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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?F£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- - ■ - - ■ -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




«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



issi«'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 — -T» Theodorsen

1 H» W« Emmons

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

3 Richard Tolman, Vice Chairman, NDRC
1 Att: H. Bethe

1 Att: D» A. Flanders

7 Bureau of Aeronautics, Navy

2 Att: Lt. Comdr. J. S. Warfel
1 Att: E« 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: H« Lewy
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1 Att: 0. Veblen

6 F. L. Hovde, Chief, Division 3, NDRC
1 Att: C. N. Hickman
1 Att: C» C. Laurltsen
1 Att: E. C. Watson
1 Att: J. Barkley Rosser

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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



•-■vV



u* *-





I



TABLE OP CONTENTS

Page

Summary ^

Introduction 1

PART I.

Isentropic Plow Through Nozzles. 4

!• 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

?• On Perfect Nozzles 36

8. CoDipressor Flow 42
9« The Stability of the Isentropic Compressor

Plow 46

lOo The Plow at the Compressor Entrance 52
11» Remarks on the Drag of a Projectile

Carrying a Conipresaor 57

Appendix*

12» Mathematical Details to Part I Concerning

Isentropic Plow Through Nozzles 64



II i^mm



Page



13» 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



^*- ^ • -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 gas»

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^:iiMB»»^';*(i-•,




^^^



.♦



\^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.



»WU.'l'«»!lfc'



■a / K '^



•r t. - V ^ i \

?^. V i I i






■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 •

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,,)^



» (°.^. = M^(q/q*)^





e*


n


_ ..2





V + 1



/^



.pS«iiei^««^



V^V'^-



.^'^



^



v-^-



lif^*



ubuv...

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The "one-dir.ensional" or hydraulic theory of
0» 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 •

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 ^^^^ » '^^^ = ^^^ °**^® •

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, © ,
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^'^- *•'»'— :-?j»







-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) = h»>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} ■ 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



^^^^^^:^^m»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 .^^



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-15-

on these da ta» '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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Online LibraryRichard CourantTheoretical studies on the flow through nozzles and related problems → online text (page 1 of 4)