1996 1.2531 1.3203 1.4066 1.6213 1.6818 1.9250-----

b = 0.030

Third 1.0061 2.0-00 1.0487 1.0921 1.1214 1.1572 1.2015 1.2568 1.3278 1.4226 1.5578 1.7762 2.2264 ------
von Kilrn 1.0051 1.0208 1.0486 1.0920 1.812 1.1570 1.2006 1.264 1.3222 1.4094 1.5256 1.6885 1.9370 ------

b = 0.035

Third 1.0051 1.0208 1.0488 1.0925 1.1220 1.1681 1.2030 1.2595 1.3321 1.4307 1.5744 1.8157 2.3449 ------
won Kanrmn 1.0051 1.0208 1.0486 1.0922 1.1215 1.1575 1.2017 1.2560 1.3245 1.4127 1.5306 1.6966 1.9514-

h = 0.040

Third 1.0051 1.0209 1.0490 1.0929 1.1886 1.1592 1.2047 1.2621 1.3371 1.4400 1.5936 1.8613 ---- -.--
van Kirmin 1.0051 1.0209 1.0488 1.0926 1.1220 1.1588 1.2031 1.2579 1.371 1.4166 1.5364 1.730O ----- -----

hb 0.045

Thitd 1.0051 1.0210 1.0492 1.0934 1.1L34 1.1604 1.2066 1.2653 1.3487 1.4505 1.613 1.91;0 ----------
von Krmanr 1.0051 1.0210 1.0490 1.0930 1.12288 1.1599 1.2046 1.2600 1.3301 1.4209 1.5430 1.7168 ----- ------

h = 0.050

Third 1.0061 1.0210 1.0494 1.0939 1.1242 1.1617 1.2087 1.268 1.3490 1.4623 1.6396 1.970 ----- -------
von rn 1.0051 1.0210 1.0492 1.0936 1.1238 1.1611 1.2063 1.2626 1.3353 1.4258 1.5505 1.7290 ------------

b = 0.060

Third 1.0052 1.0212 1.0499 1.0952 1.1862 1.1648 1.2137 1.2773 1.3636 1.4695 1.6958 1----- ------ ----
von Karn r 1.0052 1.0210 1.0496 1.0950 1.1260 1.1640 1.2102 1.2679 1.3413 1.4373 1.5681----------- ----

h = 0.070

Third 1.002 1.0214 1.0505 1.0967 1.1285 1.1685 1.2196 1.2871 1.3808 1.5218 1.76e 2 -----------------
won K rmn 1.0052 1.0212 1.0500 1.0960 1.1881 1.1673 1.2148 1.2744 1.307 1.4511 1.5695 ------ ------ ------

h = 0.080

Third 1.0063 1.0217 1.0512 1.0984 1.1312 1.1727 1.2265 1.2985 1.4007 1.589 ----- --- ------ ------.
won Kirmain 1.0052 1.0215 1.0510 1.0976 1.1289 1.1711 1.2202 1.2820 1.3616 1.4673 ------ ----- ------ ------

h = 0.090

Third 1.0053 1.0219 1.0520 1.1003 1.1348 1.1775 1.2342 1.3144 1.4232 1.6011 ----- ---- ----- ----:
van Karmn 1.0052 1.0217 1.0515 1.1001 1.1340 1.17!5 1.2263 1.2907 1.3741 1.4861 ----------- ------ -----

b 0.100

Third 1.0054 1.0222 1.0528 1.1025 1.1376 1.1828 1.2428 1.3256 1.4464 1.6482 ------------------
von Krman 1.0053 1.0220 1.0526 1.1020 1.1370 1.1904 1.2332 1.3004 1.3883 1.5076 ..-.-- ----- ------

NATIONAL ADVISORY
COMMITTEE FOR AERONAUTICS










NACA ARR No. L4G15


TABLE II
RATIO OF VELOCITIES AT LEADING OR TRAILING EDGE FOR COMPRESSIBLE AND TICOMPRESSIPLE FLOWS





N I 0.10 0.20 0.30 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.60 0.85 0.90 0.96
Approxl-


h = 0.010; (q)exact 1/1.0004

PFirst 1.0000 1.00,0 1.00001000 0.9999 0.999 .999 9 0. 999 9 0.9996 0.9998 0.9297 0.9996 0.9995 0.9991
Third 1.00 1.0000 1. 0000 .999. .99 99 .999 .9998 .9998 7 .9996 .9994 .931 .9985 .9973 .9939 .9757

