On the flow of a compressible fluid by the hodograph method


Material Information

On the flow of a compressible fluid by the hodograph method
Alternate Title:
NACA wartime reports
Physical Description:
54, 5 p. : ; 28 cm.
Garrick, I. E
Kaplan, Carl
Langley Aeronautical Laboratory
United States -- National Advisory Committee for Aeronautics
Langley Memorial Aeronautical Laboratory
Place of Publication:
Langley Field, VA
Publication Date:


Subjects / Keywords:
Hodograph equations   ( lcsh )
Compressibility   ( lcsh )
Aerodynamics, Supersonic   ( lcsh )
federal government publication   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )


Summary: The differential equation of Chaplygin's jet problem is utilized to give a systematic development of particular solutions of the hodograph flow equations, which extends the treatment of Chaplygin into the supersonic range and completes the set of particular solutions. The particular solutions serve to place on a reasonable basis the use of velocity correction formulas for the comparison of incompressible and compressible flows. It is shown that the geometric-mean type of velocity correction formula introduced in an earlier paper, part I, has significance as an over-all type of approximation in the subsonic range. A brief review of general conditions limiting the potential flow of an adiabatic compressbile fluid is given and application is made to the particular solutions, yielding conditions for the existence of singular loci in the supersonic range. The combining of particular solutions in accordance with prescribed boundary flow conditions is not treated in the present paper.
Statement of Responsibility:
by I.E. Garrick and Carl Kaplan.
General Note:
"Report no. L-147."
General Note:
"Originally issued November 1944 as Advance Restricted Report L4I29."
General Note:
"Report date November 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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frAc L-lI-


I -


November 1944 as
Advance Restricted Report LAI29


By I. E. Garrick and Carl Kaplan

Langley Memorial Aeronautical
Langley Field, Va.



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-
hically edited. All have been reproduced without change in order to expedite general distribution.

L 147




~FIL~-~l~j~j~Wli*i~iyYi~;iYl~iliyll~u ~--li*,....,.


NACA ARR No. L4l29







By I. E. Garrick and Carl Faplan

S Ui ivl ARY

The differential equation of Chaplygin's jet problem
is utilized to give a systematic development of particular
solutions of the ho':iogreph flow equations, vhich extends
the treatment of Chaplygin into the supersonic range and
completes the set of particular solutions.

The particular solutions serve to place on a rea-
sonable basis the use of velocity correction formulas for
the comparison of incompressible end compressible flows.
It is shown that the geometric-mean type of velocity
correction formula introduced in an earlier paper, part I,
has significance as an over-all tyoe of approximation in
the subsonic rsnge.

A brief review of general conditions limiting the
potential flow of an adiabatic compressible fluid is
given end application is made to the particular solutions,
yielding conditions for the existence of singular loci
in the supersonic range.

The combining of particular solutions in accordance
with prescribed boundary flow conditions is not treated
in the present paper.


This paper presents a theoretical investigation that
may be regarded as a continuation of studies initiated

NACA ARR No. 14129

In part I (reference 1). In part I an attempt was made
to unify t.-e results of Chaplygin, von Ka'rm.nh and Tsien,
Temple end Y'arwood, end Prandtl and Glauert insofar as
their results were concerned with velocity end pressure
correction factors for the correspondence of incompres-
sible and compressible flows. In addition, two new
velocity correction formulas were introduced that appeared
to nave a somewhat wider range of applicability than the
formulas of the afore-mentioned authors. Most of the
results of part I were obtained with the use of two par-
ticular solutions of the hodograph equations. These two
basic solutions correspond. to a vortex and a source in a
compressible fluid.

It .'.,as mentioned in pert I that, in order to tre'.t
the e;.:ect boundary problem of uniform flow of a compres-
sible fluid past a prescribed body, a general set of
particulsr solutions of the hodograph equations had to be
o'ct.ined. Such P study is given in the present paper,
wh;i.h incidentally helps to clarify the nature of the
velocity correction factors of part I in particular,
the one referred to as the "geometric-mean" type of
approxinsa iion. In Eddition, mnny interesting types of
flows are disclosed from a physical interpretation of
the particular solutions. A few such solutions have
already been obtained and discussed by Ringleb (refer-
ence 2).

Several mathematical approaches exist by means of
which perticulir integrals of the hodograph equations
may be obLsined. Two such approaches, mentioned in
part I, may be attributed to Chaplygin (reference 5) and
Bers and Gelbrt (reference 4) and are analogous to an
exponential and to a power-series approach, respectively.
Another method of defining particular integrals is the
integral-operator method of Bergman (reference 5). In
the present paper the differential equation, first used
by Chaplygin in his treatment of jets (reference 5),
provides the basis for the definition of complete set
of p.rticulp.r solutions.

The scope cf the present p.per is limited chiefly
to a systematic study of the fundamental solutions and
to the physical interpretation of some of the particular
flows represented by them. Th1e combining cf particular
solutions to represent uniform flow past a prescribed
tdy is not treated herein. It is believed, however,
that the present study may serve as a basis for further
development end clarification of this important problem.

NACA ARR No. 14129


x, y






rectangular coordinates in plane of flow

magnitude of fluid velocity

angle included by velocity vector and posi-
tive direction of x-axis

density of fluid

pressure in fluid

velocity of sound in fluid

Me3h number (q/a)

quantities referred-to stagnation point q= 0

velocity potential

stream function

ratio of specific heats (approx. 1.4 for air)

(approx. 5/2 for air)

maximum fluid velocity (corresponding to
p = p = a = 0)

dimensionless speed variable

2pao2 2p + M2

sonic value of T Ts 1 approx.
1/6 for ir)+ 1

For p > 3 (or 1 < y < m), the range of 7 is
0i T 1.

4 NACA ATR No. 14129


Hodograph Equstions

The linear equDcLons in the hodograph variables 8
eJ qc!, ',hich relate the velocity potertlsl a.nd the
streak: function qr for the steady twc-dlmensional flow
C' a ronviscous compressible fluid, are

oe 1 Eq ,

2 (1)

in which, for the adie b-tIc equation of state between
Fr :.-su re anQ density,

(q) ^q
(1 T )
d /,/?r,
S(q) = -d

-- +i 'T

1 (r ~ + 1)T
q(l T)l

(S3e equaticns (21) r-J. (c:) of reference 1.)

