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NEW YORK UNIVERSITY

IN ^ 1 OF MATHEMATICAL SCIENCES

LIBRARY

25 Waverly Place, New Yofk 3, N. Y.



AMP Report 137.1R
AMG-NYUNo. 128



THEORETICAL STUDIES CONCERNING THE HVDROPULSE

(IDEAL MECHANICAL PERFORMANCE CHARACTERISTICS)



WITH THE APPROVAL OF THE OFFICE OF THE

CHAIRMAN OF THE NATIONAL Dr.FcN' E RES'iARCH
COMMITTdE, JHiS REPORT IIAS SEEM DECLAcSIFIED
BY THE OFFICE OF SCIENTIFIC RESEARCH AND
DEVELOPMENT.



Prepared for the

APPLIED MATHEMATICS PANEL

NATIONAL DEFENSE RESEARCH COMMITTEE

By the

Applied Mathematics Group

New York University






X






This document contains Information
affecting the National Defense of the
United States within the meaning of the
Espionage Act, U. S. C. 50; 3! and 32.
Its transmission or the revelation of its
contents In any manner to an unauthorized
person is prohibited by law.



■eo-N-PRffliPPfrA t



aJ^^



^'



July 1945




Prepared in oormeotion
with Project NA-195
under Contract OEMsr-946



:AL



ApproTed for Distribution
l&rren Weaver
Chief, Applied Uathematios Panel



Distribution List
AMP Report 137.1R



Copy No <
1-7
8-34
35

S6

57 - 40



41 - 45



46 - 47



48-54



55 - 59



60 - 61



62



Office of Executive Secretary, OSRD

Liaison Office, OSRD

Office of Research and Inventions
Att: Lt. J. H. Wakelln

Commanding General, AAP
Attt T, von Karman

Chief, Bureau of Asronautica
1 Comdr. J. S. Warfel
1 Lt, F. Ao Parker
1 B, S, Roberts

Bureau of Aeronautics Representative in Los Angeles
Att: Aerojet Coirp,

U.S. Naval Engineering Experimental Station, Annapolis
1 Lt, Patton
1 Ensign L, B, EdeLnan

Chief, Bureau of Ordnance
1 R, S, Burington
1 R, J. Seeger

1 Comdr. Levering Smith, Re2d
1 Comdr. W, A, Walter, Re3
1 Capt, D, P, Tucker Re4g
1 Capt, E. U. Crouch R06

Conraanding General, AAF Materiel Coinnand, Weight Field
1 Lt, Col. P. F. Hay
1 Major J. P. AuWerter
1 Capt, J. Healy, Armament Laboratory
1 Lt, R, C, Bogert
1 Bell Aircraft Corp,

J, E, Jaokson, Chairman's Office, NDRC
1 Busblebee Project

Guided Missiles Committee, JCS



00.



"UL



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63 - 66 Colonel L. A, Skinner, Ordnance Department Representative, C.I.T,
1 C. Millikan
1 L, G, Dunn
1 J. V. Charyk

67 - 69 Office of the Chief of Ordnance
1 Colonel G. W. Trichel
1 Major J, F, Miller, Jr.
1 H. M. Morse

70 Commanding General, AAF Proring Ground Conanand, Eglin Field, Fla,

Att: Brig, Gen, Grandison Gardner

71 - 72 Ordnance Research Center, Aberdeen Proving Ground

1 Technical Library
1 R. J. Walker

73 - 74 F. L, Hovde, Chief, Division 3
IE. C, Watson
1 J, B, Rosser

75 J, T, Tate, Chief, Division 6

76 W. R, Kimer, Chief, Division 9

Atti J, W, Williams

77 W, Weaver, Chief, Applied Mathematics Panel

78 T, C, Fry, Acting Chief, Applied Mathematics Panel

79 - 80 R, Courant '^

81 0. Veblen

82 Q. C. Evans
85 S. 3. Wilks
84 M. Rees



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PREFACE



The present report, by the New York University
Group of the Applied Mathematics Panel, was tindertaken
in connection with Project No. NA-195 (Bxireau of
Aeronautics, Navy Department). It was carried out,
under the general guidance and editorship of Dr. J. J. Stoker,
by M. Shifftnan, D. C. Spencer, B. Friedman, E. Bromberg
and E. Isaacson in consultation with Dr. P. Zwlcky of the
Aerojet Corporation.

