2 blast waves

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QUICK ESTIMATES OF PEAK

OVERPRESSURE FROM TWO

<t

SIMULTANEOUS BLAST

WAVES

<

[EVEr

'

-

R

&'D

Associates

P.

0.

Box

9695

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Rey,

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4

' .

r :

December 1977

L-j

Topical Report for Period

October

1977-December 1977

_CONTRACT

No. DNA 001 78 C 0009

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A. TIT_§_

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OV R

QICKESTIMATES

OF

PEAK

OVERPRESSURE

FRO 0O

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19. KEY WOR S (Coagrlnut, an revere

side

if

neceay and

identify by block numb er)

Multibursta

Simultaneous

Blast Waves

Peak

Overpressure

Target Coverage

Reflected

Pressure

Blast

Waves

Shock

Interactions-

Shock-on-Shock

20

ABSTRACT (Contimn aon reverse mide

It

necesaryt

and Identify by block tnumber)

Oeveral

simple

approximations

to

the peak

overpressure

between two

interacting

4

(simultaneous)

blast

waves are

explored and

compared

to

the

LAMB

approximation

method for

multibursts.

The

best

estimates

suggest

about

a 30Z

(10%-50%)

increase

in

line-target

coverage

over that covered

by

two

nonsimultaneous

bursts,

Coverage

is restricted

to

strong

shocks

(AP

>

100

psi).

DO

47 EDITION

OF

I NOV 46 It

OBSOLETE

5 U TY

NCSLID4',

SEUIYCLASSIFICATION OF THIS

PACE (*W

- -

78

09

08

Y~



Table 1. Conversion Factors for

U.S. Customary

to Metric (SI)

Units

of

Measurement

TO CONVERT

FROM

TO

MULTIPLY

BY

FOOT

METER

0.3048

MEGATON TERAJOULE

4183

MILLISECOND

SECOND 0.001

PSI (POUNDSOFORCE/INCH

2

) KILO PASCAL (kPa)

6.894

757

fort

S hite

soctwas

WC

BUuff

sedlo 0

UOWffUED

C3 I

JWTWIC TIO

I'

. ..-...-...----..-.-.

I

~

LL



TABLE

OF CONTENTS

Page

1.

Introduction

.

. .

. . . . . . . . .

. . . . .

. . .

5

2.

Several Estimates

. . . . .

. . . . . . . . . .

.. 6

3.

Interaction

of

Unequal

Shocks . . .

. . . . . . . .

18

4.

Comparison

with

the

LAMB Procedure

. . . . . .

. . . 27

5. Summary

and Conclusions

..............

34

References

. . . . . . . . . . . . .

.

. . . . . . . . . 36

'S

T,



LIST OF

FIGURES

Figure

Page

1 Two 1-MT

Surface Bursts (Simultaneous)

reflecting

to 600

psi

..........

8

2

Geometry

for

Two Simultaneous

Blast Waves

Interacting

... ....

........

9

,

3 Peak

Overpressure vs Range

for Two 1-MT

Simultaneous

Surface Bursts

Separated

by

6860

ft .

. .

. . .

. .

. . .

. .

. .

11

4

Pressure

Profiles at Early

Times for 1-MT

Surface

Bursts

.............

12

5

Approximations

for the

Peak

Overpressure

Between

Two Simultaneous 1-MT

Surface

Bursts

4500

ft Apart

..........

14

6

Shock Interaction Notation

for

Unequal

Shocks . . .

. . . . . . . . . .

. . 18

7

Resultant Peak

Overpressure from

Two Shocks

(One

at

100

psi)

.

. .

. .

.

.

. . .

. .

22

8

Shock

Interaction

Between

Unequal

Shocks

.

23

9 Further

Approximations

for the

Peak

Over-

pressure Between

Two Simultaneous 1-MT

Surface

Bursts

4500

ft Apart

.

. .

. .

.

26

10 Comparison of

Reflection Factor

(APR/APS)

for Normal

Shocks and by the LAMB

Method;

Comparison of Reflected

Density

and Density

by

LAMB

Method

.............