I t 1he irc.-mcrfssibie cIse T--> 3, equctons (1)
can hb exrressed i-. tr e C..uc.v-Pie'mann farmT. PErtlIullFr
S -,L .Jt. :n. j = /' + '-'r cr,; be e.Y .r'ss3 td in this case rs
any. rr.lE.'tic fL:nrcti-:n of the i;'r,.. 13 v1 riable

w = + i 1:.- q(


NACA ARR No. L4129

or as eny analytic function of the related exponential
e- qe-1 (5)

Thus, an infinite sec of pprticulpr integral of equa-
tions (1), in the incompressible case, referred to herein
as "the powers set," is wk. When k is a positive
integer, the particular solutions vanish at the origin
(0 = 0, log q = 3) and, when k is a negative integer,
the particular solutions are infinite at the origin. In
the case of nonintegral values of k, the origin is a
branch point of the functions wk.

Another infinite set of particular integrels of
equations (1) in the incompressible case, referred to
herein as "the exponential set," is

(e-iw)k = qe-ik

where, again, k can take on any value integral, non-
integral, positive, or negative.

In the compressible case, the particular solutions
corresponding to the powers set wk (that is, the par-
ticular solutions which reduce to w-k in the incompres-
sible case T-->) depend on whether the coefficient of
wk is real or imaginary a consequence of the fact
that, in the compressible cse, V and 'V do not
satisfy the same differential equation. For example,
for k = 1, the two functions corresponding to w
and iw, which have been developed in part I, are

W = 8 + iL


S= i(9 + i)


L = log q + f(T)


L = log q + g(T)

NACA ARR No. 14129

End f(T) and g(T) each vanish for T = 0. (See
equations (26) and (27) of reference 1.)

the dcvelon.(,enb of other functions corresponding to
the pmvwer set fk., Cor positive integral values of k,
'ilclows according to the mrethod of Bers and Gelbprt.
(see e-xpressicn 2) of reference 1.) Since the present
r.Eper is chiefly concerned with the functions corre-
E.';:onirLg to tt Xf exponential set e "kw, the owners set
i. n-t fur-cher discLdsed.

Chaplygin Differential Equation

The Functions P1 and Qk

Corresponding to the ezponentiai sets in the incom-
pressible case

e-ikw qk cos k iqk sin k


ie-ikw qk sin k9 + iqk cos k6

there appeE.r in the compressible case functions designated,

Pk(q) cos kG i k(q) sin 1:9

Pk(q) sin kP + i Qk(q) cos kG

wh-ire the functic-ns PR(q) and Qk(q) satisfy second-
-r'd-r d ifferential equations. These equations are easily
ob~rined by substicutin. in equations (1) the product-
tyn solutions

/1 = P?(q) sin (kG) (

= p (q) cs (-kG)
k k cs J

NACA ARR No. L4129 7

In view of equations (1) it is observed that

k Pk(q) q dQ,
p dq

dP,(q) -kq -.p (q)
Cq dq \pc -I

The functions Q (q) satisfy t-.e second-order differ-
ential equation

k. + ? dq kl -2(
1q2 (1 + d ) = d )
q d q

The functions Pk(q) c'n be obtained from Qk(q) by
means of the first of equLtions (5). Equation (6) may
be reduced to a strndErd type by introducing T as the
independent variable. Put

Q,(q) = q k' Yk(T) (7)

where clerirly YL(T)--1 as T---> (inccnpressible
cese). VW'ith the use of the symbolic relations

q = 2T d-
dq dT

2 d2 = 4T2 +
dq dT
and the relation
S- T

the desired differential equation is

d2Y dYk 1
T(1-T) + F(k + )- (k+ -)T] -+-k(k+1) Y=0

NACA ARR No. 14129

Ecusti on (S), which is of the hypergeometric type, was
.it introduced by Chpalygin in his memoir on gas jets
(reference 5).

The Functions Y, and Y-k

.!.r.pl:gin treated the subsonic flow of a compres-
~i:lt Lluid through jets ,with straight-line boundaries.
..:- ? [:Lrotlem:s the hodograph variables 0 and q
cro nr-tur' v.rlaibles in the sense that the solid an-]
Cilui. b'zundrries f-re described by 8 = Constant and
Q = "or.s-tant, respectively, and only the particular solu-
ti ; s of equation (c.. with positive characteristic
fi.dex k are needed. In the present peper a complete
order. d set of particular solutions of equation (P) is
oL t-i:1cd, .which extends the results of Chaplygin into
the suriersonic rrnge and to negative values of the
inelTx k. Two types of solutions of equation (8) for
nonintegral values of k are

YI(T) = F(ak, bk, k+1; T) (9)

(T) = T-k F(a1-k, bk-k, 1-k; r) (10)

ak + bk = k p

ab=k =- (k + )P

ab ake + 1)b(b + 1) 2
?(a, b, c; T) = 1 + T+ 2 (+ 1) ...
c 2: !c(c +1)

It i- now shown tha-t only cne of the solutions need be
i.lsed. F-.r positive values of k, the requirement that
.( ;)l = 1 excludes the use of equation (10). For nega-
tc v'.. les of th e inex, the solution '_ q) = q~ k(T)
obtal ned v.ith the aid of equation (10) is, except for a
C?-:t-rt ficto-r, equivalent to the solution Qt(q =qk Yk(T)
obtained witt tle rid of equation (9). Thus


-k -
) = (T)

= q-kTk F(ak+k, b-k+k, k+l; T)

= q-Tk F(ak, bk, k+l; T)

S1 \k
C2) 1 (q)

Hence, only the solutions vIven uby equptloni (9) Pe
needed for the determination of Qk-(q) and Q9 q).

Q%(q) = qk Y(T)

-= .qk v( bk, k+-; T) (11)


Qk(q) q-= -(T)
= q" F(a-k, b-k, -k+17 T)

= q-k F(ak-k, br,-k, -k+l; T) (12)

Observe that both types of hypergeomnetric functions
appearing in equations (j) and (10) are utilized in the
expressions for Qk(q) and Q-kI')*

The foregoing discussion h2s been limited to non-
integral values of the index, positive or negative. Wheen
the index is integral and positive, equations (L) and (11)
remain valid. .,hen the index is integrel and negative,
however, equation (12) does not in general lesd to a
meaningful solution and consequently another independent
solution is to be sought. The desired solution fcr
Yk(T) in such cases contains P logarithmic term and
again is subject to the condition thet it reduce to unity
for T = 0 (incompressiblc case). The expression for
Q_-k(q) is then given by

Qk(q) = q'k Y-k(T)


10 NACA ARR No. L4129

I .
+ r-4 4- (D

I-r .- L4
a 4

1-* A *
S+ .. :

P r-- c -P e-
17- E C .
-4 o D

.,. r + .4 w
F '-I I c 0

v '- o0

I -- -H + )

Sl a
: .d i -p o

I O 0
.i 0 l I 1'-. 0 .

