Charts of pressure, density, and temperature changes at an abrupt increase in cross-sectional area of flow of compressib...

MISSING IMAGE

Material Information

Title:
Charts of pressure, density, and temperature changes at an abrupt increase in cross-sectional area of flow of compressible air
Alternate Title:
NACA wartime reports
Physical Description:
9, 5 p. : ill. ; 28 cm.
Language:
English
Creator:
Joyner, Upshur T
Langley Aeronautical Laboratory
United States -- National Advisory Committee for Aeronautics
Publisher:
Langley Memorial Aeronautical Laboratory
Place of Publication:
Langley Field, VA
Publication Date:

Subjects

Subjects / Keywords:
Thermodynamic cycles   ( lcsh )
Aerodynamics -- Research   ( lcsh )
Genre:
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Summary:
Summary: Equations have been derived for the change in the quantities that define the thermodynamic state of air - pressure, density, and temperature - at an abrupt increase in cross-sectional area of flow of compressible air. Results calculated from these equations are given in a table and are plotted as curves showing the variation of the calculated quantities with the area expansion ration in terms of the initial Mach number as parameter. Only the subsonic region of flow is considered.
Bibliography:
Includes bibliographic references (p. 8).
Statement of Responsibility:
by Upshur T. Joyner.
General Note:
"Report no. L-13."
General Note:
"Originally issued January 1945 as Advance Restricted Report L4L19."
General Note:
"Report date January 1945."
General Note:
"NACA WARTIME REPORTS are reprints of papers originally issued to provide rapid distribution of advance research results to an authorized group requiring them for the war effort. They were previously held under a security status but are now unclassified. Some of these reports were not technically edited. All have been reproduced without change in order to expedite general distribution."

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003619235
oclc - 71358152
sobekcm - AA00006264_00001
System ID:
AA00006264:00001

Full Text
f C~A-l-/3


NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS





WARTIME RIREPORT
ORIGINALLY ISSUED
January 1945 as
Advance Restricted Report L4L19

CHARTS OF PRESSURE, DENSITY, AND TEMPERA~TRE CHANGES
AT AN ABRUPT INCREASE IN CROSS-SECTIONAL AREA
OF FIDW OF COMPRESSIBLE AIR
By Upshur T. Joyner


Langley Memorial Aeronautical
Langley Field, Va.


Laboratory


WASHINGTON


NACA WARTIME REPORTS are reprints of papers originally Issued to provide rapid distribution of
advance research results to an authorized group requiring them for the war effort. They were pre-
viously held under a security status but are now unclassified. Some of these reports were not tech-
nically edited. All have been reproduced without change in order to expedite general distribution.

DOCUMENTS DEPARTMENT
L-13


I '
I.i-:.





































Digitized by the Internet Archive
in 2011 with funding from
University of Florida, George A. Smathers Libraries with support from LYRASIS and the Sloan Foundalior


http://www.archive.org/details/chartsofpressure001ang








NACA ARR No. TLL19 RESTRICTED

NATIONAL ADVISORY COr c.ITTEE FOR AERONAUTICS


ADVANCE RESTRICTED REPORT


CHARTS OF PRESSURE, DENSITY, AND TEMPERATURE CHANGES

AT AN ABRUPT INCREASE IN CROSS-SECTIONAL AREA

OF FLOW OF COPMPRESSIBLE AIR

By Upshur T. Joyner


SUMMARY


Equations have been derived for the change in the
quantities that define the thermodynamic state of air -
pressure, density, and temperature at an abrupt
increase in cross-sectional area of flow of compressible
air. Results calculated from these equations are given
in a table and are nilotted as curves showing the
variation of the calculated quantities with the area
expansion ratio in ter.i of the initial Mach number
as parameter. Only t-e subsonic region of flow is
considered.


INTRODUCTION


The well-known Borda-Carnot expression for change
in pressure when an incompressible fluid passes an
abrupt area expansion has long been used fcr estimating
the pressure changes in comnressible air flow at an
abrupt expansion. Thismethod is simple but not exact.

The expressions for pressure, density, and
temperature ratios given herein are for subsonic flow
and are in precise agreement with the exact expression
for the velocity ratio in a compressible flow at an
abrupt Area expansion developed in reference 1. In
the present report, Mach number, or the ratio of air-
flow velocity to existing sound velocity, is used as
a parameter; whereas in reference 1 the parameter was
the ratio of existing air-flow velocity to the velocity
that the air would possess if accelerated isentropically
until its velocity was equal to the then existing local


RESTRICTED







2 NACA ARR 1o. L4L19


sound velocity. This difference in parameters must be
considered when equations fro.n the two papers are
compared.

