Compound Comp Nozzle Flow Heiser

8/17/2019 Compound Comp Nozzle Flow Heiser

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A . B E R N S T E I N

1

S e n i o r A n a l y t i c a l E n g i n e e r ,

P r a t t W h i t n e y A i r c r a f t ,

E a s t H a r t f o r d , C o n n .

W .

H

H E I S E R

2

A ss is ta n t P ro fe sso r ,

D e p a r t m e n t o f M e c h a n i c a l E n g i n e er i n g ,

M a s s a c h u s e t t s I n s t i t u t e o f T e c h n o l o g y ,

C a m b r i d g e , M a s s .

C . H E V E N O R

3

A n a l y t i c a l E n g i n e e r ,

P r a t t W h i t n e y A i r c r a f t ,

E as t H a r t f o r d , C o n n .

C o m p o u n d - C o m p r e s s i b l e N o z z l e F l o w

A one-dimensional theory based upon fundam ental flow relationships is presented for

analyzing the behavior of one or more gas streams flowing through

a

single nozzle. This

compound -compressible flow theory shows that the behavior of each stream is influenced

by the presence of the other streams. The theory also shows that the behavior of com-

pound-com pressible flow is predicted by determining how changing cond itions at the

nozzle exit plane affect conditions within the nozzle. It is found that, when choking of

the compound -compressible flow nozzle occurs, an interesting phenomenon exists: The

compound -compressible flow is shown to be choked at the nozzle throat, although the in -

dividual stream Mach numbers there are not equal to one. This phenomenon is verified

by a wave analysis which shows that, when choking occurs, a pressure wave cannot be

propagated upstream to the nozzle throat even though some of the individual streams have

Mach numbers less than one. Algebraic methods based on this compound -compressible

flow theory are used to demonstrate the usefulness of this approach in computing the

behavior of compound -compressible flow nozzles. A comparison of the compound -

compressible flow theory with three-dimensional computer calculations shows that the

effects of streamline curvature on nozzle behavior ca?i be disregarded for m any practical

nozzle configurations. Test results from a typical two-flow nozzle show excellent agree-

ment with the predictions from the theory.

M c

I n t r o d u c t i o n

O ERN propulsion engines often exhaust several

different streams of gas side-by-side through a single nozzle , Fig .

1 . The se f lows can exhibit sizeable comp ressib lity effects and

they will be referred to here as compound-compressib le nozzle

flows. The purpose of this paper is to provide, for the f irst t ime,

a simple method to predict the behavior and clarify the under-

standing of such flows.

A one-dimensional analysis similar to that used in single-stream

compress ib le f low prob lems is a ppl ied here t o com poun d-com -

pressib le f low problems. Th e great advan tage of this typ e of ap-

proach is that it provides physical insight into the nature of the

flow.

Mixing between the various streams is not considered in the

deve lopment o f the b a s ic theory , b ut i t s e f fe ct on compound-

compre ssib le flow behavi or will be discussed. I t will be shown

that mixing oft en has a neglig ib le effect on the f low beha vior.

Th e usefulness of the com poun d-com press ib le f low theory is

demonstra ted b y compa ring i t s predict ions w ith b o th three -

dimensional computer calculat ions and experimental results.

T he b a s ic a pproa ch used t o deve lop the compound-compress i -

b le f low theory will be to determ ine ho w changing co ndit ion s at

the nozzle exit plane change condit ions within the nozzle . This

will be seen to be the heart of the matter and all results obta ined

in this paper are presented in this l ight . No te should be taken

of a pioneering contribution to comp ound -com press ib le nozzle

flows made by Pearson, Holliday, and Smith [ l] .