The report contains a mathematical analysis of a
variety of problems concerning the mechanics of the exhaust
stroke in a hydropulse motor. A broad and basic approach
to the problems was motivated to a considerable extent by
the desire to avoid too specific assxmiptlons with regard
to ^he engineering aspects of the hydropulse development.



R. Courant

Director of Research

Contract 0EM3r-945



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TABLE OP CONTENTS

Section 1) Introduction and sinnraary p. 1

2) Assvmiptlons leading to the hydraulic theory... p. 16

3) Formulation of the hydraulic theory p. 18

4) Dimenslonless variables and parameters, p. 28

5) Duct of variable cross-section. Approximate
formulas. Shape of duct for maxiravira impulse,. p. 30

6) The tube of constant cross-section. Formulas
showing dependence of essential quantities on

all parameters p. 41

7) The tube of constant cross-section. Perfor-
mance curves, with discussion p. 49

8) Tubes with conical portions p, 95

9) Straight tube with mass and spring for

elastic energy storage p. 100

10) Spring effect obtained from a layer of gas.... p. 115

11) The valveless hydropulse p. 137

12) Scavenging p. 149

13) Effect of finely distributed gas bubbles p. 159

Appendix I The method of finite differences applied to
the ntmerlcal solution of hydropulse differ-
ential equations p. 163

II Special method for calculation of v(t) p. 171

All graphs associated with a section are to be
fotuid immediately following that section.



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Ideal Mochanlcal Performance Gharaeterlatlea

of the qydropulae
Section lo Introduction and aummary .

In this report ^e are interested in certain aspects
of the problems which arise in dealing with jet propulsion
devices to be used in water — in other words, with the
counterparts in water of the jet devices in air, such as
rockets, the athodyd, the buzz bomb motor, etc. The
manner in which such devices function can be described in
a general way as follows: The propulsive element is a duct



V




TUBE



Valves



Figure 1.1

open at both ends (as indicated la Figure 1.1). The duct
is assumed to be traveling through the water with a certain
velocity V (presumably it is propelling a ship, a torpedo,
or other craft). At a certain Instant a chemical is
Injected into the duct which reacts with the water to produce
a volume of gas, as indicated In the flgvire. The gas, on
expanding. Imparts momentum to the water and a thrust in tha
forward direction will result if, for example, the part of



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the duct in front of the gas la sealed off by closing valves.
If the forward velocity is high enough and if the intake
part of the duct is a properly designed expanding nozzle,
valves may not be needed to insure a forward thrust and the
fuel could be injected continuously. Such a device would
correspond to the athodyd in air. However, at moderate
velocities V, it is likely that valves would be necessary
for efficient operation of such a duct device. Of course,
sustained operation of the device with valves requires
the valves to open at certain times so that the expansion
chamber can be refilled with water after the gas has expanded
and pushed a predetermined amount of water out of the tube.
Once the "expansion chamber" has been refilled, the valves
would be closed, fuel would be injected, and the cycle
would repeat.

In what follows we assume that valves will be provided,
but we do not assxime that we know when and how they function -
whether they open automatically when the gas pressure reaches
a certain value (the hydro-resonator), or are controlled by
some Independent mechanism ( hydropulae ) . Our investigations
have therefore been confined for the most part to those
phenomena which can be treated without knowing in detail



This terminology we have taken from the report of P.
Zwicky: Remarks on the Basic Theory of the Qydropulse as
a Propulsive Power Generator, Aerojet Report No. R-41
(Conf.). In fact, most of the work to be discussed In
what follows was undertaken at the suggestion of P. Zwicky
and in consultation with him. Two other reports of Interest
for the mechanical performance of the hydropulae have
appeared: 1) The Solution of the Differential Equation
for the hydropulae (Conf.), Aerojet Report No. R«-43, by
K. Poelsch and Knox T. Mlllsaps, 2) A Preliminary Study
of the 5ydropulse (Conf.), Jet Propulsion Note No. 19,
Bureau of Aeronautics Project, U.S. Naval Eng. Exp.
Station, Annapolis, Md., written by Ensign L. B. Sdelman.



