Nonstationary gas flow in thin pipes of variable cross section


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Nonstationary gas flow in thin pipes of variable cross section
Series Title:
Physical Description:
81 p. : ill. ; 27 cm.
Guderley, G
Feingold, Dave
United States -- National Advisory Committee for Aeronautics
Place of Publication:
Washington, D.C
Publication Date:


Subjects / Keywords:
Aerodynamics   ( lcsh )
Difference equations   ( lcsh )
Gas flow   ( lcsh )
federal government publication   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )


A difference method is presented based on the concept of characteristics for the calculation of nonstationary one-dimensional gas flow. The method is readily applicable to isentropic flow and can be applied to flow containing compression shocks. Numerous examples are given.
Sponsored by National Advisory Committee for Aeronautics
Statement of Responsibility:
by G. Guderley.
General Note:
"Report date December 1948."
General Note:
"Translation of "Nichtstationäre Gasströmungen in dünnen Rohren veränderlichen Querschnitts." From Zentrale für wissenschaftliches Berichtswesen der Luftfahrtforschung des Generalluftzeugmeisters (ZWB) Berlin-Adlershof, Forschungsbericht Nr. 1744, Braunschweig, Oct. 22, 1942."
General Note:
"Translated by Dave Feingold NACA."

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University of Florida
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cA Iftr1l%

--3 LI"d I

) J0 UL I I L





By G. Guderley

iM 1siLCT

Characteristic methods for nonstationary flows have been
published only for the special case of the isentropic flow up
until the present, although they are applicable in various
places to more difficult questions, too. The present report
derives the characteristic method for the flows which depend
only on the position coordinates and the time. At the same time
the .treatment of compression shocks is shown. To simplify the
application numerous examples are worked out.

*'"ichtstationrre GasstrEmungen in d6nnen Rohren verfnderlichen
Quersehnitts." Zentrale fi~r wissenschaftliches Berichtsvesen der
Luftfahrtforschung des Generalluftzeugmeisters (ZWB) Berlin-Adlershof,
Forschungsberioht Nr. 1744, Braunschweig, Oct. 22, 1942

NACA TM No. 1196


In papers by F. Schultz-Grunow and R. Sauer methods have
been developed recently for completely solving the problem of
nonstationary isentropic gas flows in a pipe of constant cross
section. An expanded view of the problem is the basis for the
present report. Flows are considered, which likewise depend only
on the position coordinate; however, the cross section of the tube
need no longer be constant and the entropy may vary from particle to
particle. The method of solution applied here has been discovered
almost simultaneously in several places, by Adam Schmidt, W. During,
and F. Pfelffer, among others.

The application of the characteristic method is possible
without a previous substantial knowledge of mathematics. Corre-
spondingly, if a derivation was desired too, one could be had
which did not make any special mathematical demands on the reader.
As a model, the Busemann derivation of the characteristic method
for two-dimensional stationary gas flows might possibly do It is
actually possible to apply this derivation immediately to the

Schultz-Grunow, F.: Nichtstationhre eindimensionale Gas-
bewegung. Forschung auf dem Geblet des Ingenieurwesen; Bd. 13'
(1942) pp. 125 to 134. Sauer, R.: Charakteristikenverfahren f'Mr die
eindimensionale instationare Gasstr6mung, Ingenieur--Archiv,
XIII Vol. (1942) pp. 79 to 89. Vorbereitende Untersuchungen sowie
Anwendungen finden s1ch in den Arbeiten von H. Pfriem. Zur Theorie
ebener Druckwellen mit steller Front Akustische Zeitschrift
Jahrg. 6 (1941) part 4. Die ebene ungediapfte Druckwelle grosser
Schwingungswelte, Forschung Vol. 12 (194) pp. 51 to 64 -
Reflexionsgesetze fUr ebene Druckwellen grosser Schwingungswelte,
Forschung Vol. 12 (1941) pp. 244 to 256 Zur gegenseitigen
Uberlagerung ungedampfter ebener Gasswellen grosser Schwingungsweite,
Akustische Zeitschrift Jahrg. 7 (1942) part 2 Zur Frage der
oberen Grenze von Geschossgeschwindigkeiten Zeitschrift f. techn.
Physik 22 (1941) pp. 255 to 260. Eine weitere Anwendung findet
sich bel G. Damk'dhler und A. Schmidt, Gasdynamische Beltrage zur
Auswertung von Flammenversuchen in Rohrstrecken. Zeitschrift
fir Elektrotechnik Vol. 47 (1941) pp. 547 to 567.

2Busemann, A.: Beitraa Gaestnamik in Handbuch der Experimental-
physik (Wien-Harms) BE. 4, Tell 1, p. 421 and adjoining pages.

NACA TM No. 1196

isentropic nonstationary flow In a pipe of constant cross section
and from this by means of some supplementary physical concepts
succeed in getting a treatment of flows in a tube of variable cross
section; this is the course which had been taken, originally. In
comparison to the mathematical theory of characteristics, however,
these considerations operate with a Jack of clarity sufficient so
that the mathematical theory for ths engineer too can be
represented as the best approach to the characteristics method.

Considerations necessary for the present problem are now
brought forward from the characteristics theory As a result,
equations for the directions of the characteristics as well as
conditions which must be satisfied along the characteristics are
obtained. Proceeding from these relations, the next sections
develop the actual method of computation. Next, the characteristics
method for the case which is familiar by now, that of isentropic
flows in a pipe of constant cross section,is deduced again and
the transformations appearing there are used. to simplify the
computation in complicated ctses, too. Since this is not always
possible, the most general form of the characteristics method is
shown in a later section. A'ter this, the formulas obtained for
the special case of an ideal gas with constant specific heat are
simplified and the consideration of boundary conditions explained.
The remaining sections deal with celculatior of compression shocks;
the known relations which connect the phase befor- and behind a
compression shock with one another are set forth in a convenient
form for the present problems and with that the calculation of a
compression shock in a flow is carried out.

The theory is illustrated with suitable examples treated in
detail. In that regard, it s-em-ed advantageous to avoid definite
problems of technical interest, in doing so gaining the possibility
of working out examples under very general assumptions without
excessive effort. It Is hoped that, nevertheless, the application
of the method to physical problems offers no additional difficulties
worth mentioning inasmuch as thr earlier publications contain such
applications. The author expresses his thanks to Dr. Hans Lehmann
for working out the examples.

3Ccmpare Courant-Hilbert. "Methoden der mathematischen Physik II",
p 291. Guderley follows the representation given by H. Seifert at the
same institute for Gas Dynamics in lectures.

NACA TM No. 1196


Consider nonstationary, perfect gas flows in a pipe with a
cross-section that varies in space and time4 in the neighborhood
of the flow tube; that is, it is assumed that the velocity and
the phase over a cross section of the pipe may be considered as
sufficiently constant. In general, this assumption is Justifiable
only if the thichIess of the tube, relative to its length, changes
slowly enough. Only for flows which have as surfaces of constant
phase parallel planes, coaxial cylinders or concentric spheres
need this limitation be ignored. 'Thse flows with plane, cylindrical,
or spherical wave propagation are included as special cases In the
present problem.
To stress the relationship to stationary two-dimensional flows,
let the axis of the pipe be vertical, the position coordinate be y,
the time be t and plotted horizontally. In this yt-diagram the
flows pre investigated. (Compare fig. 1.)

p pressure
a entropy per unit mass
0 density
v velocity
F = F(yt) the cross section of the pipe, let F be given
In a region free of compression shocks, the flow is described by
the dependency of the density on pressure and entropy, the Newtonian
principle, the equation of continuity and the statement that the
entropy of a particle is preserved, as follows:

p = p(s,p) (1)

1 pE + v L + 8 = 0 (2)

S+ (4)

Problems with time variations in the cross-sectional area
are rare; they were included, since they can be handled without
additional difficulty.

NACA TM No. 116 5

In this the derivatives of p should be replaced by the derivatives
of p and s, for this purpose

0o 1 (9)
~p 2

is introduced. Therefore, instead of equation (3)

V op y be v 1 Nr o s 1nF + znF
SB.+ v --+ p --+ -+ -- pv + p = 0 (3a)
a2 y sB 6y )y a2 ot sB at Y at

is obtained.


In regard to the system of equations (2), (3a), and (4), the
familiar question is raised from the theory of -haracteristics. In
a region of the yt-plane let the solution of this system of
equations and its derivations be finite throughout. On a curve C
placed in this range let the values p, v, and s which correspond
to this solution and, therefore, the appropriate derivatives taken
in the direction of C be known. The question is asked whether the
derivatives in other directions may be cmnnuted with the aid of the
system of differential equations, and under what conditions. To
answer this, a curvilinear coordinate system Ss is introduced
in which a curve = constant coincides with C (fig. 1). All the
derivatives with respect to along this curve = constant are
given, the derivatives with respect to are sought. This trans-
formation is carried out and terms are arranged so that the unknown
derivatives with respect to are on the loft and only known
quantities are on the right. That is

= ,(y,t)

n = r(y,t)

a..=a +. +
dt 0 dt q o t

NACA TM No. 11q6

With this form (2), (3(a)), and (4) are obtained

a]2 .Iv V + 4
_n 1vb 1( 3' 3
0 ---y+ ',-+ -

3+)v LS / ( L

v&2 )y at/ a ) 3y 3qn y 6t as P y aat

v o + v L9 + -

and the unknowns, themselves, are obtained by Cramer's rule as
the quotient of two determinants. The determinant in the denominator
is the same for all unknowns. It always ives a single-valued
solution for the system of equations, if the determinant in the
denominator is different from 0. In the other case with a determinant
in the denominator that vanishes, it is a necessary condition for the
existence of solutions that remain finite that the determinants in
the numerator also vanish. In this case, however, the solution of
the system of equations is only defined over any portion of the
solution of the homogenous system. In application to the system of
equations (6) signifies the following: The determinant in the
denominator is formed from the coefficients of the unknowns.
Considering a fixed point on C, at which p, v and s are
known by assumption, the coefficients depend on andi n fo, that

is the direction of C. If C is so directed that the determinant
in the denominator does not vanish anywhere, the P soetc. are

computable as single valued.
Of greater interest for our considerations ps the other case,
namely, that the determinant in the denominator is zero at every
point of C. Such a curve is termed a characteristic. Because of
the assumption of finite derivatives the determinants in the
the r.ssulmption of finite derivatives the determinants in the

NACA TM No. 1196

numerator also vanish. Relations are obtained thereby, in which the
right side of (6) and, therefore, the derivatives of p, v, and s
along C appear as essential ingredients. These relations represent
the starting point of the graphical numerical method of solution.

