John B. Bdzil- High-Explosives Performance

High-Explosives Performance

Understanding the effects of a finite-length reaction zone

John B. Bdzil

with Tariq D. Aslam, Rudolph Henninger, and James J. Quirk

High explosives—explosiveswith very high energy density—are used to drive the implo-

sion of the primary in a nuclearweapon. That circumstance demandsprecision in the action of the highexplosive. To predict with high accura-cy the course of energy release undervarious conditions is therefore animportant problem that we face in cer-tifying the safety, reliability, and per-formance of nuclear weapons in thestockpile. Here we survey our progresson the problem of explosives perform-ance: predicting the outcome of inten-tional detonation of high explosives incomplex three-dimensional (3-D)geometries. The problems of safety(accidental initiation) and reliability(reproducible response to a prescribedstimulus) are also under investigationbut will be only briefly mentioned here.

Explosives belong to the class ofcombustibles known as energeticmaterials, which means that they con-tain both fuel and oxidizer premixedon a molecular level. Such materialscan support a whole range of combus-tion, including ordinary combustion

such as that in a match head. Ordinarycombustion is a coupled physico-chemical process in which the inter-face separating fresh from burnt ener-getic material travels as a wavethrough the sample. Exothermicchemical reactions begin on the sur-face of the match head and burn theouter layer of material. The heatreleased is transferred through thermalconduction to an adjacent unreactedlayer until that second layer ignites,and this layer-by-layer process contin-ues until the entire sample is con-sumed. The speed of the combustionwave is relatively low, depending onboth the rate of energy transport fromone layer to the next and the rate ofthe local exothermic chemical reac-tions in each layer.

Explosives, in contrast, supportvery high speed combustion known asdetonation. Like an ordinary combus-tion wave, a detonation wave derivesits energy from the chemical reactionsin the material, but the energy trans-port occurs not by thermal conductionbut rather by a high-speed compres-sion, or shock, wave. The high-pres-

sure detonation wave streaks throughthe material at supersonic speeds,turning the material into high-pres-sure, high-temperature gaseous prod-ucts that can do mechanical work atan awesome rate. Figure 1 shows theinitiation of a detonation wave fromshock compression through the forma-tion of a self-sustaining Zeldovich–von Neumann–Doring (ZND) deto-nation reaction zone behind the shock.The power delivered by an explosivedepends on its energy density and its

96 Los Alamos Science Number 28 2003

The pressure plot on this opening pageshows a steady-state detonation wavepropagating through a cylindrical explo-sive (gray) confined by a low-densityinert material (yellow). Red is the high-est pressure; purple, the lowest. Thereaction starts along the shock (redcurve) and ends along the sonic surface(white curve). A large pressure drop atthe edge of the explosive leads to a sig-nificant lengthening of the chemicalreaction zone near the edges of the deto-nating explosive, a reduction in thespeed of the detonation wave, and thedevelopment of a curved detonationshock front.

detonation wave speed. Solid highexplosives, like those used in nuclearweapons, have a detonation speed ofabout 8000 meters per second (m/s),or three times the speed of sound inthe explosive, a high liberated energydensity of about 5 megajoules perkilogram (MJ/kg), and an initial mate-rial density of about 2000 kilogramsper cubic meter (kg/m3). The productof these three quantities yields theenormous power density of80,000,000 MJ/m2/s or 8 × 109 wattsper centimeter squared (W/cm2). Bycomparison, a detonation with a sur-face area of 100 centimeters squaredoperates at a power level equal to thetotal electric generating capacity of

the United States! This very rapid rateof energy liberation is what makessolid explosives unique and useful.

The legacy weapons codes havelong used the simple Chapman-Jouguet (CJ) model to compute theperformance of high explosives. Inthis classical, one-dimensional (1-D)model of detonation, it is assumedthat the chemical reaction rate is infi-nite (and therefore the length of thereaction zone is zero rather than finite,as in the opening figure and Figure 1).That assumption leads to the predic-tion that the detonation speed is con-stant. Moreover, the values of the det-onation speed, DCJ, as well as the det-onation pressure, PCJ, are independent

of the initiating shock strength anddepend on only certain properties ofthe explosive before and after passageof the detonation front, namely, theinitial density of the unreacted materi-al, the liberated energy density of theexplosive, and the pressure–volume(P-v) response function of the reactedmaterial (called the mechanical equa-tion of state, or EOS). In this CJ limit,the explosive performance problem isreduced to providing an accuratemechanical EOS for the gaseous prod-ucts of detonation, Eg (P,v)—seeFigure 2.

