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AMP Report 82.1 R

AMG-NYU NO. 43

THEORETICAL STUDIES

ON THE

FLOW THROUGH NOZZLES

AND RELATED PROBLEMS

WITH THE APPROVAL OF THE OFFICE OF THE

CHAIRMAN OF THE NATIONAL D?FÂ£N. E f?^'S^A"CH

COMMITTEE. TH'S REPORT HAS ^J-Lri OECLA.-Sif lED

BY THE OFFICE OF SCIENTIFIC ftEjtARCH AND

DFv^LLOi-ivt^NT.

A Report Submitted

by the

NEW YoKi-., i_rilVlKiUY

INSTITUTE C- - â– - - â– -Tc.^ SCIENCES

25 Waverlv P(acp, New York 3 iSl V

Applied Mathematics Group, New York University - - -

to the

Applied Mathematics Panel

National Defense Research Committee

This document contains information affecting the

national defense of the United States within the meaning

of the Espionage Act, U. S. C. 50; 31 and 32. Its trans-

mission or the revelation of its contents in any manner

to an unauthorized person is prohibited by law.

April 1944

' ,1

Â«OUH li

Ui^lfLnuOlflLL

Copy No. ^"^^

AMP Report 82. 1 R

AMG-NYU No. 43

THEORETICAL STUDIES

ON THE

FLOW THROUGH NOZZLES

AND RELATED PROBLEMS

WITH THE APPROVAL OF THE OFFICE OF THE

CHAIRMAN OF THE NATIONAL Of FEN E RIIsrA^CH

COMMITTEE, THiS REPORT HAS fuLtl OECLA-SlfltD

BY THE OFFICE OF SCIENTIFIC fVESEARCH AND

DEVELOPMENT.

A Report Submitted

by the

NEW YORK UNIVERSITY

INSTITUTE OF MATHEMATICAL SCIENCES

L!BR.M;y

25 WaveHy Place, New York 3 N Yi

Applied Mathematics Group, New York University

to the

Applied Mathematics Panel

National Defense Research Committee

This document contains information affecting the

national defense of the United States within the meaning

of the Espionage Act, U. S. C. TO; Jl and 32. Its trans-

mission or the revelation of its contents in any manner

to an unauthorized person u prohibited by law.

0-)

April 1944

M

issiÂ«'i*' pijff^i0'^''-'^*-":i:r.

t'W

No. of

Copies

DISTRIBUTION LIST

AMP Report 82. IR

24 Office of the Executive Secretary

6 Liaison Office, OSRD

1 Att: B. S. Smith, British Admiralty Delegation

4 Att: N.A.C.A.

1 â€” -TÂ» Theodorsen

1 HÂ» WÂ« Emmons

1 Att: R. PÂ« Praser, Imperial College of Science

3 Richard Tolman, Vice Chairman, NDRC

1 Att: H. Bethe

1 Att: DÂ» A. Flanders

7 Bureau of Aeronautics, Navy

2 Att: Lt. Comdr. J. S. Warfel

1 Att: EÂ« S, Roberts

3 Bureau of Ordnance, Navy

1 Att: R. S. Burlngton

1 Att: R, J. Seeger

1 Naval Torpedo Station

Att: Coi^iraander J, M, Robinson

4 Office of the Chief of Ordnance

1 Att: Capt. C, M. Hudson

1 Att: H. M, Morse

1 Att: Colonel G. W. Trlchel

4 Aberdeen Proving Ground, Ordnance Research Center

1 Att: HÂ« Lewy

1 Att: D. L, Webster

1 Att: 0. Veblen

6 F. L. Hovde, Chief, Division 3, NDRC

1 Att: C. N. Hickman

1 Att: CÂ» C. Laurltsen

1 Att: E. C. Watson

1 Att: J. Barkley Rosser

1 A. Ellett, Chief, Division 4, NDRC

1 H. B. Richmond, Chief, Division 5, NDRC

ONCUSSIFIED

1st, AMP Report 82. IR (continued)

No. of

Copies

1

J.

T, Tate, Chief, Dlvislo:

Att: L. Dunn

1

w.

Weaver

1

T.

C. Fry

3

R.

Courant

1 Att: K. 0, Priedrlchs

1

E.

J, Moulton

1

s.

S. Wilks

1

J.

G. Kirkwood

1

J.

von Neumann

1

A.