28

11 Approximations

for the

Peak

Overpressure

Between Two Simultaneous

1-MT Surface

Bursts 4500 ft Apart Compared

with the

LAMB

Model

...............

32

3



LIST

OF

TABLES

Table

Page

1 Conversion Factors for U.S. Customary to Metric

(SI) Units

of

Measurement......

. . . . . 1

2 1-MT

Simultaneous Burst

Interaction

(Separated

by 4500

ft) Approximations

to Peak Over-

pressure . . . . . . . . . . . . . . . . . .

.

16

3 Comparison of Range and

Area

Coverage

Increases

for Five

Different Approximations to

tht

Interaction of

Two Simultaneous

1-MT Surface

Bursts Separated by

4500 ft ........

17

4

1-MT

Simultaneous

Burst

Interaction Peak

Over-

pressure App:oximations Using

Unequal

Shock

Results

. . . . . . . . . . . .. . . .. . . .

25

5

Range

Coverage Comparisons (1-MT Bursts 4500 ft

Apart). . . . . . . . . . . . . . . . . . . . . 33

bF

4

.=77

'4'.-'-'I



SECTION 1. INTRODUCTION

Strong

shocks

in

air reflect off rigid surfaces

at

many

times (5-13 times) their original pressure.

Opposing

shocks

of equal strength

behave just

as a reflected shock, leading

to very high pressures at the initial

point

of contact. If

two shocks of 100 psi collide, the resulting peak pressure is

around 500. For two 1000-psi shocks,

the peak

pressure jumps

to

8500

psi. This suggests that the area covered by a given

overpressure

from

two

simultaneous

blast

waves may be consid-

erably

larqG:: than twice the

area

of

a

single

burst.

Yet,

while strong

shock reflection

factors are

impressive,

there

is reason to believe that such

high

values

do

not extend

far

beyond the initial

point of

contact, and, further, two

unequal shocks

may interact in

a much less

impressive

way.

Strong

blast waves are extremely transitory, and pressures,

densities,

and

flow rates

behind

each

blast

front drop

off

exceedingly rapidly.

Careful considerations

of such blast wave nteractions

lead directly to three-dimensional geometries

which

very much

inhibit

the

accuracy and practical

resolution achievable

with

canonical numerical

methods.

The

following series of estimates,

without

benefit of rigorous

modeling

and

detailed nuak-1ical

calculations, are intended

to bound the expectations for

enhanced

coverage by means of simultaneous blasts. Conparison is

also

made

with the

LAMB

procedure, as

applied

to

two

simultaneous

bursts and

the

overpressures along the line joining their centers.

A

Low-Altitude Multiple

Burst.

5



SECTION 2. SEVERAL ESTIMATES

The

peak

overpressure

from a single burst on the

surface

is well approximated

with

the formula '1]

3300

W

192Np

(1)

R

R

with W

the

yield in m gatons

and R

the distance in

kilofeet.

A

normally

reflected shock reaches a

reflected

pressure

enhanced by the stagnation of the flow, and results in pres-

sures

for a strong

shock

much more than

double the incident

shock pressure.

An

approximate shock

reflection factor

for

an ideal

gas of specific heat ratio

y is

given

by

Pr

4yP

0

+

(3y-

1)

AP

AP

2yP

+

(y-1)AP

(2)

where AP is the incident overpressure,

AP

r

is

the

reflected

overpressure, and Po is

the

ambient pressure

(14.7 psi). For

sea level air, 1.1 <

y < 1.7 (y = 1.3 for AP =

1000

psi)[1].

A more

exact fit

(within

3

percent)

to this reflection

factor is provided by

tile

formula

below,

which accounts

for

the

nonideal

gas

properties of

sea level air

[2].

This formula agrees

to within 10 percent with the accepted

average of

the

nuclear test data

which

in

turn are

90

percent

contained by

a spread

of

±50 percent at pressures above

40 psi.

