J- - 13 IEC~INICAL lIar r w AIRESEARCH MAN UF/iCTURI G C

INGLEWOOD , CALI FORNI A

/+ J( R - J. :>L,L ) 9 o.

NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

ORIGINALLY ISSUED January 1945 as

Advance Restricted Report L4Ll9

CRAR'lS OF PRESSURE, DENSITY, AND TEMPERA'lURE CRANG:ES

AT AN ABRUPT INCREASE IN CROSS-SECTIONAL AREA

OF FLOW OF COMPRE3SIBLE AIR

By Upshur T. Joyner

Langley Memorial Aeronautical Laboratory Langley Field, Va.

WASHINGTON

NACA WARTIME REPORTS are r eprints of papers originally issued to pr l.'vide rapid distribution of advance research r esults to an author ized group requiring them for the war ~ffort. They were pre-

;' viously held under a security status but are llOW unclassified. Some of these r eports wer e not tech­nically edited. All have been r epr oduced without change in order to expedite general distribution.

L-13

TEel :NICAL lI;:~ I Y .' 1

AIRESEARCH MANUF;\CTU~lr G Ci:r:.-:I

9851-9951 SEPULVEDA BLVD.

NACA ARR No. L4L19 RESTRICTED ING LEWOOD , CA IFORNIA

NATIONAL ADVISORY CGriM ITTEB FOR AERONAUTICS

ADVANCE RESTRICTED REPORT

CHARTS OF PRESSURE, DENSITY. AND TEMPERATURE CHANGES

AT AN ABRUPT INCREASE IN CROSS-SECTIONAL AREA

OF FLOW OF COMPRESSIBLE 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 i n a table and are plotted as curves showing the variat ion of the calculated quantities with the area expansion ratio in te r ms of the initial Mach number as parameter . Only t he subsonic region of flow is considered .

INTRODUC TION

The well - known Borda- Carnot expression for change in pressure when an incompressible fluid passes an abrupt area expansion has long been used for estimating the pressure change s in compressible air flow at an abrupt expan sion . Th i smethod is simple but not exact.

The expressions for pressure , density , and temperature ratios given herein are for subsonic flow and are in precise agra ement with the exact expression for the veloc ty ratio in a compressible flow at an abrupt area expansion developed in reference 1. In the present report , Mach number , or the ratio of air­flow velocity to existing sound velocit~ is used as a parameter j whereas in reference 1 the parameter was the I'atio of existing air - flow veloci ty to the veloci ty that the air would possess if accelerated isentropically Q~til its velocity was equal to t he then existing local

REs 'rR IC TED I

)

2 FACA ARR No . r4L19

sound velocity. This difference in parameters mus t be considered when equations from the two papers are compared .

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

The expressions obtained herein for pressure , density, a nd temperature chanc es in a compressible flow at an abrupt expansion are too invol ved to be of practical use . The p resen t computa tions have therefore been made and a re presented in tabular and g raphical form.

S Yl'vlBOLS

A cross - sectional area of flow , square feet

a ve loci ty 0 f souno. , fee t ge r se cond

f area ratio (A1/A2 )

M Ma ch number ( via)

p static pre ssure, po~~ds per square foot

V ve locity of flow , feet per second

y ratio of specific heat at c ons tant pressure to s pecific heat a t constant volume , dimensionless

R universal gas constant , Btu per slug pe r OF

T absolute temperature , 460 + OF

p densi ty of air , slug per cubic foot

Subscripts :

1 before abrupt expansion

2 a fter abrupt expansion

NAC A fu~R No . 14L19 3

ANALYSIS

Brom the fundamental equations for the conservation of energy, of continuityj and for the conservation of momentum, equations a re obtained that gi ve the variation of the pressure, density, and temperature ratios with the area expansion ratio f in terms of the initial Mach number as pa r ame t er .

Figure 1 shows the conditions of flow assumed for the p resent calcula ti ons . The static pressure at a ( fig . 1) is taken to be the same as at b for subsoni c flow, as has been pr oved experimentally by Nusse1t (reference 3). Unifo r m veloci ty dist ribution before and after the expansion is assumed . The ratio of s pecific heats y is taken as 1 . 405 .