X 1 4 ,- 04 0 FA ."
.+ + I -: .

,.-. I I j 3

A. .. A At
-1 Cd

II + l
A- ld 5 "- Sq t4 "

+ +

.f Tr *- W
+ +3 k- 0 i -H
1 .- -I

0 I
S I At + hO <

S f C. CQ
+ +
4 r- Ar

I ) A l r- + P t

NACA ARR No. LT29 11

Crse of y = -1:

Consider as an example the "-on K;'rman-Tsien treat-
ment of compressible flow (referc-nce ) in which the
diabetic index y = -1 or = -!-. Then

k + 1
a 2


For a negative integral index, equcti',n (15) may appear
to be applicable, in which case the expression for
Y-k(T) would be a poiynminial of degree k 1. An
examination of equ&.tio.n (12) shows, iho,-,ever, that for this
case no infinitieF:- rrise and that, wNiin the index is
negative, integral or n.-ninrgr i,

Y-kT) = ( -, -k;

The hypergeometric series represented by Y,(T) con-
verges for values 3 IT < 1. For th'- present case of
y = -1 or p = --, vr.lue7 of T cot-responding to
positive values of M lie nutsi-'le tne rEnge of conver-
ger.ce. A lecsed expression for T.iI) e n be found,
however, for this case which, by EnDsitic continuation,
is therefore v-lid for all v.lue of T. Thus

Y_-.:(T) = F 1-I-; T

+ (1 T) k

Siilry, from e2 (), wen the ide is ositve,
Similarly, from equation (9), when the index is positive,

12 NACA ARR No. i1I29

Yk(T) = FT( 1+k;

1 + (1 T)1/2-k
Cbserve th.t
Qk(q) 1 1
-k (q)

q 1/ (14)
I 2 ^1/2
1 + (1- T)

This identity for the von Ks'rm'n-Tsien case corresponds
to the identity qk for the incompressible case.

Case of k = 1:

For k = 1,

al = 1 bI = -p cI = 2

Then, for the positive index,

Q1(q) = q Y1(T)

= q F(1, -P, 2; T)

1 (1 T)P+
= q (15)
( + 1)T

For the negative integral index, it may appear at first
glance that equation (13) is needed; however, equation (12)
dtes yield a relevant and finite result and accordingly
is the equation to be used. Thus

NACA ARR No. r14T29

Lim F(ak -k bk-k, 1-k; ) 1. + T 2 2 + 1) T3
k->l 2 2 x 2 25x3

Sp2(a !- 2) ...T
2x 4+

=1+1 p -1-(1 -T)P+1
2 p + 1 J

and therefore

Q (T) = q-1 + 1 [ (1 +]} (16)

Case of k = 0:
The exceptional cSse of k = 0 is directly treated
by means of equation (G). The differential equation for
Y,(T) or C (T) then is

d J d' =d
dT (1 T)F dT!

The gereral solution of this equation can be written as

Qo(q) = 2C1 los q + C (1 T) 1 + C2

NACA ARR No. 14129

where C1 and C2 are arbitrary constants of integra-
tion. The constants C1 and C2 are determined by the
imposed condition that the expression for Qo(q) reduce
in the incompressible case simply to log q. Then
Cl 1

C2 =0
:id therefore

1 [T dT
Qo(q) = log q + F(1 T)P 1 d (17)

In a similar manner, from the differential equation for

d T(1 T)+1 dP,
dT 1 (2p + 1)T dl T

the expression for Po is obtained as

1 1' ~ 1 (2P + 1)T dT /,QV
P (q) = log q + T 1 -- (18)
J, 2 3+1 7

It is remcirked thit the functions Q%(q) and P (q)
cre identical with the elenient iry functions L(q)
asd L(q), respectively, introduced in part I (refer-
ence 1) a"nd are associated with a vortex and a source
t.'"e of flow.

The Functions Rk and Sk

A linea.r homogeneous differential equation of
order n can, in general, be reduced to a differential
cqution of order n 1 by means of an exponential-
type substitution for th. dependent variable. Chaplygin
mpde use of such a substitution to reduce the second-
order differential equetions satisfied by Pk and Qk

NACA ARR ITTo. L&29 15

to first-order equations oP the Riccati form, in order
to study properties of the functions Pk and Q in the
subsonic range for only positive values of k. In the
present analysis the Ricceti equations are also found
useful in order to extend the study of the functions Pk
arid Qk to the supersonic rpnge for both positive end
negative values of the inde:: k.

The second-order differential equations for Pk
and Qk, with T as the independent variable, are

d T(1 T)P+1 dPk k2 (1 7T)P
Pp =
CT 1 -(2 + 1)T dT T


S__ T 1 (2p
dT (1 -[ O 1, TI( T)+1

The corresponding first-Drder i_'ccati equations ere
obtained by substitutitng for P, ean.] new dependent
variables H,. aod 3S,, respectively, as follows:

T j dP.
S- k Pk dT

T ~log p (19)


: dT

16 NACA ARR No. 14129

S27 1 dQk
Sk -
S k Qk dT

27 d
k dlo g o (20)
k dT

The equations satisfied by Rk(T) and Sk(T) are

'k + (2p + 1)T p R _k 2 1- (2p+ 1)T
+ Rk + k = O (21)
dT 1 (2j+1)T 1 -T 2 1 T


dS. P k 2 1 (2p + 1)T
+ S + .- 2S = (22)
dT 2T T

Ir.itirl conditions for Rk(T) end Sk(T) are found by
examinet ion of the incompressible case T ->0. In this
cepe Fk = C = qk and, since 2T d q -, it follows
dT dq
from equations (19) and (2D) that

R,(3) = Sk(0) = 1

The following irportr:nt relation exists between the
functions 'hk(T) and Sk(T):

1 (2p + 1)T
P I(T) Sk(T) + 1
1 T

= 1 M2 (23)

Equ&tion (23) CFn be verified directly from the hodograph
eq-ations (1). It may be noted Ft this point that this
result is cf significFnce in connection with the
.n:,imetrL-:ml'ean type of velocity correction factor intro-
,rlcmd in -p.rt I end is discussed more fully in a later
s t1,*:, n .