h 0J.015; (ql)exac. = 1/1.0009

First 1.0000 1.0000 1.0000 0.9,99 0.9999 0.x999 0.9998 0.9996 0.9397 0.9996 0.9951 0.9134 0.9992 0,9968 0.9980
Third 1.0000 1.0000 .9999 .9996 .9990 .9996 .9991 .9993 .9991 .9986 .99 8 .9967 .9940 .a86" .9454

h 0.020; (q|)ext 1'1.0016

Pirt 1.000 1.000 0.9999 0.9999 0.990 0.999 0.999 0.999 0.9995 0.0994 0.0932 0.9960 0.9986 0.9979 0.99e
Third 1.0000 .999 .9998 994999991 .9988 .9983 .9976 .9964 .9941 .9893 .9.7 .9028

h 0.025; Iqlexmet 1/1.0025

Pirst 1.0000 1..000 0.9999 0.9998 0.9997 0.9996 0.9995 0.9999 0.9992 0.9990 0.9987 0.99g3 0.9978 0.9966 0.9945
Third 1.0000 .9999 .9997 .995 .9993 .9990 .9986 .9981 .9974 .9962 .i994 .990B .9833 .9620 .6480

B 0.030; lql )ect 1/1.0306

Pirst 1.0000 0.9999 0.9998 0.9 97 0.9996 0.9994 0.9993 0.9991 0.9969 0.9986 0.998 0.9976 0.9968 0.9953 0.9921
Third 1.0000 .9999 .9996 .9992 .9990 .9966 .g980 .9973 .9962 .9945 .9916 .9867 .9759 .9452 .7B08

b 0.03S; -qi)exact 1/1.0049

Pirst 1.0000 0.999 0.9998 0.99if 0.9994 0.9992 1':.990 0.9968 0.9985 0.9980 0.99 C. .996' 0.9956 0.9937 0.9892
Third .9999 .9998 .9995 .9990 .9986 .99B" .997: .9963 .9946 .9925 .9788 .9B19 .9672 .9253 .7013

h 0.040; 1qllex. t 1 '1.0064

First 1.0000 0.9999 0.9997 0.9994 0.9992 0.9990 0.1967 0.9984 0.9980 0.9974 0. 9 0. ;37 0.9943 0.991' 0.9659
Third .9999 .399979 .9993 .9968 .9981 .9974 .9965 .J992 .9932 .'903 .96,1 .97e3 .9570 .9023 .6092

b 2 0.C45; (qi exact 101.0081

Fir t 1.0000 0.9996 0.996 0.9993 0.9990 0.9986 0.964 0.9960 0.9974 0.3968 O. aa u.9946 ..997 0.9895 0.9822
Third .9999 .9997 .9991 .9985 .0976 .9967 .995 .9939 .9914 .9676 .9814 .9700 .9455 .8761 .5045

Sh 0.0C50; (q1le.,.t 1/1.01

Pirst 1.0000o0.9998: .9969, .9991 0.9988 0.998E 0 .9960 0.g97B 0.99E8 0.9960 0;.a49 0.9933 0.9910 0.9871 0.9780

= 0r .06,, q0 .9act = 1/1.'0144

First .9999 0.9997 0.9993 0.9997 0.9983 0.f9'6 0.9972 0.9964 0.99E45 .9942 0.9926 0.9904 0.9871 0.9814 C.96B3
Third .9999 .9994 .9965 .9969 .9957 .f4 .9920 .0690 .9846 .g779 .9667 .3463 .9025 .9'74 .1137

b h 0.,O'.; iql)exar S l'1.0"9

First .9999 0.99961 0.3991 0.9382 0.9977 0.09'0 0.9961 C..9951 0.9936 0.g992 0.900 .-.969 0.9824 0.9746 0.9568
Third .9998 .9991 .999" .9958 .9942 .9 .9891 .9650 .9760 .9697 .9545 .9266 .6666 .6968 ------

b 0.060; tql)exct = 1/1.0256

First 0.9999 0.9995 0.9936 0.9977 0.9969 0.9960 0.9950 0.9936 0.9919 0.9898 0.9969 0.9829 0.9770 0.9669 0.94C6
Third .9997 .9969 .9973 .9944 .9923 .9e95 .9856 .9602 .97_ .9602 .9402 .9035 .6247 .60159 --
h C.,90; (qiljct = I1/.0324

First 0.9996 0.9993 0.994 0.9971 0.9961 0.9950 0.936 0.9919 0.9696 0.9870 0.9634 0.9784 0.9709 0.9561 0.9266
Third .9997 .9986 .9965 .9909 .9902 .2866 .9o1'1 .2746 .9648 .9493 .911' .8770 .*766 .4921 ------
h = 0.100; I(qiexact = 11l.04