4

Thei r results

are consistent with the general conclusions arrived at in this

paper.

j \ l INTERFACE

FLOW

3 -

1

i

i

—

FLOW

2

1

1

1 —

1

1

1

; i i

i

i

i i

1 / —

1

I NLET

P L A N E

F i g . 1 S c h e m a t i c d r a w i n g o f a x i a l l y s y m m e t r i c c o m p r e s s i b l e - n o z z l e f lo w

O n e - D i m e n s i o n a l C o m p o u n d - C o m p r e s s i b l e

N o z z l e F l o w T h e o r y

T he deve lopment o f one -d imensiona l compound-compress ib le

nozzle f low theory follows that of Shapiro [2 ] for single-stream

flow. Th e most importan t a lteration is that the fluid stat ic

pressure is chosen as the depen dent param eter because it can va ry

o n l y

along

the nozzle in one-dimensional f low, whereas a ll other

fluid propert ies can also change from stream-to-stream across the

nozzle .

This analysis is sufficiently general to include any arbitrary

number of streams designated by the integer

n.

F or exa mple ,

at anj ' posit ion in the n ozzle ,

dx ~ ^•

1

Now , Propulsion Staff Engineer, Commercial Airplane Division,

The Boeing Company, Renton, Wash.

2

Now, Head, Turbine Research and Tech nology Group, Pratt &

Whitney Aircraft, East Hartford, Conn.

3

Now , Compu ter Specialist, Gerber Scientific Instrument Com -

pany, South Windsor, Conn.

' Numbers in brackets designate References at end of paper.

Contributed by the Applied Mechanics Division for publication

(without presentation) in the

J O U R N A L O F A P P L I E D M E C H A N I C S .

Discussion of this paper should be addressed to the Editorial De-

partment, ASME, United Engineering Center, 345 East 47th Street,

New York, N. Y. 10017, and will be accepted until October 15, 1967.

Discussion received after the closing date will be returned. Ma nu-

script received by ASME Applied Mechanics Division, November 15,

1966. Paper No. 67—AP M-L .

t = l

dx

1)

where

A

is the total flow area, /l , is the flow area of the ith s tream ,

an d x is the axial nozzle posit ion coordinate . In single-stream

one-d imensiona l theory ,

dA /dx

is arbitrarily small and this carries

over into the present case where a ll dAJdx are arbitra rily small.

The transverse pressure gradients caused by streamline curvature

can then be neglected and this leads to the conclusion that stat ic

pressure is only a function of axial posit ion.

It is a lso assumed that the f low in each stream is stead} ' , adia-

batic, and isentropic and that each fluid is a perfect gas with con-

sta nt thermod yna mic propert ies . Note tha t these a ssumpt ions

exclude m ixing effects.

5 4 8 / S E P T E M B E R 1 9 6 7

T r a n s a c t i o n s

o f t h e

A S M E

Copyright © 1967 by ASME

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n S T R E A M S

D T V _

r

0H oi ' i'

R . K N O W N

_ Ai ( J L _ i \

da; A,- V M ,

j

/ da;

(In p)

where A:, is the rat io of specif ic heats, M , is the M ac h nu mbe r (M,-

= VJy/kiRiTi) for the ith stream, and p is the fluid static pres-

sure. Equ ation s (1 ) and (2) may be comb ined to y ield

dx

(In

V

) =

dA

dx

where

i - (

-

-

hi

k

0

everywhere in the nozzle and

p

will therefore change in the same

direct ion as A throug hou t, equation (3) . In part icular, both p

an d

A

will have their smallest values at a geometric throat where

A

reaches its min imu m. No te that the integration also shows

what the ba ck pressures must be to maintain these f lows.

At the same t ime, the differentiat ion of with respect to p

yields

d§_

dp

=

s

P ^ w [

( 1

-

M

a ) i + 2

(

1 + M ,

2

> 0

(7)

F i g . 2 C o m p o u n d - c o m p r e s s i b l e f l o w i n a n o z z l e o f f i x e d g e o m e t r y

Each stream may then be separately treated as a single-stream

one-dim ensional f low (Shapiro [2 ] , Tab le 8 .2) . Cons equen tly ,

2)

which s hows th at (3 a lways changes in the same direct ion as p.