Since the solutions of a linear system of equations are no
longer single valued for vanishing numerator and denominator
determinants, the pursuit of a given solution of a characteristic
is possible in various ways. These different possibilities actually
appear on changing the initial and boundary conditions.


To find the directions for which the curve = constant is a
characteristic, the determinant in the denominator must be set equal
to zero in the solutions of the system of equations (6).

v 0


This gives

v ay at

o __



( y

+ / -0

From this are obtained the conditions

S + =


v + a) + = 0
%. Fy^ 5t


oyr + ta
3,7 at)
oy ot

NACA TM No. 1196

(v a) 3 = 0
3y cit


The slope of any curve = constant is given by

dt aS

From (7) and (8), together with this, the slopes of the charac-
teristics are

= v

Iz = V + a

_ v a



The characteristics defined by (9) are path-time curves for the
individual gas particle) they might be termed life lines of the
particles. According to (10), velocities are determined from the
slope of the other characteristics, which differ from the velocity
of the particles by ta. For stationary flows the Mach waves
correspond to these last characteristics; this designation will be
adopted. Therefore, let Mach waves of the first family be those
which spread out with the velocity v + a and Mach waves of the
second family be associated with the velocity v a.


As shown in section 3, along the characteristics, certain
conditions must be complied with by the derivatives which result

NACA TM No. 1196

from the vanishing of the determinant in the numerator. These
conditions are called consistency conditions, for p, v, and
are subject to them, if the derivatives with respect to are
to remain finite. If the right side of (6) is designated Ri,
and R in sequence, then the following is obtained for the
determinant in the numerator of the quotient for op:

v + F
3y Ft

V \

I R 0

This determinant must vanish to give
teristics. Substituting (7) gives

R1 0

R2 o
~.y 0

R O0


+ 0




the directions of the charac-


0 =0


According to this the determinant (11) vanishes
(8a), that is, for a Mach wave 1

R-a -a
1 0'

R2 0 a

3 0 -a
3 .Y

by itself.



is obtained, or

NACA TM No. 1196

[ -oaRl a2RP + O aR 0

In this, is certainly different from zero, as long as v and a

are finite and grad E 0. As the condition for Ahe Mach wave 1 is


-RI aR2 + a 3 R3 = 0

The consistency conditions for the Mach waves 2 is, if a is
replaced by -a

-oR1 + aR2 a R = 0


A condition for the life line is obtained if
determinant in the numerator in the quotient

from (6) requires

a2 )y a

v1 y



+- }

the vanishing of the
for 8 to be got

= 0



For the Mach vave this equation
for the life line is

is satisfied by itself, the condition

R3 = 0


The determinant in the nu''Lerator of could be investigated, too;
however, this vould not give any new consistency conditions.

NACA TM No. 1196

The values of Rl,
From (12a),

R2, and R3

are still to be put in.

+ 0 i i (v
1W 1-;V

+ a) + 1

= -apv ~nF a .InF
=-p y --Ft.

is obtained. The direction d' along a Mach wave 1
(10a); on that account

is given by

d. + = (v + a) + 'l

is valid for it. With that the consistency condition for the Mach
wave 1 can be written in the form

A+ dv -a tv +F + )71n\1
ao dt dt n T + (14a)

The consistency condition
substituting -a for a

for a Mach wave 2 is obtained, by

_dp dv -a v 2nF
ap at dt T

+ Jr.


From (13) for a life line is obtained


This may be integrated immediately

a = constant


Naturally this constant will differ from particle to particle, in

-H (v+ a) +
a d][d t

NACA TM No. 1196


Equations (14), (1), (9), end (10) just obtained are certainly
useful, fundamentally, as a starting point of a characteristics
method in fact, there are -examples, where it is necessary to revert
ro them comparee section 11); in most cases, however, there are
still other transformations suitable. The direction in which to
proceed for these is obtained if an attempt is made to derive the
characteristic method for isentropic flows in a pipe of uniform
cross section from equations (14) and (15) possibly in the form
applied by Schultz-Grunow. To emphasize the fundamental ideas, no
assumptions of any kind are made therein of the characteristics of
the flow medium.

On account of the hypothesis of constant entropy, equation (15)
satisfies itself. In equations (14) the right sides are omitted
since it concerns a tube of uniformm cross section. Further, on
account of the hypothesis of constant entropy the state of the gas
is still dependent as only one variable, perhaps the pressure,
or the temperature; the quantities appearing on the left side
and a are accordingly functions of this variable. It is possible,
therefore, to consider the expression LP as a differential. Let

T temperature

i heat content (enthalpy)

s entropy

By the second main theorem

T ds = di i dp

From this, on account of the hypothesis of constant entropy,

o uldT .dTm

With that, it follows that

a 1 (di' dT
pa a dTl

NACA TM No. 1196

W(T) = T d dT
aoT a 'dT


is introduced in which ao is the sonic velocity of a comparison
phase which was added to make W dimensionless. The phase of the
gas may be characterized by W from W = W(T). It follows that

T = T(W) (18a)

Further, it is valid that

With the use of



p = p(T) = p(W)

a = a(T) = a(W) etc.

W equations (14) appear in the form1

dW + dv = 0 for a Mach wave 1

dW dv = 0 for a Mach vave 2



Bringing in


S= W+ -'-

S= W. .L.


these last relations change to a form which may be integrated. This

X = constant for Mach wave 1 (20a)

S= constant for Mach wave 2 (20b)

If the magnitudes of X and j are known for a point of the yt-diagram,
the velocity is thereby completely defined as well as the thermodynamic
phase. It is, to be exact,


ao 2



NACA TM No. 1196

and on account of equations (iR)

p = p(X + u) (22a)

a = a(X + ,) (22b)

The next sections explain this transformation and the application
of equations (20) to an example of an ideal gas whose specific heat
is a function of tmriperature.




c specific heat at constant pressure

cv specific heat at constant volume

R gas constant

For an ideal gas

E PT (23)
According to the second main theorem, if p end T are considered
as independent variables

ds = 1 i dT + 1 d~ dpp (24)
T 'T T ,p pT

Since ds is a perfect differential,

S11 1 i T i 1
Bp ',T T/ BT T p T

Accordingly, si.ibstituting p fron (23), the following known fact
is obtained


NACA TM No. 1196

that is

i = i(T)

Fp = Cp

p = DC (T)

From (24) as a result

ds = -

dT R dL

and front this by integration

dT -n .L
T Po

the index o


characterizes a comparison phase.


P = e

P -

Considering c.(T) as known, p by (28) and p by (23),
are given as functions of T and s; the thermodynamic properties
of the medium can be calculated in principal, therefore. The






s so /T C,
R T/ R




NACA TM No. 11Q6

is by all

a is also defined by p and p. The computation of
means simpler if carried out in the following way.
to (5)

for which the entropy is to be kept constant. For constant entropy
from (24o)

dT R= dR
T c. p

by differentiation from (23)

d do dT
P p T

From the last two relations together with the familiar relation

Cp C = R

is obtained

a(T) =/P2 RT


The relations discovered up until now describe the properties
of the gas and must always be knownT it makes no difference which
variation of the characteristic method is chosen for the calculation
of the flow. In contrast, the introduction of the functions W, X,
and L serve only as prenarai.lon for carrying out of the charac-
teristics method in the form presented in the preceding section.
Next, to compute W. Fron (25) it follows


.'5 01
* '6F/

with (26)

, 6T3 p
8~\ c


NACA TM No. 1196 17

Setting the last equation as well as (29) into (17) .gives

iTV = Ti (30)
2 o To ET

For the present case, 9 = constant = so (28) becomes

'T R T
--L (T) = e (2a)

Now the following can be formed

T = T(% + p)

a = a(X + P)

P P(X + u)

These calculaEti.?ns were c-rr1?d. out, numerically for carbon
dioxide. The relation between the specific heat and temperature was
taken from Hi.,tte with the aid of these- values (i -,-,)a
a/ao, .P, and W can be co-1i !.:ed from 3qi.:ations (26), (29), (27a),
and (30) as functions of the temperature. (See figs. 2(.) nd s(b).)
Figure 3 shows a.a P, and T plotted as functions of
X L = 2LW.


The following problem. should be dealt with: Along a curve K
of the yt-riagra., which has at the most one point in corrimon with
each characteristic, let p,p and v/ be giv;n (fig. ). The
flow should be constructed f.or the following times as far as it is
defined b,' the portion of K iion. Ther~f'.re, it is concerned
here with the coTiputation o.f thl part of t-h- flow defined by the
initial conditions vhich by th- same rga,.mcnts appear everywhere in
the interior, too. Before t]h- construction of th: flow can be started

fitte, 27th edition, Vol. 1, p. 48, table 5, Berlin 1941,
Wilhelm Ernst und Sohn, publishers.

NACA TM No. 1196

the initial values p/p0 and v/a0 must be expressed in terms of
the variables X and p.. Since the entropy was assumed constant,
p/p0 and P coincide. With the aid of the relation presented in
figure 3, between P and X + p and equation (21b), X and p
may be ascertained without difficulties. Figure 4 shows en the right,
the yt-diagramon the left, the diagram of the assumed values P/Po
and v/ao as well as those of the computed quantities X and i as
functions of y.

Proceeding from the individual points of K if the network of
Mach waves had been brought in. the ohase at each lattice point
would be determined thereby; according to (20) X is constant along
Mach vave 1, p along Mach wave 2, and on that account, equal to
the values at those omin-s of K from which the Mach waves spread
out. By (21b) end (22) the phase is given by X rid p. To be
able to dray the nei-'ork of Mach vaves, only their directions are
still needed. These are given at the lattice points by (10);
a/!a is a functim-n of X + p in figure 3, v/ao is computed

as X .