First 0.9998 .9992 0.9961 0.9964 0.9952 0.39S36 0.9921 0.990 0.9974 0. ..40 0.9-95 0.9733 .9641 0.9462 0.9119
Third .999E .9982 .9"5E .91J2 .9897 .9813 .9772 .9687 .9561 .9269 .9051 .8469 .7221 .3660 ------
-AT RNAL ADVISORY
COMbIITTE FOR AERONAUTICS










NACA ARR No. L4G15 63


AhrLE ill
AAITTL :F AA%'.i VL'EL.'IrIE1 F f C':"F E"TIBLE At.L I 'C:OMP BE'.J fLF. fL.OAS i




Appr:.x- 1 .10 c.0 ..'0 [ .40 4.45 '-.0 G 4 ,.


h = 010: qi 1 1.04; (q14 i. ; 41. l.-: 41 04 .. = 1.404Iqi

r. t .C0 '2 1 08 .0 9 1.046 1. .0 1.0 .1; 1. ?' 1 16 1 4 ..: .0433 1 LC'4"
"ic-nd l1.000Co l..00 1.,,'2, 1.0037 l.,4, 1,0,.r4 l'.0 0 i ..'I'. 1.012.4 ).. 171 3.',;;2 1.0;99 j.. .l I... ', 1.13 4"
Third 1.y0 ] :0i l..:0t": 1..3E I -0-0 l :6j1. M I .yC' 1.033 1. 1. 3 1.7 3..: (*' 1. 44. 1...:-, .I C- '

h = *.Ci., tqi 4 1 1.06;: i : l. ., iqt ; = 1.0 : ; h e, = 1.,:.?

F i r -. 0 .' 1 :.d. 1 i.. .. :. .c. 1 ..: I. .: I I l C .1.0 14- 1 .:., *-. .. i 1 ...: l l 2. 2 4
7ec.-r.d 1.0003 1.033 3. I30 1.0 1.005 5 1 7 l. 0 1. ..'.; l 1 .1.1 1z' I. !. 1.' ; 1 1..-D3 ] i 1.:'.
Third 1. 003 1.:0i 3.C 1.:'074 I.C'OT 1.0'9 i..';V 3.016: 1.uL9 :. : l.1. I. l. t' 4 .'i.' l0 .7 1. 1.& .d.'

S= ,.0:0' ; l1 )I = l.0C.; qi = 1- ]..- ; 1.0 1;" Ill a t = 1 .031

Fir lt 1.0004 l.':'01: l.OC.'F i..j.c8 1.0068 1.0u 1: l.'7';qc l.01112 .!.'4 I. 1 i 4. 4 L I I1. 4.. -*' I.']*3
7.e'ond 1.00C4 1i.i i l .0040' l.':.0'7 1.12 l.l.. l..I"r' 1...l' .'1 1.'. 1. .'I.' : 1 .. 6 .t.961 2.16,0 1.41.3
Inr rd 1..OC.4t '.:" 1 .. 41 1..-077 1.010. 1 .. :4 ...' 1 .''..:l.I. .l 1 ...-- .:.:- 1.1 46 i 44L .41. 9

h I ..."0, 'q i 1.l0: H I l;- 1. i' *" : 1 3 = 1.1,: 4: ; n ti., :. 1.1 .I,:',4

Firn! 1.0'' l 1.r..'.. 1.', 44 ..0l 3 1.- 409 l.* *l ly. 79 ..c:'I' Z. 1a 1 3 .
Sec, nd 1.,000 .'.:" i '_0 1 .,0)' 1.0129 1..,:j" .1.. i i.:, l .:.'"8 1 .'..:. 1.,1l ,i .,:.'1.b l .-1 L'.r 1. a
Third l.00C0t 1.' 1. !l 1.:9a 1.01 30 1..' 11 .y.. 1. 1.C-7|.* I .. 1 l ..'7- I'.4 e I-

h = : -.030: 1,l I = 1.12; lqill = 1.l';. ,: 1.1 = 1.1 ]-" 4; -:0l-Je ,ct 1.1 4 4

I I I I
rirn 1.0000 1.00 1.C0282 1.Cy'49 1.0128 I.' 3..l: i .0268 1.':39 4.04 .- l.''4 H l. .'; ':.0 1.1'5" l.:7'- I