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just how the complete cycle of operations will be carried out.
Our calculations should be helpful in making decisions on
this point. Roughly speaking, what we have done is the
following: We have ignored the part of the cycle concerned
with the refilling of the "expansion chamber", (except in
Section 12) and concentrated our attention on that part of
the cycle during which the gas is expanding and pushing water
out of the tube. We consider the motion up to tie time when
the gas has expanded down to the exit pressure, and define
as the time of stroke — or simply the stroke — the time
required for such an expansion. In carrying out the calcula-
tions certain simplifying assumptions are made which are
discussed in Section 2, immediately following this intro-
duction. Briefly speaking, these assumptions lead to a
"hydra laic theory", in which the water is treated as
incompressible and the velocity distribution over any cross-
section of the tube is replaced by the average over the
section. In addition, the gas is assumed to be uniform in
state throughout its volume at each instant of time, and to
expand adiabatically in such a way that the interface between
gas and water is at all times a plane surface at right angles
to the axis of the tube. The pressure at the exit of the
tube is assumed to be constant throughout the stroke. On the
basis of these assumptions we study the motion of the water
in the tube, the impulse delivered over the stroke, and the
time of stroke — that is, we study the mechanics of the
hydropulse operation, but leave aside the chemical and thermo-
dynamic problems concerned with the production of the gas*
In order to calculate efficiencies we simply assume that a
certain mass of a perfect gas is present. These assumptions
which seem rather reasonable for the purpose of a preliminary
attack, should yield the ideal mechanical performance data
for devices of this type.

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Wo outline the contents of this report section by
section together with the main conclusions reached In eachs

(Section 2* Contents discussed above.)

Section 5« Formulation of the hydraulic theory * The equation
of motion of the water during a single stroke Is derived from
the relation for the energy balance. The formula for the
Impulse Is also obtained. (This is essentially the same as
the theoiT" derived in the previously quoted report of P.
Zwicky. ) The theory leads to an initial value problem for
a non-linear ordinary differential equation of second order.

Section 4. Introduction of dimensionlesa variables and
parameters . The essential variables and parameters are
expressed in dimenslonless form. In all cases Involving thQ
determination of quantities as functions of the time through
actual integration of the equations of motion (as in part of
Section 7 and in Sections 9 to 12 incl.) these variables ai^
very convenient.

Section 5. Duct of variable cross-section i Approximate
formulas. Shape of duct for maximum impulse . Although the
hydraulic theory as developed in Section 3 is quite elementary
from a mathematical point of view, the amount of calculation
necessary for a complete discussion of the motion of the
water, the impulse I per stroke, the ratio of impulse jjer
stroke to mass of fuel injected per stroke (the "effective
velocity" or "specific Impulse" Ig/^g)# the time of stroke t ,
and the average thrust F = I_/t_ in their dependence on all
of the essential parameters would be truly enormous. The
reason for this is that the equation of motion can not be
Integrated explicitly, so that numerical Integration methods
must be used. However, it turns out that the quantities just
now enumerated above, which are the most important ones for



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practical purposes, can be expressed by approximate

formulas which do not involve a complete integration of the

equation of motion. Section 5 contains a derivation of these

formulas, together with proofs that the formulas yield

rigorous upper bounds for I_ and f and a rigorous lower

s s

bound for t„.
s

By comparison with the results obtained through

accurate numerical Integration of the equation of. motion

in a large number of cases the formulas are known to furnish

results which are accurate within 2 per cent for I and

a
5-15 per cent for t^ as long as the initial pressure ratio

Is greater than 10 and for ducts which do not depart in
shape too much from that of the straight tube of uniform
cross-section.

Section 5 also contains a discussion of the question
of maximizing the Impulse with respect to changes in shape
of the hydropulse tube. In case the forward velocity of
the hydropulse is small, it Is not difficult to show that
the straight cylindrical tube Is as good as any other for
this purpose. This conclusion follows from the fact that
our approximate formula for I yields an upper bound for all
shapes of tubes and the fact (determined by actual calcula-
tion) that the upper bound is attained within a per cent or
two In the case of the straight tube. Bowever, If the
forward speed of the hydropulse is high, so that the velocity
of the water in the tube Is high at the beginning of the
stroke, the Impulse per stroke I^ Is affected somewhat by
changes In the shape of the tube. We have not obtained a
final result In these cases, but some conclusions from the
present section, combined with a few numerical results
from Section 8 on tubes with conical sections. Indicate that
the best shape of tube for maximum Impulse during the power
stroke at a given forward speed would be one with a flared-out



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forward part joined to a straight cylinder extending to the
exit, as Indicated In the accompanying sketch.