Before the fun,-t'.ons Rk(T) and Sk(T) are treated,
certain general observations can be nmade regarding the
functions Pj'( T CT.(), Rk(T), nd S(T). Chaplygin,
vho limited his investigations to the subsonic rpnge and
to positive values of the index k, has sncwn thEt Q
end consequently the other functions possess no roots
for any value of the independent vErlable in the subsonic
range, with bM = 0 excluded. In the supersonic range
S> 1, P,(T) and y(T) in gi eral posess zeros.
Certain relEtions obtained by means of equations (19),
(23), and (25) between P Q ~, nd S, at the
zeros of Pk and Qk are suinni.izei z s follows:

k -I ,T IT k Sk

"0 .:- x or' .* in ... -a 0

i','ax :r ";in .0 M -o

It is ret:. ; Lh t tlhe niurberc of zeros of Qk,
as a function of thc index, k, c ?, be found from an
expression Qeveloped by K).ein rnwd Hurv'itz (reference 7)
in connection with the zeros of the hy,,prgeometrie fur:c-
tion. In generEl, the niu;ber of zeros increases with
the magnitude of tie index k nrd is infinite for
k :+.

A further observation of interest c n be made in con-
nection with equation (25). Ch.pvlv'lin has show. thqt,
for positive finite vFiaes of k (a.nd t.e sa~ne is true
for negative finite vai"es of k), the functions S(T)
are not zero for the sonic vfilue T = Ts or M = 1.
From equation (2') then, it follows that ti-e functior.s
Rk(T) = 0 for ,1 = 1.

In view of the relation between, the functions Rk
and Sik given by equation (25), only SI: need be dis-
cussed. The Riccati eq'..etioni (22) n.ay be used to discuss
certain properties of t.i, functions. Sk but in general,
for numerical evaluation, the original definition (equa-
tion (2J)) in terms of the function Qk may be used

NACA ARR No. 14I29

2T 1 dQk
k k Qk dT

27 dYk
Sk = 1 +
Yk d"

In general, the functions Sk are expressible in infi-
nite series. For several values of k, however, Sk
can be expressed in closed forms. For k = 0 and
k = Sk may be obtained by a limiting process from
equation (20); however, for these special cases the
hiccati equation (equation (22)) yields the results
directly. Thus

So = (1 T)P

= [1 (2p + 1)T 1/2

( 1- 2
= 1 -M2)1/2



The cases k
closed form.
for and

= 1 and k = -1 may also be expressed in
With the aid of the equations (15) and (16)
Q-l, equation (20) yields


S1 1 (1 + PT)(I T)
S = -- 2
1 1 (1-T)+1

a nd

S = 1 -

pT(l T)P

1 ( i- (1 T)
2 p + 1


In order to illustrate the behavior of some of the
functions thus far introduced, a number of tables and
figures are given. All the calculations have been


NACA Ar3 INo. LI229 19

performed with the adUibatic index y = 1.4. Table 1
gives values of Yk as a function of or T for
several positive and negative values of the index k.
Figure 1 shows the Y- functions plotted against M.
Values of the 31. and Rk functions cre given in
tables 2 and 5 3nid are plocted against I in figures 2
and 5.

The Functions fk(T) and gk(T)

In the incompressible cass, th3 sets of functions Qk
and Pk can be reduced to a single function log q by
means of a simple opcor&tor L log. Thus

'k = P. k -


1 -
1 q = .c q

This sy'ne opsr.tion a-l'e': tn t the r'iect-ons rk and Pk
in the c f!.,.'-s i le c.E s.. erves t3 *.efi'. t.'.o r' eful sets
of functi.ons icg q + f':(T) anid I. q 'I, respec-
tively. Thus

lo log o + If(T) (23)
k +


Lo.P" = log q + gk(T) (29)

From equation (7), Iemely,

-; = r k(T)

it follows th-t

fk(T) = log Yk) (0)
k (30)

NACA ARR No. 14129

From equation (5) for Pk and equation (20), which
defines Sk,

Pk QkSk
(1 T)P

It follows that

1 Yk(T) Sk(T)
gk(T) k= log
1 Sk __
= fk(T) + log (51)
k (1 T)

For example, for k = 1 and k = -1 and with the use of
equations (15) and (16),

1 (1 T)P+1
fl(T) log (32)
(p + 1)T

(1 T)P[1 + (2p + 1)T] -1
gs(T) = o10g (533)
(p + 1)T(1 T)P

f-l(T)= log 1 + (1 (54)
2 p + 1

,1(T) =- log (3 + 1) P()+ T)( -T)P
2(p + 1)(1 T)P

For k = 0 ard k = +, equations (50) and (31) require
a limitin: process for their evaluation. Alternate forms
'or f,(T) end gk(T) may be obtained, however, by
mean3- of equations (19) and (20) defining Rk(1) and
Sk(T), which yield the results for k = 0 and k = -+
iir.ctly. Thus

NACA AaR. No. LI129 21

fk(T) = 1T (36)

1 O iT
gl(T) = I R T) (]--T (37()

where F.k(T) end Sk(T) are relrlted acc.-ding to equa-
tion (25). Then

1 'T
fo(T) = (1 T) (5S)

^T [r, ? (3 -)

1 1 2+ T 1
g(T) 1 34 -


2- T T '

It is vrortihy of special notice t.rt the functionS fo(T ),
go(T), and fL, ) e identi al. '.iit- the functions
f(T), g(), a' ), rcspecL iveil, which formed the
ubsis 0o part I 'iefi'recc ). in: action, the expres-
sion3 lo, i q + foT T, o q + o(), and log q +f+-(T)
are identical with the functions L, L, and H,
respectively, which ,vere introduced in part I.