Therefore, for curves a and b , /3 will a lso change in the same

direct ion as

A

and will a lso have its minimum value at the throat .

As the inlet pressure is decreased , the value of /3 at the th roat

will also decre ase. In fact, when the inlet pressure is chosen t o

be sufficiently small,

3

reaches zero at the throat . Wh en this

occurs, equation (3) is indeterminant and no longer serves to

determ ine the axial pressure gradient at the throat . Und er this

condit ion, applicat ion of L 'Hospital 's rule to equation (3) y ields

dx

(In

p)

(3 )

d

2

A

dx

2

M ,-

2

)

2

+ 2

8)

Th e geometry of any th roat is such that

d A/dx

is a lways >0 .

T here fore , d(ln p)/dx will be either the posit ive or negative root

of a real num ber.

Curve c represents the choice of the posit ive root while curve d

represents the choice of the negative root . Com parison of

curves c and d reveals a familiar single-stream compressib le f low

situatio n: Th e geometric thro at is a saddle poin t for tw o isen-

tropic solutions in the divergent sect ion of the nozzle . N o back

pressure between curves c and d can correspond to an isentropic

flow. I t is anticipated th at those bac k pressures which do n ot

correspond to isentropic solutions, such as that of curve e , may

be reached by means of

compound shocks

init iated at some point

on curve d.

Th e behavi or of the f low along cu rve c is similar to that of

curves a and b ; i .e . ,

A, p,

and (3 will reach their minim um values

at the throat . N ote that the posit iv e root of equation (8) is

chosen

only

when the back pressure corresponds exact ly to that

of curve c.

The implicat ions of the choice of the negative root of equation

(8) will now be considered in detail (curve d) . Since equation (7)

has shown that j3 a lways ch anges in the same direct ion as p, dfl/dx

must a lso be negative at the throa t . Acco rdin gly , /3 mus t de-

crease from posit ive to negative as it passes through zero at the

throat . Furtherm ore, with /3 negative entering the divergen t

sect ion (where

dA/dx

is > 0) , simultane ous examination of equa-

t ions (3 ) and (7) shows that the local values of both

p

and /3 must

continue to decrease through the divergen t sect ion. No te that

the integration of equation (3) st il l shows what the back pressure

must be to maintain this flow.

No isentropic solutions exist for

inlet

pressures corresponding

to values less than that of cu rve d becaus e /3 would reach zero

upstream of the geom etric throat . This would result in an infinite

axial pressure gradient , equation (3) .

J o u r n a l o f A p p l i e d M e c h a n i c s

S E P T E M B E R 1 9 6 7 / 5 4 9



3/7

Some interest ing conclusions can be made by examining the in-

fluence of back pressure,

p„ ,

on the inlet pressure. A back pres-

sure greater than that of curve c will a ffect the pressure at the

inlet plane and will thus influence the f low rates of the individu al

streams. An y back pressure less than that of curve c will a ffect

neither the inlet pressure nor the f low rates. This cond it ion will

be referred to as compound-choking. Under such condit ions , the

nozzle geometric throat controls the behavior of the f low.

Also, since

dp/dx

is a lways < 0 for curv e d, Ber noulli 's equa-

t ion shows that a continuous acceleration of the f low takes place

through out the nozzle . Ap ply ing this to equatio n (3) , it can be

seen that , for contin uous acc eleration of the f low, |3 > 0 whenever

dA/dx

< 0 and (3 < 0 whenever

dA/dx

> 0 . Thu s, for con -

t inuous acceleration of the f low in a

single-stream

convergent -

divergent nozzle , exam ination of /3 reveals that the f low m ust be

subsonic in the convergent sect ion, sonic at the throat , and

supersonic in the divergen t sect ion, equation (4) . In the follow -

ing sect ion, it will be shown tha t for compo und -com press ib le f low

an analogous situation exists: Th e f low must- be co mp oun d-su b-

sonic in the convergent sect ion, compound sonic at the throat ,

and comp ound -super sonic in the divergen t sect ion. I t will a lso

be shown that these regimes are differentiated by the com-

pound-flow indicator, /3 .