In this article, we focus on anotheraspect of the performance problem:creating accurate 3-D detonation mod-

Number 28 2003 Los Alamos Science 97


L1

Distance

L2

(a)

(b)

(c)

Pre

ssur

e (M

bar)

Pre

ssur

e (M

bar) 0.3

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0

0.5 1.0 1.5 2.0

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0

–5 0 5 10

Time (µs)

Distance (mm)PZND = 0.36 Mbar

ZNDreactionzone

PCJ = 0.27 Mbar

t0 t1 t2 t3

Figure 1. Initiation and Propagation of a ZND Detonation Wave(a) A schematic 1-D (planar) experiment is shown at different times. In theexperiment, the impact of a plate thrown on one face of a cube of explosive(t = t0) produces a planar shock wave (t = t1) that gradually accelerates (t =t2) to a steady-state detonation (t = t3) as the shock sweeps through theexplosive and causes chemical energy to be released to the flow at a finiterate. (b) The corresponding pressure-vs-distance snapshots show the evolu-tion of an essentially inert shock wave at t = t1 growing into a classical 1-DZND detonation structure at t = t3, namely, a shock, or pressure, discontinu-ity at the ZND point followed by decreasing pressure through the reactionzone, ending at the CJ point, the pressure predicted by the simple CJ model(see text). (c) Pressure-vs-time plots for material particles originally at the shock front locations in (b) show the particle pres-sure (or velocity) histories in the form measured in actual experiments (see Figures 5, 6, and 7). Only at the location of the right-most particle has a ZND detonation fully formed. Note: The point of maximum acceleration of the shock, called the point of deto-nation formation, coincides with the shape change in the pressure profile and the first appearance of a choked flow condition(sonic condition). Refer to Figure 3.

els that account for the effects offinite chemical reaction rates (andtherefore a finite reaction-zonelength behind the detonation front).The finite length of the reaction zonehas many effects. For example, it canaffect the detonation speed andtherefore the power level at which adetonation engine operates on inertmaterials. It also places limits on theminimum size of the explosive andthe minimum input pressure that willlead to detonation, especially ingeometries that cause detonation

waves to go around corners, say, neara small detonator. The models wehave been developing are specificallydesigned for adaptation to the legacycodes and to the Advanced Simulationand Computing (ASCI) high-fidelitycodes used to study weapons perform-ance. Known as detonation shockdynamics (DSD), these are subscale(or subgrid) models that capture thephysics of the reaction zone withoutexplicitly modeling that zone and,therefore, without requiring enormouscomputing time. Although they are

state of the art for modeling 3-D deto-nation flows, our models predict deto-nation propagation only in homoge-neous explosives under standard con-ditions. That is, they do not fullyaccount for the effects that the hetero-geneity of the real explosives we usetoday have on detonation. We there-fore conclude this article with ourvision for the future of detonationpropagation modeling—one thataccounts for that heterogeneity yetremains practical for weapons per-formance studies.

The Detonation Process

How does a detonation wave reachand then maintain such enormouspower levels as it sweeps through theexplosive? The enormous pressures (afew hundred thousand atmospheres, ora few hundred kilobars) and tempera-tures (2000 to 4000 kelvins) behindthe detonation front originate from thevery rapid release of chemical energy.Reactions are 90 percent complete inless than a millionth of a second. As aresult of this rapid release, the reac-tion zone is very short. But how arethe pressures sustained?

As shown in Figure 3, the reactionzone is bounded by two surfaces thatisolate it from the regions ahead andbehind it and thereby maintain itsextreme pressure. First is the shocksurface, which initiates the reaction.Because it travels at supersonic speedrelative to the unreacted material, itprevents any leakage of pressureahead of the shock. Second is thesonic surface (labeled choked-flowstate), which moves at the localspeed of sound in the frame of themoving shock front. To explain theeffect of this surface, we consider anobserver riding with the shock andlooking back. The observer sees anincreasing amount of energy releaseback into the reaction zone as a func-tion of distance. This energy release



0.3

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0.05

00.5 1.0 1.5 2.0

Pre

ssur

e (M

bar)

Volume (v/v0)2.5 3.0 3.5 4.0

CJ detonation process (Rayleigh line)

Initialstate

CJ detonation state

Fully expanded stateof detonation products

EOS of detonation products (PSCJ(v) isentrope)

Figure 2. Maximum Work Obtainable from a CJ DetonationThe Eg(P, v) mechanical equation of state (EOS) is required to model explosive per-formance. Most often, it is measured along a restricted curve—an isentrope, PS(v),or a shock Hugoniot curve, PH(v)—in the state space defined by the thermodynamicvariables Eg, P, and v. To characterize the maximum work that a detonation can per-form (shaded area above), we need determine only the principal or CJ expansionisentrope of the detonation products, PSCJ(v), given that we know DCJ, and PCJ. Thetwo curves shown above are the detonation Rayleigh line, shown in red (detonationprocess), and the detonation products expansion isentrope, PSCJ(v). The area underthe isentrope (to some cut-off pressure) minus the area under the Rayleigh line(work done by the shock in compressing the explosive) is the maximum mechanicalwork that can be obtained from the explosive. For our high-performance, mono-molecular explosives, such as HMX, this work compared with the available explo-sive energy can be very high (more than 90%). We perform experiments to measurethis isentrope and then construct the Eg(P, v) mechanical EOS for the products ofdetonation, which is an essential ingredient in every model of how detonations dowork on their surroundings. We are working on both better theoretical (Shaw 2002)and experimental (Hill 2002) methods for determining the Eg(P, v) EOS.

serves to accelerate the flow awayfrom the shock front and reduce thepressure, in much the same way as arocket nozzle accelerates the gasejected from a rocket and therebypropels the rocket forward. As thereaction is completed, the flow speedat the end of the reaction zonebecomes equal to the local speed ofsound in the frame moving with theshock, CCJ. As a result, the flowbecomes choked and thereby stopsany further pressure decrease in thereaction zone. Collectively, these twoeffects are referred to as inertial con-finement.

