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/ AW'DNA453
QUICK ESTIMATES OF PEAKOVERPRESSURE 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 AGENCYUNDER RDT&E RMSS CODE 0310078464 P990AXDB00136 H2590D.
DDCPrepared for r n
Director OCT I=
DEFENSE NUCLEAR AGENCY
Washington, D. C. 20305
78
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SECURITY CLASSIFICATION OF THIS PAGE (When flate Entered)
READ INSTRUCTIONSREPORT DOCUMENTMTON PAGE 13EFORE COMIPLETING FORMI. REPORT NUMB3ER .GOVT ACCESSION N . CP "TS CATALOG NUMBER
DNA 4503TA. TIT_§_ PEROD COVERE
QICKESTIMATES OF PEAK OVERPRESSURE FRO 0O Topical Xepget,
/0 . L.1Jsrode N
P.O. Box 9695 SubafsLEQAX.OOI-36
15I. SECURITY CLASS (of this report)
IMONITORING AGE Y NA ltdifferent from Con troling Office) NIASFE
S '15a. OECLASSI FICATION/ DOWNGRADING ISCHEDULE
16. DISTRIBUTION STATEMENT (of #hia Report)
Appioved for public release, di )utio0Wjs~_
~5,D-r $a' / l 4'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 WORDS (Coagrlnut, an revere# side if neceay and identify by block numb er)
Multibursta Simultaneous Blast WavesPeak Overpressure Target CoverageReflected Pressure Blast WavesShock Interactions-Shock-on-Shock20 ABSTRACT (Contimn aon reverse mide It necesaryt and Identify by block tnumber)
Oeveral simple approximations to the peak overpressure between two interacting4 (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 nonsimultaneousbursts, Coverage is restricted to strong shocks (AP > 100 psi).
DO " 47 EDITION OF I NOV 46 It OBSOLETE 5UTY NCSLID4',SEUIYCLASSIFICATION OF THIS PACE (*W
- - 78 09 "08 Y~
Table 1. Conversion Factors for U.S. Customaryto Metric (SI) Units of Measurement
TO CONVERT FROM TO MULTIPLY BY
FOOT METER 0.3048
MEGATON TERAJOULE 4183
MILLISECOND SECOND 0.001
PSI (POUNDSOFORCE/INCH2) KILO PASCAL (kPa) 6.894 757
fort
S White soctwasWC BUuff sedlo 0UOWffUED C3I
JWTWICATIOI' .'..-...-...----..-.-.
c.
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 WavesInteracting ... .... ........ 9
, 3 Peak Overpressure vs Range for Two 1-MTSimultaneous Surface Bursts Separatedby 6860 ft . . . . . . . . . . . . . . . 11
4 Pressure Profiles at Early Times for 1-MTSurface Bursts ............. 12
5 Approximations for the Peak OverpressureBetween Two Simultaneous 1-MT SurfaceBursts 4500 ft Apart .......... 14
6 Shock Interaction Notation for UnequalShocks . . . . . . . . . . . . . . . 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-MTSurface 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 Densityby LAMB Method ............. 28
11 Approximations for the Peak OverpressureBetween Two Simultaneous 1-MT SurfaceBursts 4500 ft Apart Compared with theLAMB 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 (Separatedby 4500 ft) Approximations to Peak Over-pressure . . . . . . . . . . . . . . . . . . . 16
3 Comparison of Range and Area Coverage Increasesfor Five Different Approximations to thtInteraction of Two Simultaneous 1-MT SurfaceBursts Separated by 4500 ft ........ 17
4 1-MT Simultaneous Burst Interaction Peak Over-pressure App:oximations Using Unequal ShockResults . . . . . . . . . . . .. . . .. . . . 25
5 Range Coverage Comparisons (1-MT Bursts 4500 ftApart) . . . . . . . . . . . . . . . . . . . . . 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 nteractionslead directly to three-dimensional geometries which very muchinhibit 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 enhancedcoverage by means of simultaneous blasts. Conparison is also
made with the LAMB procedure, as applied to two simultaneousbursts and the overpressures along the line joining their centers.
A
Low-Altitude Multiple Burst.
5 ,r
2l
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 egatons 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 4yP0 + (3y- 1) AP
AP 2yP + (y-1)AP (2)
where AP is the incident overpressure, APr is the reflected
overpressure, and Po is the ambient pressure (14.7 psi). Forsea 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 acceptedaverage of the nuclear test data which in turn are 90 percentcontained 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 -6AP2
+ (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 blastpressures 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 distancewere increased by a factor of two, the area coverage would be Icorrespondingly increased by a factor of four.
7
06
LOA 04J
4J
4J,
4J
.9-go
aI
A'U10, 7-
I.
I
p Ptncl / 3 P s to 1/lk P
i s
* I S
0 RREINOF REFLECTION
Figure 2. Geometry for Two Simultaneous Blast Waves Interrcting
I.-
9~ KY-. *;I v ~ ---.-
- - - - - -- - - - - - - - - - - - - - - - - - -
througn Pch other, they continue to expand spherically, andso 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 of3200 ft ,ould experience two shocks--both less than 600 psi.
rit 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 opposingblast w,,,ves the transmitted shock starts at the peak reflection
v,:lue, the initial value "; well defined (Equation 3), but theree-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). Infact, 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.