The fundamental equations are th.e equation for the conservation of energy

V 2 = -L +

2

the equati on of continuity,

and the equatlon for the conservation of momentum

P2 A2 V22

- PI Al v12 = - A2 ( P2 - PI)

From equation (1)

or

y - 1 2 M12 + 1

( 1 )

(2 )

4 NACA ARR No. 14119

From equation (3)

P2 = 1 +Yf :1

2 (1

PI '\

From equation (2)

= 1 + yfM 2~ 1 t

\

== --P, = ""'::'f P2

( 6 )

W"'1en equa ti ons (5) a nd ( 6) are subs t i tuted in equatIon (4 ),

y - 1 2 -'----HI + 1

2

or

Wn en equation (7) is solved for Pl/P2 and the resulting equation i s inverted ,

==

In order to obtain an express i on f or t :te pr e ssure ratio , equation (8 ) is substituted in e quation (5) and the following equation results:

1 + ynI12 + yV 2yfM12(1 - f) + 1 - 2f2 r':12 + f2 u14

= y + 1

------ "----"-""--

( 8 )

( 9 )

NACA ARR No. r14-L19 5

By differentiating P2/PJ. wi t!;. respect to 1', it can be shown that the maxi:num static-pressure recovery for any value of l.rl - is obtained when

l' =

,/y - 1 -y+ v -2 yl2+l

2 (y + l) - M12 (10)

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

In equations (8) , (9) , and (10), the sign of the radical has been chosen so that the results obtained are in the region where the ass1l.'1lptions are valid.

The temperature ratio is obtained by use of the general gas law and the C0'11puted values of pressure and density ratio as

Pl RTl - =

Pl

P2 RT2 - =

P2

T2 P2~ ( 11 ) =

Tl P2/Pl

Figures 3 and 4 show the variat i on of density ratio and temperature ratio with area expansion ratio.

In order to make the results shown in fi oures 2 to 4 usable in cases for which only the conditions after the expmlsion are known, the value of M2 in terms of Ml and f is gi ven in figure 5. If M2 and f are known, M1 can be determined from this figure. The relation plotted in fioure 5 is developed as follows:

6

By use of equation (6),

:; (V2/a2)2

( Vl/al)2

~~ = YP2/Pt:;

V12

YPl/P1

~ C~)2~ ::

}ffiCA ARR No . L4L19

RESULTS AND DISCUSSION

The calculated values of pressure ratio, density ratio, and temperature ratio are gi ven in table I. The value s have been computed to e decima l p laces because of t he fo rm of the equations, in which small differences in large quantities are invol ved . 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 calculat ions to 12 decimal p l aces in order to obtain smooth curves for the quantities c a lculated.

NACA ARR No . L4L19 7

As in the case of incompressible flow, the calcu­lated changes occur gradually after the abrupt increase i n c r oss - 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 gi ves the pressure ratio calculate d on the basis of incompressible flow for the same initial conditions that are a s sumed for compressible flow at an initial Mach number of 1mity. It is evident that the effect of compressibility is vffi1ishingly small fo r values of the a r ea ratio of expansion below about 0. 25 . The short - dash line in figure 2 shews 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 pres s ure ratio has been measured at an abrupt exuansion with compre ss ible gas flow at high Mach number . These data were obtained for an area expansion rat io of 0. 246 , however , for which the difference between compressible a nd incompressible flow is insignificant . These experimental 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 0.8 would be conclusive evidence of the difference in pressure ratio obtained with compressible flow from that calculated by the Borda - Carnot formula for incompressible f l ow .

An experimental investigation of the changes in pressure , density, and temne rature at an abrupt increase in cross - sectional area with c ompressible flow would serve to determine corrections fo r the effect of nonun iform velocity distribution and friction on the idealized results obtained from the present calculations .

Lang ley Memorial Aeronautical Laboratory l\Tat ional Advisory Committe e for Aeronautics

I ,angley Fie Id , Va.