Thira I.00':'O 13.74.2 ..0.4:t 4.'.'0'. 8' .'l l I .I:'!9 4 ., 1..:. 1.030 1i l *.*H 1.1- I '... 'c 1 .564.' ;..7 ;

rn 0.,' : (i = 1.1; qil = i.]44 :* q ) 1.144: I I 144E

Firn 1.008 1.07> I.JK'. 1.0112 1.0147 1.417' 1.Ce42.10'? 1.7 4.y42 F.... 1..3.4 11.!1
2ecjnd 1.00 1.0I1 1.0073 1.01.39 1.0184 1.04 1 .0-, .040- 1.050 4.,.84 1.0. 16 ..49 .10t 1. 2, ..-.4'?!
Frs.r.la 1 .00. .,,., I" 4 1.014_._ .. ._'."9 1 : 4 i C4 31 Q ,.6 i .1 : i 1 i0-0 :1. 4 i I' -i'

h .= 'a.Cl: I I I = 1 .161 1 2 = ;. ls I i 4 = I. .: I.t 1 .:. 3l e

Fir.t 1. :. ..* 1 1.016 i 1.. 1. 1 .0 ... .: 1.1 1.1- 1 *
eccnd 1.0009 I.u'.' .''0 4 1.160 1.0213 1. '0E 4.'34li. 0'?'4.*E*" 1. *%. 3... I E. ... .._ .. 1.4 :..7
Thira 1. '; 1. 1 '.01840 1.15;, |1.I9 1. 41.3 8r l ,,, -. 3 .4
Iral = I: 0 : 1 1 u 1 I It 1 I I .*lq: 7l] = 1 .1 4: i leI r l. ].1 l4

irst 1.000 3.0 ;.i ." 4 1.01, 1.18; (q 1.03 3. I'.. 1. l9 1.04 '4 I. Ill ..'> 1.1 1.1'' 0 3. 1.






4 E ,
Tr rnd 11.0011 1 .'Ii 1 .7 l..16 1..0270 3.7'S I. I.. '' l. 1 C: 1... 1... 1....4 ..1 .49 4 I. '.





Ir ir l.'jOll .1'. 1 '3 .i' = 1.20' t qll. 1. 3m. 0 : | 1 '' = 1.7 -4 4 1. 1 .4l i- .: 71 ; l I



Fir 1 1 1 2 l i 1 1 1 1. 1 1 4 .
I 11 ..4C- .0 1 1 .0' 1.027 1J.4 i .J..i 1.. 1 I. 4 i 1. .
Trr. d 1. 0 l141 1. ', 1 1.',.'1O 1.0:.Ei3 1.u4- .' 4, l..0 .b 3 1. E3 .l 1 .l ii4 1.7.9 l..'l 1.l.2 1 'l 'l 'l. "







= .i ;: t1. 41 = l .2 ; 1: .1 I '4, ; i q l. = 1 l =; 'qi slset. 1 .H.S I
Iirsl |1 .0012 114.36 1..:'\ l.'j T 1.C232 1.'- : l .14C' 1.0 11 ..4 13.. 1Li i .43 4.3 ..1 ; 1. ; .4
e ared ll."13 1Q 4 V'Q .019 1.^4"8 1:.: 1.4.'3c 1 :. 75 1.1724 1. 1- l.134e 1.7 13. I .:-44 *. ." .. 6
IntrS '1.':.1 1 .. h 1.- [ 18 1, 1 .1.'S4I. J 1.:. ..ji i 4 :14 .14 ." 1i. 1. 1 I *J l ;.r. h '2 '. .'- .
h -7 ....-j: iq l l.: ; q = I.. I i T.ic 9 6:.. Z













T n-t l.',l1. 3..,-, 1 .'" l 1.0'1 1.0.' 17 1..' 4 3.' l ll ; l L 1"- ; l."l''l'' l. i l ; I'. :3., .': :
r.,n.d 1.'0 1._l. l. l._' S l.:''34 1.,91 3.u"__. 3. .-- 1. 11i7: 5 ..''/ 1 .i 4.l ] ol. o 13.' l. .'" 14. 1
T i rd 1.001: 1. 1." 1 1. I071 1.216 1..'' ..7 I .14 7 < t .1 ''47 I. 1 1. I ..lR3.'. n E' ''I .-1


n- 1. ..'' 1Iq I 3 I 1.4 1i 2l, I 41 E l 1. 1_ 6. 1 !4- :9I
F 1 2. '


Ft r't 1.' .34 z .' 1. 1.2401 1.043171.: 44.7 3 54 1.1%24 1.090 1.1; E .t 3.1 ;0 .l r ".I",





7', r l. 2 1I 1.090;11. l.34.''"7 1.' .I.74 i. '71 l1.1 2 !4. 14 i 1i .r : ..
T r' 1. l l..: l 1. :" !i.t!S' | l.1 : S l.l '4 l 1 0 I 1 1. l i .. .