A numoer of functions fk and gk have been
calculated, v:ith y = 1.), for several positive and
negative values of the index k, and the values are
given in tables 4 and 5 and plotted in figures 4 and 5-

22 NACA ARR No. L29

The opportunity is taken here to note that, for the
von Karman-Tsien case = -1 or = ,

S= = + (1 )1/2
fk =k = log

= log 2}--
1 + Vl M2

end that the sets of functions Pk and Qk, as in the
incompressible case, are reduced to a single function by
the operator 1 log; namely (compare equation (14)),

log q + log 2 2
1 + V ?M

In fact, the complex flow potential I + id can be
expressed as an analytic function of a single complex

variable 9 + i log q 2-- Tsien has made use
1 + v M2
of this complex variable in his hodograph treatment of
the compressible flow past an elliptic cylinder (refer-
ence 3).

Velocity Correction Factor

The solution of the problem of an expect corre-
spondence between the flow past a prescribed body in an
incompressible fluid and the flow pest the same body in
a compressible fluid is a difficult matter. This
problem cen be solved exactly for certain types of flow
p-.tt.-rns (not pest closed shapes), such es flows inside
or outside angles or channels, and for certain flow
singularities such rs a vortex, source, and doublet -
t:':rs of flow which can be associated with the particular
solutions Q. Some of these types of flow are illus-
trated by examples in the following section. Combining

NACA AR. No. Ll429

particular solutions to represent uniform flow past a
prescribed body is E complicated process, since the treat-
ment of infinite series in the functions iQ for both
positive and negative vluez of k is involved. Further-
more, the process of :eturning to the physical-plans
vrrisbles from the hodogreph-plane variables hinges on
nonelementary prirts no differential geometry. Certain
types of jet problems can be properly treated in the sub-
sonic rnge by series in Qk with k positive, rs was
shown by Chpplygrin (reference 5). Thus, it appears that
much work remains to be done in order to render feasible
exact and practical solutions for unif'crm flow past pre-
scribed bodies i:i a corrpressiole fliid. BecEus-? of the
difficulty end complexity of the general icroble:., of 'low
in a compressible fluid, attempts have beiei n.ad by a
number of investigators to obtain results oy means of
velocity correction formulas tn.t serve to place in
correspondence velocities i:n an incoLrpresz.ible and in a
compressible filed.

In part I the volocit,' corrcctioin factor was dis-
cussed with perticuLar reference to tie tvfo functions L
and L (i, an:L P oP' te present paFpert ) aSsoci SCd
with a vortex end source type of flow, respectively. The
main justification for the results of part I was the
yielding and the unifying of t-ie results of Chap.lyin,
von Kariman and Tsien, Te-iple arn, Ysr..ood, and Pracndtl
and Glauert. The know.leJge of tl-,e inlinite set cf funrc-
tions Pk dand QC oisc'ussed in, th': preser.t paper can
now servo to establish further on a reas-on.ble basis the
concept of a veloit;, correction formula.

In oraer thet a single v.?locit-, correction factor
be feasible, even for a flow associated with a particular
solution, it is necessary that Pk Qk. Consider, for
example, the functions Qk and Pk insofar as the first
poser of the vEriable T is concerned. It can be shown
easily that

Slog Qk = log q + fk(-)

Slo q -PT

2L NACA ARR No. 14129


log Pk = log q + gk(T)

log q -pT

Thus, to the first power of T and independent of k,

fk(T) = k(T)


T'e n

Pk = Qk

= (qe OT

T-c nature of the correspondence between the incompressible
flow ard the compressible flow is such that

1 C (L4)
i = c

Without going into any details here of the field point
correspondence or of the boundary distortion, the veloci-
ties in the incompressible end compressible cases may be
rl :eG in correspondence as follows:

(log qi = (log q 2PT
-i T
qi = qce 2(h2)

NACA AViR No. I1429

This result implies ctht the complex variable
9 + i log q PT) in the compresstble case corresoonds
to the complex varisb'.e 6 + i log q in the incompres-
sible case. Equation (42) represents the approximation
of Temple and Yr-wod discussed iri part I.

Consider nov. the functions rk and Pk inscfsr as
largs values of the index k are co-icernad. It is
recalled that, as the index k ,

Rh ---> S.

and that

1 1
log C -4> -
k 1k k k

-> l1g q + h(T)


h(T) = foo(T) = g.D(T)

Then, as 1k -- +,

k = k

= hq(T)]!<

The funcbtop hvT) is expressed i in .rtegrl frmi in
equation (40) rnad ihcs been evsiuated anil tabulated in
part I. (See also table 4 an. fig=. 4 of the present
paper.) The correspondence of velocities in the incom-
pressible end the comrpressitle cEse is given by

q = qceh 5)(T)

NACA ARR No. 14129

Equation (43) constitutes the geometric-mean velocity
correction formula introduced in part I and is limited
to thl-. subsonic range 0 M < 1. It is observed that,
for positive values of k, h(T) lies between fk(T)
and gk(T) in magnitude. Moreover, the deviation of

en'(T) from ek(T) and ek() is quite small in the
subsonic ranie. (See table 6.)

The foregoing remarks, together with the fact that
the geometric-mean type of approximation contains the
results of Chaplygin, von Ka'rm&n and Tsien, Temple and
Yarvwood, and in the limiting case of smnll disturbances
to the main flow the exact Prendtl-Glauert rule, lead
to the suggestion tnet it may be adopted as an over-all
type of approximation in the subsonic range.

Flow Patterns Corresponding to the Particular Solutions

Before the flow patterns corresponding to the par-
ticular solutions k Pnd *k given by equations (4)
for the compressible cnse are discussed, it is instruc-
tive to examine the incompressible case. Consider the
complex velocity potential

S= f+ i1

= Uzn


where U and n are constants and z = x + iy. It is
vell known thrt, if n = where a is an engle between
0 and 2Tr, equation (44) represents the flow in a shrrp
panle. For example, the flow inside a right angle is
obi,.inea with n = 2 and the flow outside a right angle
i2 ?bt ired with n = -. Again, the value n = 1 or
5* 1
a = n corresponds to a uniform flow and the value n = -
or a = 2n corresponds to the flow around a semi-
in:finite line. CieErly, all the angle flows are obtained
with values of n between 1/2 and co. Other types of
flows are given by other values of n. For example,
n = -1 corresponds to a double Fnd the remaining

NACA ARR No. 14129 27

negative integers are associated v:ith singularities of
higher order than the dcublet. in addition to the flows
described by the powers z", tiheee are the two funda-
mental flows, the source and the vortex, associated with
the function log z. If, now, it is desired to obtain
generalizations for the compressible crse of the fore-
going particular flows, the procedure is first to express
or W for tne inconmpressible flow as a function of
the hodograph var'iables q and 6 and then to replace
qi by Pk or Qk' respectively. Several examples will
best illustrate this procedure:

(1) Consider the ccmoLessible generaliz~tion of the
angle flows. By mrern of the relation

di -'w
2- e-

where w = 9 + i log q, the hodograph complex variable w
is introduced as nde.,eendent vari.&bl in place of z.
From equation (.l4)

= nUzn-1
-- e


z 1 (i e- -l


Q U i-wn-1
0 = -


23 NACA ARR No. LI29

U n-1 i n
q sin- 9
n n-l

If n is replaced by k, the compressible generali-
zrtion of the angle flows is given by

= k sin kO (45)
(U k
S k-li

The inside angle flows are given by values of k in the
rpnge2 1 < k < and the outside angle flows, by values
of '< in the range 1 -k < m. For example, k = 2
for the flow inside a right angle, and k = -2 for the
flew outside a right angle. Other types of flow are
,iven by values of k in the range -1 < k < 1.

The case k = 1 or n = 0 is exceptional and, in
fact, corresponds to the incompressible flow

e = ez (46)

-.here c is a constant.

(2) Consider the compressible generalization of the
doublet. The complex velocity potential for the incom-
-ressitle doublet at the origin is


The reflected-velocity vector is

d 1
dz z2


NACA ARR No. L4 29 29

z = le

0 = -ie

The stree'r furction fr tn- ir3n.oinpress.ble 'doublt is
then given by

The compres- ible generalizt.ion of the dcublet is
others fore

i/2 = QI/2 cos 19

(5) Consier the ccmopirssbl1. i. gnerElizt iLon of the
source. Tht complex vplc-itry potential for e unit source
at tne origin is

2 = log z

The reflected-velocity vector is

.fl- z

= e-


z = e


Q = iw

J3 NACA ARR No. J4129

It.e velocity potential for the incompressible source is

P = log q

T c.;r'essible generalization of the source is then
given by

A- 7

(1) Ccnsider the compressible generalization of a
-oit vortex. The complex velocity potential for a
vortex of unit strength at the origin is

0 = i log z

The reflected-velocity vector is

dO i-
dz z


e ence

z = -iei

End, except for en additive constant,

0 = w

Thc streri f'zction for the incompressible vortex is

I = log q

The compressible generalization of the vortex is then
givrn by

I, = Lco


Trensformation from the Hodograph
to the Physical Variables

Given the velocity potential aqnd the stremn
function 4i in terms of the hodogrrph vrriables e
and q, it is possible to e::pres: the ccordinates x
and y of the physical plan in terirs ~f 0 end q.

From the bss.c flow equations

9,( o ,,j
tx pA 7

6o = Pc 6'
by p &'x

it follows (see equ-tionr (S ) of refere:ece 1) that

dz = ei6(d/ + i P2 d;,
The rei an imairy t of tL on yield d
The real and imaginary parts of tLtis equeaion yield

dx = cos

+ 5 CC'

dy s= ^ sin 0 +

+ sir

-3 si n 9) rq
p eq

e9 ,- o si

S cos 6
p Oq /

S+ 6. cos03
p 6e

Equations (47) relate the differential line elements in
the physical x, y pione and the hodograph 9, q plane.

e) du


) deb


NACA ARR No. 1429

'hn expressions for and \ as functions of 0
F.d q are known for a given flow, the integrals of
equations (47) are the equations of transformation of
i.e 6, q to the x, y coordinates. It may be remarked
.at the hodogrsph flow equations (1) are the integra-
uilit' conditions for the differential equations (47).
i.e right-ha.nd sides of equations (47) are therefore
-er'cot differentials.
Consider one set of particular solutions from equa-
tions (4)

S= Pk(q) cos kG

= Qk(q) sin k.

where k = l1 and k = 0 are excluded. By the use of
equations (5), it can easily be verified that

x k (Pk p_ 0 QJ cos0k + 1)9
2 q pq k+ 1

+ ( + Q cosk 1)] + Constant
S pq k k 1

k P p sin(k + 1)6
k q pq k + 1

Pk+ !1oC sin(k 1) + Constant
+ -- +Constant
q pq 'k k 1

The equations of transformation corresponding to the
other set of particular solutions from equations (4) are
obtained by replacing in equations (48) the cosine by
ti-:c s-ne a,-nd the sine by the negative cosine.
The excluded cEses k = 0 and k = 1 are now
tre-ted. For k = 0, one set of particular solutions
corresponds tc a source and is


o = Po

oo =a

Equations (47) then yield

x = cos

y = rsin 6

The other set of particular solutions corresponds to o
vortex and is

'6 = 9o
S= Qo

Equations (47) then yield

x = sin b

y = cos e

For k = 1 with

S= P1 cos 9

1'i = QI sin 9

equations (47) y'eld

1 = P ( 1 dP dQo
x. cos 29 + 7 + dq
q p9 12 dq pq dq

5t NACA ARR No. L4I29

yl = Q Psin 29 + 6
1 q pq 1/ 2 \q pq 2

Vith the use of equations (15) and (5),

x =- IT cos 28 + log q
(P + 1)T(1 T)

T 1 2P, + 1 1
g(T) 1 ii -i > (49)
P + 1 2 p + 1 ( T)P

1 1 (1 T)]+1
Yl = 1 1 sin 29 9
=(2 + 1)T(1 T)

where g(T) = go(T) by equation (59) and is evaluated
in p-.rt I (reference 1).
For k = 1 wi th

S= P1 sin 9

=1 = 1 cos 9

i = 2' sin 29 + 6
( + I)T(I T)'

1 (1 T)P+
1 = cos 29 + log q (50)
(F + 1)T(1 7)^

S g(T 1 2 + 1 1
P + 1 2 p + 1 (1 T)P
p + ~.2 (3 I-i L T)'

NACA A.-R No. 14T29

For k = -1 with

-1 = P-1 cos 6

^- = Q-I sin e

equations (1'7) yield

+ Icos
p. /

1? I+
29 + 1
2 q


Pc dQ-l
1 dq
pq dq

o n-1
+ j sin 29 -
0q /

i oq

With the use of e.uatLcne (16) and (5),

4q1 ; +V 1 -
-1 2 1 IT ( T)H U1

+ -- log

+ + 2
e(p + 1)