By use of the same three fundamental equations
used herein, a somewhat similar equation showing
pressure and density changes across a shock loss in a
pipe of uniform cross section was developed by Hugoniot
and is given in reference 2.

The expressions obtained herein for pressure,
density, and temperature changes in a compressible
flow at an abrupt expansion are too involved to be of
practical use. T'\e present computations have therefore
been made and are presented in tabular and graphical
form.



SY",:DOLS



A cross-sectional area of flow, square feet

a velocity of sound, feet per second

f area ratio (A1/A2)

M ..i ch number (V/a)

p static pressure, pounds per square foot

V velocity of flow, feet per second

Y ratio of specific heat at constant pressure to
specific heat at constant volume, dimensionless

R universal gas constant, Btu per slug per OF

T absolute temperature, L160 + OF

p density of air, slug per cubic foot
Subscripts:

1 before abrupt expansion

2 after abrupt expansion







NACA ARR NTo. ITL19 5


ANA LYSIS

Prom the fundamental equations .or the conservation
of energy, of continuity, an. for che conservation of
momentum, equations are obtained tnat give the variation
of the pressure, density, and temperature ratios with
the area expansion ratio f in terms of the initial
Mach number as parameter.

Figure 1 shows the condition of flow assumed for
the present calculations. The static pressure at a
(fig. 1) is taken to be the same as at b for subsonic
flow, as has been proved exneriirentally by Nuaselt
(reference 5). uniform velocity, distribution before
and after the expansion is assi1.t'n-.. The ratio of
specific heats y is taken as 1.L0G5.

The fundamental equations are the equation for
the conservation of energy:

V 2 _1 _)22 Y 2
V12 +1 122 + Y P2
--+ + (1)
2 Y 1 Pl 2 Y 1 P2

the equation of continuity,

PA1V1 = p2A2 (2)

and the equation for the conscrx ition of momentum

P2222- pAlvl12 = -A 9 "- Pl) (5)

Prom equation (1)


Y -1 2 2 Y 1722 2
2 1 a1 2 Vl 1 P2

or


2 M + 1 -= -2 ( + 1+ (4)






YACA ARR No. I4L19


From equation (5)

P2= 1 + yf 2 V )
P-- 1 -+

= 1 + yfM2 12 P

Prom equation (2)


V2 P2 = P2
VI P2 A2 P2


When equations (5) and (6) are substituted in equation (4),


y -1 + 1 =Y 2 l1 2 i2 -2 I
Y-2+ 1 = 2- f2 + 1
2 2 eP2^ P2


+ _L 2 2 2 'p 2
P2 2


Pl\2 2 + 1 P122 + 12 _M2 + 2
2) i 2 P2/ 2 2o


(7)


When equation (7) is solved for pl/p2 and the resulting
equation is inverted,


P2
Pl


f21l2(y + 1)

1 + yf.1l2 -2yfMl12(1 f) + 1 2f2 .12 + f211


(8)


In order to obtain an expression for the pressure
ratio, equation (8) is substituted in equation (5) and
the following equation results:

p2 1 + YfM12 + y\2yf..:!2(1 f) + 1 2f2:'.i2 + f11
I- = (9)
Pl y + 1


(5)


(6)







NACA ARR Io. LLL19 5

By differentiating p2/p-' wIth respect to f, it can
be shown that the maximum static-oressure recovery for
any value of 1 is obtained when

Y +J 2 12 + 1
f = (10)
2(y + 1) N12


The locus of maximum pressure ratio is shown in figure 2.

In equations (3), (9), and (10), the sign of the
radical has been chosen so that the results obtained
are in the region where the assumptions are valid.

The temperature ratio is obtained by us. of the
general gas law and the computed values of pressure and
density ratio as

pl
P1

P2
= rT2

T2 E2 /
-- = p/ (11)
?- P2, /P

Figures 5 and 4 show the variat'-.on of density ratio and
temperature ratio with area exoa'nsion ratio.