H, Taub

1

H.

Weyl

1

M.

Rees

â€¢-â– vV

u* *-

I

TABLE OP CONTENTS

Page

Summary ^

Introduction 1

PART I.

Isentropic Plow Through Nozzles. 4

!â€¢ Basic Relations and Hydraulic Treatment 4

2. Refined Treatment 8

3. Critical Remarks 14

4. Remarks on Jet Detachment 18

5. Thrust 21

6* Ezaznples 30

PART II.

On Perfect Ejchaust Nozzles and Compressors 36

?â€¢ On Perfect Nozzles 36

8. CoDipressor Flow 42

9Â« The Stability of the Isentropic Compressor

Plow 46

lOo The Plow at the Compressor Entrance 52

11Â» Remarks on the Drag of a Projectile

Carrying a Conipresaor 57

Appendix*

12Â» Mathematical Details to Part I Concerning

Isentropic Plow Through Nozzles 64

II i^mm

Page

13Â» Construction of a Perfect Three-dimensional

Plow from any Expanding Plow with Axial

Syminetry 70

References 84

Additional Literature on Plow Through Nozales 86

Further Literature on Cori5)resslble Fluid Flow 88

-I-

SUMl^IARY

The present report sxxmmarlzes the results of

a study which the New York University Group of the

Applied Mathematics Panel has undertaken upon the

request of the Bureau of Aeronautics, Navy Department

(Project No. llA.j;167), and which was carried out under

the responsibility of Professor K.O.Priedrichs with

the assistance of Dr. Chas. R. DePrima and other mem-

bers of the group. Our group is greatly indebted to

Mr. E. S. Roberts of the American Cyanamid Corr5)any

for his stimulating advice.

The original request was for assistance in the

design of unconventional nozzles for rocket motors,

with a view toward attaining shortness while preserving

high efficiency. Since the opening angle of such

nozzles is necessarily wide and the expansion rapid,

it was necessary to improve the classical (hydraulic)

theory of the de Laval nozzle and to widen the scope

of the investigation further by consideration of gas

dynamical phenomena more generally. Therefore, the

present report is not concerned solely with the original

problem of exhaust nozzles, but it contains methods

and results of potential interest for other problems j in

particular, attention is given to the design of "kinetic

compressors", i.e., of nozzles serving, not as means

m

..... .^LU

i^^itjf

^*- ^ â€¢ -II-

for ejecting a supersonic Jet, but for receiving a

parallel flow of gas at high supersonic speed in order

to compress and arrest the incoming gasÂ»

More specifically, the contents of the present

report can be summarized as follows;

Part I contains a three-dimensional treatment

of flow through nozzles as a refinement of the customary

one -dimensional theory, under the following ass\iraptions:

1) The fluid is ideal and homogeneous, and the flow

isentropic, steady and irrotational. 2) Viscosity and

heat conduction are ignored. 3) It is further assumed

that shocks and jet detachment do not occur in the

nozzle; however, conditions for the absence of these

phenomena are analyzed*

A simple formula for the thrust produced by such

a flow through a nozzle is derived and applied to various

types of contours; particular emphasis is given to widely

divergent short nozzles. Among families of nozzles yield-

ing the same thrust, the shortest nozzle is determined.

Tables I to III at the end of Part I (p. 35) show numeri-

cal results in a condensed form.

Pa rt II concerns, first, the construction of "perfect"

exhaust nozzles which yield maximum thrust and a parallel

;\ V- .. K

jj;;7^:iiMBÂ»Â»^';*(i-â€¢,

^^^

.â™¦

\^V

-5-

heat conduction are ignored. This appears to be justi-

fied for the exhaust flow (Cf. T. Stanton [ 1] )^''""^ except

that viscosity will play a role in determining where Jet

detachment and shocks occur. The formulas to be given

are valid as long as the flow remains free of shocks

and as long as the jet remains attached to the nozzle

wall*

The isentropic character of the- gas flow is ex-

pressed by the relation

(1.01)

p ^ = const.

p being the pressure, x> the density (mass per unit volume)

and f the adiabatic exponent. The sound velocity

(1.02)

= ( f p ^ -1)V2

is connected with the flow speed q through Bernoulli's

law which we write in the form

(1.03)

l-^)c +^q =c^=q^

* Numbers in brackets refer to the bibliography at the

end of this memorandum.