The direction for the portion of a Mach wave between two lattice
Dnin+s is approximated as the average value of the corresponding
directions at the lattice points.

The construction becomes especially L-nple if the Mach waves
are drawn for equidistant values of X and p. The directions
of the Mach waves appearing can be co.nputed beforehand and possibly
prepared in the form of table I. The interval between adjacent
values of x or u was selected as 0.1, the size of the interval
depends on the accuracy desired. In the table the upper column
headings and signs refer to Mach wave 1, the lower to Mach wave 2.
The numbers entered in the table represent the average values for
(v + a)/a, and (v a)/ao. For Mach wave 1 for which X = 0.3
and which leads from a point with p = 0.2 to a point with 4 = 0.1,
in the column with the heading X = 0.3 the value is to be taken
from the row p = 1.5, that is, (v + a)/ao = 1.103.

In the flow diagram the values of X valid there are entered
to the lift of the lattice point and the values of p to the
right. To determine, for example, the position of C from the
points A and B since the phass of C is given beforehand by
X = 1.1 and = 0.5 the average directions of the Mach waves
(v + a)/a0 = 1.422, (v a)/as = -0.778 can be taken from table I
and dra'n in the yt-diagram. The auxiliary diagram on the left in

NACA TM No. 1196 19

figure 4 can be used for this. There the direction of a Mach wave
for which (v + e.)/ao 0.8 is drawn in. Similar diagrams can
be used as aids for the following exa:,ples, too. The portions of
the Mach ,vae goin0 out from K really require a special co~ioutation
since the average values of X or for them do not agree, in
general, with the of ta3be I. The small deviation was
tolo'rable, however.


If the cross section of the pipe is net constant, the right side
of equations (14) from which it is necessary to start out, here too,
are reserved. With that, there is the possibility of undertaking
that integration along the Mach waves which led to equations (20).
Nevertheless, the introduction of X and L still re.nains useful.

+ = M (31)


dxk -84 (32a)

is obtained as the- consistency c-ndllon f-ir M-ch wave 1 and

.a --a 1. (32b)
dt o
f-r Mach '-ave 2.

The consistency conditions in 'he fori. of (0i) contain at any
given time the differen.ial of only one of the Lnknoun quantities
X or p while the differentials of both p and v appear in (14)
already. This implies an appreciable improvement in the numerical

The construction of the flow rests on the fact that equations (32)
are considered different equations. LAt GA be the value which a
quantity G assumes at the ooint A, C-GA the difference GB GA
and GmBA an average value of G takun between A and B.

NACA TM No. 1106

Applying equations (32) to commute from two known points A and B
the phase at a third, C, which is on the same Mach wave, in the

XC XA = 'XC,A = -MACao tA C(

PC LB = = .CB -MBCao tCB

For the determination of the flow XC and PC have to be computed
by memns of these last aquat-ins and, at the same time, the position
ascertained of the points sought in the yt-diagram by the use of
equations (10). The calculation process might be explained by an

The flow is cnsid-ered as givcn along a curve of the yt-diagram
and, admittedly by X and 4 (fig. 5, table TI). In addition, the
pine cross secti'-n must be r known fimncion of y and t. For that
it is only necessary to require that F can be differentiaLed
'-ith respect to Tnsltion and tine, a premise which is always fulfilled
In Practice. For this exa sle F is taken in the form

F = F t

From ('1) for M

M = ( 2- + -
a\ ao0 y a0

The positions y = 0 and t = 0 for which M goes to infinity do
not belong to this region of flow where such singularities appear
(for example at the center of spherical waves); it is necessary to
make special investigations which cannot be entereA into in the
present report .

The best way to follow the calculation is by means of the
systematic calculation in table II. To facilitate comparison with
the description the columns are numbered. The first column contains
the designation of the point which is to be computed, the second
column gives the known point which, in common with the point to be
computed, is on Mach wave 1. Column 3 contains the corresponding

7Compare G. Guderley. "Starke kugelige older zylindrische
Verdichtungsstisse in der IHhe des Kugealittelpunktes oder der
Zylinderachse." Luftfahrtforschung, Bd. 19 (1942).pp.302-312. Th-e
concerns itself with a complicated special case of such a singularity.

NACA TM No. 1196

point for Mach wave 2. The first five rows reproduce the initial
values as vell as some further values that hold at the given points
which are necessary for a later calculation. The calculation of a
new point is carried out in the form of an iteration method; as
an example the point 4 will be explained. Next, the values for
X4 and u14 are estimated (columns 4 and 5). In order not to
use too favorable an estimate, it is assumed that )4 -= 1
and P4 = p2 The quantity (X + 0)4 is determined for these
magnitudes and fro'i that, with the aid of figure 3 and,
\ ao.4 /
farther on ( (columns 6 and 8). With these values v + a
.o4 ao ,4
anda are computed columnss 9 and 10) Now the average
\o J4
directions for Mach waves 1 and 2 v + a and v a
\o ml,4 a4
are formed (columns 14 and 19) and the Mach waves are plotted on
the yt-diagram. From this Y4 and aot4 (columns 12 and 13) are
obtained. With these values M4 (column 11) and the average
values Mml,4 and Mm2,. (columns 15 and 20) are computed. To
continue for Mach waves 1 and 2 Aaot4,1 = aot4 aot1
and Saot4,2 = aot4 aot2 have to be computed (columns 16 and 21)
and can be substituted in equations (33). The quantities AX4,1
and Ap J. as well as X4 and g4 (columns 17, 18, 22, 23) are
obtaineds If the values X and p calculated in this manner do
not agree well enough with tho. original estimate, the calculation
must be repeated in which X and p just calculated appear in
place of the earlier estimates. Naturally, the Mach wevee must be
plotted over again, too, in the yt-diagram for this. These figures
only show the final form at any instant. For that reason all the
steps in the iteration method are put in the tables. A good view
of the results of the calculation as well as insight into the
estimates to be carried out by the iteration method is obtained, if
the flow is followed simultaneously in a Xp-diagram, as well as the
yt-diagran (fig. 5, right). There the X-axis was selected
slanting up to the right at 450 and the p-axis downward at 45. With
a suitable vertical scale X t, and therefore v/aos is obtained
in ediately on a horizontal scale X + i or W and with the use
of unequal distributions a/a and P and, for isentropic flows
D/p, too. The X- and 4-aics were inclined 450 to obtain the
quantities of physical interest v/a a/ao, etc. in a coordinate
eystein with the conventional arrangement.

22 NACA TH No. 1196


The introduction of X and g with the object of obtaining
equations in only one unknown, at any time, with the iteration
method for the determination of the flow was possible up until now
because the expression dL with constant entropy might have been
considered as the differential of a function W independently of
the characteristics of the incident gas. Naturally, that is no
longer possible with variable entropy. The computation of the
flow must, in general, therefore, return to (14). The ideal gases
constitute an excOption. Here, as recognized in (30), the function

W which essentially agrees with f IT for constant entropy,
depends on the temperatilre alone, and no longer on the entropy.
If the expression 1d is considered, therefore, in the, case of
variable entropy as dependent on the variables T and s the
effect of change in entropy is separated, tht-n the rnst can be
written here as a differential and X and ii ccn be introduced as
previously. The change in the entropy along the Mach waves must
naturally be regarded This is oossibl3 without especial
difficultiio since the entrony is constant along the life lines. The
transformation re carried through in the following mf-nner. From
the second law

T ds di 1 do
kingg into acco nt (26 and (29)

2- d T as /Cpc dT /v d'
a a a RT /

Inrroducinc -t, X, and p as before, the consistency conditions
are nb-ained in the form

__ = j lF + a + /"v T J de (34a) for Mach wave 1
dt ao t Vc R CI dt

du a (vtnF + Ir-F + /c T 1 _s (34b) for Mach wave 2
dt .o )y At / R a, dt

NACA TM No. 1106 23

The differential quotients ds/dt formed along the Mach waves
interfere: The following transformations are possible. Analogous
to the flow function of two-dimensional stationary flows, a
function is introduced, VI is constant along the life lines.
This can be achieved by requiring that

l4 F _2_ (35a)
E -Fo PO

F .P (35b)
ay Fo o0

Along any curve of the yt-diagram

+ dt (36)

Along a life line d = v therefore

SF o F a v
y" Fo o, F Po v

that la is actually, constant along the life line. At each point
of the yt-diagram \ itself can be defined by a line Irjtngrn] that
leads from a fixed point A at which might be zero to B.

Sdy + T -(37)

The physical significance of can be recognized as follows:
Let C (fig. 6) be the intersection point of the life line through
A with the line t z constant through B. To begin with, the path
of integration is along the life line fromn A to C and, from
there, out along the line t = constant to B. Along the life
line AC, is constant

S= A = 0

24 NACA TM Nc. 11Q6

along the section CB, dt is equal to zero, accordingly

IB = / -LF-L d
B C F., 00

From this, it is evident that I represents the mass which is
enclosed between the par+icles at an instant in time for which J
is zer.

The fact that s is constant along a life line can be written
with the use of I1 in the forn


S = s(4)

For ds/dt then

ds ds d
dt W a dt

for which dX. is to be taken, just as
the Mach wave considered.

From (35) and (10)

Wt 2 P a
dt F0 pO

d Fo 0O
d' F, P

ds/dt previously, along

for Mach wave 1

for Mach wave 2

Substituting these in equations (34), allowing for (23), (28), and

(29) replacing is according

to (27b)

by -1d and po/o
7 d 0

by -o a0 yields the f'-lloving consistency; conditions:

NACA TM No. 11q6

For Mach -.ave 1

X_ = a a v nl-nF
dt n a, \ y

For Mach wave 2

at a v
dt o \ ^

Here P is a function of X + j (fig. 3),
a function of y and t. From (38) and (

F/IF is known to be
27b) it follows that

n = n()

and from this

Id li (*) =
d, d^'

constant for a life line

For the sake of corntactness, introducing

C F d
N -the for

Then (39) goee over into the forn

dX a (-M + N)
dt o

S=ao(- N)

(41a) for Mach wave 1

(41b) for Mach wave 2

Equations (40) and (41) supplant the previous consistency conditions
(15) and (14).