441 .IF ] I I.'.I t
..1.-"l 1.T0 r. 1 1-1'. 1C2'
q lqlll..AL AD-; I I%)
"3: E e450 '. TC









NACA ARR No. L4G15


TABL. I

RATIO OF MINIMUM VELOCITIES FOR C..MPRESE'IIBLE ND INCO'PRE:'SIBLE L.*:


Sqmlre qminl

Apr xi- 1 0.10 0.20 0.30 0.40 0.45 0.50) [0.5 0.60 0.65 ('.70 0.7E ..C, 2..85 0.90 0..35
aflt Ion

h 0.0.10; (lq11 0.961 (qil = ,.9604; Iq1)8 3 0.96'04 (qlIpsec, = 0.36)4

Firs. t 0.999 0.991 0.90 0.92 0.90 0.993 ,.9318 C.989 0.9868 0.9632 0.9' :.. 0.3 0.9461 ..9082
Second .9998 .9212 .9981 .9965 .9954 .9941 .9925 .9904 .9882 .9852 .191 .3'68 .97r0 .9661 .9841
Third .9998 .9992 .9981 .9965 .9954 .9940 .9924 .9904 .9881 .9650 .3612 .9759 .9!83 .9525 .8451
h 0'.012.; i1 0.94 (q; -0.94091 Iq) 0.9409: (q1, 0.9409


Second .9997 .9986 .99"2 .9948 .9932 .1913 .9900 .9860 .9829 .37988 .9 79 .68 0 .9619 .9:. 1.0336
Third .9997 .9188 .9972 .3948 .9932 .9912 .6889 .98-B .9B25 .9780 .9":26 .960 .9-29 .9211 .5551
h = 0.E0:; lqt) = 0.921 (q')2 0.9.1; Iq1)3 = 3 0.921; :ll,. = 0.9)i7

rire 0.9996 0.9982 '0.998 0.9921 0.9896 0.982 e 0.9828 : .9783 0.972' :.-.62 '.35i 0.-420 C.2219 0.88-5 0.905
Second .9998 .9384 .9964 ..9799 .911 .18 .57 .9818 .971 .94 .1 .368 .9" ------
Third .9996 .9984 .9963 .9931 .9910 .9885 .9854 .9813 .972 .9 .97 .9624 .86" .250 .80.8 ------
h = 0.025: tqi1 0.901 (qi) 0* o.9g ; 1)il3) = 0.9026: qe ct 0G.906

First 0.9994 0.9977 '.9946 0'.9899 0.9867 C.9 28 0.9"B1 0.9722 0.9149 O.5Is 0.9421 j.3209 0.>02 0.10 62 0. 3
Second .9995 .9981 .995E .9917 .9892 .9862 .9827 .9779 .9738 .9682 .962 .'356"' .969 .98" ------
Third .2995 .9981 .9925 .3915 .9880 .9858 .9821 .9769 .9721 .9651 .96g5 .9418 .912E .7914 ------
h= 0. .30; qi)1 0.881 (qi)2 = 0.8836: Iqit) = 0.8838; (ilel.et = 0.5983

FIret 0.9993 0.0.9"89 :'.27231 0.6859 0.9569 0.9454 9.21C2 0.9091 9.9'"" 0.. 3 09 .6997
Second .9994 .9' 94.994 .9902 .9873 .9839 .9799 .9742 .9700 .9642 .-9: ."544 .9592 1.:096 .--..
Third .9994 .9977 .9946 .9900 .9869 .98B3 .9"89 .9725 .967(C .9587 .9472 .928) .8830 .6"21 ---. -
h = 0.035: iqiq 0.865 (q112 m 0.8649; tql)3 = 0.8652: qilexact 0.892E,

First 0.9992 0.99 0.91 0.9852 0.9808 0.9"48 0.96'9 0.9593 0.9488 0.9349 C'.917 0.891 0.95t: 9.7893 u.6415
Second .9994 .99'4 .9C40 .9B89 .9856 .9818 .*973 .9709 .9668 .9610 .959 .9542 .3680 1.043 ------.
Third .9994 .9973 .9938 .9865 .9849 .980" .97" .9681 .9620 .9521 .94"" .411l .8429 .59.6 ------
n 0.040; Iql)1 =0.84 (qi)2 C..8464:; 'q1) = 0.849 qilext = 0.8469

First 0.9930 0.3161 0.9908 'j.9827 0.9772 0C.970: 0.924 0.9524 0.1398 0.935B 0 :.9 5 5..8"'3':' 0.889 0.'S 'j..,05
Second .9993 .3971 .995i .9876 .9840 .3798 .9"11 .9879 .9642 .95" .9549 .'5"'3 .9815 1.994 ------.
Third .9993 9970 9930 .9970 .9830 .9""9 .97'0 .9637 .9569 .9451 .9.73 .6926 .8'3. .2645 ------
r = 0.045; *qi) 0.821 (qil2 = :.8281; itq)3 = 0.8628;: lqilexact = .: 88