Â»WU.'l'Â«Â»!lfc'

â– a / K '^

â€¢r t. - V ^ i \

?^. V i I i

â– JU2S ILL

with the abbreviation

^l^V

ft id "^

>^ ^ + 1

The significance of the "critical" speed q^. = c,^

Is that the f lov? speed Is critical when it coincides

with the local sound speed* Subsonic and supersonic

flow can then be distinguished by q < q^ and q > q^

since, as Is easily seen, q > q^ implies q > c â€¢

With the aid of Bernoulli's law one can express

the quantities c, p,/o in terms of q/q^. ;

using the abbreviation

V =

^ - 1

(1.04)

(1.05)

(1.06)

V

1 - /x2(q/q,,)^

Â» (Â°.^. = M^(q/q*)^

â€¢

e*

n

_ ..2

V + 1

/^

.pSÂ«iiei^Â«Â«^

V^V'^-

.^'^

^

v-^-

lif^*

ubuv...

-7-

^...1L

The "one-dir.ensional" or hydraulic theory of

0Â» Reynolds (1885) assumes, as a first approximation,

that the flow speed q and hence also c,p,^ are

constant over cross -sections perpendicular to the

nozzle axis. The mass flux per unit time across a

cross-section of area A is then given by A /o q; hence

this quantity is a constant, or

A Aq

(m = q/c being the Mach number) from which follows the

hydraulic approximation.

(1.08)

M = (M^ - 1) ifl .

A q

Ml

s

IfUV

-8-

This relation shows that a flow with Increasing

speed q is possible for q < q^,. or q < c only

when A decreases, and for q > q^,. or q > c only

when A increases. In other words, in the hydraulic

approximation, flow with increasing speed is possible

only when the critical speed is just reached at the

throat.

2. Refined Treatment

In order to refine the hydraulic treatment just

explained it is necessary to employ the basic partial

differential equations for an irrotational isentropic

flow. We introduce coordinates; x, along the axis in

the exhaust direction, and y, the distance from this

axis; further we introduce the angle 6 of the flow

direction with respect to the x-axis, the potential

function ^ and the stream function i/r â€¢

The stream function Y ^^ to be so normed that

2

tr y is the rate of mass flow per unit time carried by

the stream tube yr < const. The basic equations for the

four dependent variables ^, ^^ , q, and 6 can then

be written in the form

(2,01) ^ = q cose ,^ = q sinG

w* r-

\i

-9-

(2.02) -W^^ = - /oqy ^^^^ Â» '^^^ = ^^^ Â°**^Â® â€¢

The density p Is to be expressed in terms of q

by means of relations (1.06) and (1.04).

It would be natural to prescribe nozzle contour

and state In the chamber and to ask for the resulting

flow. Mathematically this would mean solving a boundary

value problem for equations (2.01) and (2.02). To sim-

plify the task we reverse the procedure: We first pre-

scribe the velocity distribution along the axis. -

(2.03) q = %M f or y =

and then determine possible nozzle contours from the re-

sulting stream surfaces. To this end we Introduce the

stream functions ^ and ^f/ as Independent variables in-

stead of X and y, consider the quantities tl, y, Â© ,

and q as dependent variables, and develop them with

respect to powers of yi*^ It is sufficient here to report

* One may also Introduce q and V as independent vari-

ables: this might seem to offer advantages since the non-

linearity would then refer only to the independent variable

a. Nevertheless it was found that the present scheme is

better, at least for the expansions up to the second order

as given here.

tjsiKj^'^- *â€¢'Â»'â€” :-?jÂ»

-10-

only the results. Details will be carried out in the

Appendix, Section 12.

To fonaulate the results we introduce the impor-

tant dimensionless quantity h by

(2.04)

h = ^f^^q^TJq.

the significance of which is that h = A/A^,_ in hydrau-

lic approximation. Since by (1.06) the density a is

a given function of q the same is true for hj we have

h = 1 for q = q.,.. Frora (1.07) we obtain

(2.05)

2 dh/h = (M^^ - 1) dq/q .

On the axis, where we had assximed q = q (x) the quantity

h becomes also a function of x

(2.06) h = h^{x) = (q,,yq)

1/2

1 - }-^Wi.,f

1 -

^'

-2

[Cf. (2.04), (1.06), (1.04)] which is given inasmuch as

q.QM is given. Also M, defined by (1.08), becomes a

given function M^(x) along the axis when q = q (x) is

inserted.