C o m p o u n d W a v e s

T he compound-choking phenomena j ust descr ib ed ca n b e ex -

plained by examining the effects of small pressure disturbances

on the f low. A diagram of such a disturban ce is shown in Fig . 3 .

I t is consistent with one-dimensional theory to take the f low area

as constant in wave calculations. I f a weak plane pressure dis-

turbance is impose d on the f low, Fig . 3 (a) , this disturbance canno t

propagate at different absolute velocit ies in each stream without

violat ing the condit ion that the stat ic pressures at the stream

interfaces be equal. Ther efore, the wave must be continuous

and must travel as a single compound wave, Fig .

3(b).

A l though

the wave is not necessarily plane, the pressure rise across it can not

vary from stream to stream.

As indicated in Fig . 3 (6) , the absolute terminal velocity in the

upstream direct ion of the compo und wave is designa ted by a . I t

follows direct ly that :

a

> 0 correspo nds to

compound-subsonic

f low ;

a

= 0 corresponds to

compound-sonic

flow;

a

< 0 corre-

sponds t o

compound-supersonic

flow.

An analytical expression for the compound wave velocit j^, a ,

can be derived by treating each stream separately as a f lexib le

tube and conserving mass, momentum, and entropy across the

compoun d wa ve in the f ra me of re ference o f the compound wa ve .

It fo llows that

AA

f

A-i

A^

kiPi

1

M =

, . Y

A V h R i T i 7

(9)

where A signifies the change across the com pou nd wave . Since

the flow area is constant,

3 a I N I TI A L P L A N E P R E S S U R E D I S T U R B A N C E

cy

FLOW 3

FLOW 2

FLOW I

FLOW 3

F


4/7

As ha s b een shown, the b eha v ior o f compound -compress ib le

flow is determ ined b y the relat ionships of

k

{

, R

h

A,-, w

it

Tm ,

po;, and

p.

Equ ation s (1 ) and (5) can be com bined to y ield

£ uu Vr~*i / M

k <

12 kj \

i

_

i = 1 Poi \pj

r> \ki -

1 L

\PoJ

ki-V

b

= E ^ = (13)

=

Using equation (13), the follow ing expression m ay be written for

two streams at any point in the nozzle :

•A

w t

VT,

ir, VT,

A

u *

fcl+1-

where

i r i i ' / » i (

_ ( V ) * '

1

_ ( p . )

h

f W j ( h - 1 \

\ p j L \ P o. / J )

[R>k\

\A-2 — 1 /

fa+r

I C ^ ) -

\), similar solutions can be generated for

any other values of these parameters.

U n d e r choked f low condit ions, it has been shown that the f low

behavior is determ ined by the nozzle geometric throat wh ere

ft = 0 . Equ ation s (4 ) and (6) can be combined t o y ield the follow -

ing equation for choked c ondi t ions:

v

2

V l

w,

V t

Tm ^ /h

Po

2

\ JA-i - 1

/p_\

.

\A-1

p j I

2 L w

1 - i i

- 1 1

X

C 0 2 ) ' f t (a-2 - l )

1 -

l-h

C 0 2 )

h-

r

hi

- l

X

1

r fa - 1

2

(

fcl \

1 -

( )

h

\p

0./

'i - 1J

1 -

( )

h

\p

0./

16)

Equations (14) and (16) may be simultaneously solved, by tria l-

and-error, to determine the relat ionship between P02/P01 "-'2

VT

m

/

« ' 1

V T

0

i, Ai*/Athroat and pthroat/poi at chok ing for any given gas

prop erties . Th e relations liip betw een P02/P01, W 2

VT\n/

w

i V Tqu

and /liV-lehroat is shown in Fig. 5. It can be seen that, fo r any

given value of

w

2

VTm/

w

t VT

\

u

ther e is a un iqu e P02/P01 cor re-

spo ndi ng to each .Ai*/ylthroat. Th ese values appear as horizontal

lines in Fig . 4 because the back pressure does not a ffect the

choked solution.

