NATIONAL ADVISORY COMMITTEE
* *.I’
FOR AERONAUTkS
*
TECHNICAL NOTE
No. 1592
Lmgley
Aeronautical Laboratory
Langley Field, Va.
w!gv
Washington
.
NATIOWL ADVISORY COMKIXTEEFOR PERONAUTICS
TECHNICAL NCXCENO. 159
COMPFUZSSlBLEFlOWTABLESFORAIR
By Marie A. Burcher
A tabUl&iOn
is presented of functions
of th8 Mach number which
are frequently
US8d in high-Sp88d aerodynamics.
The tab188 iIElUd8
valu8s of Mach nImib8r from 0 to 10.0 in increments of 0.01 and the
functions
are based on the assumption that air is a perfect gas having
a Specific-heat
ratio of l.h-00.
Il?'IRODUCTION
i
In hi&-speed
research, frequent Us8 must be YEad Of th8
theoretical
r8titiOnShipE
existing
between th8 Mach nurmber and various
flow parametere, for example, the Str8am-tub8 area ratio,
pressure,
temperature,
and density ratios referred
to stagnation
conditions,
and the normal shdck 'relations.
The pUrpOS8 of the tabulation
is to
present values of the most cammonlg needed functions
for Ma&number
increments (0.01) small enough to provide for linear interpolation
without loss of accuracy for the Mach number range from 0 to 10.0.
The quantities
shown are fhctions
solely of the Mach number with the
eXCeptiOn Of a f8W dimensional q,Uantiti88 Computed for Specified
stagnation
conditions.
Tables I and II cover the subsonic and supersonic Mach nrrmber ranges, respectively.
An earlier
unpublished Compressible flow table which has been
in 13~38at the Langley Laboratory of the National Advisory Cortmittee
for Aeronautics
since 1945 was COmpUt8d by Mis6 Vivian P. Adair of
the Fluid and Gas Dynamics An.al+is
Section.
The present tabulation
extended this preview
work from M = 3.0 to M = 10.0 and added a
number of functions
not included in th8 earlier
table.
Y
*
.
A sketch illustrating
the notation
ditions
in subsonic and supersonic flow
adopted for the stream conis shown as figure 1.
A
cross-sectional
area of stream tube
A cr
cross-sectional
area of atream tube for
Ml = 1.0
. ----
2
NACA TN No. 1592
a
0peed of sound in air
FC
compressibility
M
Mach nUmber
P
Mach mgl8
P
absolute
9
dynamic pressure, pounds per square foot (coquted
for
arbitrary
stagnatIon pressure, p. = 2l21 I.b/sq ft)
factor
(V/a)
(.ti-l$
pressUr8,
pOUnda per S&r8
T
temperature,
v
airspeed
P
angle between oblique shock and flow
(see fig. 2), degrees
9
ratio
%lax
maximum possible flow-deflection
oblique shock, degrees
Em8x
maxham acute angle between oblique
undisturbed
stream (see fig. 2),
P
Subscripts:
0
1
2
3
cr
heats,
To = 520~ F absolute,
7
Y
an
$P+?
(.I
absolute
CoRQut8d for
of specific
fOOt
f80t
direction
per second
behInd shock
-
,
taken as 1.400
angle
(s&fig.
2) for
shock and direction
degrees
of
expansion angle required to Change Mach number from 1.0
to Ml (8863 fig. 3), de@T888
_
mam
amsit
bf
al+
at sea level
.
_.
--
Stations in assumed stream tub8 (fig.
I).
For US8 Of th8
tables In subsonic flight
problems Ml should be taken as
the flight
Mach number; then subscript
1 refers to atms-ph8riC conditions
and subscript
o refers to stagnation
COIIditiOnS on th8 aircraft.
similarly,
for SU$8rSOIIic flight
problems Ml is the flight
Mach number, and subscripts
0
and 1 are, respectively,
the ideai stagnation and the
atmosph8ric conditions.
-.-
I
Y
Y
3
NACA TN No. 1592
ACCURACY
The values given in the tables were computed individually.
Most
of the steps in the various calculations
were carried to seven
Over half of the values
significant
figures on comlmH.ng machines.
The rest of the values
given have been checked by recalculation.
were checked by en examination of the differences
between readings.
The values are believed to be correct to the number of places shown,
with the possible exception of the values of $ax, which may
occasionally
be in error by one point in the last place (O.Ol").
.
DISCCSSION
The fomulas
used in calculating
the tables are, in general,
well-known and their derivation
may be found in modem text books
A possible exception is the equation for
&
on fluid mechanics.
which was first
given in reference 1. For convenience, all the
equations me listed
in the appendix=
It has been found convenient to use the stream-tube notation
shown in figure 1. The theoretical
relations
involved are, of
course, equally applicable
either to stream-tube problems such as
those encountered in wind tunnels and internal-flow
systems or to
.flight
problems.
Attention
is called to. the fact that the subscript
d refers to the ideal stagnation
conditions
(sta@mtion
conditions
for isentropic
deceleration
to zero velocity).