8 NA CA ARR No . LL~L19

REFEttENCES

1. Busemann , A. : Gasdynamik. Handb . d . :Cxperimentalphys ., Ed . IV , 1. Teil, Akad . Verlagsgese llschaft m. b . H . (I,c i p zig ), 1931, pp. 403 -L~07.

2. Acke r e t, J.: GasdYl1amik. Handb . d . Phys ., Bd . VII, Kap . 5, Julius Springer (Berlin) , 192 7 , p . 325.

3. rus s e lt, W.: Del' Druck Ln Ringquerschnitt von Rohr en mit p lotzl i chel" Erwei terung be i m Durchf l uss von Luft mi t hoher GeschvlTindigke it. Forschung a. d. Geb . d. Ingenieurwesens , Ausg . B , Bd . 11, Heft 5, sep t.-Oct. 1940 , pp . 250- 2 55 .

~ 0

.1

.2 :, .5 .6

:A .9

1.0

0 .1 .2 :, J :~ ·9

1.0

0 .1 .2

:t ·5 .6

:~ .9

1.0

TABLE I.- PRESSURE RATIO, DENSITY RATIO, AND ABSOLUTE-TEMPERATURE RATIO FOR VARIOUS VALUES

OF INITIAL MACH BUMBER AND AREA EXPANSION RATIO

0 0.1 0.2 0.3

1 1 1 1 1 1.0012~39 1.0022481l 1. 002~190 1 1.00505 36 1.0050~6~ 1.011 24}2 1 1.011}721~ 1.0202 561 1. 02666591 1 1.02029')8 1.03601048 1.04755~07 1 1.~1G.4961 1.0

A624622 1.07459 75

1 1. 5 ill3 1.0 11 3A

57 1.107&0 31 1 1.0 17 2 1.1~2 05 1.147 23g8 I 1.08~3~ 1.1 9750 1.1,3892 3 1 1.101 00 3 1.1 306256 1.2 682673 1 1.12551775 1.22624052 1. 3.0657959

1 1 1 1 .9979~9~ .99926195 1.00030258 1.0011~72 .9

S19 5 .99708680 1.00120830 1,00~a23a

.9 210121 .~~~31 1.0026,844 1.009 62

.96861682 • 49 59 1.0047 6al 1.017452ii

.95181440 • 9824,10}1 1.007}13 9 1.02A

03 .9}2053~1 .91595678 1.010e~F 1. 03 51 'A09731 6 .9 1)817~ 1.01} 9 1.05175~ • 8526

A12 .9~7612 1.01769 29 1.066~3

.85908 08 .9 02960 1.02187692 1.082 5I3

.83160083 .93777649 1.026~0629 1.100326

1 1 1 1 1.00202500 1.0020048, 1.00194500 1.00184}~ 1.00810000 1.00A9228 1.00777701 1.0~p5' 1. 01822500 1.01 0 662 1. 01749995 1.01 61~ 1.0~2~0000 1.03207618 1.~111627 1.02l58 8 1.050 2500 1.o;OOlg12 1. 857745 1.gt a~32 1.07290000 1.0721I n 1.070~3~ 1. 6 1 50 1.09922500 1.0981~~ 1.09~3 , 1.09115250 1.12960000 1.1281 5} 1.12 59 10 1.1192355l 1.16402500 1.16220066 1.157$3488 1.1~lttlg2 1.20250000 1.20{J198}o 1.194 0951 1.1 7 97

0.L. 0.5 0.6 0.7 0.8

P2/P1

1 1 1 1 1 1.00~~7575 1.00,51948 1.00~~8250 1.00296,~~ 1.00226171 1.01}t477O 1.01 16152 1.01}65613 1.01201 7 1.009214i4 1.0}0 4~46 1.0321~4a4 1.0~l1r65 1.0276~199 1.021380 1.gst90 99 1.0579 3. 5 1.0566 ~72 1.~06 60~ 1. 039'l,4071 1.0 54401 1.09206623 1.09cS Ui 1. 2~703.2 1.065 042 1.125,5163 1.1~51~3~ 1.1349a 1.124318~9 1.10132226 1.173 p05 1.1 792 07 1.1901 al~ 1.179669M 1.14~~Z04 1.22&4 329 1.25117950 1.2574.§ 6 1.247279 1.21 2 10 L2l ,47~5 1.,2563900 1.,39 264 1.,328S501 1.2~~7517 1.3 01 7 1. 1197583 1. 3608912 1. 374 166 1.4 0987

P2/P1

1 1 1 1 1 1.00167088 1.00199195 1.