Referring to Fig . 4 for any given com bination of A i* /y i

8 I

u and

A]* / A throat in a single nozzle, the intersection o f the corresponding

lines indicates the onset of choking (cu rve c in Fig . 2 ) . For

values of poi/p

a

smaller than that at the intersection, the flow

behav ior is g iven by the unch oked curv e. For values of poi/ p«

greater than that at the intersect ion, the f low behavior is g iven

by the choked straight l ine.

Therefore, the entire flow behavior of a

compound -compressible nozzle can be described by a single line.

For examp le, the dashed line in Fig . 4 represents a nozzle w ith

-4 1 * / . 1 throat = 0 .4 3 1 a nd A S / A ^ n = 0 .2 2 6 . (T hese a re the a c -

tual dimensions of the test model described in the experimental

sect ion of this paper. )

T he b eha v ior o f Mi a nd M

2

at the throat fo r choked flow as a

funct ion o f

w

2

VT^/Wi VT

m

and fixed -4i*/Athroat and gas

propert ies is shown in Fig . 6 .

Note that neither stream is sonic at

the nozzle throat.

-1.4

2 .5

R,, Rj = 53.3 FT-LB.SLUG- =R

;

0 . 4 3 1

K|. K

2

= I* R,. R

2

= 53.3 FT-LB SLUG -

:

R

P02/P01

W 2 y T o

2

/ W

l v

/ T o i

F i g . 5 R e l a t i o n s h i p o f fl o w p a r a m e t e r s d u r i n g c h o k e d f l o w

0 1 1 1 1 '

0 0 .1 0 .2 0 .3 0 .4 0 .5

W

2

v % / w ,

F i g . 6 S t r e a m M a c h n u m b e r s a t t h e n o z z l e t h r o a t d u r i n g c h o k e d f l o w

S T R E A M I N

M A C H

1 0

N U M B E R

A T T H R O A T ,

1.0

J o u r n a l o f A p p l i e d M e c h a n i c s S E P T E M B E R 1 9 6 7 / 5 5 1



5/7


6/7

T H E B A S I C N O Z Z L E A N D T Y P I C A L V A R I A T I O N S

E X P E R I M E N T A L R E S U L T S

O N E D I M E N S I O N A L T H E O R Y

A

E X I T —

S H O R T E N E D

1— H

"- T H R O A T | E L O N G A T E D N O Z Z L E

N O Z Z L E

;

- E X I T

B A S I C N O Z Z L E

B A S I C N O Z Z L E D I M E N S I O N S :

A ' r '

A

T H R O A T = 0 . 4 3 1

M / * E X I T - 0 2 2 6

U H R O A T / D T H R O A T = 0 . 37 4

L E X I T - E X I T

0 . 6 9 5

Kj=K2= 14

R l = R 2 = 5 3 . 3

F i g . 8 N o z z l e s u s e d i n t h r e e - d i m e n s i o n a l c o m p a r i s o n

O N E D I M E N S I O N A L

A X I A L L Y S Y M M E T R I C

0.6

P02/P01

0 . 4

W j ^ / W . / f o , =0 . 0 5 2

C O M P O U N D

C H O K E D

W

2 V

/ T

0

/ W , Y F O , 0 . 31 5

P02

P 0 1

W 2 y T o 2 / w , y T m = 0.0 52

C O M P O U N D

U N C H O K E D

Pm/P» 4 0

W

2

/ W W i v / T o i = 0 .3 1 5

P0 1 / R 0 2 7 5

W 2 A 2 W , / T

0

, = 0 - 3 1 5

P02/P01

Pfl2 /Poi

W

2

y T

0 2

/ W , y T |

1

, = 0 . 2 0 9

w

2

y f

0 2

/w

lv

/ r

0

i = 0.052

—0 o-

0 0 . 4 0 . 8 1 . 2 0 0 . 4 0 8 1 . 2

L T H K U A T ' D T H R O A T L T H R O A T / D J H R O A T

F i g . 9 C o m p a r i s o n o f o n e - d i m e n s i o n a l a n d t h r e e - d i m e n s i o n a l r e s u lt s