Wh

Li

-11-

We place the origin x = 0, y = 0, at the

throat; more precisely, we place it such that

(2.07)

^o^O) = q,

or what is the same thing, such that l^'(O) = 0. We

then have

(2.08)

h^(0) = 1, and M (0) = 1 .

Instead of and '\^ , two other parameters, |

and 07 , are introduced by the relations

(2.09)

o

(2.10)

^= /e^v

Clearly, ^ and Oj have the dimension of a length and

\ reduces to x on the axis V = 0,(ji5 = for x = y =

being assumed)*

In terms of the quantities just introduced, the

expansions of x, y, q, 6 , h and p with respect to powers

of >? (instead of y ) are. If h^( ^ ), h^( ^ ), hj( \ ),

Mq( I ), and Pq( | ) are written singly h^,h^,hJJ,MQ, and

p^ respectively:

JllUi

(2.11) :: = ^ -i Vi'^^

(2.12) y = h^77 1 1 + 1 [ (m2- 1) h^h;; - (h.)^],^^!

(2.13) q = q^ jl+^h^hj 07 ^j

(2.14) = hÂ»>j

(2.15) h = h^ I 1 + ^ (M^ - 1) n^h;; 7,2|

(2.16) p = p^ {i-^M^h^h;; 7i^\

(For a detailed derivation see Section 12).

It is to be noted that the siirfaces ^ = const.

and y} â– const, are the potential and stream s\irfaces

respectively. Thus the two equations (2.11) and (2.12)

yield parametric representations for potential and

stream surfaces. If q ( ^ ) is chosen, and then h ( ^ )

and Mq( I ) are determined from (2.05) and (1.08), these

stream and potential surfaces are easily drawn. Each

stream surface may serve as nozzle contour.

2

If the terms involving 07 were neglected one

would obtain x = ^ '^"^o^^^''?' ory = h(x)07 . This

-13-

relation Is identical with the result of the hydraulic

theoryj indeed, since x = corresponds to the throat

2 2

and h (0) = 1 we would have A/A^^ = (y/y..^) = ^q^^^'

It Is thus clear that the formulas above represent a

refinement of the hydraulic theory.

The relation (2.13) for q yields an improvement

over the hydraulic assumption that the speed is constant

on the potential surfaces. To form an idea about the

magnitude of the deviation from constancy we may intro-

duce the radius of curvature R of the stream contour

*>) = const. R is obtained approximately from

y = h^(^ )] , X = ^ , as

R = l/hj'oi.

Therefore, relation (2.13) can be written in the approximate

form

(2.17)

q = q.

,(i*H)

This formula shows that the deviation of the speed from

that at the axis depends on the ratio of the width of the

nozzle to the radius of cxirvature of the nozzle contour.

Unless the end section of the nozzle curves inward

as would be the case for h^( ^ ) < 0, the speed Increases

If' ii f" ^

3

idOirfLiJ

^^^^^^:^^mÂ»H^-(^\ |0

-14-

and thus, on moving away from the axis, the

pressure decreases along the potential sturfaces*

This is equivalent to saying that the lines of

constant speed and hence the lines of constant

pressure bend backwards as shov/n in Table I,

Our formulas thus indicate that the pressure along

the nozzle wall assumes ^Blues that are less than

those calculated from the hydraulic theory * This

behavior could be expected since a widely divergent

opening will give the gas an opportunity for quick

expansion.

3. Critical Remarks

Before deriving formulas for the thrust and

before considering exanples, we shall discuss several

possible objections to the preceding method.

On mathematical grounds it may be considered

improper to prescribe a quantity such as q (x) along

the axis, as we did, whereas the proper procedure

would have been to prescribe the nozzle contovir and

the state in the chamber, in agreement with what is

prescribed in reality. A way of prescribing data

mathematically is said to be proper if for such data

there exists a unique solution, which depends continuously

^^mmm-kmil^

^^mmmm^ii .^^

i'L,U

-15-

on these da taÂ» 'Prescribing the quantity qQ(^) on

the axis is indeed not proper in this sense. It is

true, if q (x) is an analytic function, a unique

solution of the differential equations exists in the

neighborhood of the axis but one does not know how far

this solution extends without developing singularities.

