Ignace I Kolodner.

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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
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


1 3