Curves o a nd d demonstra te the com pound-u nchoke d f low

regime. Again , agreeme nt with the one-dimen sional results is

excellent . Th e discrepancies for short nozzles are not so pro-

noun ced here since the exit condit ions, a nd not the throa t c ondi -

t ions, are dom inan t. Som e departure from the one-dimensiona l

line is seen for the longer nozzles. I t is fe lt that this ma y be d ue

to accumulating inaccuracies result ing from the f inite difference

techniques used. For long nozzles, these errors ma y becom e

significant by the t ime the exit plane is reached.

The agreement between the one-dimensional and the axially

symmetric solutions is excellent for the range of nozzle variat ion

exam ined. Three-dim ensional effects have lit t le influence on the

level of P02/P01 for th e nozz le geom etries consid ered.

E x p e r i m e n t a l R e s u l t s

Exten sive test program s cond ucte d 011 a wide variety of nozz le

types and geometries have shown excellent agreement between

the one -d imensiona l compound- f low theory a nd exper imenta l re -

sults. These test progra ms have included converge nt , cylindrical,

and convergent-divergent nozzles with both two and three

streams.

Th e success of the one-dimension al the ory is not sur prising.

T he prev ious sect ion indica ted tha t the b eha v ior o f compound

flow nozzles is reasonably insensit ive to three-dimensional effects.

For nozzles with small wall frict ional effects, and fa irly undis-

torted inlet f low, only the effects of mixing can cause the on e-

dimensional model to be inaccurate when applied to real nozzles.

I t is reasonable to assume that , since the f low is turbulent , mixing

is confined to a shear layer between adjacent streams which grows

with axial posit ion at an angle of less than 8 deg [3 ] . I t is there-

fore clear that the shapes of the and the nozzle length are the

majo r factor s which determine the degree of mixing. Th e ten-

dency of the mixing will be to pump the low-velocit j ' streams and

2 4 6 8 10

Poi/Pco

F i g . 1 0 C o m p a r i s o n o f c o m p o u n d - c o m p r e s s i b l e f l o w t h e o r y w i t h

e x p e r i m e n t a l r e s u l ts

to retard the high-v elocity streams. Th e extent of the influence of

mixing depend s primar ily on the flow rates of each stream. Th e

behavior of streams with proport ionately low rates will be greatly

affected by mixing while those with higher f low rates will ex-

perience only small effects.

Note, how ever, that in the important

case of choked Uow, mixing effects downstream of the nozzle throat

can exert no influence whatsoever upon the behavior of the flow.

Thus, for many nozzle applicat ions, mixing will influence only a

small portion of the flow.

Because of the space limitations of this paper, a fully compre-

hensive comparison of one-dimensional theory with experimental

da ta f rom a w ide va r ie ty o f compound- f low nozz le types a nd

geometr ies i s impra ct ica l . Ac cor din g^ , the b as ic nozz le prev i -

ously discussed was selected for comp arison. Being conv ergen t-

divergent , it is a representative two-stream nozzle in that it can

displa y b o th the choked a nd unchoked reg imes o f compound

flow.

Tests were run over a wide range of temp erature-co rrected

mass f low ratios while varying pm/p^ from approximately 2 to 10 .

This a llowed the nozzle to exhibit both choked and unchoked

beha vior at each mass f low ratio . Fig . 10 is a compar ison of ex-

perimental results with predict ions based upon the one-dimen-

s ional compoun d- f low theory . T he choked f low reg ime is the

straight port ion of the theor et ical l ines and, as expec ted, occur s

at the higher

poi/p

a

The unch oked flow regime is the curved

port ion of the lines which occurs at the lower primary nozzle

stagnation pressure rat ios.