First 0.9989 .9 .979999 .4 .900 0.9737 .60 0.9 0.9451 0.930" 0.911 C.ee" 0..53' 0.68 C.7159 ...15
Second .9992 .9968 .9326 .9864 .9826 .."81 ."3.'1 .9653 .9622 .9274 .9226 .9629 1.0005 -------------
Third .9992 .9967 .9922 .9855 .9811 ."'58 ."695 .959. .9516 .1377 .9154 .88 ."'51-----...- ......------

b = .OO; 0 qi: a 0.801 (q ) = O.81l0:O; Hl4 3 = 0.8110; i qilect ,.J109

First 0.9987 0.3148 0.8"1 0.9727 0.9701 L.9815 '.O.~ ''*.9375 0.3210 0.8993 *.822' 0.6!53 0.-"24 .'.676s 0.4414
Second .9991 .9968 .3320 .1854 .9813 .9"66 .712 .3631 .9 10 .2 5" .9Qcu .9-15 1.0253 ------ .-----
Inrd .9991 .9963 .9914 .941 .9793 .7;4 .64 .545 .9461 .996 .916 .95 .692 -............--- ------
h = 0.060; (qi11 0.768 (qi)1 = 0.'"44; ..7 ; l t .'"9

First 0.984 C0.f935 0.9i8 0.J'12 0.9822 C.9 ,3 0.1'" y. 211 :.900: 0.89"6 :'.8"4 0.'95 0."1683 0.5913 0.':45
Second .9990 .9960 .9909 .9936 .9792 .9"44 .9993 .9645 .3608 .6y:4 .384 .9964 l. ..... ------..--
Third .9989 .1957 .990: .9813 .97.8,i .968i .9401 .6491 .339 .9105 .E66 .'600 .3964 -------------
h = 0.0": (qi)1 = 0.72; (qI)1 0. C.396; I1)3 0."4231 Iq1)eec = :."'419

First 0.980 C'0.9 00 ,:'.98121 0.96486 0.97534 C-. 2 ':.32; 0.i028 0.8"72 :0.445 0.801.0 0.7407 0.5:7 0.496" 0C.143-
Second .9389 .9S56 .99.01 .9824 .9779 .7.'- .96e 3 .3652 .9441 .E.99 ."7' 1i.0397 1.19'3 ------ ------
Inird .9988 .2 9E1 .98851 .9786 .37191 .9.M 4 .096l .915 .8823 .8181 .43 .0 ------ ------......
b 0.080; 1:)l 0.658 1qil = '-.'.v-: iqC); 0 '.7097; Uleeact 0.70,0

First 0.9976 0.'990; .3 ".'. 9571 0.94386 0.2:" l'.93:'"I 0.6824 0.8513 v.8118 0.751i 0.6663 0'.5"73 0.2910 365
Second .9988 .'955 .9895 .9818 .9775 .273l1 .9':00 .9685 .9712 .3633 1.'0 l:'1.i3"6 .------ ----. --------
TIbrd .9986 .?~945 .99 "' .9'58 .9681 .9.a15 .1429 .128; .9016 .8537 ."S15 .4"9. -----. --. -- -----
b = 0.090; () 0.6 (0.84.6814; 9qil13 = 0.8182: Rqil8ect = (.6771

First 0.9972 0.9;4 .:.. "8 1..3448 0.9326 .910.:.8.. 0 c.8594 0.822' : .7748 0.'l:1 0.6l50 0.494" o0."--0 0.2386
Second .9986 .951 .9893 .9819 .972 .9" .97 .749 .3831 1.C046 1.052 1.42 ------ -----..................--
Third .9985 .,939 .3857 .730 .92 .98 .9"'i"5 .9152 .8790 .811r .6611 .5'20j------ ------ ------
h = 1.00':; 1q91l 0.601 (qi = 0.E640.; 1qi)3 = 0.6480; (ql react -= .6462

First 0.9966 0.9863 0.9678 0.9393 0.9202 0.6969 0.8684 0.8333 0.7894 0.7I31 (..6588 0.55, 0.4011 0. 13"--0.4684
Second .9988 .351 a .89 62 .9829 .9800 .9"a ."Jl .9848 1.0000 1.0.38 1.10I.:. ---- ------ ------ --I...
Third .9384 .93 .9943 .9700 .99 .9464 .926 .8990 .8504 ."559 ._40" -.-.-- --.--- ---.- .-...-