In order to make the results shown in figures 2
to 4 usable in cases for which only the conditions after
the expansion are known, the value of h!2 in terms
of MN and f is given in figure 5. If N2
and f are known, Y1,l can be determined from this
figure. The relation clotted in figure 5 is developed
as follows:






6 IACA ARI No. L4L19


/ (v2/a2)2


V22

YP2/P2



V2 P 1 /P 2

1 V 2l/ P

By use of equation (6),


I P2 P2
Pl PI


Of
fM
M2 =
2 f2



RESULTS AND DISCUSSION

The calculated values of pressure ratio, density
ratio, and temperature ratio are given in table I.
The values have been computed to 8 decimal places
because of the form of the equations,in which small
differences in large quantities are involved. In the
region where M and f were both small, that is,
0.1 or 0.2, it was necessary to carry some of the
calculations to 12 decimal -.-laces in :or.er to obtain
smooth curves for the quantities calculated.







NACA ARR No. L4L19


As in the case of incompressible flow, the calcu-
lated changes occur gradually after the abrupt increase
in cross-sectional area of flow, and the calculated
and measured results are in best agreement at a distance
the order of 6 to 10 diameters of the large cross
section downstream from the abrupt area increase.

The comparison of pressure ratios for compressible
and incompressible flow is shown in figure 2, in which a
long-dash line gives the pressure ratio calculated on
the basis of incompressible flow for the same initial
conditions that are assumed for compressible flow at
an initial Mach number of unity. It is evident that
the effect of compressibility is vanishingly small
for values of the area ratio of expansion below about 0.25.
The short-dash line in figure 2 shows the pressure ratio
to be obtained with isentropic expansion and an initial
Mach number of unity.

The experimental points from reference 3 shown in
figure 2 were obtained from the only experiments known
to the author in which pressure ratio has been measured
at an abrunt expansion with compressible gas flow at
high Mach number. These data were obtained for an
area expansion ratio of 0.246, however, for which the
difference between compressible and incompressible flow
is insignificant. These exnerimental results agree
well with the calculated results but are by no means
conclusive. Agreement of experimental with calculated
values at an area expansion ratio of 0.7 or 0.3 would
be conclusive evidence of the difference in pressure
ratio obtained with compressible flow from that
calculated by the Eorda-Carnot formula for incompressible
flow.

An experimental investigation of the changes in
pressure, density, and temperature at an abrupt increase
in cross-sectional area with compressible flow would
serve to determine corrections for the effect of
nonuniform velocity distribution and friction on the
idealized results obtained from the present calculations.


Langley Memorial Aeronautical Laboratory
'Tational Advisory Committee for Aeronautics
langley Field, Va.






8 A CA ARR No. .L L419




1. Busemann, A.: Oasdynamik. Har.db. d. Experi-'entalphys.,
Bd. IV, 1. Teil, Akad. Verlagsi.esellschaft m. b. H.
(Leipzig), 1951, pp. 405-407.

2. Ackeret, J.: Gasdyrnamik. Handb. d. Phys., 3d. VII,
Kap. 5, Julius Springer (-erlin), 1927, P. 525.

3. 'lurselt, W.: Der Druck im Ringquerschnitt von Rohren
mit plitzlicher Lrw:iterung beim Durchfluss
von Luft mit hoher Geschwiidigl:eit. Forschung
a. d. Geb. d. Ingenieurwesens, Ausg. B, Bd. 11,
Heft 5, Sept.-Oct. 19l10, pp. 250-255.









NACA APR No.


-.:- I -I' .. .-. I


co -hrO Nrlj O1O -m
hr"0 r'a-o c,'- --o .- .-i
o3-rj .-u in .ta'






0NN N-4 -.. ,
0 o\C .-l 0 oN-'-N r











-4 -,.i 3,o o c-r-
nj mo-. a ,. 0w ."co-



"- -N 4- -'




















O m N,
,a -" I- N 0 N N0
0'1 0 \" l< 0 ,o 0 -M


0 N CP 6<8
4- rr 0 4


CO -f, Ol -rD .4





Lr ''o"t0 o nT














0000 0 9 C
H i--, a- r
0 _010 0" i 0 0,.r




DPcOWo 0aN F
0 SIBr cir rflcc jNN





0 N40000. ."-do N -\


PO CN'.D U aJ, -0 c NC
0) Oi-C -t1,lfl f --I
frO -- hr u-5 r -3n -3 o





0 30 N N 0'-




-"s-.ol cyO, -q^Fu. F
N .40 N-CN"0%r i'
.-----------------------LT








0 04 r-IJOrqu r4 -: r ~

'-iir1fli0.3tN0ti
NO nN in4 N ?Oin
.4 r It%.-l4O.-inrdi-l-i-


-rj N :" -r -co a 0
0


-.---4 -- -4-r -4 -4


oo WMO 0 C- CZ
ar00 0.-0 N

0000000 "j


- r-4 r" CN
r- !l.L4% JrC-C N '0 'IOX
rj J GIW\vr W -% r4% P^ M
u-n N %'a-c 01 N N
"OOL 2C '- '*