6



RF=

0.002655AP

+ 2

1

+

0.0001728AP

+

1.921

x

0-AP

0.004218

+

0.04834AP

+

6.856

x

10

-

6

AP

2

+

(3)

1 + 0.007997AP + 3.844

x 10-6AP2

At the point

where

two

simultaneous spherical

blast

waves

meet, the peak

overpressure can be fairly

precisely described

with tkese two

approximations (Equations

1

and

3). For

example

(as in Figure

1), for two 1-MT simultaneous

surface

explosions separated

by

6860

ft,

the

blast

waves meet when

each has a peak

overpressure

of

116

psi

(Equation

1),

and

these

shocks

reflect

at

the

point

of contact

to

a

pressure

of

600

psi (a reflection

factor of

5.18)(Equation

3).

The value

of 600

psi from

a single

1-MT surface

burst

occurs

at

1840 ft.

For nonsimultaneous

bursts, a separaticn

of 3680

ft would

cover a

line target

with more than 600

psi

everywhere.

The

separation

distance at

which the interacting

shocks reflect

to

600 psi

(6G6C

ft)

is nearly

twice

as

long.

The area associated

with the region

of

enhanced

blast

pressures

is

a

thin

lens

about

the point of contact

between

spherical

(or hemispherical)

shocks, and is only a

small

fraction

of

the area covered by the

individual

blast wr--es.

However,

the use of the larger distance

(6860

ft),

where

the shocks reflect

to

just 600 psi

as

an

effective

kill

distance

for

a line

target,

is quite

incorrect.

The geometry

of

the

situation

(Figure

2)

suggests

that

as the

two

shocks

pass

It is

very wrong

to

ansume that

if

the

peparation distance

were

increased by

a factor

of two,

the

area coverage

would

be I

correspondingly

increased by a factor of

four.

7



06

LOA

0

4J

4J

4J,

4J

.9-go

aI

A'U

10,-



I.

I

p

tncl

/

3

P

s

to

1/lk

P

i s

*

I

S

0

R

REINOF

REFLECTION

Figure

2.

Geometry

for

Two Simultaneous

Blast

Waves

Interrcting

9~

-.

*;I

v

~

--.-

- -

- - - - -

- - -

-

- - -

-

- - -

-

- -

- - -



througn

Pch

other, they

continue to expand spherically,

and

so

dr.,p

rapid,

.n pressure from that peak value of first con-

tact (1:. .

1).

If

their peak inturaction gives

only

600

psi,

then all L.iv region between burst

points

that lies beyond a

radiAo.s

from

each burst point of

1840 ft

would

see

less than

600 "

(Figure

3).

That is,

as in Figure 3, a

total

of

3200 ft

,ould

experience

two

shocks--both

less

than 600 psi.

r t creite,': interest

is the

sepe

ration

distance

that

exposes

the nt :ire

target to more than

600 psi (or some

other peak

.vei'prC3s.re

of interest).

For this we

need to know how

rapidly

these diveraing

shocks decrease

in

overpressure

as

they

expand

into

each other. Since,

on first contact

with the opposing

blast

w,,,ves the

transmitted

shock

starts at the

peak

reflection

v,:lue,

the

initial

value "; well defined

(Equation

3 , but there

e-s not exist

an

equally

simple or

direct formula for predicting

the

subseqaent

decay rate

as the two

blasts

penetrate

each

other.

Numerical blast calculations

provide detailed

descriptions

of

the pressure

field

into

which each transmitted

shock

is

expanding

and

how it

behaves

in

both space and time

[3,4,5].

For most applications

to hard targets

(e.g.,

for

600-psi

trenches), a

simple strong shock

model would suffice.

In

any

case, while

peak pressure

decays with distance

(as

in Equation

1)

approximately in proportion

to

the

inverse

cube of the distance,

the interior of the

fireball/blast-wave

follows

with

a pressure

about one-third

to

one-half

of the shock value (Figure

4). In

fact,

both peak

pressure

and

interior

pressure

decrease in

pro-

portion

to the inverse of the time measured

from

the instant

of

explosion.

An approximate

relation is given

by

12]

Two simpler

but less

exact

forms

for

the pressure-time

rela-

tion in a nuclear

blast wave

may be found

in

Reference

1,

pp. 180-181.