Y-_ 41
y-1 ~42

,L 5- 2 1 (1+
P. 1 (1 -T)3 C' 1

(1 + () cos 2e


sin 26

+ --e

X-1 =



q + -_51__tk g(T)
,' + i ) O 2

2p 2 (1 T- -
2~+l [1 T)3-1


y- =
-1 4

7, NACA ARR No. L4129

For I: = -1 with

-1 = P-1 sin 8

-1 = Q-1 COS 9
I= Q-Cos 8

=-- P+2 p- (1+ pT) sin 28

4, 2

( lP p+ 1 (1+ + ) cos 2e (52)

( 1 PP + 1

+ log q + 53 + (T)
4aao 4(p + l)ao

+ 3 +' 2 + 1 1 1
-(P + 1) a"0 (1 7T)

Ringleb (reference 2) gives an example of the flow
of 9 comoressible flaid around a semi-infinite line. An
e(r.] nation of RinSleb's stream function 1/ = sin
sLows th&t it is a linear combination of $1 and -1'
tl-t is,

S= I~Q1 + Q sin 8

NACA ARR No. LIl129 57

In fact, all the exterrgl angle flows (1i -k < m) are
nonunique; for, in view of the discussion preceding
equation (11), a general form of 4f. is

k = q-: [A k (T)+ Y() in k8
-k L k -k- J

where A is an arbitrary constant.

Observations on Limit Lines

In the present section t'he-e are reviewed briefly
certain conditions, discus::ed bj Tollmien (reference 9)
and Ringleb (reference 13), with regard to possible limi-
tations on the potent-al flow o0' an adiebatic compres-
sible filaid.

Consider tle family of str:-a:rlines

Then along a s re-r'line

d = c d' + 1'-dq = 0

and, from equations (L-), the irne elements along a
streamline ere

> (53)

dy o d 2 i dq

Singular pcitbs long a streanlkine are characterized by
the vanishing. of tne comanon factor of equations (55):

NACA ARR No. 14129

2 2 0 54)

(Stagnation points at which '' and vanish, the
vortex for which = 3 and the source for which =0,
ere excluded from this discussion.) Observe now from
Q-'i ions (147) that the Jacobian of the transformation
r-i:i th'. hoaogreph variables 0 and q to the physical-
Slane variables x and y is given by

J( Po( ^
pe2 pq2 6q WOq6

= ( ( 2 )M2 (55)

Thus, the vanishing of the Jacobian is equivalent to the
condition for the existence of a singular locus for the
fairly of streamlines

9(8,q) = Constant

This singular locus consists of points at which the
stre.m.lines undergo an abrupt change of curvature and
means, physically, that the acceleration q of a
fluid ;.article is infinite at such points.

Both Rinleb and Tollmien have shown that the
S l1,:ul3r, locus for the screamlines is also the envelope
ci the Mj4ch lines in the plane of flow. The Mach lines
r-e3 related to the streamlines in such a way that the
co:r ornent cf the fluid velocity normal to a Mach line is
cqi.Jal u the ircal velocity of souna. The Mach lines
-.-n identical vith the so-called characteristic curves
of tlhe second-order partial differential equations
forn and 1' and are the integral curves of the ordi-
n'.ry differential equation

NACA ARR No. L4129 59

d62 -M ) dq2= 0


dB = -Vli 1 dq (56)

The real solutions of this differential equation inter-
preted in the physical x, y plane yield the asch lines
for a given flow. The solution of eqietion (56) is
r -i( 'r- t ,_ -jn'
9 9o = tan- V- v l) t- 2 (57)

where T -= j-- and whert 0 assU.T.es t'he values
s3 + 1
of 9 eron. t- N": f, = 1 lIne vn flow.

It is recallEd th;t L !.: fanition

= 1l- q + htT)

introduced in. o:.:- I ',:f.2--rces l in connection with
the georet-'ic-:!rar. tr -- O: ty; c-)riection formula,
is a 3slutr ott r t 1-e ,ll' .-r i l Equation

dH = '1- dq

in tne .uo.so'nic ran.-. Cbs'-rve thet a continuation of
the fancti-Ln H intj the supiersonic r.-nge is given by
ecuati-'rn (. a's
d9 = \ d dq

In the superscnic rsnge, the function H = 8 8o can
thus bu itr..r"',eiC; s-j th'1es odograph of the Mach lines
for a giv-.1 fiow.


The differential line elements dx and dy for the
Mach lines in the physical plane are now given. Fro-n
equations (47) and (56), the line elements along a Mach
line for a given flow are

dx = 1 cos 9 sin 2 )dq
S\~ \ (58)

dy = 619 1 sin 9 +cos v dq
pdy q q 9/

Singular points long a Mach line are characterized by
the vanishing of the coImmon factor of equations (58)

6p- V2-1 0 (5961
q q 5T (59)

Equation (59) represents in the plane of flow two pos-
sible singular loci or "limit lines" for the two families
of '.'ach lines associated with the plus End minus signs
in equation (56). Clearly, the two singular loci cannot
oczur simultaneously since the two conditions cannot be
satisfied simultaneously. Observe that equation (59) is
equivalent to the vanishing of the Jacobian given in
equations (55). Thus, the vanishing of the Jacobian is
not only the condition for the existence of a singular
(cusp) locus for the streamlines but also the condition
for the existence of a limit line (envelope) for the
Mach lines.

The existence of a singular locus may be looked upon
as being equivalent to the vanishing along a curve of
the Jacobian J (X') of the transformation from the
hodograph-leane variables 6 and q to the physical-
plane vericbles x end y. It is remarked that
sin.uler solutions exist for which the Jacobian J( x-.
of the transformation from the physical-plane variables x
and y to the hodograph-plane variables 6 and q
vanishes identically in a region of the physical plane.
In this case, as Tollmien pointed out, 0 and q are

NACA ARR No. L4129

no longer independent variables and the flow cannot be
described in the hcdograph pltne. Examples of these
missedd flows" are the solutions of Meyer (reference 11)
for supersonic flow inside end outside sharp angles.