It is also known that if

L'

Copy isin /7S

AMP Report 82.1 R

AMG-NYU NO. 43

THEORETICAL STUDIES

ON THE

FLOW THROUGH NOZZLES

AND RELATED PROBLEMS

WITH THE APPROVAL OF THE OFFICE OF THE

CHAIRMAN OF THE NATIONAL D?FÂ£N. E f?^'S^A"CH

COMMITTEE. TH'S REPORT HAS ^J-Lri OECLA.-Sif lED

BY THE OFFICE OF SCIENTIFIC ftEjtARCH AND

DFv^LLOi-ivt^NT.

A Report Submitted

by the

NEW YoKi-., i_rilVlKiUY

INSTITUTE C- - â– - - â– -Tc.^ SCIENCES

25 Waverlv P(acp, New York 3 iSl V

Applied Mathematics Group, New York University - - -

to the

Applied Mathematics Panel

National Defense Research Committee

This document contains information affecting the

national defense of the United States within the meaning

of the Espionage Act, U. S. C. 50; 31 and 32. Its trans-

mission or the revelation of its contents in any manner

to an unauthorized person is prohibited by law.

April 1944

' ,1

Â«OUH li

Ui^lfLnuOlflLL

Copy No. ^"^^

AMP Report 82. 1 R

AMG-NYU No. 43

THEORETICAL STUDIES

ON THE

FLOW THROUGH NOZZLES

AND RELATED PROBLEMS

WITH THE APPROVAL OF THE OFFICE OF THE

CHAIRMAN OF THE NATIONAL Of FEN E RIIsrA^CH

COMMITTEE, THiS REPORT HAS fuLtl OECLA-SlfltD

BY THE OFFICE OF SCIENTIFIC fVESEARCH AND

DEVELOPMENT.

A Report Submitted

by the

NEW YORK UNIVERSITY

INSTITUTE OF MATHEMATICAL SCIENCES

L!BR.M;y

25 WaveHy Place, New York 3 N Yi

Applied Mathematics Group, New York University

to the

Applied Mathematics Panel

National Defense Research Committee

This document contains information affecting the

national defense of the United States within the meaning

of the Espionage Act, U. S. C. TO; Jl and 32. Its trans-

mission or the revelation of its contents in any manner

to an unauthorized person u prohibited by law.

0-)

April 1944

M

issiÂ«'i*' pijff^i0'^''-'^*-":i:r.

t'W

No. of

Copies

DISTRIBUTION LIST

AMP Report 82. IR

24 Office of the Executive Secretary

6 Liaison Office, OSRD

1 Att: B. S. Smith, British Admiralty Delegation

4 Att: N.A.C.A.

1 â€” -TÂ» Theodorsen

1 HÂ» WÂ« Emmons

1 Att: R. PÂ« Praser, Imperial College of Science

3 Richard Tolman, Vice Chairman, NDRC

1 Att: H. Bethe

1 Att: DÂ» A. Flanders

7 Bureau of Aeronautics, Navy

2 Att: Lt. Comdr. J. S. Warfel

1 Att: EÂ« S, Roberts

3 Bureau of Ordnance, Navy

1 Att: R. S. Burlngton

1 Att: R, J. Seeger

1 Naval Torpedo Station

Att: Coi^iraander J, M, Robinson

4 Office of the Chief of Ordnance

1 Att: Capt. C, M. Hudson

1 Att: H. M, Morse

1 Att: Colonel G. W. Trlchel

4 Aberdeen Proving Ground, Ordnance Research Center

1 Att: HÂ« Lewy

1 Att: D. L, Webster

1 Att: 0. Veblen

6 F. L. Hovde, Chief, Division 3, NDRC

1 Att: C. N. Hickman

1 Att: CÂ» C. Laurltsen

1 Att: E. C. Watson

1 Att: J. Barkley Rosser

1 A. Ellett, Chief, Division 4, NDRC

1 H. B. Richmond, Chief, Division 5, NDRC

ONCUSSIFIED

1st, AMP Report 82. IR (continued)

No. of

Copies

1

J.

T, Tate, Chief, Dlvislo:

Att: L. Dunn

1

w.

Weaver

1

T.

C. Fry

3

R.

Courant

1 Att: K. 0, Priedrlchs

1

E.