WI

Aw

10



I-U

co-

Uj

U

CD

4-

I--

121

_

_ _ _

4.-



AP t,T)

8.000 t)

0

417

+

0.583

]1 t-T

f(t) (4)

in which t is the time in

msec (after burst),

T is the shock

arrival time (msec)

for

1

MT, (t T),

D

is the

positive

phase

duration (msec) (the

time

during

which

the

blast pressure is

greater

than ambient), and f(t) is an empirical

adjustment

fit

of secondary importance.

k2

t

0

+

6.72t

+ 0.00581t

2

fltM

(5)

100

+

18.8t

+

0.0216t

2

The essential time

behavior

of the pressure

is

illustrated

by this dpproximation:

The

dominant behavior

is a

decay

almost

linear in time (more

precisely

as

A-

- '

15)

with

a

very

sharp

drop just

behind the shock front to a value of about 40 percent

of

that

at

the

front

(

t

-

6

).

One possible approximation is

to

assume that

the

transmitted

shock

continues

to

generate the same peak

reflection factor as

it continues to

expand

and decrease

inside the other blast wave.

This is

likely a gross overestimate of the off-peak pressures,

since one

might better

use reflection factors

appropriate to

the

transmitted shock as

it

continues to

expand and decrease.

The original

blast that

it

is running into

also

continues to

expand and decrease

in pressure. A more correct approximation

will account

for this double decay,

but,

for the

moment, consider

several

simple

approximations

for

the

transmitted

peak

over-

pressure.

Figure

5

shows

several

choice*

for

these

transmitted

pressures

in

the particular

case of two

simultaneous

surface

.

13



2400

2200

'CO'

1800

CI'

~1600I

1400

00

800

L-

S

1000

3

200

40

0

200

1400

1600

1M8

900

2200200

60

2800

DISTANCE

FROM

BURSTS

(ft)

3100

290

2700

2500

2300

-2100

1900

1700

Figure

5.

Approximations

for

the Peak

Overpressure

betweein

Two Simultaneous

1 MT

Surface

Bursts

4500

ft

Apart

14



bursts (1-MT) separated

by 4500 ft. Table 2

illustrates the

development of numerical values

for

each case.

(1) The peak

interaction

is

defined

as the local

pressure

in front of

the transmitted

blast wave

multiplied by

the initial reflection

factor

for

the colliding

shock fronts

(reflection factor

RF

=

7.0).

(2) The peak

interaction pressure

is defined

as the

local

pressure

in

front of

the

transmitted

blast

wave

multiplied

by

the reflection

factor appropriate

for

a shock of strength

equal to

that of the second

shock if

it

had expanded unreflected)

to that dis-

tance beyond the initial contact

point.

(3) The peak


is defined

as

the

local pressure

in front of

the transmitted

blast

wave

multiplied

by

the reflection

factor

for a

shock

of peak pressure equal

to

that pressure.

(4)

The

peak


is

defined as

the

unreflected (incident)

shock

pressure multiplied

by

the

reflection factor appropriate

to the time

of first contact

(RF

= 7.0).

(5) The

peak interaction pressure

is defined

as

thie

unreflected

incident) shock pressure

multiplied

by

the

reflection factor

appropriate

for

that

reduced

shock strength as it

expands in an undis-

turbed

sea level

atmosphere.

-Z

15

'77777



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( x LO II

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V-4

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4

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-

4-

Z; w

w.

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ww

4 4

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(A o

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v v a

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-t

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-.4.. ,



SECTION 3.

INTERACTION

OF UNEQUAL

SHOCKS

The

preceding

discussion

has

dealt

with

the case of two

simultaneous

bursts

(equal

strength shocks) interacting.

Interactions between shocks of

unequal strength are also of

interest,

since timing of

multiple bursts may

not

be exact

and the first blast

may be considerably weaker by the time

the

second

blast meets

it.

Suppose

two shocks meet when their

respective peak

over-

pressures are AP

1

and AP

2

,

and

suppose the first

is stronger

than

the

second

(AP

1

>

AP

2

).

These shocks,

on

interacting,

lead

to an

overpressure

APR, a density

PR

and

a resultant

particle velocity UR

.

Figure 6

identifies the

nomenclature

in which two

initial

shocks

have met and have

resulted

in

two

transmitted shocks which have

velocities

URl

and

UR2

directed

away

from each

other,

and a

particle velocity UR which

is

positive

to

the right if AP

1

AP

2

.