Another way to understand inertialconfinement at the sonic surface is tonote that the postreaction-zone flow(left of the sonic surface) in the ref-erence frame of the shock is super-sonic. Consequently, the reactionzone is essentially isolated from dis-turbances originating in the flowbehind it. Insulated from its surround-ings, detonation is self-propagating,depending only on what is happeningin the reaction zone.

Real vs Idealized Explosives

If an explosive is to be useful inengineering applications—be theymining, nuclear weapons, or modern“smart” munitions—its chemical reac-tion rate must be essentially zero atthe ambient state and must becomeextremely fast once passage of ashock wave substantially increases thepressure and temperature in the mate-rial. As mentioned above, in the clas-sical CJ model, the chemical reactionrate after the shock front has passed isinfinite, the reaction-zone length goesto zero, and the detonation wave trav-els through the material at a constantspeed and pressure. In reality, theexplosives we use in practical applica-tions do not behave like the ideal CJmodel but have finite reaction rates.This situation is indeed fortunate. Ifthe reaction rate were infinite and thereaction zone of length zero, then sub-jecting even a tiny region of theexplosive to a high pressure or hightemperature would initiate detonationof the entire sample. The extreme sen-

sitivity of explosives such as nitro-glycerine is legendary in this regard.

Because the reaction rate and reac-tion-zone length of practical explo-sives depend significantly on pressureand temperature, a sample subjectedto a weak initial shock will experiencetransients during initiation of detona-tion. If the sample is a slab of finitethickness (L1 in Figure 1) but infinitelateral extent (L2→∞), the shock canpass through the slab in a short timecompared with the duration of thetransient, and no detonation occurs.Conversely, to initiate detonation in asample of finite thickness with finitereaction-zone length, the shock musthave a finite strength. That decrease insensitivity caused by a finite reaction-zone length is what makes practicalexplosives safe enough to handle.

The fact that real explosive sam-ples have a finite lateral extent (L2 inFigure 1) also contributes to reducingsensitivity. Some of the energyreleased in the reaction zone leaks outof the sides of the explosive trans-verse to the direction of detonationpropagation and thereby reduces thesupport for the forward motion of theshock. If that energy loss is too great,detonation dies out. Thus, the longerthe reaction zone in practical explo-sives, the more difficult they are todetonate—or, in other words, themore insensitive (and safer) they are.

One way to control sensitivity isto control the “effective,” or global,reaction rate as opposed to the localreaction rates. Alfred Nobel usedthis technique to turn the liquidexplosive nitroglycerine into dyna-mite. Nitroglycerine is an extremelysensitive explosive because its highviscosity allows it to form bubbleseasily. When these bubbles collapse,they generate localized high pres-sures and temperatures calledhotspots. The hotspots serve as initi-ation sites for localized, rapid reac-tion, leading to the establishment oflocalized detonation that spreads



Chemical energy releasezone

Chemicalinduction

zone

Shock state(ZND)

Choked-flow state(CJ)

ηrz

U = – CCJU = –DCJ

Supersonic inflow

U

U

Figure 3. The Finite-Length, Self-Sustaining Reaction ZoneIn many respects, the self-sustaining detonation reaction zone operates like arocket motor. The reaction zone is bounded by the shock surface at the detonationfront and the choked flow-state surface some distance behind. Those two surfacesisolate the reaction zone from the regions in front of it and behind it and therebymaintain its extreme pressure. Looking backward in the frame that moves with thedetonation shock front, one observes that an increasing amount of heat added tothe flow with increasing distance into the reaction zone acts like a nozzle in arocket, accelerating the flow to sonic speeds, CCJ.

through the otherwise cool materialand consumes it all. By adding ahighly porous silica to nitroglycerine,Nobel turned the material into apaste, thereby suppressing small bub-ble formation and dramaticallyreducing its sensitivity.

At Los Alamos, we have followedthe reverse path. We start from a veryinsensitive explosive and increase itssensitivity by using it in the form ofsmall granules that serve as centers forinitiation of chemical reaction hotspotsand subsequent detonation. A typicalexample of these insensitive, high-mass, high-energy-density solid explo-sives is HMX. To detonate a singlecrystal of this material, a few centime-ters on a side and free of most physi-cal defects, requires an input shockpressure of hundreds of kilobars(Campbell and Travis 1985). To

increase the effective, average globalreaction rate, we formulate a mixtureof small, heterogeneous HMX gran-ules and polymeric binder and thenpress the mixture to a densityapproaching that of pure, crystallineHMX. By controlling the size of thegranules and the final pressed density,we can vary the sensitivity of theexplosive. The granular HMX explo-sive PBX 9501 requires only tens ofkilobars of pressure to initiate detona-tion within a fraction of a centimeter.