It is of special interest to apply the condition
for the vanishing of the Jacobian to the particular
solutions I and k treated in the early part of
this paper. The expression for the Jscobian for a
particular solution

S= P,C cos :G

In,- = -Q., sin kG

is, with the use of equations (5), for k / 0,

pq2 b q 6 q 69'

k2p2 sin2k6 + ()\2 -) 2 ccs 2k (60)

Clearly, this expression for J is positive in the sub-
sonic range Ni < 1. At the scn.- value M = 1, Pk / 0
(see tatle following equat-on (25)) and J is again
positive. At tne first zero of Pk in the supersonic
range M > 1, Q. ? 0; hence, J is negative. The
values of ., for all the pairs of values 6, 1. for
which the Jacobian J vanishes, therefore lie between
M = 1 and the value of M at the first zero of Fk
(or Sk) in the supersonic ran'.e.

By means of the relation

Pk =- Ok
P k

NACA ARR No. 41299

t-e vanishing of the Jacobian yields

cot k = Sk (61)
R 1

Equation (61) is the relation for pairs of values 9, M,
whic'- interpreted in the physical x, y lane constitute
tl.e liniit line for the particular flow k' k. The
values of I. tiht satisfy equation (61) accordingly lie
between M = 1 and the value of M at the first zero
of Sk in the supersonic range.

This paper is closed with the following remarks on
limiting values of M in connection with the use of
velocity correction formulas. The limiting local values
of 2i in the case of uniform flov. past a prescribed
boundary, in general, depend on shape parameters. The
use of a velocity correction formula, however, yields a
constant limiting value of M that depends only on the
particular correction fonmula used. The geometric-mean
correction formula yields the value i = 1; the approxi-
!i.ction of Temple and Yarwood yields M = 1.55; and the
r1 tithietic-mean correction formula given in part I
(reference 1), which is based on a linear combination of
D: .ni.rce (limiting value M = 1) and a vortex (limiting
value ii = ) or a spiral flow, yields the value M = 1.15.

Langley Mer.iorial Aeronautical Laboratory
National Advisory Committee for Aeronautics
Langley Field, Vs.

NACA ARR No. L'129


1. Garrick, I. E., and Kaolan, Carl: On the Flow of a
Comoressible Fluid by the Hodograph Method.
I Unification and 2xternsion o f Present-Day
Results. NASA ACR No. L4C24, 144. (Classification
changed to Restricted, Oft. 19)4.)
2. Ringlab, Frirdrich: Exakte L'sungen der Differential-
gleichun-en einer adiabatischen Gasstr'dmung.
Z.f.a.M.M., Bd. 20, Heft 4, Aug. 1940, pp. 185-198
(available as British Air Ministry Translation
No. 1609).
5. Chaplygin, S. A.: On Gas Jecs. (Text in Russian.)
Sci. Ann., 'os-zow Imperial Univ., Math.-Phys. Sec.,
vol. 21, 100c, -,p. 1-121 (available as NA>A TM
Vo. 1065, 1944).
4. Bers, Lipman, and Oelbirt, Abe: On a Mlass of Dif-
ferenitial iquations in Mechaniis of Continua.
Quarterly ,rpl. Math., vol. I, no. 2, July 1945,
pp. i68-lS3.
5. Bergman, Stefan: T-e Hodogr=Arh Mlethod in the Theory
of Compressible FluLi. Advanced Instruction and
Research in Mechanics, Briov.n Univ., Summer 1942.
6. von Ka'rmn, Th.: Ccmrressibility Effcts in Aero-
dynamllics. Jour. Aero. Sci., vol. 3, no. 9,
July 1941, rr. 537-356.

7. Hurwitz, Adolf: Mathenatiscite Werke. Pd. 1 -
Funktionentheorie. Emil Birkh'auser 9- Cie.
(Easel), 1;52, np. 514-520 and 52-654.
8. Tsien, Hsue-Shen: Two-Diime si.onal Subsonic Flow of
Co.noressible Fluids. Joar. Aerc. Sa3., vol. 6,
no. 10, Aug. 1959, on. 3499-07.

9. Tollmien, W.: Grenzlinien adiabatischer Potential-
str'murngen. Z.f.a.IN.., 3d. 21, Heft 5, June 1941,
po. 140-152.

10. RinRleb, Friedrich: troer die Differentialgleichungen
einer adlabatischen Gasstromun,5 und den Strmrungs-
stoss. Deutsche C'athematik, Jahrg. 5, Heft 5,
1940, pp. 377-537.


11. Meyer, Th.: Uber zweidimensionale Bewegungsvorgange
in einem Gas, dae mit Uberschallgeschwindigkeit
strbmt. Forschungsarbeiten a. d. Geb. d.
Ingenieurwesens, Heft 62, VDI-Verlag G.m.b.H.
(Lcrlin), 190S, pp. 51-67.

12. Kraft, Hans, and Dibble, Charles G.: Some Two-
Dimensional Adiabatic Compressible Flow Patterns.
Jour. Aero. Sci., vol. 11, no. 4, Oct. 1914,
pp. 283-298.

This reference, which has appeared since the present
paoer was submitted for publication, contains
graphical flow patterns associated with the
particular solutions of index k = 1 and +2.

NACA ARR No. L4129

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15- -


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r--- .--

-- --

.4 .6 .8 1.0 1.2 1.4 1.6 1.8 2.0

PFiure 1.- The Yk functions against M for several values of the Index k.

- -.


__1____ __T__C

- I

tc~tf--ct-c ~,rl I ~ In-i i Irlll I I 1 111 1 1 I

Fig. 1









Ik and Rk






_ -:



- -

- N I 4 4 4 l l.

-- .-.. -- ...... __ .N

... ....l I --, --'-

--- 4-- I I -- --- ---- -
~ f ,- -- -
-]-- i ,i--



- -1,-- i -0--- '- '
.... --

---I- t --- ---



iro3t MiSoRi

0 .2 .4 .6 .8 1.0 1.2 1.4 1.6 1.9 2.0

Figure 2.- The Sk and Lk functions against M for several positive values of the index k.

-cc-~c-e-c-l-c-~-k-cc-ct-cc-cc-l -- --c-t~-~- -~--c--~- c

Fig. 2


and RI,








.2 .4

Fig. 3


x- ^'1xz -4-A


_ .-. -j LM T_ __Na t

==:=::r=:======-ami WES^^

- -7-- -^r -- -- -* ^ -- I I I

.6 .8 1.0 1.2 1.4 1.6 1.8 2.0

Figure 3.- The Sk ad 4 I rmaolonu agltnut I ftr seweral negative values or the Indx k..


NACA ARR No. L4129



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P.O. BOX 117011



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