Cv F p
Cpo F
p0 o





3?nF, +
t )



P dL




NACA TM No. 1196

Before starting the characteristic construction, the problem
arises here, too, of computing X and p, and now da besides,

from the initial values. Along a curve K of the yt-diagram let
the velocity be given by v/ao, the phase of the gas by P/Po
and T. From T with the aid of figure 3 X + p is obtained,
from v/ao, X i; with this X and p are known. Since p/Po
are given, and P as a function of T is to be gathered from
figure 2, n is obtained iLmmediately from (28). As a result of
plotting it against the values of y from the curve K and
differentiating is obtained. From (36) and (35) together with
(23) d_ for the curve K may be computed for the curve K and,
finally, with that

V dy dy

is deter-lined. In many cases 1hese computations are superfluous;
if entrnoy differences arise fr'm compression shocks, the
determination of dT X and p includes their calculation. The
way the computation of f3ow has to be carried out is shown in
figure 7 and table III with points 4, 5, 6 as exunples. The
related Xp-diagram is right center. (The points included, in
addition, in the table and the figures relate to a later section.)
Along the curve K (points 1-3) X, p, and dA are assumed as
known, in the auxiliary diagram dn has been reproduced as a
function of y. The computation of a new point take point 4 as
an example begins, here too, with an estimate of X and i
(table III, columns 4 and 5). After that, as before, the following
ev/ v + a\
are computed (X + p)4; (a/ao)4; (v/ao)4 ; ( Lv4Y a(
o /4 a /4

( ac ; va i 1 ; (columns 6-10, 19 and 24), the position
a ml,4 a0 /m2,4
of 4 is indicated in the yt-dlagram and y4 and at4 in the
table (columns 11 and 12) assumed. The determination of d6 with

the aid of the life lines enters in as something new. It should be
sufficient for this to draw in a multitude of life lines, simultaneous

NACA TM No. 1106

with the construction of the Mach waves and going back over these
to learn the desired value 1 from the auxiliary diagram. The
kX-diagram is useful for a quick determination of the direction of
the life lines. The position of the intersection points of the
life lines with the Mach waves may be estimated there without
difficulty, and then the average velocity learned. (Compare points 14
and 15 in the yt- and in the Xp-diagrarn.) After dL has beenn found

and, in addition, P has been learned from diagram 3 (columns 13
and 14), MI ani N, as well as (-M N)4 and (-M + N)4 may
be co'-Dured (c.lumns l'-18), the average values (-M lN)i 4 and
(-M + N)2, for the Mach waves be forced (colurrns 20 and 25) and
vith 'he aid of A a t (columns 21 and 26) from equations (4l)
com-ute AX and A& and, ulitimately with that X and i.
(Colu-ns 22, 23, 27, and 29.) Where the original estimates were too
bad, ihe computation 'as reTpeated.


An outline shall be giv:n of how to proceed if the simplifications
given above are no longer possible or if the flow is so small that
the prepared computations as given at the end of section 7 do not pay.
As an example, let the caipuitcation of the point 4 be carried through
from the points 1 and 2 of figure 7.(See fi.. 8.) The quantities

pl/p = 1.44; nl = 1.2; Vl/a = 0.425

p2/Po = 1.866; r2 = 1.332; v2/ao = 0.400

correspond to the initial values assumed there. For the medium to
be investigated p and a must be giv=n %s functions of p and it.
In this case P is obtained, first of all, from (28) and from that
and fiGure 2(b), T. Then c/o and a/a are obtained with (23)
and (29). H.:nce

al/ao = 1.021; 01/00 = 1.375; = 1.46
a ao '71

a2/ao = 1.037; o02 / = 1.710; = 0.637
\ o .2

28 NACA TM No. 1196


M1 = 1.747; M2 = 1.442

can be computed. Here, too, an estimate is made in computing a new
point. For example

I1 + '2
4/po = p/p = 1.44; v4/ao = v/ao = 0.425; 14 2 1.266

With this P4 = (p4/Pol/4 = 1.137 is obtained, whence

T = 282.5

Continuing further

a4a = 1.014; 04/0 = 1.390; ( + a4ao 1.439

74 a) o = -0.589; (v + a)l, a/o = 1.443; (v a)m2, / = -0.613

With that the position of point 4 in the yt-diagram may be found,

Y4 = 1.446; a0t4 = 1.258; (aot)4,1 = 0.048; A(aot)4,2 = 0.092

and, after further calculation

M4 = 1.403
The average values are found to be

/)l, = 1-.33; a/aon14 = 1.0175; l, = 1.439

(/o) = 1.550; a/a 1.026; M,4 = 1.423
O ,/mn2,4 0 m2 ,4

NACA 2M No. 1196

Considering equations (14) as difference equations then7

00 1, 4 mlnJ,4N

1 o3

,n 4
d m2,k

a 0 4
\ a 2,4Po 0 1Po -

a 0

aot r

= -a tk "ot)ml,4

= -(ao t aot)m2,4


by 2

from (29) and (23) gives

C 0 A 7
)p 4
Po0' l,4 o

C -0

CO m2, a /m2,k





\o \ a p1
CPO\ -/mlp\ a )ml,4po

- ,4(aot Aot)

So 0 a P2
S m2, ma 2,4 T0

-M a t a
2 ,4 0 4 o 2

For ideal gases the first tern of the left. side of (14) may
be written 1 d"np a, then o/o does not have to be computed
k dt o
separately. To permit the procedure to be applicable in more general
cases, this simplification is not used here.

+ ao -

NACA TM No. 1196

Putting in numerical values gives as a result

0.547 P4/po + v4/ao = 1.1430

0.o84 p-/p v4/a0 = 0.3715

P4/Po = 1.470

v,/a = 0.342
4 O

From the velocity computed above v4/ao and the velocity at
a point 4', estimated for the present, of the connecting line 1.2,
the average direction of the lif line passing through 4 is obtained-
by an approximation method. If this is proceeding from 4 backwards,
the more accurate position of 4' is obtained. By interpolation
between 1 and 2 V = n = 1.243 is obtained. Since the values
Pi v4
P-, -, PT4 do not agree sufficiently well yet with the originally
Po a'
esti-nated values, the computation must be repeated with the magnitudes
just obtained as starting values. This gives

S/p = 1.478; v/a = 0.3383; 4 = 1.243


Generally the flowing medium is an ideal gas with constant
specific heat or at least can be considered as such, as an approxd-
.nation. In such a case appreciable simplifications are possible.

k = C /C

k 1
C ; C R
P k-i k -i

NACA TM No. 1106

From equations

(27s), (29), and (30)

(T 5 aL


a = */. kPT,
0 / o0

W += + 2 / a
2 k 1 \ -,
2 k-1\0

With this, it follonE that

and from that

a = 1 + I- ( + i)
& 4



P= 1 + ( + )I k-1


The directions of the characteristics are obtained from (9) and (10)
in the for!n

d= ao -- -

_ a(- + Ltd x
dt 4

dy ao -- k + 1

for the life lines

- -- ) for 4pech waves 1

+ -.. Xk for Mcch waves 2
4 /

2 .
k lao

k- I a


r \



NACA TM No. 1196

The consistency conditions for the Mach waves remain unchanged in
the form (40) and (41). M and N are expressed as follows, now,

M I= i + + (X + )' I z F 1 + inF
4 -a2 y at


1 F =+ h: 1( + k-1
T4 4 Ur,

The directions of the characteristics may now be found very
conveniently graphically. A construction which is suitable if the
simultaneous treatment of the flow in a Xp-diagram is avoided is
the contribution of Adam Schmidt.(Seo fig.9.)For the determination
of the direction dy/dt for a life line, two vertical scales at a
distance of 1 apart are used with plotted on the right one
and t5 on the left one as above. A life line for a phase which
is given by X and LI has the direction of the connecting line
of the points concerned on the function scales. Similarly, there
are scales to use for a Mach wave 1, which give 3- k on the
k 1 on the
left and 1 + k- 1 on the right. For Mach wave 2 k + 1
3 ]4
has been plotted on the left and -1 + ----X on the right. In
figure 9, the direction of Mach waves 1 and the life line is given
for X = 1.1 and 0.6.

If the chases in the course of the construction of a Xp-diagram
are followed up, the following "ethod-is suitable (fig. 10, right).
A vertical line is sent through the 0-point of the Xj-system and
the poles Pl' Po2 and PL are determined, where PL is on a
level with the origin of the Xp-system and *2 avay from it. P1
and P2 are directly below and above PL, respectively, and
likewise the distance /2 froa it. To find the direction of the
characteristics for a given phase, a horizontal ray and two rays
slanting upward and downward at an angle arc tan k 1 are drawn.

These intersect the vertical line through the origin of the Xp-system
at the points Q1, Qr, and QL. The connecting lines PiQ ,

P2Q2, and PLQL are the directions of Mach waves 1 and 2 and the
life lino. In figure 10 the construction for point 4 is carried out.

NACA TM No. 1106

This construction is especially convenient with a triangle having

an angle arc tan k .. Figure 10 and table 4 give an example of
an application for the same initial values as in figure 7 and with
Op/Cv = constant = 1.4.