Despite our control over the manu-facturing and thus the reproducibilityof explosive detonation, we cannotpredict the effective reaction rates insuch heterogeneous HMX explosivesfrom first principles. One reason isthat we have been unable to measurethe chemical route by which the solidexplosives decompose to gaseous

products under the extreme condi-tions of detonation (about 0.5megabar in pressure at temperaturesof 3000 kelvins). Only recently didwe acquire appropriate techniques toaddress those questions. In particular,we can now generate and characterizeplanar shocks using a combination ofultrafast lasers and interferometers. Inthe future, we hope to use ultrafastlaser spectroscopy to observe, in realtime, the chemistry behind thoselaser-generated shocks (McGrane etal. 2003).

We also have little understandingof how the fine-scale substructuresand hotspots in the detonation reac-tion zone affect detonation initiationand propagation. Figure 4(a) shows aphotomicrograph of the granular sub-structure of PBX 9501. Research onthe complex, micromechanical, hydro-dynamic interactions that developwhen such a material is subjected to ashock wave is still in its infancy—seeFigures 4(b) and 4(c).

Measuring Reaction-Zone Effects

Because the length of the effec-tive reaction zone affects the sensi-tivity to initiation as well as the det-onation speed and extinction rates,we would like to predict its size.Since we cannot predict the reactionscale ab initio, we have taken a morephenomenological approach. That is,we have performed macroscale con-tinuum experiments to measure theeffects of the reaction-zone length,and we have developed continuumtheories and models that, whenforced to match those measurements,allow us to infer the global reaction-zone lengths and reaction rates.

The experiments are done on sam-ples whose dimensions run from a fewto many centimeters. Some experi-ments subject the explosive sample to1-D detonation hydrodynamic flows



TsPsρsUs

Reacted “gas”

Reactionsheet “fire”

Unreacted“solid”

50 µm

PgUg

Tg ρg

543210Length (mm)

Wid

th (

mm

)

Piston

800°

700°

600°

500°

400°

300°K

0

1

Figure 4. Substructure of Heterogeneous High Explosives(a) The photomicrograph shows the granular substructure of PBX 9501 (Skidmore etal. 1998). (b) A numerical simulation shows the temperature distribution that devel-ops in a heterogeneous material subjected to rapid, compressive loading (Menikoffand Kober 1999). (c) The drawing shows a detailed view of the hotspots that developwhen explosives such as PBX 9501 are subjected to a shock wave. We are not yetable to accurately model such complex, micromechanical hydrodynamic interactions.

(a)

(b)

(c)

(planar impacts in simple explosivegeometries), whereas others generateand measure fully 3-D flows (result-ing mostly from complex explosivegeometries). Although the reaction-zone length can be quite short—about 0.01 millimeter for some of thesensitive explosives—its effects canbe detected because detonationhydrodynamics tends to amplify anychanges in initial or boundary condi-tions. (This property can be seen inFigure 1, where the transients occurover a distance of many reaction-zone lengths and later in Figure 10,where the overall displacement of themultidimensional detonation shock ismeasured in a number of reaction-zone lengths.) Still, the experimentsmust be capable of nanosecond timeresolution in order to characterize thehydrodynamic response of theseexplosives.

In the high-resolution 1-D experi-ment shown in Figure 5, a nested set of10 magnetic velocity gauges made ofthin conducting wires is embeddedobliquely, relative to the faces of anexplosive sample, and the sample isplaced in a magnetic field (Sheffield etal. 1999). A planar projectile impactsone of its faces as shown, and the qui-escent sample begins to react and ulti-mately detonates. The active elementsin the gauge package, shown in red,maintain their shape as they move withthe explosive flow in the direction ofthe detonation front. As they move,they cut through the magnetic fluxlines, producing a voltage directly pro-portional to the velocity of the flow ateach gauge location. Thus, the 10active elements track the history of 10different particles in the explosive. Thegauge package has a thickness of 60micrometers and is capable of a timeresolution of 20 nanoseconds. Notethat, because the multiple-gauge pack-age is mounted obliquely to the princi-pal flow direction, the gauges fartherupstream are not perturbed by thosedownstream.



Impactor

Magnetic field(B = 750–1200 G)

Lexan projectile

Gun barrel

Targetplate

Voltage = BLup

packet

Explosivesample

Figure 5. Magnetic Gauge Measurement of 1-D Detonation Flows (a) A nested set of 10 magnetic gauges made of thin conducting wires is embeddedobliquely relative to the faces of an explosive sample, and the sample is placed in amagnetic field of strength B. (b) The impact of a planar projectile initiates detonation.When the active elements in the gauge package, of length L and shown in red, aremoved by the explosive flow, they cut the magnetic flux lines, thereby producing a volt-age directly proportional to the velocity of the flow, up, at each gauge location.Thus,the 10 active elements track the history of 10 different particles in the explosive.Thegauge package has a thickness of 60 µm and is capable of a time resolution of 20 ns(Sheffield et al. 1999).