J, Moulton

1

s.

S. Wilks

1

J.

G. Kirkwood

1

J.

von Neumann

1

A.

H, Taub

1

H.

Weyl

1

M.

Rees

â€¢-â– vV

u* *-

I

TABLE OP CONTENTS

Page

Summary ^

Introduction 1

PART I.

Isentropic Plow Through Nozzles. 4

!â€¢ Basic Relations and Hydraulic Treatment 4

2. Refined Treatment 8

3. Critical Remarks 14

4. Remarks on Jet Detachment 18

5. Thrust 21

6* Ezaznples 30

PART II.

On Perfect Ejchaust Nozzles and Compressors 36

?â€¢ On Perfect Nozzles 36

8. CoDipressor Flow 42

9Â« The Stability of the Isentropic Compressor

Plow 46

lOo The Plow at the Compressor Entrance 52

11Â» Remarks on the Drag of a Projectile

Carrying a Conipresaor 57

Appendix*

12Â» Mathematical Details to Part I Concerning

Isentropic Plow Through Nozzles 64

II i^mm

Page

13Â» Construction of a Perfect Three-dimensional

Plow from any Expanding Plow with Axial

Syminetry 70

References 84

Additional Literature on Plow Through Nozales 86

Further Literature on Cori5)resslble Fluid Flow 88

-I-

SUMl^IARY

The present report sxxmmarlzes the results of

a study which the New York University Group of the

Applied Mathematics Panel has undertaken upon the

request of the Bureau of Aeronautics, Navy Department

(Project No. llA.j;167), and which was carried out under

the responsibility of Professor K.O.Priedrichs with

the assistance of Dr. Chas. R. DePrima and other mem-

bers of the group. Our group is greatly indebted to

Mr. E. S. Roberts of the American Cyanamid Corr5)any

for his stimulating advice.

The original request was for assistance in the

design of unconventional nozzles for rocket motors,

with a view toward attaining shortness while preserving

high efficiency. Since the opening angle of such

nozzles is necessarily wide and the expansion rapid,

it was necessary to improve the classical (hydraulic)

theory of the de Laval nozzle and to widen the scope

of the investigation further by consideration of gas

dynamical phenomena more generally. Therefore, the

present report is not concerned solely with the original

problem of exhaust nozzles, but it contains methods

and results of potential interest for other problems j in

particular, attention is given to the design of "kinetic

compressors", i.e., of nozzles serving, not as means

m

..... .^LU

i^^itjf

^*- ^ â€¢ -II-

for ejecting a supersonic Jet, but for receiving a

parallel flow of gas at high supersonic speed in order

to compress and arrest the incoming gasÂ»

More specifically, the contents of the present

report can be summarized as follows;

Part I contains a three-dimensional treatment

of flow through nozzles as a refinement of the customary

one -dimensional theory, under the following ass\iraptions:

1) The fluid is ideal and homogeneous, and the flow

isentropic, steady and irrotational. 2) Viscosity and

heat conduction are ignored. 3) It is further assumed

that shocks and jet detachment do not occur in the

nozzle; however, conditions for the absence of these

phenomena are analyzed*

A simple formula for the thrust produced by such

a flow through a nozzle is derived and applied to various

types of contours; particular emphasis is given to widely

divergent short nozzles. Among families of nozzles yield-

ing the same thrust, the shortest nozzle is determined.

Tables I to III at the end of Part I (p. 35) show numeri-

cal results in a condensed form.

Pa rt II concerns, first, the construction of "perfect"

exhaust nozzles which yield maximum thrust and a parallel

;\ V- .. K

jj;;7^:iiMBÂ»Â»^';*(i-â€¢,

^^^

.â™¦

\^V

-5-

heat conduction are ignored. This appears to be justi-

fied for the exhaust flow (Cf. T. Stanton [ 1] )^''""^ except

that viscosity will play a role in determining where Jet

detachment and shocks occur. The formulas to be given

are valid as long as the flow remains free of shocks

and as long as the jet remains attached to the nozzle

wall*

The isentropic character of the- gas flow is ex-

pressed by the relation

(1.01)

p ^ = const.

p being the pressure, x> the density (mass per unit volume)

and f the adiabatic exponent. The sound velocity

(1.02)

= ( f p ^ -1)V2

is connected with the flow speed q through Bernoulli's

law which we write in the form

(1.03)

l-^)c +^q =c^=q^

* Numbers in brackets refer to the bibliography at the

end of this memorandum.