If the flowing gas column is not infinite, the variation of
the flow is determined by the phase at the start, in addition,
also by conditions at its bour.darles. For example, a gas can be
closed off by a Diston or rigid wall, flow out into a space with
a given pressure, or be sucked out of th same. Generally, the
boundary conditions may be formulated so that relations between the
phase magnitudes of thu gas and its velocity along a curve of the
yt-diagream are prescribed. The number of conditions which are
needed for the boundary curve corresponds to thi number of charac-
teristics which run out from there into the interior of the flow.
For example, the gas flows out of the end of the pipe into a snace
with constant pr-ssure, with v< n, th-n tbe- line y = constant
is the curve for the nipe for vhich the boundary conditions are
given. A family of M-ich waves spreads out fro.r it inward, while the
other family and the life- lines reach this curve, approaching it
from within. In this case the condition can be prescribed that
the pressure in the exit section be equal to the outside pressure.
If the gas is sucked in from outside, Mach waves of the one family
proceed front the curve of the boundary conditions as well as the
life lines. Accordingly, two conditions must be given. The one
states that the entropy of the entering particle is the same as
the entropy in the outer space, as a second it would be required
perhaps that the phase. of the gas in the entrance section be related
to the phase in the outer space through Ecrnoulli's equation8.
(An exact formulation is difficult, since the flow at this location
is no longer one-dimensional.) If the characteristics of all three
families cf a given curv. lead out into the iterior of the region
to le computed, there ar. three, conditions to jrsecribo; this is
the initial value problem already treated. The other extreme, that
at the boundary of the region of Interest, generally, no condition
can b. fulfilled,is physically conceivable too. For example, if a
gas with v> a flows in a space at constant pressure, generally no
characteristic goes inward from the outflow section.

compare Schultz-Grunow, loc. cit.

NACA TM No. 1196

Actually here disregarding boundary conditions which force
compression shocks no effect on the flow variation in the interior
is possible frao outside.

The treatment of boundary conditions is explained with two
examples which are connected with the flow in figure 7. The oaf-
putation is entered in table III, as far as possible. The first
example includes points 7 to 9 and, admittedly, it has been assumed
that the gas column is bounded by a piston whose life line is
represented in the yt-diagram as the curve 3, 7, 9. (Whether it is
practicable to realize such a piston in a tube of variable cross
section is unimportant for carrying out the computation.) The
Xp-diaCram referred to is in figure 7, upper riSit. To begin
with, an estimate is made of the phase at 7 which has been chosen
7 = X = 0.00C, 7 = 13 = 0.050. Since the line 3.7 is the life

of a particle, (dT is alread;- known ar-d is uqual to ( .

With this the values in columns 6-10 and 19 are calculated. As a
result of drawing in the Mach wave 5.7, Y7 and aot7 (columns 11
and 12) are obtained and besides v7/ao from the direction of the
life line at point 7 which has been reached. (This quantity is
found in column C under the value computed from the initial estimates.)
Now the quantities in columns 14 to 18 and 20 to 23 may be computed,
the value v7/ao obtained from the boundary conditions will be used.
With that X7 is already known. The quantity 17 is obtained from
the relation

v X-p
ao 2

Inserting numbers

0.323 = 1/2. (0.47 n7); n = -0.229.

Since the first estimate was too poor, the computation must be

Point L is computed from 6 and 7 by the method explained
in section 9. From 6, point 9 is obtained in the way Just described.

This method of calculation is useful for any laws of motion
of the pipe; a special argument is necessary only if a discontinuity
appears. The discontinuity in the velocity is to be considered
attained on transition of the boundary from a continuous velocity
variation at very large acceleration. In the yt-diagram that means

NACA TM Na, 1196

that the life line of the piston which has a bend at the instant
of the velocity discontinuity is rounded off immediately. Then the
flow may be drawn accurately just as previously. To obtain
sufficient accuracy, enough points must be taken on the rounding off
so that the velocity of the pistcn does not change excessively from
point to point, and at each point a Mach wave of the first family
may converge and a Mach wave of the second family may divergt from
there. First of all X must be computed for the converging Mach
wave and than from X and the velocity at the incident point I
determined for each Mach wave. If the rounding off becomes smaller
and smaller, these points on the rounding off draw closer and
closer. With that the values of ) approach a single value, which
may be computed from the field before the bend. The Much waves 2
spread out in the share of a fan from the bjnd end the fan includes
all values of p which lie between the values of p for the
velocity before and after the velocity discontinuity.

For the second example, there is at the position y = yl an
op en pipe end, through which gs is sucked in from outside and for
which two conditions '-ust. bo specified along the boandary-condition
curve. The curve is the curve 1, 30, 13 in figure 7. In the outer
space let i = il; for the entering particle therefore di = 0.

This is one boundary condition. As tha second boundary condition
there is tha require nent th'nt the phase in the inflow section be
related to the phase in the outer space by th? Bernoulli equation.
This condition may be satisfied, already, :t point 1, accordingly

i + v2/2 = 1i + 2

or also

i- 1 vi2 11 -i 1( ll 2
S 10 v\2 -ao (l = constant

To determine these constants, from figure 3 the temperature T1 is
taken for (X + 1), from figure 2(a) for T1, (i1 io)/ao-.

il 10 V'1-2
+o -- +- 0.292
a2 a

NACA TM No. 1196

Since (i io)/ao2 is a function of T and, therefore, pf i + n,

-L = --L this boundary condition can be plotted as the curve K
o 2
in the Xc-diagram (fig. 7, lower right). At best, the computation
of point 10 begins anew with an estimate for X and V so that
the boundary conditioi e are already satisfied (columns 4 and 5).
With this, the qurantities in columns 6, 7, 8, 10, and 24 are computed
and Mach wave 2 drawn in with that. The quantity a tl1 is
obtained in column 12, the values yl0 = yl and = 0. (Columns 12
and 11 are given beforehand.) Now the quantities in columns 14 to 18
can be obtained.

To determine, with this, the quantity (-M + N) in column 25
it is to be noted that (-M + N) for the particle originally in
the pipe has the value, perhaps, at point 4 and changes dis-
continuously for the particle recently sucked into the quantity
(-M + )10.

On that account the life line is drawn, which separates the
particles in the interior originally fron those particles flowing
in from outside. This intersects M-ch wave 4, 10 at point 11.
Then the following is obtained (column 25)

(-M + N)m4 10 = Lrt(aot)4,11(-M + N)

+ L(aot)11,10(-M + N)10]

The quantities in columns 26, 27, and 28 may be computed now. As a
result of inspecting the curve of the boundary condition in the
Xu-diagram with the value of i found, X is obtained (column 23).
The computation is repeated with the values found in this way.

From points 6 and 10, point 12 is obtained in the manner
described in section 9. In connection with that the difficulty just
described appears again in finding the average value for (-M + N).
From 12 and the boundary condition, point 13 may be computed by
the method just presented.

The XiL-diagrams of the two last exa-ples were k3pt separate
from thr Xp-diagram draTn for points 1-6 for the sake of clarity.

NACA TM No. 11q6

If the various figures are visualized as being joined the upper
diagram connected to the middle one ut the line 6, 5, 3, the middle
one with the lower one at the lne 1., 4, 6- it is recognized that
the plane is covered with several sheets which are connected long
the figures of the characteristics. There is such a superposition,
already, in the low1r Xp-diasram; tncre arc to be imagined inclosed
the quadrilateral 10. 4, 6, 12 along 10, 4, the tri.-ngle 1, 4, 10,
along 10, 12 the triangle 10, 3, 12.

In addition to the boundary conditions, transitional conditions
can also appear in the interior of Lhe flow. In the example just
discussed just that woula have been the caee. if in the outer
space it vere different front at. At the location of such a
discontinuity for n arpenent of pr3ssuro and velocity must be
required. To go into such questions with greater detail lies
beyond the scope of this report.


The flow in a given part of the yt-plane is defined by the
initial and boundary ccnditicn3 end tc calculabl.- ty the methods
derived up until. now. It 13 possibl- that it might hcppon during
th3 construction that regions of the yt-pl'-rz- are covered with
phase quantities several times. This is the sign for the appear-
ance of compression shocks. The entropy is no longer.constant
after tho passage of a compression shock. On that account the
computation of compression shorks simultan nously includes the
determination of the function es() or d.(), too, for the
region of the yt-plane b-ehind the compression shock.

For the mathematical tr-eatmient, a compra essin shock is to be
considered a curve along which two flows collide, which are related
to one another and to the direction of this curve by transition
conditions. It will be th. problem of this section to deriv,
these (known of themselves) transition conditions in a conveni-nt
form for the present purpose.

Proceeding front a setaionary compression shock, that is from
a conrression shock whose front is at rest r-lative to the coordinate
system selected, let the index I lc-signae the phese before th'.
Conpare Ackeret for instance Beitrag Gasiriamik in Handbuch
der Physik, Md. VII, p. 324 and following pages, Berlin 1927.

38 NACA TM No. 1196

shock, the index II the phase after the shock. The additional
index a might point out that this concerns the calculation of
a stationary shock. Then the momentum and the energy theorems
as well as the equation of continuity are written in the form

DIs + IsVI = PIIs + PIIVI 2 (43a)

S+1 2 = + v 2 (43b)
Is 2 Is 'Is 2 IIu

o v = v (43c)
Is Is =IT ITs
Furthermore, the characteristics of the ga. concerned must be known,
possibly in the form

p = p(I, o) (43d)

If the quantities in advance of the shorck i., olT, and VIs are
known then +he co,'-pressi'on shock is therevith calculable. Actually
all threE quantities enter into the general gs laws, too- RB
parameters. TIn order to carry out the computation practically, in
such a case, z from (43) an from 443b) have to be expressed
as functions of vIis and the known -uantitles and then substituted
in (43a). With that, an account of (43d), p too, is a
function of vIT and the known quzntitics in advance of the shock.
In this manner an equation for vlls alone is obtained which must
be solved numerically in a suitable manner. For an ideal gas for
which c is not constant, equations (43) transform with the aid
of (23) as follows:

I1 RTIs + v v (4a)
IlIs UIs

(Ts) s= (Tis) v 2 (4b)

Is (44c)

NACA TM No. 11Q6

Since os appears here only in the combination p /pII only
TIs and v still remain as par- eters upon which thc phases
behind the shock depend. To calculate the shock curves nmnericslly,
it is useful, first to regard Tjs and TIIs as parameters and
determine v., from this subsequently. The computation process is
the following: From (44a) ard (44c)

- + I -I
+ s Ie = I s VIIs
Vs '"!Is


As a result of squaring this

R2TI 2
- + RTI + VIs2

- TTI + 2PT 2 v
- 2I + 2IIs IIs



Ai = iTis is

I2 2 2\I

from (44b).