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ticle

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ocity

, up(

mm

/µs)

1.2 1.6

A

B

C

Figure 6. Magnetic Gauge Particle Histories for 1-D Detonation Flow Shown here are the results from a magnetic gauge experiment on the insensitivehigh explosive PBX 9502. The input pressure was 0.135 Mbar. The experimentaltraces follow the transformation of a planar shock wave into a detonation. (Point A isthe input state, point B indicates an interior velocity maximum, and point C is theZND point. The shape change in the particle velocity profile (from an interior velocitymaximum to a maximum at the shock) coincides with the first appearance of achoked flow condition, or sonic condition (Sheffield et al. 1998).

(a)

(b)

Figure 6 shows 10 particle-velocityhistories, one from each magneticgauge, for the insensitive high explo-sive PBX 9502 subjected to a planarimpact with an initial pressure of0.135 megabar (Sheffield et al. 1998).The series of particle histories (fromleft to right) reflects the transforma-tion of a planar shock wave supportedby the energy from the projectileimpact into detonation supported bythe energy release in the explosive.The gauges nearest the impact sur-face (leftmost trace) record theprogress of what is essentially aninert shock wave passing through theexplosive, whereas the later gaugesshow what, at least at first glance,resembles a classical ZND detonationstructure (a shock followed bydecreasing particle velocity throughthe reaction zone, as in Figure 1).The shock speed, also recorded inthese experiments, shows an initialconstant-velocity shock that thenaccelerates rapidly to a new, higherspeed (approaching the detonationspeed). The point of maximum accel-eration, called the point of detonationformation, coincides with the shapechange in the particle-velocity profileand the first appearance of the condi-tion of choked flow (sonic condition).

Modeling the DetonationReaction Zone

To infer more specific informationon the global heat-release rate and thedetonation reaction-zone length fromthese and other hydrodynamic meas-urements, we must model the detona-tion process. We first discuss the stan-dard modeling paradigm. By compar-ing its predictions with experiment,we show that it can model 1-D flowsfairly well but has serious shortcom-ings when applied to 3-D flows.Finally, we show how we have alteredthe standard paradigm to create theDSD model that not only solves some

of those shortcomings but also is com-putationally efficient and suitable foruse in the ASCI codes.

In the standard models, a detonatingexplosive is described as a continuousmedium that obeys the conservation ofmass, momentum, and energy for anEuler fluid:

1) (1)

(2)

(3)

where I is the identity matrix, u is theparticle velocity in the laboratoryframe, P is the pressure, ρ = v–1 is thedensity, e = E + u ⋅ u/2, and E is thespecific internal energy of the reactingexplosive as a function of density andpressure. The energy E as a functionof pressure and specific volume, E(P,v), is the particular constitutive law (alaw determined solely by the intrinsicproperties of the material) that weintroduced earlier as the mechanicalEOS, and it must be provided as inputto the fluid equations.

Because the much used Chapman-Jouguet theory requires as input amechanical EOS of the form, Eg(Pg,vg), where the subscript g denotes det-onation product gas, some realizationsof Eg(Pg, vg) are available. To obtainan analogous expression for unreactedsolid explosive, Es(Ps, vs), one canstart from a simple Mie-GruneisenEOS and calibrate it to replicate themeasured jump-off (shock state) valueof the particle velocity seen with themagnetic gauges (as in Figure 6) andthe measured shock velocity. To con-struct a mechanical EOS for the react-ing mixture of solid and gas, one typi-cally assumes pressure equilibriumbetween the solid and the gas, P = Ps= Pg. Then, to interpolate between theequations of state for the gas and the

solid, one assumes that the internalenergy and density of the mixture aregiven by

by (4)

and

(5)

where λ is the mass fraction of reac-tion product gases.

Closure is brought to this system ofequations, namely, Equations (1)–(5),with two additional assumptions.First, by extending the mechanicalequations of state to include a simpletemperature dependence and thenassuming that the temperatures of thetwo phases are equal, T = Ts = Tg, onecan relate vs and vg and thereby elimi-nate these intermediate variables fromthe problem. Second, one assumesthat the rate of conversion of solid togas in the reaction zone is given by anaverage, effective global heat-releaserate law of the form

(6)

where the heat-release rate functionR(P, v, λ) is constituted so that thegauge data in Figure 6 and other rate-dependent data are reproduced.

The Lee-Tarver Ignition and Growthmodel (Tarver and McGuire 2002) is anexample of the standard modeling para-digm. It uses an empirical EOS foreach of the components and takesappropriate account of the detonationenergy in the unreacted explosive. Italso uses an empirical form for theglobal heat-release rate function inEquation (6). The sets of constants forthe equations of state of each explosivehave been calibrated to a suite ofhydrodynamic experiments performedon each explosive (see the box on theopposite page).