000000 I

C! C
"0,%0 0 a'a'Lnl o0 Ler
9 N'0 0 c-n Ot

om a -O04 -8




rJ co- 0 .41flC0 wse a- a-'






rooo^> > O -rfMdn
E000- N ro
a.-c'...o ...a'
hc 00 0






fe- (V0r NU O "-
l .OF
pr4-r4w4p4r4.4- r4 .4 r4r4







10 a V% _t WS
Lr o co rm CD r- ir% r < _:I r ^-
ao0^02 O ao r
0 0 O4 Nq' C-g a 0'W% 0


r O rO -rr
a'lf NOD N-I 'U' 0 n'o


'D % N lD
rT- a (T<0C 0 ^C-^r-rM
0.0 -C'C rd.'4N^

1 0.-ID 4 -4- 4r 4 4 r-

0 0 r0'4 t'0.






OO O ~ -i' f


-N 'auf
0 C
r 00. o o rra 0






000000000.44.
0 wvo W040 m0m
r4 ..-4.4M.-.4.4.4.4.4


ID e -% "%M P
0C \0 -1 W %.40 ni.N
:ooodaoo r4&82







N0 ia0 a 'a 'a ^S


OmN^ON rfeme O^c
^1-^COei0 (1s-1 0 K



S00r r-iO

800


rOONO "l(7Dr O DC''< C
OO 'CI-0'
,cc( N tO' M rS
aP I- a'ccc-
a a- a'-o'a-o'a'a'a'


a.O .4'S 0 0.
o 0 -4 4rtn-4D 0oc

Ca-LO,-4- iDO f[-VQ 0
C- 'DO a 'nrc (7N NOa
N C-, f l u-^ca ui-'-
C' r n'-a irr0 CO hrE
o a a 0.1 0o ancDn
-' Ny "TS tnf c-- 0' 0~fs

0 "I?*in -N ^o K-4


3 -T J 0\


.- ," C '
"' D-:'" ., '- .- ,
--C'-, P- )O:2--' uj. .'.a a




* rj b' ajc I4 r
7r o jo .- 1



--z
oa '-%,a'-"o Lft
3 .-> 0 -0 r 0 0 .-.



00000-o00 .-:


O0 .99 hr C- Q' in.9

00 100 0. 000 -i .4





..- .. r r'-- _. '
,'N3 mp 0 r '- .1 p 1- 1 _'.



0N j A- N

0000 O .- .- .e,
N %o T D o 00 -
c' r D- .r 0 u.C- r- ._co
g' w, rj o j ,







f N c o -IC 0 ., O











000 000 -,-
4N a- am I









N FMr O c- N O'-r-a**?
0OOO,--IN 0u0
00000 o0.- ,.-
r 0 r-D *- 0 0 -J





rzt< o o r'o I,

0 0 NOOO, C18 N.4L-N







99N.9..9..
00. "'t co .- onn0

.4- -4- --' '-4-c
I 000m000000-V 07GO
oa-* 0 0 a- a r. 0 a' C'







0 -. j a7 j1fl C ii-
C O O ^ ^ r .


L4L19


I







Fig. 1 NACA ARR No. L4L19






o



0 0
--,








,-.-I



0
cd








o
02 .






Ct-


0


0




,--
0








co





.,t4


5-.
'-I <
5^ -H


'-4







NACA ARR No. L4L19


IT


ititi

i -.- --L--- -- --l l~
>1^ ^ L
i-Ce- -


I I I, t ,4 1
S i -- r -- -- -- .. ", 1
III I
S- I I



I .i I


.... rL : 2,I it-';; 2


_---; .... .
-i .-
H -I "










-- N -n .- T .





F-. -- I -- 14
rr \ ^ \ i '' I
-- ---- ^ ^ ^ \ ~~~- :- N ~ \ \ "

="= -_3 4-.- --^ \ ,

_. -.-.. ...r. 1^ I K .- '---V---
\.---t T ',, ,

--k- =:E^,_ _3^


o .1~~


1:4
I,
xzw~JxLwrn


'__"- .-- _


o 0 0 "
I N PJ-
-s H


F. -ii


0


H,4




-Id
.' m
CO


o I,





*~i,-.
0.-
1. 0

00.

trjo

to

0. 0

C*it-0
a,-
(-C
O C'

*"$ 0 P

a6. i


Fig. 2


II






NACA ARR No. L4L19


'I -I.