Putting this in
obtained as

VIs 1 A 2 (TIIs

(h5b), the desired equation for

* TI + IBS2 -412 + 4FAi TIIs
l i

-2 ( T 22
l l s

2\1 -2R290 2L1 =
- TIs 0

If vIs is determined, then vIIe and PIs /PIs are computed in
turn with the aid of (46) and (44c); flrially

IIs/Ps = ITs/ Is Ts Is/T



- TIS)

NACA TM No. 1196

For an ideal gas with constant specific heats, the following trans-
formations may be undertaken. According to the familiar relations

k 1

a? = kRT

Equations (45a) end (44b) are v.ritte-n in the form

aTs2a -
is Is
Is Viis

2 2 2 2 2
Tk +1 v I k 1 IIs+ i13

k 'a I+

2 'Is

als 1
si 9 )

:. ^

'11s '~is

- k I + IVTs
k 1

By this, als,/als andl V ls/ale and, with that, the other quantities,
too, depend on the parameter v /'als alone.

To compute Vls./als, aIIs/aIs)2is eliminated:

k + VIIs I
Ials )

v /
-s k


r v \2
1 + l = 0
-. -2 a -)

is obtained as a result.




a Is\
+ +

NACA TM No. 1196

The solution of this equation is fund, immediately, if it is
borne in mind that on account of the form of (47) a solution is
represented by

"IIs/ais = VIs/aIs


2 als
k + 1 vI

+ k 1 Vs
2 a

Using this, the following is obtained from (47b)

a' '*2
( alls "
be ;

1k 1

oIIs/ Is Is /a. I IIS)

IIs In =

= s'als als /VIs
Is' Is Is' U~s

alS/al B

The change of entropy is of interest, as well; with the aid of (27)
and (28), these expressions result

SI 1s S s k -
R k 1


Rie I

= ?k __1_
k 1 a s

k2 i I1
k 1 ais

- 2n -I1

-n I--

+ 7n VIIs 27n aII
Sis al

+ In IIs


olls/ fTlis /T I

NACA TM No. 1196

From this

(ITs 8 k-1 v IskI
^ VIIs Is Is
TIe als als VIIs

Arbitrary compression shocks result from the stationary compression
shocks Just calculated because a velocity is superimposed. In doing
so, the thermodjnamic phase quantities before and after the shock
for which accordingly LhE index s can be omitted are retained and
moreover the velocity differences. Since the phase in advance of
the shock is alr-eady given in the construction of flows, before the
shock is computed, th.e relative velocities with respect to the phase
in advanc.- of the shock -are formed. Lut

u absolute velocity of shock front

,u relative velocity of shock front with respect to particles
in advance of shock


-v v a
Au Is. V V 2 Is I
n1 a II II I k + 3 a

The signs opp':aring in this are not astonishing. A stationary
compression shock in a gas which :noves in the positive direction
propagates itself in a negative direction relative to the material
ahead of ths shock, and in so doing, produces a change in velocity
in the direction of its propagation velocity, that is, in the
negative direction, toe. Naturally, compression shocks, which travel
in the positive direction in the material at rest are also possible,
the signs of the velocities hare to be changed for these. The
thermodynamic phase quantities of this ar3 not touched upon. Corre-
sponding to the distinction which had been met in Msch waves, these
last compression shocks are designated compression shocks of the
first tyDe, those which propagate in th3 negative direction as
compression shocks of the second type. In figure 11 the pressure
ratio, for an ideal gas with k = 1.405 the propagation velocity of
the compression shock and the change in entropy (-.xpreseod by
it /(l) has boon presented as a function of the velocity change
Lv IT. For compression shocks of the first type Au and Avi

are to be taken with positive sign, for compression shocks of the

NACA TM No. 1196

second type with negative sign. The fundamental numerical values
appear in table V. Such a diagram would have to be used to apply
the characteristics method in the form given in section 11 in the
computation of compression shccks. How are these relations for the
compression shock expressed in terms of X and p? If two
compression shocks which only arise separately from superposition
of a velocity they are distinguished by the inlexss a and P -
are represented in a Xp-diagram, that is, if the phases in advance
of the shock X1 ; ila X 10; I IB, and the phases behind Che
shock ars plotted, then here, too, the expression must be iarived
at that the thermoilynamic phases in advance of and behind the
shock, as well as 'he velocii.y differences for both compression
shocks are the samTe. Accordingly,

Ia I. + u,

IIa+ PTI, = I I I, 11


- ( x P = (x
II,a/ !,a I,a) ( IIP,

By subtraction of the fi:'st two equations

- .) II,a- I,a)


- IP) +

("IIP I)

Rearranging terrm in the third equation gives

-,a \ I -I, I
Ia) IIcL Ia (lip

From the last two equations it follows that

Ii,az I,0

II,' I,p
1143 143

xII ,a Ia = II,P I,

IP ) IIP 1)

" II,p)

- %TO I)

NACA TM.No. 1196

that is, the changes in X and
tainted in the superposition of a
are designated by

I in a compression shock are main-
velocity. Accordingly, the shocks

X11I,I = %II -" x

11,1 = CII LI

The following relations hold for
heats, according to (42)



k -

ideal gases with constant specific

v V
II _



2 =I
k 1 a 0

a | 2 ( \
a k -1 a1
0 1I

by (42.z). For the

V -V
T _

a /ao is to be computed from ,I and L
expressions in curved brackets

AX =

-2 II
k 1 aI

a II V
a a a
o/ o o

- 4-


NACA TM No. 1196

are introduced. These quantities, as well as Au/aT, Ey/fi,
and p.l/n1 denend only on vs /a according to relations
previously developed. They are plotted in figures 12(a) and 12(b),
and. admittedly, the -.upper designations refer to the compression
shocks of the first type, and -he lo-:er designations to compression
shocks of the second type. Figure 12(b) represents an increased
section of figure 12(a), with the appropriate numerical values in
table V.

The following example shows a first application of this diagram.
In a .I-e of constant cross section there is a quiescent gas of
constant entropy and constant pressure, the sonic velocity is taken
to be aI = a Suddenly, a piston is driv-n into the pipe at a
uniform speed of 0.5ao. What is the ensuing flow like? Figure 13
shows the yt-diagram. The starting point of the piston motion lies
at the origin of the coordinate system. The life line of the piston
is shown with hatching. A compression shock forms in front of the
piston, which imparts the velocity of the piston to th. particles, so
that the particles behind the compression shock move with constant
velocity. Corresponding to the phase in front of the compression
shock is

1 = 0; pI = 0

The velocity behind the compression shock is

II = 0.5a


xIT ):= 0.5

II II = 1

From this, on account of X = 0 and 4 = 0


Since ei/ao = 1 this gives

A -Ai = 1

NACA TM 1o. 1196

As a result of causing this straight line in the Xts4-diagram
(fig. 12(b)) to intersect the shock curve, the following is obtained:

AX = 1.022; A~ = 0.022; 4" = 1.346; 'ii/i = 0.970

II = 1.022; Ii = 0.022; u = 1.346

From II and rp, P is computed by (42d), from this by (28)

pII/pi = 1.970

The goal would be reached somewhat quicker in this by application of
diagram 11.



It is the object of this section to show first of all by what
data a compression shock in a flow is determined, and, secondly, to
give a method by which the computation of ouch a compression shock is

AP can be readily show, the velocity of a compression shock is
larger than 1he velocity of a Mach wave in the material. This means,
that the flow field in advance of the compression shock remains
unaffected by this and can be computed independent of it. It will be
assumed to be Imown vhat follows. For the field behind the shock,
a compression shock of the first typo represents on the one hand the
start of life lines and Mach waves 2, on the other hand the terminal
of Mach waves 1. It follows, from this, that the flow behind the
shock and the shock itself are mutually related and can only be
computed together. This is the reason, therefore, that the computation
of the compression shocks becomes, essentially, more complicated
than the computation of other parts of the flow.

Next will be shown how examples can be conceived of flow fields
with compression shocks. If in the yt-diagram (figs. 14(a) and 14(b),
the flow field in front of the compression shocks and the portion CD
of the life line of the compression shock is given, then the phases
behind the shock are also determined. From the slope of the life

NACA TM No. 1196

line the propagation velocity of the compression shock is given,
namely for each point of CD. Beside, the phases in front of the
shock can be learned for the points of CD; with this the phases
behind the shock are calculable. From the phases behind the shock,
a portion of the flow field behind the shock, namely the region CED
(fig. 14(a)) may be computed, or if the entropy is known for the life
lines at the lower end of C. The region CFD (fig. 14(b)) as well.
It is necessary to go forward along the life lines and Msch waves 2,
backwards along Mach waves 1. Imagine in figure 14(a) that'the
computed life line CE is realized through the motion of a piston,
then there is a flow in which a compression shock appears and which
satisfies a boundary condition (if not prescribed, too). In
figure 14(b) it is necessary to imagine another flow field ad.oined
continuously at the lower end of CF; here the compression shock
and the flov determined by it satisfy the condition that it is
compatible along the Mach wave CF with another flow.

Fro-i these flo fields the following is recognized; tho
compressIon shock through the portion CE of -he life line of the
piston or CF of the Mach wave is defined as far as it is reached
by Mach vaves of its tyno (here the first, therefore). A change of
the life line of the piston outside of CE or the Mach waves
outside of CF propagates along Mach wava 1 in the yt-diagram, to
be exact, and neglecting cases in which a second compression shock
arises, attains the compression shock at the upper end of D,
certainly. On the other hand a change brought about between C
and E or between C and F in the bound-ry or junction
conditions takes effect at that position on the compression shock
where tre Mach wave 1 concerned reaches iL, that is, the portion CD
is certainly changed.

If the life line of the piston is known beyond E to G or
the Mach wave beyond F to H, then a further portion of the flow
field is thereby determined, without the necessity for knowing the
continuation of the compression shock beyond D; it concerns the
regions CEGJD or CFHFD.