Â»WU.'l'Â«Â»!lfc'

â– a / K '^

â€¢r t. - V ^ i \

?^. V i I i

â– JU2S ILL

with the abbreviation

^l^V

ft id "^

>^ ^ + 1

The significance of the "critical" speed q^. = c,^

Is that the f lov? speed Is critical when it coincides

with the local sound speed* Subsonic and supersonic

flow can then be distinguished by q < q^ and q > q^

since, as Is easily seen, q > q^ implies q > c â€¢

With the aid of Bernoulli's law one can express

the quantities c, p,/o in terms of q/q^. ;

using the abbreviation

V =

^ - 1

(1.04)

(1.05)

(1.06)

V

1 - /x2(q/q,,)^

Â» (Â°.^. = M^(q/q*)^

â€¢

e*

n

_ ..2

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The "one-dir.ensional" or hydraulic theory of

0Â» Reynolds (1885) assumes, as a first approximation,

that the flow speed q and hence also c,p,^ are

constant over cross -sections perpendicular to the

nozzle axis. The mass flux per unit time across a

cross-section of area A is then given by A /o q; hence

this quantity is a constant, or

A Aq

(m = q/c being the Mach number) from which follows the

hydraulic approximation.

(1.08)

M = (M^ - 1) ifl .

A q

Ml

s

IfUV

-8-

This relation shows that a flow with Increasing

speed q is possible for q < q^,. or q < c only

when A decreases, and for q > q^,. or q > c only

when A increases. In other words, in the hydraulic

approximation, flow with increasing speed is possible

only when the critical speed is just reached at the

throat.

2. Refined Treatment

In order to refine the hydraulic treatment just

explained it is necessary to employ the basic partial

differential equations for an irrotational isentropic

flow. We introduce coordinates; x, along the axis in

the exhaust direction, and y, the distance from this

axis; further we introduce the angle 6 of the flow

direction with respect to the x-axis, the potential

function ^ and the stream function i/r â€¢

The stream function Y ^^ to be so normed that

2

tr y is the rate of mass flow per unit time carried by

the stream tube yr < const. The basic equations for the

four dependent variables ^, ^^ , q, and 6 can then

be written in the form

(2,01) ^ = q cose ,^ = q sinG

w* r-

\i

-9-

(2.02) -W^^ = - /oqy ^^^^ Â» '^^^ = ^^^ Â°**^Â® â€¢

The density p Is to be expressed in terms of q

by means of relations (1.06) and (1.04).

It would be natural to prescribe nozzle contour

and state In the chamber and to ask for the resulting

flow. Mathematically this would mean solving a boundary

value problem for equations (2.01) and (2.02). To sim-

plify the task we reverse the procedure: We first pre-

scribe the velocity distribution along the axis. -

(2.03) q = %M f or y =

and then determine possible nozzle contours from the re-

sulting stream surfaces. To this end we Introduce the

stream functions ^ and ^f/ as Independent variables in-

stead of X and y, consider the quantities tl, y, Â© ,

and q as dependent variables, and develop them with

respect to powers of yi*^ It is sufficient here to report

* One may also Introduce q and V as independent vari-

ables: this might seem to offer advantages since the non-

linearity would then refer only to the independent variable

a. Nevertheless it was found that the present scheme is

better, at least for the expansions up to the second order

as given here.

tjsiKj^'^- *â€¢'Â»'â€” :-?jÂ»

-10-

only the results. Details will be carried out in the

Appendix, Section 12.

To fonaulate the results we introduce the impor-

tant dimensionless quantity h by

(2.04)

h = ^f^^q^TJq.

the significance of which is that h = A/A^,_ in hydrau-

lic approximation. Since by (1.06) the density a is

a given function of q the same is true for hj we have

h = 1 for q = q.,.. Frora (1.07) we obtain

(2.05)

2 dh/h = (M^^ - 1) dq/q .

On the axis, where we had assximed q = q (x) the quantity

h becomes also a function of x

(2.06) h = h^{x) = (q,,yq)

1/2

1 - }-^Wi.,f

1 -

^'

-2

[Cf. (2.04), (1.06), (1.04)] which is given inasmuch as

q.QM is given. Also M, defined by (1.08), becomes a

given function M^(x) along the axis when q = q (x) is

inserted.