For
the stagnation
conditions
at an impact
subsonic-flight
application,
tube, for exemple, are represented by the subscript
00 In the
supersonic case, however, because of the presence of a normal
compression shock ahead of the 3qac-t tube, the stagnation pressure
shown by an impact tube will not correspond to the value represented
by this subscript
o but rather to that represented by the subscript
3, which refers to the sta@ation
condition
behind a normal
shock. The atmospheric conditions
(p, T, and p) in the flight
case are represented by the subscript
1.
Y
The assumption that air is a perfect gas with a value of 7'
of l.&o.is
valid for the condftfons
usually encountered in the
subsonic and lower supersonic speed regions for normal stagnation
conditions.
For Mach numbers greater than about 4.0 or for unusual
stagcation
conditions,
however, the behavior of air will depart
appreciably
from that of a perfect gas if the liquefaction
condition
is approached, and caution should be.used ia applying the results
in
the tables at the higher Mach numbers. The data, of reference 2
*
.
.
.
.
A._.
NAZA TN No. 1592
4
afford-a
mean8 of estimating
the conditions
under which the constituents
of air will Uquefy at the relatively
low pressures
usually encountered in supersonic a-erodynam3.c problem.
Langley Aeronautical
Laboratory
National Advisory Committee for Aeronautics
Langley Field, Va., January 8, 1948
NACA TN No. 1592
FORMULASITSEDFOR CALCULATIONS
Subsonic Calculations
The significant
Mach number functions
for subsonic applications
are given in the column headings of table I. The notation
used is
illustrated
in figure l(a).
In flight
appiications,
if Ml is the
flight
Mach number2 pl represents the atmospheric pressure and p.
the s:agnation
pressure indicated
by an ideal pitot tube.
Similarly,
Tl and. pl are atmosphsric temperature and density,
respectively.
Tne following
formulas,
require no discussion:
p1
-=
PO
which are functions
.1 + L+!$
(
A
- cr = Ml
A1
, r
of
Ml
and
7,
I--;7
*
( )
l&y
l+qq
f
6
NACA TN No. 1592
The values of Pl presented
temperature
To =.--520° F absolute.
in tabls I are based on a stagnation
The form,ti
is
0
al
v==M1ao
Tne value of
per second.
a,.
for
The -values of
To
ql
ao.
was taken 8s lLU7.372 feet
-
= 520' F tib&Iut&
were obtained
from the ftJi&la
91=$.f$*p-Pl
0
PO
2
= l484.7M,
u
0
taken as 2121 pounds per square foot.
The
where p. was arbitrarily
dynamic -pressure far any other
p, (regardless
oC&
v-alw of To)
can consequently Be obtained by rc&l$iplying
the tabulated
values by
the ratio
~~2121.
.-.
Thti'general
formula for the compressibility
factor is
l?Or
Small values
of
Ml,
F,
was computed from
F, = 1 + $412 +' &Ml4
+ .&Ml6
NACA TN No. 1592
7
Supersonic
Calculations
The functiona
pertinent
to Problems involving
supersonic flow are
shown in the column headings of table II.
The stream.-tube notations
applicable
to this case are iUustrated.in
figure l(b).
The formulas
P~/P,,
al/a,,
Vl,
and ~1 are the same for
for
Pl/Por
supersonic calculations
as for subsonic calculations.
The value
of Po was taken as 2121Pounds per square foot and the value of a0
W&B taken as lJl7.372
feet per second.
for To = 520° F absolute
Formulas having especial significence
in supersonics,
however, are
T1/To,
Al.
-=
Acr
-.zL
)
1 IL + -2
If1 1 + *
2(y-1)
<
112
-1
vA+
CO8
,[m]1’2
c )
Y
R
--=
6-
+p,-90o
-P
where
E-=
Sill -1
1
Y+Q
I ml2 [
-
7 + 1) 1 + yM12
(
+
1'
-1
414
8
-NACA TN No. 1592
3 Pg3
( >( 1
PO =
P,
P2
-
2
p1
p2=
M1
01
>
(5x?2
,
NACA TN No. 1592
1. Crocco, Luigi:
Singolari&
della corrente'gassoea
lmperacmtica
L'Aerotecnica,
vol. XVII,
nell'intorno
di una prora a diedro.
fast. 6, June 1937, pp* 519-534.
of
2. Dodge, Barn&t F., and Dunbar, Atherton K.: An Investigation
the Coexisting
Liqui& and Vapor Phases of Solutions
of Oxygen
Jour. Amera Chem. Society, vol. 49, no. 3,
and Nitrogen.
197, PP. ml-610.
I
9
.I
.
10
NACA TN No. 1592
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11
NACA TN No. 1592
.
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NACA TN No. 1592
32
Isentropf c
expansion
I
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0
(a) Subsonic.
1
T
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(b) Supersonic.
FYgure l.-
Sketch showing stream-tube
notation.
,
.
,
Ml
wFIgwe 2.- Oblique-shock notation.
aA= 1.0
Flgure 3.- Expansion-angle notation.