It will now be shown how to procede funaamentilly to compute
a compression shock for specified boundary or junction conditions.
As a concrete example assume the compression shock to be produced
by a piston which experiences a sudden jump in velocity. (Soc fig. 15.)
The starting point of the compression shock is that point of the
life line of the piston at which the velocity jump appears. The
phase immediately behind M can be ascertained immediately by the
method applied to the example of the last section. The compression
shock as in previous examples of Mach waves i computed in
individual sections, which are so ,sall that the phase quantities

43 NACA IM No. 1196

for 'hen may be regarded as varying linearly. As just carried out,
the phases behind the compression shock are calculable, if the
velocity of the shock is known. The velocity at M is known.
Along the portib.n of the compression shock to be computed, M, N,
the nhase change and, with i*, the change in propagation velocity
of rhe conrreesion shock, too, are regarded as linear. Accordingly,
for all possible shocks which satisfy the transition conditions,
the nortinn M, N, of the compression shock depends only on a
single paraneter, the velocity change between M and N, to be
exact. As a result of coipu*ing the field behind the compression
shock for various values nf this Drraneter, by interpolation, that
shock iey be ascertained which is consisl.ent with the specified
piston iLovelent. At best, for this N is permitted to travel on a
fixed life line in the field in advance of the shock. Let C be
the point on the life line for which the Mach wave 1 passing
through N proceeds. Now the region OPQN may be computed in a
familiar manner. For the determination of the extension of the
compression shock NR the phase behind the' compression shock at
the point N may be regarded as given everywhere along the entire
Mach wave NQ. On the other hand, that value of velocity changes
between N and R has to be determined by interpolation, which
relates to a flow field that continuously joins the known field
along NQ.

With these two typEs, ntmely the computation of a compression
shock going out from a piston or wall and the computation of a
compression shock continuing into or arising in the interior of the
flow, the most important problems have been mastered that can appear
here. The interpolation methods described become pretty tedious;
instead of them, iteration methods will be used, which actually lead
to the goal more quickly. The interpolation method was mentioned
previously, however, since it affords better insight into the basic



Exa-ples rill be given of how the problems formulated in the
mnrceding section can bj solved by -aeanp of iteration methods. Let
the flo'" be thai computed in figure 10 and table IV. As the start
of the nev portion of the conpression shock to be computed, point 1
is chosen in every case, accordingly it is identified with the
point M (fig. 15) once and with the point N a second time. The
nev portion of the compression shock to be computed that corresponds
to M or NR, accordingly, is assumed to end an the life line 8, 9

NACA TM No. 1196

of figure 10. The phases in front of the shock for N or R are
obtained as a result of interpolation along this line. For these
calculations it is necessary, on that account, to have the knov-
ledge of the flow field in front of the shock at the points 1
(M or I) and 8 and 9. In table VI which has the same arrangement
as table IV these values hare been recorded. While it sufficed to
know !d fnr the construction of the flow field, here i itself
d .!r
must be known. These quantities for points 1, 8, and 9 are located
in column 26. In the designations, in these examples, the only
deviation from figure 15 is that only points on the compression
shock are characterized by letters. Numbers are used for points of
the flow field, corresponding to previous use.

We begin with the more elementary problem of continuing a
compression shock in the interior of the flow. For this the phase
behind the bhock at the point N and the phases along the Mach
wave N 11,10 (fig. 16(a)) may be considered known. The phases
at NI and at point 10 appear in table VI, phases in between are
found by linear interpolation; moreover, for NII the velocity of
the compression shock and n have been given (columns 25 and 26).
Besides 2d for the life lines lying below may be viewed as
computed. It was entered for point 10 in the corresponding column.
If the distances between points on the compression shock are not
chosen too large, it is sufficient to regard d. between them as

as constant. In the following this has happened throughout. Since
NI and 10 lie on a Mach wave, the consistency condition must
naturally be satisfied.

Tn connection with .h flow calculation rhe existing data are
to be taken from the preceding calculation steps. The real
ccnoutaiion begins with the fact that the different n from
its value at the starting point of the portion of the compression
shock to be computed (N here) is ascertained for the life line
up to vhich the compression shock is to be. comnjut'ed (8, 9 here).
This computation is carried through along the curve of the initial
values in figure 10, the life line 8, 9 used here passes through
point 7 there. By (37)

NACA TM No. 1196

S- v dt)
F o 00

\- 0 v

a t7,
o 7,N

By (23) and (42a)

= r1 + k -(1 +
l + +

Just as for figure 10, F has the Dorm

F = Fo t

For point 7

y = 1.450;

at = 1.180;

, = 0.66;

4 = -0.16;

7 = n = 0.849
7 8

For N the corresponding values appear in table VI. With this the
following is obtained:




P- = 2.690;

N\ 2.170;


" F

-0 J ,7

= 1.110

= 0.930

= 1.020

% so)
i F 0 a

= 2.42r x 0.07T + 1.020 X 0.03 = 0.2122

7,F o

S .7 7,N



: 7,N


NACA TM No. 1106

In figure 10 kd had already been given, it must be the same as
that found from the quantities just computed. In fact

7- = .Oh.2 = 0.230
( 7,N A7 0.212

This is the average value of 1T. as can be gathered for the
stretch 1.7 from the auxiliary diagram in figure 10. After these
preparations, the actual iteration method i2 reached. To begin
with, the phases at the points Ri and 11 are estimated, in
that 11 is the intersection point of the Mach wave 1 leading
backwards from R with the giren Mach Wvve 1,]0. Since no better
reference point exists for the estimate, these phases are equated
to the phase at 1N .* Moreover, otill another estimate is needed
for S behind the shock; for this, the same value that prevails
at the lower end of I is. chosen. With these assumptions, the
figure ay beigurer, 1 n be .w 16(a). Starting with the
life line of the cormrssEion D:-ock I:P, whose, direction here is
the same as the direction of the compression shock at Tl (t9ble VI),
P is obtained as the in,.erseetion point with the life line 8, 9.
Then the Mach rqave E,11 is irawn in proceeding frorr, E ba',7kwards.
The direction of this Mach wave ,,as taken in the familiar manner
fron a Xj-disgrai (nor given here). Fron this figure I.he position
of P in advance of the is learned ly interpolation along
N,10 the ohase ai 11. (See table VI.) Prom this may be obtained
the values entered further on in the r.soective lines which ara
necessary; for later conpu 'stion. YProcec-ding from XI1 by neans
of the consistency conditions, the quantity PTRIT is comiputed for
the Mach vsve (l1,RI). For this the initial estimates for the
phase in R1T are taken as a basis and then colurn s 6 to 13, 17,
15, 16 and 18 Lo 20 compu'ed. For XRII so obtained the
properties of the compression shock ar.; taken from the shock
diagram 12(b). The following computations are ess--ntiel to this

AX XRI R = 0.95

= a- 1II,/(a/l) = 0.9/1.040 0.918

NACA TM No. 1196

From the shock diagram

A = 0.0130;

]RII/RI = 0.978;

From this it is computed that

= 0.0135;

R = 0.090;

r = 0.830;

= 1.307

-R = 1.360

u o = VR /a + u,/ao = 1.661

A portion of these results are given in table VI (columns 24 to 26).

dr PII NTI = 0.830 0.781 = 0.-30
d ~ B N 0.2122

To improve these values, IeL a second iteration step be carried out.
First, the figure N,R,11 has to be drawn again for the values
just obtained. The average direction of the compreselon shock is

U.R (uN + R) = 1.733

Then RI and 11 are crbttined b. in-crpolation, XpRI
consistency condition for the Mach wave 11,RI .

from the

To find the characteristics of the shock, it is necessary to
carry out the following computation

RII,I 1.4-6 0.493 = 0.963; L = 0.927

From the shock diagram

S= 0.0130; RII/T" R = 0.980;

"-- = 1.310

From this is obtained


= -0.087;

it I= 0.828; u a0 = 1.657;
RIT R I3 o '


du = 0.220

NACA TM No. io6

An additional iteration step is not necessary any more. In the
second example (fig. 16(b)) the compression shock is produced ty
the sudden velocity change of a piston. The point of the yt-diagran
at which this velocity jump takes place let it be designated M
in agreement with figure 15 is to coincide with point 1 of
figure 10. From the point M the piston has the velocity corre-
stoinding to the life line in the field in front of the shock, in
particular the velocity at M in front of the velocity jump is
0.425ao. At M the velocity changes, suddenly, to the value
VM = 0.95ao and rises until the instant aot = 1.3 to the

magnitude 0.97",9. This and the flow field as determined by the
initial conditions and the niston Tnotion uo to the point M is given.
Next the phase behind the shock at the point M is computed.

ao aa!7 4,aII

= 0.483

S X = 0.483

As a result of this line in the shock diagram 12(b) intersecting
the shock curve, the following is obtained

= 0.996; T1 = 0.020

$1,II/I' = 0.977; AuM = 1.333

From this

M,II = 1.620; ,II= -0.229

= 0.781; 1.80o

'1L NACA TM No. 1196

The chase at M i is known ":ith that. (table VI.) Now the
difference must be cormputed., over again, from the life lire of
the piston for the life line up Lo which it is desired to compute
the coapr.ssion shock. It is desired to allow the compression
shock to end at the life line 8, 9, here too and take the phases
in 8 and 9 (teble VI) froi the preceding example and

nM = = 0.2122
8,M N,M
The computation of the compression shock makes use of figure M,S,
II, 11. (See fig. 16(b)). M, S, N Is the life line of the compression
chock; N, 11 is the Mach wave 1 returning from N; 11, S is the
Mach wave 2 retui-ning from 11. To begin, an estimate of the phase
at the points N I, 11 and SII is mad d and this is chosen equal
everywhere to the phase at M I. In addition, an estimate for

is necessary. Let k- = 0.230 as a start. Figure M, N, 11, S
may be drawn with assumed values. The' ord.r in which the
points were named correBnonds to the order in which they came up
in the drawing. For th- positions of T and 11. obtained thereby
the phase in front of th,; shock (se.e tabl.-. VT) or thc velocity of
the life line is oltain-d! by; interpolation. The iteration method
begins at point 11 and it can be shown that 11 can be only
slightly different from STI because the line ele'nant S ,l11 is
-'TI I
smsll relative to the other dimensions. The quantity pS can
differ from "MII only slightly, since it originat-es in linear

interpolation betweeMn M1 and T, and N li,-s v:cr close to M.
Therefore pl1 PMII is chosen as a starting point. If the
11 M,II
velority of the Distor at 11 that is known from the boundary
conditions is use(P for :his kXl1 'na. be comnuted. From the
consistency condi tion for Ihe Mach vave 11,N XTIIy is obtained.
NTov the following co.ipuui:ation

11NI-I = 1.017 1,1 = 0.988

and from the shock

Ai = 0.020; I/i = 0.973; 334

NACA TM No. 1196

from this

,11 = -0.077; II 0827; = 1678

Further it is calculated that

dn = 0.217

The phase at S is obtained by interpolation between M and N.
With the aid of the consistency condition for the Mach wave S,11ll,1
is.finally obtained, and )ll from the boundary condition for
point 11. The first iteration ste-p nds with that. It is necessary
to check whether the quantities Xll, 1l1 Ni 4NIIJ) tII,
and computed agree sufficiently with the original estimates.