WIAw" 10
PEAK REFLECTED PRESSURE EQUALS
4'icc
j / ~%(APPROXIIWATE)A
- SINGLE001 (UNREFLECTED)
SHOCKS
___________________REGION______WHERE______AP<___600_________
Figure 3. Peak Overpressure vs Range for Two 1 MT SimultaneousSurface Bursts Separated by 6860 ft
-1777-
'I-U
co-
Uj U
CD
4-
I--121
_ _ _ _ 4.- F7
AP(t,T) 8.00 0 t) 0 4 1 7 + 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 phaseduration (msec) (the time during which the blast pressure is
greater than ambient), and f(t) is an empirical adjustment fit
of secondary importance.k2t 00 + 6.72t + 0.00581t2flt M (5)100 + 18.8t + 0.0216t
2
The essential time behavior of the pressure is illustratedby this dpproximation: The dominant behavior is a decay almost
linear in time (more precisely as A- t - ' 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 approximationwill 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 transmittedpressures in the particular case of two simultaneous surface
. 13
2400
2200
'CO'
1800
CI'
~1600I
1400
00
8 100L- S
1000 3
200
40-
0 1200 1400 1600 1M8 900 2200200 60 2800DISTANCE FROM BURSTS (ft)
3100 290 2700 2500 2300 -2100 1900 1700
Figure 5. Approximations for the Peak Overpressure betweeinTwo 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 localpressure in front of the transmitted blast wave
multiplied by the initial reflection factor for
the colliding shock fronts (reflection factorRF = 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 blastwave 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 timeof first contact (RF = 7.0).
(5) The peak interaction pressure is defined as thieunreflected (incident) shock pressure multipliedby the reflection factor appropriate for thatreduced shock strength as it expands in an undis-turbed sea level atmosphere.
-Z
15
'77777
II
( x LO II r. ls GO f r.
-C * V-4 - "4 v-4 V-4 "
0N0 0 4)a
.0c 040 a "4a a
4 J -4-
Z; w w. 00 0 O 0Y C~4w r-4 1-0. 40 Ln0 r M0 M
ww
4"4
2
(A o " < 4'0 " - PI n v v aIt 5 -t " 9-4
Of-~
., "4 @ O 5. 0 U44 N:1,"4 " 4 4 "
4 ~ 4J
2. ~ c r- 4
t ,ctr I, r. U;.f LA~ rr'A~f 4 -. 4 4.. ,
None of these approximations account for the expansion
of both shocks as they pass through each other. The most
reasonable approximations are (3) and (2), but the others
are shown for comparison (Figure 5).
The extra line target coverage for this example (1-MT
bursts separated by 4500 ft) is listed in Table 3.
Table 3. Comparison of Range and Area Coverage Increasesfor Five Different Approximations to the Interaction
of Two Simultaneous 1-MT Surface Bursts Separated by 4500 ft
PEAK SINGLE DISTANCEOVERPRESSURE BURST ADDED BY RANGE
APPROXIMTION COVERED RANGE INTERACTION INCREASENUMBER (psi) (ft) (ft) (%)
3a 540 1915 670 17
2 610 1830 840 23
1 640 1800 910 25
5 990 1540 1410 46
4 1120 1470 1550 53
_ __ _
aMOST PLAUSIBLE APPROXIMATIONS.
The last two are clearly overestimates of the effect,
but the first three may also predict more enhancement than is
real.
Further estimates are given in the folloving sections.
17 *
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 AP1 and AP2 , and suppose the first is stronger
than the second (AP1 > AP2). These shocks, on interacting,
lead to an overpressure APR, a density PR and a resultant
particle velocity UR. Figure 6 identifies the nomenclaturein 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 ispositive to the right if AP1 AP2.
-I I I2
A1 uR 2U 1 2
U p2
URl PRI PR2 UR2
Figure 6. Shock Interaction Notation for Unequal Shocks
Actually, two states of density and temperature separated bya contact discontinuity exist in the shock interaction region(shaded area, Figure 6), but pressure and velocity are thesame in both states, viz., APR, .
-'- A
From standard shock relations, Ulf U2, Pl. and P2 are
expressible in terms of AP1 and AP2 and are considered as
known here.
U =API((XYA.)AP 1 + YPo)
U Al 2 1AP + YP 0) vPO
AP1 = p [(4)AP1 + YP]((~) A715- +
1I\ + 1[P2~ AP' + 1PPi -Po 2, Yo] [(Yo)
= o [ AP2 AP - 2 + P (6)
Expressing the conservation of mass flow rate through each
shock, one may write
PRl(URl + UR) = P1 (Ul + URi)
PR2 (UR2 - UR) p 2 (U2 + U R2) (7)
The momentum change related to the pressure jumps at the
shocks can be expressed as
APR - A 1 = PlU (U1 + URI) - PR1UR(UR + URi)
AP R - AP- P2 U2 (U2 + UR2) + PR2UR(UR2 - UR) (8)
19
Eliminating URI and UR2 from Equations 6 and 7, one
can write
I2APR = Ap1 + - (U1 - UR) 2
PRI"Pl
and
APR = AP2 + PP 2 (U2 + UR) 2 (9)
The init 4al energy density (E) jump conditions at each
transmitted shock require
ER - E 1 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 + 2yP0
71 (Y-11)APR + (y+l)AP 1 + 2yP0and
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 U2 in tezms of AP1 and AP2 (Equation 6), one
can derive
20
TMA,
APR 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
_AP 2 2+(X AP2 +Ypo) P2 (U2 +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' AP1 ' and AP2).