Dt recti on or motion



Vol



vea



Section 6« The tube of constant cross -section: Formulas
showing dependence of essential quantities on all parameters *
The main bulk of our calculations concern the tube of constant
cross-section. Our objective is a complete discussion of the
behavior of all Important quantities when any or all parameters
are varied. We have already stated that approximate formulas
are available for tubes of any shape. These formulas are
developed explicitly in Section 6 for the special case of
the straight tube In such a way as to make a complete dis -
cussion possible for the full ranges of all essential
parameters . In particular, formulas for the Impulse I ,
the effective velocity Ig/*^-* (which is a measure of the
efficiency of the device), the time of stroke t„, and the
average thrust P = I_/t_ are obtained. These formulas
exhibit explicitly the dependence on the following parameters:
1) the Initial pressure ratio P^/Pq* 2) the ratio M yM of
the mass of gas M Injected per stroke to the initial mass
JL of water in the tube, 3) the initial velocity U. of the
water in the tube, 4) the adlabatle constant 7,5) the

The initial velocity TJ^ of the water in the tube would be
approximately equal to the forward velocity of the hydro-
pulse.

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Initial temperature and molecular weight of the gas.

At the end of Section 6 graphs and a table are given
which compare the results obtained by the approximate
formulas with those obtained by accurate numerical integration
of the equation of motion. The approximate formulas are
found to be amply accurate for our purposes.

Section 7> The tube of constant cross-section. Performance
curves, with discussion . This section is perhaps the most
important one in the report. The behavior of the essential
quantities with respect to variations of the parameters is
given in the form of graphs placed at the end of the section.
A brief summary of the main results follows:

a) The effective velocity Is^g, or ratio of impulse
per stroke to mass of fuel injected per stroke . The effective
velocity increases quite slowly once the initial pressure
ratio Pi/Pq becomes greater than about 30, particularly for
the higher initial velocities. Thus there would be no great
advantage from the point of view of fuel conservation in
going to pressure ratios much above 50 . The effective
velocity decreases steadily to zero as M increases steadily
to infinity. On the other hand, as M tends to zero, the
effective velocity increases steadily and tends to a definite
limit which is finite if the initial velocity JJ^ is not zero,
but infinite if U. is zero. It is therefore clear that
efficient operation is promoted by keeping the amount of fuel
injected per stroke as low as that which is compatible with
the attainment of other objectives . It is clear that M must
be large enough so that a sufficiently high thrust can be
attained. As the initial velocity TJ^ increases, Ig/^g decreases
and has the limit zero. This is to be expected, since the
thrust augmentation is lowered with increase in the initial
velocity of the water.

The actual numerical values for the effective velocity

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in practically reasonable cases are of the order of several
hundred thousand feet per second. These are high values
compared with those achieved with jet devices In air, the
difference being due to the very great density of water as
compared with air.

b) The time of stroke tsJ The time of stroke is not
very sensitive to changes in the initial pressure ratio P^/Pe

if this ra tio exceeds 50 or 40 « When the fuel ratio M /M

.^.»— — — — — — — — — — — — — — — g/ w

alone is varied the following behavior of t_ is found j As

s

M M tends to zero so als o do es the time of stroke tg, while

° /M

t tends to infinity like /«S. when M /M tends to infinity.

S V 'vj o

That t should tend to zero with M seems at first sight
s g

rather strange, but it is readily explained by the fact that

the initial pressure ratio is assumed to be held fixed while

M tends to zero. This would be difficult to achieve in
g
actual practice.

c) The average thrust P^ = 13/tg. The average thrust

increases steadily with increase in the initial pressure

ratio, as we would expect. When the fuel ratio M /M^ alone

is varied, the average thrust P_ behaves as follows: P

M

decreases steadily from a finite value for jp* - °° and tends

w

as M /M -»• to another finite value which is half the value
g/ w

for M M. - 00 , Again this somewhat paradoxical behavior of

F. as the quantity of fuel M injected per stroke approaches
s g

zero is explained by the fact that the Initial pressure ratio

is assumed to be maintained as M^ -^ 0, and this could not be

g
achieved in practice. Nevertheless, in view of the fact that

the efficiency as measured by the effective velocity I./^I-

has its maximum for M yM = 0, the following important

conclusion is strongly indicated: The hydropulse should be

operated at high frequency with injection of small amounts



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of fuel per stroke. The quantity of fuel In.jected pr stroke
should be kept as lov? as Is compatible with maintenance of a
minimum Initial pressure ratio (of the order of 30 or 40 If
possible) 30 that the average thrust will be maintained at
the value necessary to overcome the drag on the hydropulse *

