25 Waverly Pbc«, N«w Yorit S, KY}

NEW YORK UNIVERSITY

INSTITUTE OF

MATHEMATICAL SCIENCES

IMM-NYU 233

JULY 1956

Underwater Explosion Bubbles IV. Summary

of Results and Numerical Computations.

IGNACE I. KOLODNER

AFSWP-101 5

PREPARED UNDER

CONTRACT No. Nonr-285(02)

WITH THE

OFFICE OF NAVAL RESEARCH

REPROi

IS PERMITTED F

0£ IHE UNITED STATED GOVtKiSMiiNT.

IMM-NYU 233

July 1956

UNDERWATER EXPLOSION BUBBLES IV. SUI^MARY

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OF RESULTS AND ITOI4ERICAL COMPUTATIONS.

by

lernace I. Kolodner

AFSWP-1015

This paper represents results obtained at the

Institute of Mathematical Sciences, New York

University, under the auspices of the Office

of Naval Research, Contract No. Nonr-285(02) .

New York, 1956

PREFACE

In this concluding report of a series of four on "Underwater

Explosion Bubbles" we present results of numerical computations for

specific underwater explosions. The work on these computations

began in 1953 and x^ras completed, except for various drawings, in 1955.

Vifialle the organization of the work and the responsibility for

its accuracy are assumed by the ^Inderslgned, actual work has been

largely carried out by others. Professor E. Isaacson was frequently/

consulted on the computational techniques. Miss K. Reissm.an,

Mrs. S. Hahn, L. D. Grey and Miss L. Wertheimer did all the computa-

tions. Drawings were prepared by Mrs. S. Hahn and Miss L. Wertheimer

and tables were extracted by J. Smith and Miss E. Kramer.

July 1956 Ignace I. Kolodner

TABLE OP CONTENTS

Par,e

I. Introduction 1

II. Matheraatical Formulation 2

III. Dirr.ensional Analysis h

IV. Pomulae for Bubble, Migration, and Potential

Coefficients 7

V. Choice of Paraneters and Corcputational Procedure 13

VI. Discussion of Results 17

Tables 1-6 19-36

Figures 1-6

References 37

-2-

has a fully developed cavity, and its shape indicates a possible

change into toroidal shape. However, it is expected in this case

that the bubble will perform several oscillations before chan/:.:ing

its topological structure. Altogether we found it difficult to use

the numerical results for more than one oscillation period. In

order to obtain a description of the bubble motion for two periods,

we had to reduce the value of the parameter o- to .00125, and this

corresponds to an explosion of 1.05 grama (I) of TUT at a depth of 125

ft.

II. Mathematical Formulation .

Our study is based on the following siraplifying assumptions.

1. The ocean water is incompressible and inviscid.

2. The water flow is laminar and irrotational.

3. The bubble gas has no internal motion and expands

adiabatically.

h. The bubble is initially spherical and has no radial

velocity,

5. The ocean bottom is a rigid surface.

6. The bubble is sufficiently far from the water surface

to allow the use of a linearized surface condition.

The object is to find the water velocity potential, i(x,T,z,t),

the migration of the center of gravity of the bubble, B(t), measurld ^

from the center of explosion and the equation of the bubble surface,

r = R(e,t), The equation for the bubble surface has been written in

spherical coordinates with origin at the moving center of gravity

,.:' brjK ^-.^cliv '

•r r

.1

■.v-'i

jji ■-

.' J

t'loil'

-3-

of the bubble, and the polar axis pointing upwards.