-I

I I2

A

1

u

R

2

1

2

U

p

2

URl

PRI

PR2

UR2

Figure

6. Shock Interaction

Notation for Unequal Shocks

Actually, two

states of density

and

temperature

separated

by

a

contact

discontinuity

exist

in

the

shock

interaction region

(shaded area, Figure 6), but

pressure and velocity are

the

same in

both states,

viz.,

APR,

.

-'-



From

standard

shock

relations,

Ulf

U2,

Pl.

and

P

2

are

expressible

in

terms

of AP

1

and AP

2

and are

considered

as

known

here.

U

=API((XYA.)AP

1

+

YPo)

U

Al

2

1AP

YP

)

vPO

AP1p [ 4)AP

1

+

YP]((~)

A7

-

+

1I\ +

[P2~

AP' +

1P

Pi

Po,

Yo]

Yo)

= o

[ AP

2

AP

2

+ P

(6)

Expressing

the conservation

of mass

flow

rate

through

each

shock, one

may write

PRl(URl

+ UR

=

P

1

(Ul

+ URi

PR2

(UR2

-

UR)

p

U

2

+

U 2)

(7)

The

momentum

change

related

to

the pressure

jumps

at

the

shocks

can

be

expressed

as

APR

-

A

1

= PlU

(U

1

+

URI) -

PR1UR(UR

+

URi)

AP R - AP- P

2

U

2

(U

2

+ UR2) + PR2UR(UR2

- UR)

(8)

19



Eliminating

URI

and

UR

2

from

Equations

6 and

7, one

can

write

I2

APR

=

Ap1

+ -

(U

1

-

UR

)

2

PRI Pl

and

AP

R

=

AP

2

+

PP

2

(U

2

+

UR )

2

(9)

The init

4

al

energy density

(E)

jump conditions

at

each

transmitted

shock

require

ER

-

E

2

.

P-

and

E

-

E

2

2

(

2

-R2

If

one assumes

an

ideal gas with ratio of specific heats

(y),

then E

-P/[(y-l)p].

Solving

for

pR

leads to

PR

(Y+l)

APR

+

(Y-1)API

+

2yP

0

71

(Y-11)APR +

(y+l)AP

1

+

2yP

0

and

PR2

(y+l)APR

+ (y-l)AP

2

+

2yP

O

-R2

yI)AP

R

+

(Y+1)AP

2

+

2P

0 (11)

Using

Equations

10 and

the

Hugoniot

relations

to express

Pit

P,,,

U,,

and U

2

in

tezms

of

AP

1

and

AP

2

(Equation

6),

one

can derive

20

TMA,



AP

R

A _A

2

+

C

(12)

R|

and

APR =

D ±D

+ F

13)

in

which

A

AP + Pl(Ul

-

UR)

2

C-E-API12

+

(Yi

API

+

yPo)

P1(UI

-

US)

2

_A P

2

2+(X

AP

2

+Ypo)

P

2

(U

2

+UR)

These equations can

be solved

for APR

by

iterative

selection

of

the

velocity

UR

all

other

values being pre-

determined by the values

of

yPo' AP

1

' and AP

2

).

A set of example

values

of

the

resultant

peak overpressure

APR

for

two

shocks meeting when

one of them is

100

psi is

illustrated in

Figure

7

for two

values of the

specific heat

ratio. For

example,

a 100-psi and

a 500-psi shock meet

to

give

more

than

1500

psi for y

-

1.4 and about

2000 psi

for

y-

1.2.

1

Figure

8

illustrates the

peak pressure amplification

from

the

meeting

of two

shocks

with

a

plot

of the

ratio

of

the

resulting

peak overpressure to

the

sum of the

two

incident

peak pressures. The

amplification

approaches

unity if one of



3000 ]

N

2500

-

2000-

w.J

1500

1000 AP-100

psi

1

500 2

I

I ,, iI

I I I , I

300

600 900

INITIAL OVERPRESSURE

(psi)

Figure 7. Resultant Peak Overprnsire from

Two

Shocks

One at 100 psi)

22

____ _T__777_



-

C-

(D

9n

0

0 1

*

'4

0

41

p..

idl

0

4n

m cm

)

41t

1

23

i 1J

i0

14 ,I Iv '.401

i

(Zd'

LdU

i d0



i

the shocks

is weak, and

seems to

reach a peak when

the ratio

of

the

two incident overpressures lies between

3 and 5.