The authors and Ashwani Kapila of

∂λ∂t

P+ ⋅ ( )u ∇∇λ λ= R v, , ,

1 1ρ λ λ= −( ) +v vs g ,

E P E P

E P

, , ,

, ,

v v

v

λ λ

λ

( ) = −( ) ( )+ ( )1 s s

g g

∂∂ρ ρe

te P+ ⋅ +( )[ ] =∇∇ u 0 ,

∂∂ρ ρu

uu It

P+ ⋅ +( ) =∇∇ 0 ,

∂ρ∂

ρt

+ ⋅( ) =∇∇ u 0 ,



Rensselaer Politechnic Institute (privatecommunication—2003), used the Lee-Tarver Ignition and Growth model topredict the results of the magnetic gaugeexperiment for PBX 9502 shown inFigure 6. To solve Equations (1)–(6),these authors and others have developedsolution algorithms and adaptive meshrefinement codes (Aslam 2003, Fedkiwet al. 1999,Quirk 1998, Henshaw andSchwendeman 2003). Here, we used aminimum of 1000 computational zonesto model the reaction zone. Figure 7compares the simulation results with theexperimental data for PBX 9502. Thewave profile that develops far from theinitiating piston surface (that is, to thefar right) clearly shows a nearly steady-state reaction zone. We see that this phe-nomenological model—a simple homo-geneous fluid model with a global reac-tion rate—mimics a 1-D experimentreasonably well. However, it does notdescribe the complicated interactionbetween hydrodynamic hotspots andfundamental chemical processes, aninteraction that produces the heatrelease rate in effective, global, hetero-geneous explosives.

Application of StandardModels to Multidimensional

Flows

The class of models just describedhas been applied to problems withfully 3-D geometries, but our studiesshow that the solutions contain fea-tures that are unsatisfactory for usein real performance codes. As anexample, we consider the propaga-tion of a detonation wave in a stackof right-circular cylinders of explo-sive—see Figure 8(a). The object isto predict the progress of detonationas it diffracts from a smaller to alarger coaxial cylinder. We simulatedthis experiment with a model similarto that described above but with asimpler EOS and a simpler rate lawfor the reaction zone. The EOS for



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Experiment—dashed lines

Calculation—solid lines

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ocity

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EOS and Rate Law for the Ignition and Growth Model

The empirical Jones-Wilkins-Lee forms are used for both the solid (i = s) andthe gas (i = g)

EOS

where Vi = ρ0/ρi and ωi, CVi, Ai, Bi, R1i, and R2i are calibration parameters.

The internal energy is solved for using the thermodynamic constraint

The heat-release rate law is given by

Rate Law

where 0 ≤ λ ≤ 1 describes the progress of the global heat-release reaction(λ = 0, is unreacted, and λ = 1 is fully reacted), H(λ*

i – λ) is the unit stepfunction, and I, a, G1, G2, and λ*

i are parameters.

Figure 7. Comparison of Model with Experiment for 1-D DetonationPredictions of the standard modeling paradigm are compared with the measuredparticle histories for PBX 9502 shown in Figure 6. The calculations were done usingthe Lee-Tarver EOS and a recalibrated rate law. The computing mesh had a mini-mum of 1000 computational zones in the reaction zone. The agreement is reason-ably good, but there are noticeable discrepancies.

TC

P A R B RV

= − −( ) − −( ){VV Vi

i ii i i i iωexp exp ,1i 2 }

Pe T

P

e

P

TT

P P

−

+

=ρ

ρ ρρ ρ

2 2∂∂

∂∂

ρ ∂∂

∂∂

.

, ,

,.

R P IH a

G H P G H P

v λ λ λ λ

λ λ λ λ λ λ λ λ

( ) = −( ) −( ) − −( )

+ −( ) −( ) + −( ) −( )

∗

∗ ∗

17

1 20 111

2 33

1 1

1 1

V 1

the unreacted and reacted explosive isobtained by setting Ai = Bi = 0 and ωs= ωg = 2 and by simplifying the ratelaw to

(7)

where k is a constant and µ is set to µ= 1/2. All constants were selected tomimic the condensed-phase explosivePBX 9502. In the experiment, theexplosive is embedded in a low-densityplastic. The plastic does not affect theflow in the reaction zone; rather theexplosive behaves as it would if it weretotally unconfined (floating in freespace). We simulated the experimentusing the Amrita (Quirk 1998) compu-tational environment, which providesadaptive mesh refinement, simulationscheduling, and documentation of theresults. We also used the Ghost Fluidinterface-tracking algorithm (Fedkiw etal. 1999) to keep a sharp interfacebetween the explosive and the confin-ing inert plastic and a Lax-Friedrichssolver to update the flow.