Under the above assiomptions, ^ , B, and R are found to satisfy

and to be determined by the following conditions:

(2.1) A 2 = * in water,

(2.2) VI'Vp+P^"^' ^°^ P = (kinematic bubble surface

condition) ,

(2.3) Pq - p[|. + |(VS)^3 - Pgz = KV"Y, for P = (dynamic

bubble surface condition),

(2.1+) 1, = * for 2 = - H (rigid bottom condition) ,

(2.5) 2 = * i'oi' z = z (linearized free surface condition),

(2.6) R(9,0) = Aq = constant,

(2.7) R^(0,0) = , f ^

^ / Initial

,^a\ ^,^\ ^,^^ ^ \ conditions / •

(2.8) B(0) = B(0) = , \ /

Here,

p • '4vvn. density of v/ater

p - atmospheric pressure

z - depth of explosion center

H - distance of the bottom from the explosion center

Pq = Pq + PSZq - hydrostatic pressure at the explosion center

'kr-

2 - B

P(x,y,z,t) = z - B(t) - R(e,t)cos 9 , cos G =

/x2+y2+(z_B)2

V = Y^ I R^(e,t)sin d9 = volune of the bubble.

o

Y - adiabatic exponent.

X - adiabatic gas constant

W - weight of the explosive

K = xv;^ .

A - Initial radius of the bubble,

o

When the effects of the boundaries can be neglected, the conditions

(2.1j-, 5) are repla^ced by a requirement of regularity of "^ . One

obtains the sane result by solving the problem with finite H and

z , and letting H — > co , z — s^ oo .

o ^ ' o

III. Dimensional Analysis .

The number of parameters appearing in the formulation is re-

duced by introducing dimenslonless units and parameters. It is

convenient to introduce for units of length L and time T the

following:

(3-1) ^ = {^0ff' .

(3-2) I = Ly^ •

o

Here,

-5-

(3.3) E = ^ A^tipAf. P„ -^-^(^2 A3,-r) .

is the total constant energy that the system water bubble would

have in the absence of boundaries. VJhen boundaries are present,

the total energy is still constant but different from E -by a terra

E . However, one shows that even in this case E = if the initial

t

radial velocity of the bubble, A = .

The units used here are different from the imits introduced

by Friedman [2], and adopted in the literature on Underwater

Explosion Bubbles. If L and T are the Friedman units, we have

the relations

I

For Y = 1.25 , a value generally accepted for explosion products,

see [1] ,

(34') L = .585 L ,

T = .716 T .

L is the equilibrium radius of a bubble in a system of energy E ,

while L has no exact physical meaning, though it is close to and

always somewhat less than the theoretical maximum radius of the

bubble. The choice of T is motivated by convenience.

On introducing dimensionless variables, four independent para-

meters appear, namely:

-6-

i3.B) X = A pI'^ e - ^ ,

y-l O

(3.6) CT'=^

2o '

(3.7) V = ^

(3.8) ti = ^ .

Here, Z = z + p /pg , is the hydrostatic depth ("head") of the

explosion center.

In a scaled experiment the values of x, cf, \jl, and v r.iust

be preserved. Using (3.3) in (3-5), and assioming A = , (see

assiomption 1|, Section II) one can show that

(3.9) X = x(l + x)"Y ,

where

(3.10) ^^pTTTT •

Here, P is the raaximun pressure of the bubble which depends only

on the type of explosive, but not on its anount. See also [7],

p. l\.. Using the sarae notation.

(3.11) E = P^V^Cl + x) .

-7-

Even for large values of z , the internal energy of the

bubble is large compared to its potential energy. This is equivalent

to the assumption x » 1 , whence the simplified approximate

formulae

{3.9') X- y? - -^ cC Z^-l

(3.10') E ~ P^V X = e W ,

where e is the specific energy of the bubble.

Formula (3 '9) shows that a scaled experiment with the saine

explosive is impossible. On the other hand, it is possible to pro-

duce scaled experiments with different types of explosives, pro-

vided that the pressure above the free surface is reduced. Denote

by prime the quantities referring to a scaled experiment. Let

f t t I

P = pP ,8 = qe , p = rp . Then the choice: p = pp ,

f -1 ' -1 » k -1 -3 .

z = pr z , H = pr H , and W = p^q r -'^W will preserve the

values of all the parameters, while X = pr" L , T = ^/pr" T .

