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/W DNA453
QUICK ESTIMATES OF PEAK
OVERPRESSURE FROM TWO
<t
SIMULTANEOUS BLAST
WAVES
<
[EVEr
'
-
R
&'D
Associates
P.
0.
Box
9695
Marina
del
Rey,
California
90291
4
' .
r :
December 1977
L-j
Topical Report for Period
October
1977-December 1977
_CONTRACT
No. DNA 001 78 C 0009
APPROVED FOR PUBLIC
RELEASE;
DISTRIBUTION
UNLIMITED.
THIS
WORK SPONSORED BY THE DEFENSE
NUCLEAR
AGENCY
UNDER
RDT&E
RMSS
CODE
0310078464
P990AXDB00136
H2590D.
DDC
Prepared
for
r n
Director
OCT
I=
DEFENSE
NUCLEAR AGENCY
Washington,
D.
C. 20305
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UNCLASIFIED
SECURITY CLASSIFICATION
OF THIS
PAGE
(When
flate
ntered)
READNSTRUCTIONS
REPORT
DOCUMENTMTON
PAGE
13EFORE
COMIPLETING
FORM
I. REPORT
NUMB3ER
.GOVT
ACCESSION N . CP
TSATALOG NUMBER
DNA
4503T
A. TIT_§_
PEROD
OV R
QICKESTIMATES
OF
PEAK
OVERPRESSURE
FRO 0O
Topical
epget
/0
.
L.1Jsrode
N
P O Box
695
SubafsLEQAX.OOI 36
15I.
SECURITY CLASS (of this
report
IMONITORING
AGE Y
NA
ltdifferent
from
Con
troling
Office)
NIASFE
S
15a. OECLASSI
FICATION/
DOWNGRADING
SCHEDULE
16.
DISTRIBUTION
STATEMENT
(of #hia
Report)
Appioved for
public release,
di
)utio0Wjs~_
~5,D-r
a'
/ l
'dp3Sr AD ,JOO
34/
17 DISTRI13UTION
STATEMENT (at the ahatract entered
in nleck 20, It different from
Report)
14 SUPPLEMENTARY NOTES
This
work sponsor'ed
by
the
Defense
Nuclear Agency
under KDT&E
RHSS
Code
B310078464
P99QAXDBOO136
H2590D.
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~
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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
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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,
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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
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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
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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
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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
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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
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06
LOA
0
4J
4J
4J,
4J
.9-go
aI
A'U
10,-
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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
~
--.-
- -
- - - - -
- - -
-
- - -
-
- - -
-
- -
- - -
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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
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I-U
co-
Uj
U
CD
4-
I--
121
_
_ _ _
4.-
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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
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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
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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
interaction pressure
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
interaction pressure
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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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,
.
-'-
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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
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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,
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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
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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_
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-
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
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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.
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-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
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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
-
----
-
-
, "
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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,.
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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
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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°
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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
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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
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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
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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
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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
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The
LAMB
procedure
may,
in
some cases, underpredict
the peak
pressures
for
interacting blast
waves.
F
35
:.';,'35
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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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U.S. Army
Engr. Waterways Exper. Station
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Livermore Laboratory
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Science
& Ter4.
Ctr.
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(og) ATTN: Research & Coacepte Branch
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System
Engineering Org.
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T.
Neighbors
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DEPART'ffENT
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Conti.nud)
U.S. Army
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°
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DEPARTHENT
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(Contined)
DEPARTMENT
OF ENERGY (Continued)
Commander
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-~A
6.
DEPARThENT
OF DEFENSE COhrkAC'£ORS Continued)
DEPARTMENT
OF
DEFENLE
CONTRACTORS
(Continued)
University of
Denver
Nathan
H. Newmark
Colorado Seminary
Consulting
Services
Denver Research
Institute
ATTN:
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Newmark
ATTN:
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Officer
for J.
Wisotaki
Pacifica
Technology
EG&G Washington
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BJork
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Card, Inc.
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R & D Associates
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Port
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lIT
Research Institute
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ATIN:
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Ltter
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Brode
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International
Martin
Marietts Corp.
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ATTNs
0.
Fotieo
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Corp.
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K.
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DEPARTMENT
OF DEFENSE
CONTRACTORS
(Continued)
DEPARTMENT OF DEFENSE CONTRACTORS
(Continued)
Tvarra
Tek,
Inc.
Universal
Analytics,.
nc.
ATTN:
Library
ATTN:
E. Field
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A. Jones
ATTN.
S.
Green Eric
H.
Wang
Civil
Engineering
Rsch. PAC.
Tetra
Tech., Inc.
ATTN:
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Baum
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Weidlinger
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Enginere
TRW
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Baron
ATTAr: R. Plebuch
ATTNt
P.
$hutts
Weidlnger Assoc., Consulting Engineers
ATTN: Technical
Informatior.
Center
ATTN: J. Isenberg
ATTN:
D.
Beer
ATTN:
I.
lber Westinghouse
Electric Corp.
2
cy
ATTN
P.
Dal
Marine Division
ATTN:
W.
Volz
TRW
Defense
& Space Sys. Group
San
Bernardino
Operstions
ATTN:
E.
Wong
II
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