Figures 8(b) and 8(c) are compos-ites. Each shows two solutions forthe pressure—one before and oneafter the detonation passes into thewider (acceptor) section of the explo-sive. These two figures differ in thatthey show solutions for two differentenergy-release rate functions R—Equation (7)—proportional to thesquare of the pressure, n = 2, and thecube of the pressure, n = 3, respec-tively. The top and bottom halves ofeach figure show results for differentresolutions in the steady-state ZNDreaction zone, 72 and 18 points,respectively, for the n = 2 solution,and 18 and 9 points, respectively, forthe n = 3 solution. For both ratelaws, the location of the detonationfront depends significantly on numer-ical resolution. Also, for the n = 3rate law, the detonation is highlyunstable: There are very large pres-sure and high-frequency structures inthe acceptor explosive (wider sec-

R P kPn, , ,v λ λ( ) = −( )1 µ



Interface at time 1(heavy black lines)

Detonation front at time 1 Detonation front at time 2

Fine grid Medium

Interface at time 2(heavy white lines)

(b) Release Rate Exponent n = 2

(a) Experimental Setup

Hig

h re

solu

tion

(c) Release Rate Exponent n = 3

Detonation front at time 1Detonation front at time 2

Low

res

olut

ion

Hig

h re

solu

tion

Low

res

olut

ion

Donor explosive

Acceptorexplosive

Igniter

Fine Medium

CoarseMedium

Fine

CoarseMedium

Fine

Figure 8. Standard Modeling of Detonation in 3-D GeometriesIn (a) we show the setup for detonation wave propagation in a stack of right-circularcylinders, and in (b) and (c) we show the results from the standard modeling para-digm with different energy-release rate laws, n = 2 and n = 3 with µ = 1/2 in Equation(7). In each case, top and bottom figures display the results for different resolutions:72 and 18 points in the reaction zone for n = 2 and 18 and 9 points in the reactionzone for n = 3, respectively. In both cases, the results change markedly with increas-ing resolution. Also, with the more pressure-sensitive rate law, n = 3, the reactionzone shows high-frequency structure that is an artifact of this modeling paradigmfor heterogeneous explosives.

tion), and the details of the instabilityare very much resolution dependent.

Short et al. (2003) have analyzedthe stability of the classical, steady-state ZND reaction-zone structure tosmall, multi-dimensional disturbancesand shown that it is unstable to evensmall perturbations whenever n isgreater than 2.1675 for the model thatwe have described here. Figure 9shows the results of this stabilityanalysis. Also, the addition of a noz-zling term to Equation (3) (that termmimics the energy loss from the reac-tion zone because of multidimensionalflow effects) leads to a further destabi-lization of the reaction zone to 1-Ddisturbances. (Note: The high-resolu-tion n = 2 simulation also shows somesigns of instability.) The root of thisinstability can be understood with thefollowing argument. Detonation in thishomogeneous-fluid model is a balancebetween shock-initiated energy-releas-ing reactions and the acoustic transportof that energy to support the shock.When a pressure perturbation in thereaction zone affects the reaction ratemuch more than it does the soundspeed, then small pressure fluctuationscan disrupt the balance between ener-gy liberation and transport, and insta-bility can be the result.

Problems with the Reaction-Zone Modeling Paradigm

Both the dependence on numericalresolution and the appearance of thehigh-frequency structure in the accep-tor region of the explosive represent asignificant problem for this modelingparadigm. In independent studies ofthis simple model for the n = 2 case,we have shown that to predict the det-onation speed in the donor section towithin 10 m/s requires 50 or morepoints in the ZND reaction zone.(Aslam et al. 1998). This numbertranslates into a very computationallyintensive problem for typical 3-D

engineering scale problems, wherethe reaction-zone length is very muchshorter than the dimension of theexplosive piece. At any instant, about1010 relatively small computationalnodes would be needed in the reac-tion zone, and the computational timeon a large parallel processing com-puter would be about 100 days.Second, the high-frequency, acousti-cally based transverse wave structureis an artifact of this simple homoge-neous-fluid model and is notobserved in our heterogeneous explo-sives. The substructure associatedwith the granular structure of real het-erogeneous explosives derives fromthe material particles, not acousticwaves. In fact, the granularity inhibitsthe formation of large transverseacoustic waves. Thus, although thesimple homogeneous-fluid modelreproduces reasonably well the lead-ing-order features of detonation, suchas the ZND structure, it fails todescribe higher-order features of realheterogeneous explosives.

DSD, a Subscale Model of Detonation

We have championed an approachto the performance problem that isphilosophically different from the stan-dard paradigm just described (Aslam etal. 1996). In the DSD approach, weexploit the fact that the explosivepieces of engineering interest are largecompared with the reaction-zonelength and substitute a subscale modelfor the detailed model of the reactionzone. To construct this subscale model,we consider how the detonation reac-tion zone is influenced by weak curva-ture of the shock front and derive aconstraint equation relating the speedof the detonation front to the shape ofthat front. We then derive a boundarycondition on that equation that relatesedge effects to the detonation waveshape. Thus, on the scale of the explo-sive, the reaction zone becomes a front,a discontinuity, separating fresh fromburnt explosive. In this way, DSDfocuses on the two primary goals ofthe performance problem: accurate pre-diction of (1) the local detonationspeed and detonation arrival times in aweapons simulation and (2) the P-v(pressure-volume) work that the high-pressure detonation products can per-form on the inert materials with whichthe explosive is in contact. As we willsee, this approach also filters out thehigh-frequency features and vastlyreduces the computational require-ments by comparison with the standardmodeling paradigm described above.

