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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.

â– 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

CONFIDENTIAL

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

CONFIDENTIAL

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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 â€” 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. RÂ«-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 â€” or simply the stroke â€” 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 â€” 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 5Â« 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 tâ€ž.

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 6Â« 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 tâ€ž, 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 Â« When the fuel ratio M /M

.^.Â»â€” â€” â€” â€” â€” â€” â€” â€” â€” â€” â€” â€” â€” â€” â€” 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

Â° /M

t tends to infinity like /Â«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* - Â°Â° and tends

w

as M /M -Â»â€¢ 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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9

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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 â€” 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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10

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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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11

CONFIDENTIAL

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.

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.

â– 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

CONFIDENTIAL

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

CONFIDENTIAL

CONFIDENTIAL

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

CONFIDENTIAL

CONFIDENTIAL

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 â€” 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. RÂ«-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 â€” or simply the stroke â€” 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 â€” 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 5Â« 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 tâ€ž.

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 6Â« 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 tâ€ž, 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 Â« When the fuel ratio M /M

.^.Â»â€” â€” â€” â€” â€” â€” â€” â€” â€” â€” â€” â€” â€” â€” â€” 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

Â° /M

t tends to infinity like /Â«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* - Â°Â° and tends

w

as M /M -Â»â€¢ 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 â€” 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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