To increase the accuracy a second iteration stev might be
carried out. On ths basis of the values just computed, tn-- figure
is redesigned and the computation is carried out in the manner .jur.t
described. The vailu- fir P11 just computed is taktn as a beginning.
The following calculation is obtained for the determination of the
characteristics of the shock

lX = 1.003; LX = 0.967

From the shock diagran
t u
S= 0.0189; 'i I =0.97; I -= 1.327

from this

rI = -0.0o3; I = 0.828; 1.678; : 0.221

The computation is continued in the mariner givcn until the phase
at point 11 is obtained, again. An additional iterhtion step is
not neccsstry.

NACA TM No. 1196


The differential equation system for norstationary, one-
diqersional flo"s possesses three families of characteristics; the
thermodynamic and the flow phase are described by three variables.
As a result of setJing up consistency conditions for the charac-
teristics passing through the pcin+ for which the conditions have
been set up, there? equations are obtained from which the phase
may be obtained. In that a possibility for the computation of the
flow has been given fuindn.mentally. The report carries out these
idees, in generl-, and brings the simplifications which are possible
under special assumptions, as well as detailed examples. Compression
shocks appear, in this, as transitional conditions in the interior
of the flow and are likewise investigated in detail.

Translated by Dave Feingold
N-itional Advisory Committee
for Aeronautics

w s w A & t I ot-na 0
< I il il il il fl il r- rr < l rI1 rt w oIn ol <

** *. *. *

AS 0 R 4 M S g M M a
A 4 4 r l H H1 H4 H

9 M o
4 a l r: l l r4 l 4l l 4 4l 4l l r 4
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C WN a en



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I H 1 I

R O 2 Ct 1 i r; CR t a 1 a 1
a 0 aU at A

NACA TM No. 1196

5 B

! g

; Q4

4 S 4 q B0 UN

o 0

u u 0 O4 0
WN HC m4

00 00 *0

- i* jS r4 .-

I.I I I- *.1 1 II
I" Cd .i ."

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A4 .4.4 .4.
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a l 4 ll l 4 14 1114 .41 .41

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4.9 1;;l rla

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|u W% en 01 on

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r4 Mo 4 -1 r4 H

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Ha U Cd d i in

.4 400 .4 CM ( .t Ifi '0

NACA TM No. 1196






) ( t m m e m f mt m Q
.0 .. ,. .

N" S 1s ss Rse

..,. ., *
A Qp.44!7 M-0
9 99

u I I I I I II q

0 0__ __ __ W _

t .. .
i.s a A ..
"T^' iiitt iT 'T m ; | tt
.4 g S S 8 9 a

In 8 .

0 I c o N
4. .4 .4.4 .4 4 4-, 4 .4,,
a r 3
SI s gg
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4 J .44 M 14 I 4 A 1 4 .4 A .4

= ^0 & S & R & % 9 g
01 ri .4i CC? ..

mm N C. o u
-- ---- -

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-- ----l -- o,- C 2,0


.4 71odt aO C C E 0

NACA TM No. 1196








I c

R 8 I If

a a i

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-- gal 1 h ~ -- "~ -- '- -
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NACA TM No. 1196








NACA TM No. 1196



/Au A/a A ,I, /a1 I/z ,/AI T. A

1.00 1.00 0.0 1.00 1.00 0 0
0.98 1.02041 0.03360 1.04817 .0.99998 0.06721 0.00001
.96 1.0167 .06791 1.09938 .99991 .13588 .00006
.94 1.06383 .10298 1.15391 .99970 .20615 .00019
.92 1.08696 .13884 1.21203 .99927 .27816 .00047
.90 1.11111 .17556 1.27406 .99862 .35206 .00094
.88 1.13636 .21319 1.34037 .99748 .42806 .00167
.86 1.16279 .25180 1.41137 .99587 .50633 .00273
.84 1.19048 .29146 1.48748 .99369 .58711 .00419
.82 1.21951 .33224 1.56925 .99080 .67062 .00615
.80 1.25 .37422 1.65722 .98708 .75713 .00869
.78 1.28205 .41751 1.75201 .98240 .84694 .01193
.76 1.31579 .4620 1.8 545 .97659 .94038 .01599
.74 1.35135 .50840 1.96527 .96951 1.03781 .02101
.72 1.38889 .55625 2.08544 .96101 1.13965 .02716
.70 1.42857 .60588 2.21608 .95091 1.24634 .03458
.68 1.47059 .65745 2.35841 .93899 1.35843 .04345
.66 1.51515 .71114 2.51387 .92506 1.47647 .05418
.64 1.5625 .76715 2.68413 .90903 1.6011 .06684
.62 1.61290 .82570 2.87113 .99067 1.73320 .08180
.60 1.66667 .88704 3.07714 .86974 1.87350 .09941
.58 1.72414 .95147 3.3"1484 .84602 2.02302 .12009
.56 1.78571 1.01931 3.55735 .81954 2.18291 .14430
.54 1.85185 1.09094 3.83845 .79006 2.35447 .17260
.52 1.92308 1.16680 4.15261 .75746 2.53923 .20563
.50 2.0 1.24740 4.50520 .72182 2.73895 .24414
.48 2.08333 1.33333 4.90276 .68305 2.95572 .28905
.46 2.17391 1.42529 5.35333 .64131 3.19120 .34141
.44 2 .2773 1.52410 5.86637 .59679 3.45070 .40251
.42 2.38095 1.63073 6.45517 .54980 3.73531 .47385
.40 2.5 1.74636 7.13401 .50078 4.05005 .55732
.38 2.63158 1.87242 7.92319 .45026 4.40174 .65517
.36 2.77778 2.01063 8.84701 .39889 4.79144 .77018
.34 2.94118 2.16314 9.93888 .34751 5.23210 .90582
.32 3.125 2.332&6 11.2418 .29698 5.73172 1.06644
.30 3.33333 2.52252 12.8138 .24825 6.30261 1.25756
.28 3.57143 2.73716 14.7347 .20234 6.96o61 1.48630
.26 3.84615 2.98225 17.1156 .16016 7.72656 1.76206
.24 4.16667 3.26542 20.1167 .12251 8.62809 2.09725
.22 4.54545 3.59705 23.9721 .090026 9.70301 2.50891
.20 5.0 3.99169 29.0416 .063089 11.0043 3.0a088
.18 5.55556 4.47032 35.8933 .041760 12.6082 3.66757
.16 6.25 5.o6445 45.4722 .025786 14.6295 4.50062
.14 7.14286 5.82359 61.6909 .014596 17.2482 5.60104
.12 8.33333 6.83019 80.9708 .007388 20.7642 7.10387
.10 10.0 8.23285 116.672 .003219 25.7176 9.25191
.08 12.5 10.3285 182.394 .001133 33.1878 12.53078
.06 16.6667" 13.8101 324.386 .000288 45.6935 18.07325
.04 25 20,7568 730.082 .000040 70.7902 29.27672
.02 50 41.5634 2920.83 .000001 146.254 63.12755
.00 o oo o 0 o oo

NACA TM No. 1196

co i uln I doinst
muy pmmo

DIoIUI r oj ld a
U1r winCs


9 7 .4 7 E

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ft C. ft f t i1 ft 1Cd ft a C1

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C Cl a -

-- C --- -- -------- ---- -

NACA TM No. 1196

Figure 1.- Curvilinear coordinate system t .


64 NACA TM No. 1196


Figure 2a.- Relation between i and T for CO2.

NACA TM No. 1196

3 115


is -- i Z5

0 055 2,5

0 0
0 50 100 150 200

Figure 2b.-

-; P; W as functions of the temperature

for CO2.


1105 /

5B- ea 5---------
*LUL~~~4' ) ^- r -.-^ -\ -I

Figure 3.- -a; P; T as functions of W,

NACA TM No. 1196




or A+,, for CO2.

NACA TM No. 1196

j I I I I



_ _ ~ I ft

68 NACA TM No. 1196



/ / />-a

VX V. rI
a a

o Ca


,4 o,-




0 '4 .
'*-K --- 5 ---- S ---- '! --- "Is a) ^



TNACA TM No. 1196

The physical significance of .


pethline of a particle

Figure 6.-

NACA TM No. 1196


Figure 7.- Any flow of an ideal gas with variable specific heat. Treatment
nf hninrinr rnnrfitinn..

NACA TM No. 1196

1,1 1,2 13

Figure 8.

NACA TM No. 1196

0,5 --0

0 ---

- 1,









Figure 9.

I -qs

NACA TM No. 1196

--LA -I -

* UI U













:~ :

;- ;

NACA TM No. 1196

Figure 11.- Characteristics of compression shocks K = 1.405.

NACA TM No. 1196


\ I
t.. In
-_ s
-S -- \ -- -- S

______ __ __ __ .


NACA TM No. 1196

NACA TM No. 1196 77

path line of a particle

Figure 13.

78 NACA TM No. 1196




6 C



Figure 14b.

Figure 14a.

NACA TM No. 1196 79





Figure 15.

NACA TM No. 1196

Figure 16a.

NACA TM No. 1196

Figure 16b.

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Full Text
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