This solution for

the resultant

peak pressure

from the

collision

of two unequal

shocks suggests

yet

another

approxi-

mation to

the

peak

overpressure distribution

for our example

of

two simultaneous

1-MT bursts

separated

by 4500 ft (Figure

5).

The interior

overpressure (behind the shock

front) that a

transmitted

shock

encounters can

be

assumed

to

act similarly

to

a shock of

that

overpressure, and

the

resultant overpressure

computed

(by means

of Equations

12

and

13).

This

approximation

is

tabulated in Table 4

and

illustrated

in Figure 9 (labeled

Case

6).

This

approximation, using unequal shock

properties,

is

not

very different

from

the

simple average of the peak

over-

pressures from the earlier bounding

cases--Cases 3 and 5--

which

use the interior

blast

wave

pressure multiplied

by the

normal

shock

reflection factor

for

that pressure

(Case

3)

and

the

single shock peak

overpressure multiplied

by its

normal

reflection

factor

(Case 5).

This average is also shown

in

Figure

9 as

Case

7.

24

.

.1.



-4

U)I 0 U) Le)

LO 0 In)

w *rx L 10

9 4

r..

r-. im

co m

c C.V

+

-j 411- U) 4

Om e to

0a

I

0

am.

N .e

C0C

r%. V4 -4

M0 In

a.

A

0o

W-

O 1

n

W*

m~ r%

C.

r..

4Y4

C%

jN

4A

Q>4w

L

I

n In P%.C

Eu

m

m

m

m

.4

.4 4 4

4J:3

cm 5

0 0 0D

(C

0

in f .

0.. A U P. N

4

0 In

IA

40

14

NY w 04 04 04 (m 4

25

Nix~-'I



I

00

i

/

2200

/

\I

18 I

CL

16

00

0

0

0

100010

20

70250

20

10~

,.

ue

.FuterAprx

/in

fo

h

I

Ovrrsuebte

Boo

1010

T00-e s

206

010I

200:

I ''

I

.

.I

. .

i

_

.

_

.i

, i

,

,

TDISTANCE

PAN7MR$TS ft)

Figure 9. Further

Approximations

for

the

Peak

Overpressure

Ibttwwnt'g.

7

Two

Simultaneous

I14lT

Surface

Bursts

4500

ft

Apart

l2

-

----

-

-

, "



SECTION 4.

COMPARISON

WITH THE LAMB

PROCEDURE

A

more

careful accounting

for the

usual conservation

of

mass, momentum,

and

energy

during these shock interactions

requires some

further

assumptions

and

more geometry

and

arith-

metic.

Such an

attempt is

the basis of extensive

calculations

and

predictions

at

the

Air Force Weapons

Laboratory

with a

program

referred

to

as LAMB

[6].

The essential features of this procedure are

as

follows:

Mass

conservation;

assume:

n

pc

+ i

Api,

(p

0.5

p

0

)

(14)

in

which p

is

the

air density, p

the

ambient (pre-shock)

air

density, and

Api is the over-density

(pi - pc) in

each blast

;

wave,

wv.Recognize

that densities in strong shock fronts

may

rise

to more

than ten times

the ambient

density,

but

may also fall

to less

than one-tenth

the ambient

value

inside the

fireball.

This prescription

predicts the

peak density of

two equal

crlliding

strong

shocks

to be

only

about

half

of

the

correct

Hugoniot

value.

(See Figure

10.)

The program,

originally

designed

to

approximate over-

lapping bursts at altitudes

between

10

and

30

kft

(in a

missile

defense

role),,

quite

arbitrarfly

restricts

the under-

denpities (negative

overdensities)

to

half (or

more) of

the

ambiant

density. Such

a restriction

provides

a

reaxonable,

....

if

unsatisfactorily

empirical

accounting

of other

disnersive

7

S,.



o

PR p

co

Q-j

a-S.