d) Gas pressure and exit velocity of the water aa
functions of the time . In this part of Section 7 we report

a few of our calculations on the behavior of the gas pressiire
and velocity of the water as functions of the tlmfe . These
results are of Interest In themselves. They also lead to the
development of the approximate formulas. The principal ob-
servation to be made is that the gas pressure decreases very
rapidly Initially, while the velocity of the water Increases
very rapidly at the beginning. Both quantities then remain
nearly constant after the first short interval of time at the
beginning of the stroke. A number of graphs exhibiting this
behavior in a few typical cases are given at the end of
Section 7. In Sections 9 and 10 two possible means of smooth-
ing out these initial abrupt variations in pressure and in
velocity of the water are discussed.

e) Effect of linear buildup (in time) of the gaa
pressure . In all but this portion of Section 7 we assume
that the initial high pressure of the gas la created instan-
taneously. Actually the gas pressure would probably be
built up gradually to its maximum after injection of the fuel.
In what manner the pressure would rise in actual practice

is not known — it would depend on the chemical reactions and
the thermodynamic a of the process. In this part of Section 7
we give the results of a few calculations assuming the gaa
pressure to rise linearly in time to its maximiun, after which
the gaa expand a adiabatically down to exit pre a sure (as in
all of our other calculationa ). For the caae of zero initial
velocity of the water in the tube curvea are given from which



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the gas volume and rate of expansion are determined for any
given rate of increase in pressure. The time required for
the pressure to reach its maximum was taken as about 30 per
cent of the entire stroke in one numerical case and 40 per
cent in another. The time of stroke and impulse per stroke
were computed in a number of cases and the results compared
with cases in which the gas pressure reaches its maximtim
instantaneously. The results, though inconclusive, are
interesting. In these few cases we find a gradual buildup
in pressure to be favorable for the operation of the hydro -
pulse, since the impulse delivered per stroke for a given
quantity of fuel appears to be substantially larger than in
the case of instantaneous rise of pressure. (However, our
method of comparing the quantities of fuel may be open to
serious objection on physical grounds. For details, see
Section 7e).

If the last conclusion were to be found generally
valid, it would mean that our efficiencies (which were all
computed on the assumption of instantaneous pressure rise)
are on the conservative side.

Section 8. Tubes with conical portions . The results of
calculations giving the Impulse and time of stroke for three
different types of tubes having conical portions In front
are presented and compared with those for the straight
cylindrical tube. The purpose and the main results of this
section have already been stated in the course of our remarks
above referring to Section 5.

Section 9. Straight tube with mass and spring for elastic
energy storage . One of the results of the calculations of
Section 7 is that the gas pressure (and consequently the
thrust) and the velocity of the water in the tube change very
rapidly at the beginning of the exhaust stroke. In order to
promote efficiency it seems likely that a more even distribu-

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tion of th9 thrust and of the exit velocity of the water
would be desirable. One possible way to do this would be to
provide for an elastic storage of energy during the early
part of the stroke which would then be delivered again in the
later part of the stroke. Just what means should be taken
to accomplish this in practice is far from clear. What we
have done is to investigate a mechanical model in order to
find out if possible what can be accomplished by such means.
The model Is obtained by assuming that the left end of the
exhaust chamber is closed off by a piston of a certain mass
which is attached to a spring (instead of a rigid wall at the
left end.) The calculations were carried out for various
values of the mass. It turns out that the time distribution
of the thrust and the velocity of the water can be very
considerably influenced through proper choice of a mass and
spring. In particular, the abrupt initial variations in these
quantities can be smoothed out rather successfully. (It
might be noted that a spring acting alone without a mass is
not effective. )

It is found that the operation of the hydropulse might
be rather erratic for certain values of the parameters since
the time of stroke may be discontinuous for special values
of the parameters.

Section 10. Spring effect obtained from a layer of gas . One
method of approach to the problem of providing elastic
storage of energy during part of the stroke might be to inject
the fuel in such, a way as to create two gas volumes separated
by a mass of water. Various cases of this kind are treated
in Section 10. The number of parameters which could be varied
in these eases is rather large, and our calculations cover
only a small number of the possibilities. However, they do
show that such a method could be used to modify to some extent
the time distribution of thrust and velocity.


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Online LibraryJames Johnston StokerTheoretical studies concerning the hydropulse → online text (page 1 of 7)