See [5], p. 11.

IV. Formulae for Bubble, Migration, and Potential Coefficients .

We define ({)(r,9,t), X(9,t), b(t) by

(I;.l) B(t) = Lb(|)

(4.2) R(e,t) = LX(0, |)

2„-lr:/t^ z

(1^.3) |(x,y,z,t) = LV-'[b(^) I + 2S

3.01^25

3.0525

3.0615

3.0695

3.0765

3.0835

3.0885

3.0915

3.0935

3.0955

3.0970

3.0985

3.0995

3.1005

3.1015

3.1020

1.61

1.65

2.12

2.[^2

2.76

3.11+

3.59

[j-.ll

i|.67

5.26

5.87

6.51

1.1k

7.91

8.tb

9.3i|

1.02 (1)

1.08

1.15

1.25

1.3i|

1.53

1.63

1.76

1.91

2.10

2.33

2.60

3.00

3. ill

2>.lk

i|.00

1+.29

k.Sk

i-.82

5.01

5.20

5.i|0

5.50

2.26

2.S2

2.82

3.18

3.60

i|.12

1^.78

5.65

6.72

8.02

9.61

1.15 (1)

1.37

1.69

2.06

2.i|6

3.01

3.52

k'22

5.21

6.26

1.11

9.18

1.11 (2)

1.39

1.80

2.ij.0

3.27

6.82

9.62

1.20 (3)

1.39

1.59

1.7ii-

1.88

1.9i|

2.00

2.02

2.03

.111+

.195

.288

.396

.S22

.667

.838

I.OU

1.26

1.50

1.76

2.03

2.30

2.6i|

2.99

3.30

3.69

U.Ol

I+.39

i+.86

5.30

5.84

6.28

6.79

7.43

8.20

9.07

9.96

1.09

1.17

1.18

1.12

1.03

8.91 (0)

7.i|6

5.i|8

k'll

2.kS

.886

(1)

- .200

.111

'hlk

.89)4-

1.38

1.93

2.56

3.30

14..06

ij..86

5.67

6.IJ.8

1.21

8.20

9.10

9.90

1.08(1)1

1.16

1.25

1.36

l.it-^

1.57

1.66

1.78

1.91

2.09

2.30

2.85

3.28

3.73

i|.09

k.3>l

k.io

11.91

S.21

5.1+8

5.69

5.91

6.02

ibou:-tj.:f^:'iu) r

A i ! -

Y.

i

V .

3.1025

3.1035

3.ioi|.5

3.1055

3.1070

3.1085

3.1105

3.1125

3.1155

3.1205

3.1275

3.l3i4-5

3.li|25

3.1515

3. 1615

3.1715

3. 1815

3.1915

3.2065

3.2215

3.2i|l5

3.2615

3.2815

3.3115

3.3i;l5

3.3815

3.i;3i5

3.i|8l5

3. 51^15

3.6115

3.6915

3.7815

3.6815

3.9815

i|.08i5

I+.1815

i+.28i5

i^..38l5

1^.6530

-23-

Table

3 (continued)

I ^1

^1

^10 1

P;

5.61 (1)1

2.02

(3) - .886 (0)

6.11; (1)

5.81

2.00

-2.i|5

6.36

6.00

1.9k

|-i|.ll

6.57

6.20

1.88

-5.i|8

6.78

6.i^7

1.7i|.

-7.1+6

7.08

6.72

1.59

-8.91

7.35

7.01

1.39

-1.03 (1)

7.68

7.27

1.20

-1.12

7.96

1 7.60

9.62

(2) -1.18

8.31

8.01

6.82

-1.17

8.76

8. in

i4..55

-1.10

9.20

8.68

3.27

-1.00

9.50

8.91

2.1^0

-9.13 (0)

9.7li

9.10

1.80