002081 1.00192483 1.00152701 1.00667901 1.00800080 1.008380 1.007Al062 1.00621571 1.01511640 1.018~428 1.01440652 1.017 ~26 1.0~391} 1.0273

a396 1.0327 162 1.03 9')0} 1. 0}26 80 1.02 635p

1.0415 974 1.05~0698 1.~~457} 1.0526~ 09 1.~38}5 0 1.~970572 1.07' ~357 1. 192} 1.0787 lA5 1. A23726 1.. 090755 1.1011 915 1.11197350 1.11192 9 1.09 46}63 1.10503550 1. 13294Z41 1.14952359 1.15M~g6 1. 13941§46 1.13187~28 1.16897 ~4 1.~}24636 1.20 9 0 1.19193 64 1.16ll6 16 1.20902157 1. }02399 1.26 55 5 1.257288

T2/Tl

1 1 1 1 1 1.001

A0202 1.001525~9 1.001299~6 1.0010~690 1.0007~~A7

1.006 2~12 1.006111 2 1.0052~204 1.00419159 1.002AA9 9 1.01g2~978 1.0~2982 1. 01iI!07~5 1.00960513 1.006 t6~ 1.02 8 134 1.0 0228 1.02~01 1.01~10~1 1.01~2 07 1.g't~15929 1.03916545 1.0~ 91 1.02 23763 1.02 8913 1. 2513~9 1.05~On21 1.050 717 1.~20565 1.~193760 1.0856~6 3 1.07 7 41 1.07033592 1.0 0S237~ 1. 6595~ 1.11261881 1.1b4~5Z97 1.09429215 1.081 2576 1.0 ,}7115 1.14~44Q44 1.13401 95 1.12255~29 1.108~m 1. 09013,63 1.17 7601 1.16786653 1.15531 90 1.1400 1.12051 81

- - ---

NATIONAL ADVISORY COKKIT7BE FOR AERONAUTICS

0.9

1 1.00127458 1.00522267 1.01224395 1.023101

S0

1.039101 2 1.06240899 1.0~65~76 1.1 71a 06 1.2220 512 1.3~036914

1 1.00088642 1.0~6~511 1.0 506g8 1.016018 1 1.027040l2 1.~926 7 1.0 61 762 1.1000§332 1.14§1 ~60 1.21 29 25

1 1. 000~8781 1. 00158180 1. 00~70584 1.00697161 1.011Z435t 1.018 120 1. 028a1065 1.~ 37a5 1.0 ~4~5 0 1.0919 969

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1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1

z » ()

» » :-0 ::0

z o

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<.D

•

Flow

Vi' Pi' Pi' Ai' Mi

•

NATIONAL ADVISORY COMMITIEE fOR AERONAUTICS

Flo w

V 2' P2' P2' A 2 , M2

Figure 1.- Flow conditions assumed for calculations.

~ r-" oq

~

z »­o · »-»­~ ~

Z o

r tl:>­l"' i-' to

I

r

P2/Pl

~

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o .1 .2 .~ .lj ,5 .6 ·7 .8 ·9 f

Figure 2.- VarIation of p,'essure ratto with area ratio and initlal Mach number at an abrupt expansion of compressible air flow. (Experimental pOints from reference 3.)

,,--. __ . - - -- -~ -- - - --' --- ----

1.0

z :> C)

:>

:> ::0 ;D

z o

r

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'Xl ...... aq

(\)

~

P2/ Pl

--- - - - --- - - - --

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