INATIBA AflDV-B0Y
CMMIn'TTE. FOR AERONATICS

















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TABLE VT
VALUES OF CRITICAL STREAM MACH NUMBER FOR VARIOUS
VALUES OF CAMBER COEFFICIENT
Mlor
h Approximation

First Second Third
0.02 0.848 0.832 0.825
.o4 .770 .716 .738
.06 .716 .682 .672
.08 .670 .628 .620
.10 .625 .585 574



TABLE VII
VALUES OF MAXIMUM VELOCITY FOR CORRESPONDING BUMP
AND CIRCULAR ARC PROFILE


NATIONAL ADVISORY
COMMITTF.E FOR AERONAUTICS


qmax

u Camber coefficient h Thickness coefficient t

0.02 0.04 0.06 0.08 0.10 0.052 0.100 0.1L5 0.186 0.226

0 1.0815 1.1659 1.2527 1.5315 1.4320 1.c816 1.1660 1.2527 1.5341 1.4320
.2 1.o835 1.1701 1.2597 1.3520 1.IL66 1.8534 1.1701 1.2595 1.53513 1.4454
.53 1.0859 1.1759 1.2695 1.3668 1.4673 1.0859 1.1757 1.2689 1.5651 1.461.
.4 1.0899 1.1851 1.2855 1.53913 1.5024 1.0900oo 1.1847 1.2840 1.53876 1.495o
.5 1.0960 1.1997 1.53116 1.4324 1.5627 1.0959 1.1988 1.3084 1.-245 1.5167
.6 1.1056 1.2239 1.3572 1.5078 1.6780 1.1052 1.2217 1.53492 1.4879 1.6357
.7 1.1223 1.2705 1.4530 1.6780 ------ 1.1213 1.2640 1.4298 1.617 ------
.8 1.1594 1.3q79 ------ ------------ 1.1557 1.3701 ------ ------ ------
.9 1.2055 ------ -------------------- 1.1960 ---- ------ ------ ------















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(a) h=o.ol.______














NATIONAL A VISORY
O_____MMITT:E FOR tERONAU11 ICS


0 .2 .,4. .6 .8 /.0


Figure 2.- Ratio of circulations for compressible and incom-
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Fig. 2a





NACA ARR No. L4315


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NATI( NAL AD IlSORY
O ______MMITTE FOR ARONAUTI S


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NACA ARR No. L4G15


I I I I I


______ /
2


ProndIl


- Glouert


von Kormon


- 3 Results of present paper


I
3 1Z/




f


(C) h= o.o3.








0/
1000


NATII
OMMITTI


)NAL A[
:E FOR A


iISORY
R ONAUT


Figure 2.- Continued.


2.2


2.0



/.8



/.6

C/4'


Fig. 2c


I





NACA ARR No. L4G15


2.2


4/?


3 21

/ Prondl// G/oauer
2 von Kdrmnfon
3 Resu/l/s of present paper





(d) h =o.o.___















NATI NAL A VISORY
(OMMITT E FOR ARONAU CS
0 .2 .4 .6 .8 -0.


Afl,


Figure 2.- Continued.


Fig. 2d






NACA ARR No. L4G15


Prondt/ Glouert
von Kdrman


3 21


3 Results of present paper





(e) h=o.s> _______

















NATIONAL A ISORY
1_ (OMMITT E FOR A RONAUT CS
- _


Figure 2.- Continued.


_____ /
2


2.2


2.0




/.8


/.0


Fig. 2e






NACA ARR No. L4G15


/ Prond// G/ouer/
2 von Kdrman
3 Results of present paper
I I I I I I I


I I
3 2


_ (f) h=0o.o06.___



----II



--y/-/


NAT
COMMIT


ONAL A
EE FOR


IVISORY
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.2. .4- .6 .8 ,


MA


Figure 2.- Continued.


2.4-


2.0



/.8



x6

4/7.


0


Fig. 2f






NACA ARR No. L4G15


I I I I I I I I I


I I I I I I


PrandIl G/oueri
von Korman


2.4




2.2




2.0




/.8




/.6
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_(9) h = 0.07._







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NAT
COMMIT


ONAL A
EE FOR ,


2-I


IVISORY
ERONAU


.2 .4 .6 .8 .


Figure 2.- Continued.


_/
2


- 3 Resu/ts of present paper


O5


Fig. 2g





NACA ARR No. L4G15


/
2


2.2


2.0


Prondt/ G/ouert
von K'rm0n


_3 Results of present/ paper


= 0.08.