I t can be seen that the one-dimensional theory shows excellent

agreement with the experimental data , part icularly in the choked

flow regime. Correla t ion with experime ntal data in the un-

chok ed flow regime is som ewha t less accurate . How ever, in v iew

of the mixing effects previously discussed, this was to be expected.

During choked flow, a ll mixing downstream of the nozzle throat

can have no effec t 011 the f low. Thu s the effect ive mixing length

for choked flow is merely the distance from the inlet plane to the

throat . On the other hand, during unchoked flow, a ll mixing

downstream of the nozzle throat will have a very definite effect

on the f low behavior. Wi th unchok ed flow, then, the effect ive

mixing length is the entire length of the noz zle .

As anticipated, we also observe that mixing has lit t le effect at

the higher mass f low ratios but exerts increasing influence as

mass f low ratio decreases. Furthe rmore , the imp ortan t effect

of mixing is to pump the secondary f low and therefore reduce the

required

pm/poi-

It is noted that the model tested was not an ideally designed

nozzle ; i .e . , the primary stream was independently choked and

slightly unde rexpan ded at the inlet plane. How ever, the mod el

used in the experiments was operated sufficiently near to its

isentropic design condit ions that stagnation pressure losses in the

supersonic stream were probably not important , and shock losses

J o u r n a l o f A p p l i e d M e c h a n i c s

S EP TE MB ER 1 9 6 7 / 5 5 3



7/7

are impossib le in the subsonic second ary stream. Thi s was indi-

cated in the previous sect ion, where the three-dimen sional solu-

t ion showed no significant coalescing of the primary stream Mach

lines under condit ions for which the tests were run. Alth ough

Poi/Pos is measured up stream of the inlet plane, these argu-

ments just ify our neglect of total pressure losses in a ll ca lcula-

tions.

C o n c l u d i n g R e m a r k s

A new one-dimens ional theory describ ing the behav ior of c om -

pound-compressib le nozzle f lows has been developed and its

implicat ions have been examine d from a num ber of v iewp oints.

The theory yields simple a lgebraic methods for ca lculat ing the

operation of com poun d-com press ib le nozzles. Com parison of the

algebiaic results with those of three-dimensional f low field co m-

putations indicates that the effects of streamline curvature are

not importa nt fo r ma ny pract ica l nozz le con f igura t ions . Co m-

parison of the a lgebraic results with experimental data for f lows

with unimportant mixing effects shows that the theory can

accurately predict the behavior of real devices.

A complete definit ion of the range of applicability of the simple

theor y requires a great deal of experimental experience. For e x-

ample, l it t le information is available about the rate of growth of

the mixing zone between the high-velocity streams, and no in-

form ation is available about the detailed nature of com pou nd -

shock waves. Furthe rmore , no predict ion s about the thrust or

nozzle eff iciency can be made when the f low is choked until the

losses ca used b y a compou nd-shock wa ve a re known. Conse -

q uent ly , i t a ppea rs tha t the compound-compress ib le nozz le pro -

vides a numb er of interest ing and imp ortan t areas of research.

R e f e r e n c e s

1 Pearson, H., HoUiday, J. B „ and Smith, S. F., "A Theor y of

the Cylindrical Ejector Supersonic Propelling Nozzle," Journal of the

Royal Aeronautical Society,

London, Section 62, 1958, p. 746.

2 Shapiro, A. H.,

The Dynamics and Thermodynamics of Com-

pressible Fluid Flow , Ronald Press Co., New York, 1953.

3 Sclilichting, H., Boundary Layer Theory, 4th ed. , McGraw-Hill ,

New York, 1960, pp. 598-600.

5 5 4 / S E P T E M B E R 1 9 6 7

T r a n s a c t i o n s o f t h e A S M E

Compound Comp Nozzle Flow Heiser

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