0 wR

..

° Cc

ILA

.0

i in

~V .

028

i:+,++' ,+.0.

W,4 M

11 1.I

4,

Q4J

2



effects likely

to

occur

in the

high temperature

(low

density)

interior of a nuclear fireball, but it cannot be justified

rigorously (from

first

principles).

Actually,

some

such limit

is

needed

to prevent an

undesired

increase in net kinetic

energy for a shock in

the

very low density interior of a strong

blast wave

(fireball)

that

would

result otherwise from this

prescription.

Recognize also that

this

expression

is

not

one

of mass conservation, but prescribes density, or

mass-per-unit

volume.

It has

long been a favorite

piece

of magic

for

simpli-

fied

blast

solutions (many

published

as

serious contributions)

to make seemingly innocuous assumptions about density

distribu-

tions

and

then

proceed

to

unfold

a

marvelously

consistent

picture

of some blast wave. The

rabbit

is always

already

in

the hat here, however, since density distributions are, in

fact,

integral representations of the entire movement history

of the blast, and the movements

so

described are

a

direct

consequence of

the acceleration or force (pressure gradient)

history. So any density profile

contains

the

blast wave

history

to

that

instant, and is

not a trivially

adjustable

parameter

of small consequence.

Conservation

of momentum; assume:

n

i-I1

where pi

is the

density

in the ith blast

wave, vi

is the

particle velocity of

the

i

th

blast wave#

and

p

and

v

are

the

resultant density and particle velocity in

the

blast interac-

tions. This represents

a

local

momentum

density vector sum

without

consideration of pressure impulse

contributions, which

is

not

quite consistent with

the usual

assumptions of inviscid

gas

dynamics

for

blast

wave characterization.

29

4°



Conservation

of eaergy; assume:

in which AP

i

= Pi

- Po is

the

overpressure

in the

ith blast,

and

AP is

the

resulting

overpressure

in

the interacting

blasts.

This prescription

purports

to convert

excess kinetic

energy

from

the

opposing

flows into

pressure Energy, a proce-

dure consistent

with

normal

hydrodynamic

flow characterizations.

However,

it

is

inexact

as

an

energy

equivalent

to

add

1/2

pv

2

terms

to overpressure. The

dimensions

are correct,

but

the

compressibility

factor

1/(y-1)

is

missing.

As a consequence,

the reflection factor

for

two equal

shocks resulting

from

this prescription

is slightly

in error.

For

an ideal

gas:

(r 2yAP

+

4yP

0

\

/.

-1)

AP

+

2yP

o

AP

LAMB

(yy

(7

More

rigorously

(as in

Equation 3),

LP (3y-l)AP

+

4yP(

AP

R

=

(y-1)AP

+ 2yP

0

(18)

These

two

formulae

are

compared

in

Figure

10

for

1

<

AP

< 104,

showing

the

LAMB .procedure

to

be

low

by

a factor of

7/8

at

high

overpressures

(y

1.4).

1L

The

LAMB procedure,

when applied

to the

previous example

(two simultaneous

1-MT surface

bursts

4500

ft

apart), gives

even

less coverage

with

high

pressure

than

the

previous

30

. 30



estimates using peak reflection

factors. As

mentioned,

the

LAMB method

misses

the peak

pressure by a

factor

approaching

7/8,

and drops

away

from that peak

value

very rapidly (Figure 11).

Table

5

compares

the range enhancements

from the approximations

previously

discussed

with

that

for the

LAMB model.

The curves

in

Figure

11

and the basic data i.n

Tables 2

and

4

are

derived

from

Reference

1,

but

any of the descriptions

in References 3,

4 or 5

would serve

as

well.

The

added

line

coverage

for this

example

with

the

LAMB

procedure is about 11

percent,

a

smaller

effect

than

any of

the

approximations given in the

previous section. Good agree-

ment with

experiments is claimed

for the LAMB model

when

applied

to

HE tests or

to

the shock reflections

on

a nuclear test

such

as

PLUMBBOB-PRISCILLA. However,

the arbitrary assumptions in

the

model

are

neither intuitive

nor physically

correct,

and

their effect

on results far

from

the

point of

initial shock

contact

remains unclear.