N TIONAL ADVISOR)
COMMI EE FOF AERONA JTICS
T,


"I


Figure 2.- Continued.


3 2

/e


I I I I I I I I I


Fig. 2h


/,0c





NACA ARR No. L4G15


2.4


2.2


2.0


NATIM
OMMITTE


FINAL AE
E FOR A


VISORY
'RONAUT


Figure 2.- Continued.


I I I I
- / Prond// G/auer/ 3 2 /
2 von Kormnon
3 Results of present paper



(Oh=o.o9. _








E:9


44


/.2


Fig. 2i






NACA ARR No. L4G15


.24


2.0





Za


A1


Figure 2.- Concluded.


Fig. 2j






Fig. 3


2.7
Z5.8



2.6



Z.3 ------- --- -5




z2.0




2.0 y____ I










1.03 4 05 06 07 09 /0







Figure 3.- Ratio of circulations for compressible and incom-------
--- ---_ __--- -- --- -- ____-_ .405
I.I .== = = 4o


1.0 "___==_ =___=__==_ _===--=
0 .0I .OL .03 -04 .05 .06 .07 -08 .09 *10

Figure 3.- Ratio of circulations for compressible and Incom-
pressible cases as a function of camber coefficient.
NATIONAL ADVISORY
COMMITTEE FOR AERONAUTICS


NACA ARR No. L4G15






NACA ARR No. L4d15


(a)


I Io
A = aoz.


Approxinalion
I3 1


-/









NATIONA ADVISE RY
___ __..___ __ _'___ COM IHTEE FR AERO AUTICS


Figure 4.- Ratio of velocities for compressible and incom-
pressiole cases as a function of stream Mach number.


.08

.06

.04-

-02


0 ./ .Z .3 .4 .5 ,6 .7 .8 .9 1.0


Fig. 4a


0






NACA ARR No. L4G15


.24

-ZZ


0 .1 .Z


.3 .4 .5 .6 .7 .8 .9 /.0


Figure 4.- Continued.


-- -
-(b) h-o.o-._--

Approx im oa on
-3



Sonic
















F NATIONAL ADVISO Y
______ _____ COMMITTEE FOR AERON UTICS


.14


( max


-08

*Ob

.04

-OL


Fig. 4b






NACA ARR No. L4G15


.32

.30

.18

.16

*Z4





*18

.16

cax .14

.12

.10



.06

.04

.0O

0


0 .1 .21


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Figure 4.- Continued.


Approxim al/on





S2'
_ (c) h= 0.06.___





Son ic





^I












TIONAL ADVISO Y
___COMITTEE FO i AERON UTICS


Fig. 4c





NACA ARR No. L4G15


.28

.26

*Z4

.22

.10

.18

.16


/9c -I
T ]wax


.10

.08

.Ob

.04

-0.

0


Approximarn on
3




_(d) h = .08 Sooc


















.ATIONAO ADVISORY
___ COM tIlTEE F R AERO AUTICS
0 .1 .2 .3 .4 .5 .6 .7 .9 /.0


Figure 4.- Continued.


Fig. 4d





NACA ARR No. L4G15


Approxmnohon
3








_(e) ho.o.___


___ __Sonc C___0__















"NA IONAL ADVISORY
_.S_ ____ ____COMMIt EE FOR ,ERONAL TICS


.1 .3 .4 .5 .b .7


.S .q /.0


Figure 4.- Concluded.


.24

.ZZ2

-20

.18


(LX -14


0 .I


Fig. 4e





NACA ARR No. L4G15


.//


.10


.09


Approxim ofion
32 I









3^_r~\


Figure 5.- Critical Mach


"v"Cr NATIONAL ADVISORY
COMMITTEE FOR AERONAUTICS
number as a function of camber coefficient.


Fig. 5





NACA ARR No. L4G15


Circular arc
Bump


0 .1 .Z .3 .4 .5 ,.6 .7 .8 .9 /.0 1./ /.2


Figure 6.- Maximum and minimum velocities as functions of
stream Mach number.


Fig. 6






NACA ARR No. L4G15


/.2J-


6/pper


Fig. 7






I
surface


I I


NAK IONAL ADVISORY
__ COMMIT EE FOR AERONAL TICS






__ Upper surface ___ \














_ Lower Srf ace 6


x

Figure 7.- Velocity distribution at upper and lower surfaces
of circular arc profile, h = 0.05, for various values of
stream Mach number.







NACA ARR No. L4G15


0 0 0 4
0. Q-. eq N,


0
(f.i

IN
...........




-~~- ---5
Qb
T

II


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_______ ______ 00
--- --- --- <
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Fig. 8





































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


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