[ -_ .... -I ...., ", !...-- L^ ...-- .-I^ /'----
-i I I -


S. : .. ... "...- -4 -i ... 4
r I i I ..I [
i---------t ...... .... ,-L .. --- .



. -- ... ; -i il-_ L _L _... i- .. .
. -i ,--- P--:- --r-- *--- -





!- -' -'i T r-, "^ 1..u.-e- "_- .-:: ...

... .:__ ', ,: .^ / I" \ / .. '. .. _. L "7_ -
,. I i 4 -- -"<- :/ ] 4: i


S -- i_, -4.K "-" .







; -_ ,- ... l._ _. ... .. _.. .... +....
P I' .' -- '-. : '- / -/ -- l j
__ l ,/ I _- 7 (









.. ._ -^ ^- ^ -.--, ; ._ ._.. 1 ^ ^ ^ i
(7 /___ry t__,]










--: --[ j "-
_ I L,, *.. ,--- --1_
I I 1 1


I-4 v tu.-I- I) ,' -i -I



... t T__d I. _. .]_ __
I-- -... -- + -
' -, 7 r \ \ ij/ '



i I
I I 4- k I t "
i .. .- + .-:- ; .
.. ', -J I -, : -- I -I-
.. ..
I -_ I _+ ___


4- i1 (2?-[ i -, 7 -


- !


-,.2 -
- *1-Ki-

^.4-


I I I

I~I
-. 4-


-- -- --
_ !-


+



* "IA
--I-I--I
* --'-A-
*---I *-I'--i


IK


C


- 7 I-





-I-t
T T-'-\ -


- I J -




*-- -I


-~ K
I j
t
ti t4.
I i
rV

I I I 'I'll


,-. --1-9-v--
. .I '_i 'j* __

\. .- ] .
I *--" -- .
,. Il -->+ -.*: -..... .-h^ -S -
'.!^ .. .'-L !


'- I -
I I


.0
-* C







Ii
-- .






.... ', g
0







-.--, 1-
t- Cr


- *4fl
'r "
-"I--^ $4 a

I 4 --40 -

'~~ ~ 0- *


&
^-- KN C



-I- -i
r-H

H1 _


C,


a is-'
C (I U)


Fig. 3







NACA ARR No. L4L19


- -4-+i


-ii

- I


7Th
I I
* I


-~ -
* -
* I .
~~1~- I-

.1-I *'-~
__ ;rj



,/ I -


-1
/




1/


_-ii- _--)
5I-





/ <
-- -- -


i--.- -:-- -----t-i:-: ::

, I *j... ... I-

W i__- i- _


,ipHJ__
i. ILL i17


L11:1


J- f_


-- I -


/-


-KITTI


* I I






1~


-_i-! i
747<.,;


..... I---


o,,,
C-


0








043
-L















ol
01
Is-
43












F.-
,, F, --'
S





, -4
o0
o.
05






ID
*ri
a a
"p
IX
4..B
4,-^
0~ B
< 0O
P 0~
ow
U,*
0 44
OS*t CF--
OS F
a
-at

C
w*


4J C, CU .0
a] -


-; *j C


C .. -3C 0
C: 0 G


4


I


Fig. 4


F. i








Fig. 5


- I.




LI


'I'


.11

I ~. I


* >4211.
--I


- I


-1--


A
__:L -- _- __ tt-- r n- ----
. .


- ---t --s---j--- i-.-+ N '- --- __

4. I

tJj*~-1-i

I L I
I I
S -s I
-N--
-4--- 'S. -. 7F
1 -'---N- -

4-- ~4~5 N -I

$,j~ 71 I
p '-~-------'--- *N--.4~ N 1\ I




~ ~ N

rAf I- ~

______________________________
0 C~ (Li ~- ~0 Li .45 L's -4 0


NACA ARR No. L4L19


-N-\ 'i ;---- --+4.--t -,
-'- t I -

,. -- ,, ... ...... -- -4 .
.. .. I


0




0'>












ax
4.
S>
E




fr4
* tIo
is.




~4,

0 0
U- JI ,0
-c



* a C.

Es.
'453









UNIVERSITY OF FLORIDA

3 1262 08104 981 8


ijr '.'ERSITY OF FLORIDA
DOCUMENTS DEPARTMENT
1 0 ,,ARSTON SCIENCE LIBRARY
0. EOX 117011
GAINESVILLE, FL 32611-7011 USA












t













i.-