Wh

Li

-11-

We place the origin x = 0, y = 0, at the

throat; more precisely, we place it such that

(2.07)

^o^O) = q,

or what is the same thing, such that l^'(O) = 0. We

then have

(2.08)

h^(0) = 1, and M (0) = 1 .

Instead of and '\^ , two other parameters, |

and 07 , are introduced by the relations

(2.09)

o

(2.10)

^= /e^v

Clearly, ^ and Oj have the dimension of a length and

\ reduces to x on the axis V = 0,(ji5 = for x = y =

being assumed)*

In terms of the quantities just introduced, the

expansions of x, y, q, 6 , h and p with respect to powers

of >? (instead of y ) are. If h^( ^ ), h^( ^ ), hj( \ ),

Mq( I ), and Pq( | ) are written singly h^,h^,hJJ,MQ, and

p^ respectively:

JllUi

(2.11) :: = ^ -i Vi'^^

(2.12) y = h^77 1 1 + 1 [ (m2- 1) h^h;; - (h.)^],^^!

(2.13) q = q^ jl+^h^hj 07 ^j

(2.14) = hÂ»>j

(2.15) h = h^ I 1 + ^ (M^ - 1) n^h;; 7,2|

(2.16) p = p^ {i-^M^h^h;; 7i^\

(For a detailed derivation see Section 12).

It is to be noted that the siirfaces ^ = const.

and y} â– const, are the potential and stream s\irfaces

respectively. Thus the two equations (2.11) and (2.12)

yield parametric representations for potential and

stream surfaces. If q ( ^ ) is chosen, and then h ( ^ )

and Mq( I ) are determined from (2.05) and (1.08), these

stream and potential surfaces are easily drawn. Each

stream surface may serve as nozzle contour.

2

If the terms involving 07 were neglected one

would obtain x = ^ '^"^o^^^''?' ory = h(x)07 . This

-13-

relation Is identical with the result of the hydraulic

theoryj indeed, since x = corresponds to the throat

2 2

and h (0) = 1 we would have A/A^^ = (y/y..^) = ^q^^^'

It Is thus clear that the formulas above represent a

refinement of the hydraulic theory.

The relation (2.13) for q yields an improvement

over the hydraulic assumption that the speed is constant

on the potential surfaces. To form an idea about the

magnitude of the deviation from constancy we may intro-

duce the radius of curvature R of the stream contour

*>) = const. R is obtained approximately from

y = h^(^ )] , X = ^ , as

R = l/hj'oi.

Therefore, relation (2.13) can be written in the approximate

form

(2.17)

q = q.

,(i*H)

This formula shows that the deviation of the speed from

that at the axis depends on the ratio of the width of the

nozzle to the radius of cxirvature of the nozzle contour.

Unless the end section of the nozzle curves inward

as would be the case for h^( ^ ) < 0, the speed Increases

If' ii f" ^

3

idOirfLiJ

^^^^^^:^^mÂ»H^-(^\ |0

-14-

and thus, on moving away from the axis, the

pressure decreases along the potential sturfaces*

This is equivalent to saying that the lines of

constant speed and hence the lines of constant

pressure bend backwards as shov/n in Table I,

Our formulas thus indicate that the pressure along

the nozzle wall assumes ^Blues that are less than

those calculated from the hydraulic theory * This

behavior could be expected since a widely divergent

opening will give the gas an opportunity for quick

expansion.

3. Critical Remarks

Before deriving formulas for the thrust and

before considering exanples, we shall discuss several

possible objections to the preceding method.

On mathematical grounds it may be considered

improper to prescribe a quantity such as q (x) along

the axis, as we did, whereas the proper procedure

would have been to prescribe the nozzle contovir and

the state in the chamber, in agreement with what is

prescribed in reality. A way of prescribing data

mathematically is said to be proper if for such data

there exists a unique solution, which depends continuously

^^mmm-kmil^

^^mmmm^ii .^^

i'L,U

-15-

on these da taÂ» 'Prescribing the quantity qQ(^) on

the axis is indeed not proper in this sense. It is

true, if q (x) is an analytic function, a unique

solution of the differential equations exists in the

neighborhood of the axis but one does not know how far

this solution extends without developing singularities.

It is also known that if

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