~*

1

31

-JJ



2400

2200

2000

1800

~14005

1200

-

100

/

II

II

I

1000

I

LMSt

I APPROXIMATION

600

400

%

2W0

1

4

m

U

=

=

0Z

2400

260

a

SISTAKNI

FIN

MSTS

Ift)

X,

ZW U

N 250

230

21

0

1900

1700

Figure

11

Approxlmttons

for

the

Peek

Overpressure

between

Two

Slultanemus

1 MT

Stface

lursts

4500

ft Apart

C

ared

with

the LM

Obdl

32



Table

5.

Range

Coverage Comparisons

(1-MT Bursts 4500 ft Apart)

PEAK

SINGLE DISTANCE PERCENT

APPROXIMATION

OVERPRESSURE

BURST ADDED BY RANGE

CASE COVERED

RANGE

INTERACTION

INCREASE

NUMBER (psi)

(ft) (ft) (1)

LAMB 450

2030

430

11

3 540 1915 670 17

6 710

1

1730

1040 30

5 990

1540 1410

35

33



SECTION

5.

SUMMARY

AND

CONCLUSIONS

A

reasonable

lower

limit approximation

to

the combined

overpressure

would

appear

to

be

to

multiply

the

pressure at

a point

inside a

single blast and in front of

the transmitted

shock

from a second

burst

by

the

reflection

factor

for a shock

of

that

interior

pressure

(Case

3). A similarly simple

upper

bound should be provided

by multiplying the single blast

peak

overpressure

by the normal reflection

factor

for

that pressure

at distances

beyond

the

point of first contact for

the two

shocks

(Case 5). A further procedure,

almost

as

simple, pro-

viding

an

intermediate

value, uses the pressure

predicted

for

the interaction of

unequal

shocks, based on the pressure

ahead

of and behind

the

transmitted

shock

(Case 6).

The

LAMB procedure is

not

rigorously

correct,

but

it

provides

a

more

general and a

more detailed

treatment of

interacting

shocks. Unfortunately,

it is

not clear whether

it

overestimates or underestimates

the

resulting

pressures,

although

all the predictions for

this example

lie above the

LAM-derived

values. Accepting all

the apparent uncertainty

in

these

approximations

to

multiple

shock interactions,

one

can still conclude

that:

*

The

region of

enhanced

overpressures from tw o

simultaneous

separate

blast waves is a small

fraction of

the area ccvered by

each individual

blast.

9

The

extra coverage

of a line

target with

high

overpressures

(AP

5

> 300 psi) is of the order

of

10-50 percent,

and, ,more

probably,

less than

30 percent.

34



The

LAMB

procedure

may,

in

some cases, underpredict

the peak

pressures

for

interacting blast

waves.

F

35

:.';,'35



REFERENCES

1. H. L. Brode,

Review

of Nucle-r

Weapons

Effects,"

Annual

Review of

Nuclear Science,

Vol. 18,

1968,

p. 180.

2. H.

L. Brode,

Height of Burst

Effects at

High Overpressures,

DASA 2506,

The

Rand

Corporation,

RM-6301-DASA,

1970.

3. H. L.

Brode,

Point

Source Exlosion

in

Air,

The

Rand

Corporation,

RM-1824-AEC,

1956.

4. H. L.

Brode, The

Blast Wave

in Air

Resulting

from a Hi h

Temperature, Hih

Pressure

Shere

in Air,

The

Ran

Corporation, RM-1825-AEC,

1956.

5.

C. E. Needham,

M. L. Havens and

C. S.

Knauth,

Nuclear Blast

Standard

(1 kT),

Air Force

Weapons

Laboratory,

Krtland

A-r

Force

Base,

New

Mexico,

AFWL-TR-73-55,

(Revised)

1975.

6 C.

E. Needham, Private Communication,

Air Force

Weapons

Laboratory,

Kirtland

Air

Force

Base,

New Mexico,

November

1977.

36i

If

I

I|



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-39



-~A

6.

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II