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 togive more than 1500 psi for y - 1.4 and about 2000 psi for
y- 1.2. 1
Figure 8 illustrates the peak pressure amplification jfrom the meeting of two shocks with a plot of the ratio ofthe resulting peak overpressure to the sum of the two incident
peak pressures. The amplification approaches unity if one of
.if
3000 ]
N
2500 -
2000-
w.J
1500
1000 AP-100 psi
1!
500 2I I ,, iI I I I , I300 600 900
INITIAL OVERPRESSURE (psi)
Figure 7. Resultant Peak Overprnsire from Two Shocks(One at 100 psi)
22
____ _T__777_
- 2C- (D 9n
00 1 *
'4 0
41
p..
idl
0
4n
m cmU) 41t 1
23
i 1Ji0
14 ,I Iv '.401
i (Zd' LdUi 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) andthe 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 *r ix L 10 9-4 r.. r-. im co mc C.V + -j 411- U) "4 Om e% to %0a I
0 am. C N .e"
C0C r%. V4 "-4 M0 Ina. A 0o W- CO1 n W* m~ r%
C. r.. 4Y4 C% jN N
4A
Q>4w 1L w
%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
44'
25
Nix~-'I
"I00
i /
2200
/ \I
1800I
CL 16 00 0 0 0
100010 20 70250 20 10~
,. ue .FuterAprx /in fo h I Ovrrsuebte \ I
Boo
1010
T00-e s
206 .010I
200: I!'' "
I . .I . .i _ 1 _ . _ .i , i , ,
TDISTANCE PAN7 MR$TS (ft)
Figure 9. Further Approximations for the Peak Overpressure Ibttwwnt'g. 7JTwo Simultaneous I14lT Surface Bursts 4500 ft Apart -.
l2
- ---- --' -, " " ..
' " " ,<. ' : ... '< "**'" " ; ''k' h/,"., .r +< , -.,">. ".< l " , ., / .'7 ,, -"V. T.7'7--=77--f, "''
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 aprogram referred to as LAMB [6].
The essential features of this procedure are as follows:
Mass conservation; assume:
npc + i Api, (p 0.5 p0 ) (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
27
VWF
S,.
o PR/pO
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, infact, 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 wavehistory 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 ith blast wave# and p and v are theresultant 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.
294°
Conservation of eaergy; assume:
in which APi = 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 pv2
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 + 4yP0\ P/. (-1) AP + 2yPo
AP LAMB (yy (7
More rigorously (as in Equation 3),
LP (3y-l)AP + 4yP(
AP R = (y-1)AP + 2yP0 (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).
1LThe 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 approaching7/8, 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 derivedfrom 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 / I IIII I
1000 I LMSt
I APPROXIMATION600
400 %
2W0
1 i 14 m W U I= = 0Z 2400 260 aSISTAKNI FIN MSTS (Ift)X, ZW U N 250 230 0 2100 1900 1700
Figure 11. Approxlmttons for the Peek Overpressure between"Two Slultanemus 1 MT Stface lursts 4500 ft ApartC ared with the LM Obdl
32
Table 5. Range Coverage Comparisons(1-MT Bursts 4500 ft Apart)
PEAK SINGLE DISTANCE PERCENTAPPROXIMATION OVERPRESSURE BURST ADDED BY RANGE
CASE COVERED RANGE INTERACTION INCREASENUMBER (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 ata 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 peakoverpressure 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 aheadof and behind the transmitted shock (Case 6).
The LAMB procedure is not rigorously correct, but itprovides 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 theLAM-derived values. Accepting all the apparent uncertaintyin these approximations to multiple shock interactions, one
can still conclude that:
* The region of enhanced overpressures from twosimultaneous 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 (AP5 > 300 psi) is of the order of10-50 percent, and, ,more probably, less than30 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," AnnualReview 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 RandCorporation, RM-1824-AEC, 1956.
4. H. L. Brode, The Blast Wave in Air Resulting from a Hi hTemperature, Hih Pressure Shere in Air, The RanCorporation, RM-1825-AEC, 1956.
5. C. E. Needham, M. L. Havens and C. S. Knauth, Nuclear BlastStandard (1 kT), Air Force Weapons Laboratory, KrtlandA-r Force Base, New Mexico, AFWL-TR-73-55, (Revised) 1975.
6 C. E. Needham, Private Communication, Air Force WeaponsLaboratory, Kirtland Air Force Base, New Mexico, November1977.
36i
If1I
* II|
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