Figure 10 shows a detailed view ofthe reaction zone and shock front,with unburnt explosive above andburnt explosive below. The reaction-zone length is short compared withthe explosive charge dimension, L. Inplace of Cartesian coordinates, we usecoordinates that are attached to theshock surface (see the upper inset inFigure 10). The constant η is the dis-tance through the reaction zone nor-mal to the shock surface, and ξ is the



µ = 0.33

µ = 0.67µ = 0.5

Unstable

R = kPn(1 – λ)µ

6

5

4

3

2

10 2 3 4 5

Stable

Transverse wave number

nFigure 9. Stability AnalysisResults for the StandardDetonation Modeling ParadigmThe vertical axis is the pressure expo-nent of the heat-release rate R, andthe horizontal axis is the wave numberof the perturbing transverse distur-bance. For any state above the curve,the ZND detonation is unstable. For µ= 0.5, a ZND detonation is unstable totwo-dimensional disturbances when-ever n > 2.1675.

distance along the shock measuredfrom the centerline of the explosive.Thus, curves of constant η are parallelto the shock, and the lines of constantξ are in the direction of the local nor-mal to the shock surface. Because thefeatures of interest are on the scale ofthe explosive, we define a dimension-less scale, ε = ηrz/L << 1, where ηrzis the scale of the detonation reaction-zone length. This scale aids in the der-ivation of the subscale model.

DSD assumes that the detonationreaction zone departs from its 1-D(planar) steady-state ZND form by asmall amount. That small departure isdetermined by both the size of theshock curvature κ measured in unitsof the reaction-zone length scale ηrz,(ηrzκ = O(ε) << 1) and the departureof Dn, the detonation speed in thedirection of the shock normal vector,from DCJ, the detonation speed for a1-D steady-state wave. The relevant

scaled detonation speed is (Dn/DCJ –1) = D = εD

~.

To construct an asymptotic solu-tion—a solution in the limit that ε <<1 —for the multidimensional detona-tion reaction-zone flow, we first intro-duce into the standard detonationmodel, Equations (1)–(3) and (6), thefollowing slowly changing, scaled,independent variables:

(8)

where φ is the shock normal angledefined in Figure 10. We then expandthe solution vector Y = (ρ, uη P)T as

(9)

(10)

The leading order term in the solutionvector (designated with a superscript0) represents the 1-D ZND solution,whereas the terms proportional topowers of ε bring in the effects ofmultidimensionality and time depend-ence. A compatibility constraintemerges on the solution that forces arelationship between the shock curva-ture κ and various derivatives of thescaled detonation speed D

(11)

where F (D ) is a decreasing functionof D with F (0) = 0, A (D ) > 0 andB (D ) > 0. A term-by-term examina-tion of the right-hand side of thisequation shows that (1) increasing κ(shock curvature) slows the detonationwave, (2) the acceleration termA (D )(DD /Dt) acts like inertia andresists changes in the front speed andshape, and (3) the dissipative termB (D )(∂2D /∂ξ2) damps high fre-

F D A D D

B D D

( ) = + ( )

( ) +

κ

∂∂ξ

D

Dt

. . . ,2

2−

u uξ ξε= +3 2 3 2 . . . ,

Y Y Y Y= + + +( ) ( ) ( )0 1 2 2ε ε . . . ,

~ ~~ , , ,t t= = =ε ξ ε ξ φ ε φ1 2 1 2



Inert

HE

inte

rfac

eS

trea

mlin

e

φs

φc φs = in this example

HE–inert material interface region

η

κ > 0

ξChoked-flow

surface

Shock surfaceStreamlinesξ = constant

Dn

φn

s

ηrz

^

End of reaction

High explosive (HE)

Shock curve

φ Dn

L

Choked-flow curve

η= constant

Figure 10. Multidimensional Reaction Zone and the DSD Shock-Attached Coordinates A multidimensional reaction zone in a cylindrical detonating explosive (gray) isweakly confined at its edges by a low-density inert material (yellow). As shown, thereaction zone is typically short compared with the dimension of the explosivecharge, L. The upper inset depicts a segment of the 3-D reaction zone with theintrinsic shock-attached coordinates used in DSD analysis. In the DSD limit, theratio of the reaction-zone thickness to the scale of the explosive charge is very, verysmall, ηrz/L = O(ε) << 1, and a subscale front model takes the place of the detailedreaction-zone model. The crosshatched area, approximately the width of the reactionzone and straddling the explosive–inert material interface, defines the region wherean analysis of the boundary region is performed. As explained in the text, thatanalysis leads to boundary conditions for the subscale model.

quencies and thus the formation ofkinks on the wave front. The disper-sion relation for the linearized form ofEquation (11)

(12)

reveals that, at low transverse frequen-cies (small values of the wave numberk), Equation (11) predicts dissipative,transverse waves moving along thefront. On the other hand, at hightransverse frequencies (large k),Equation (11) is purely dissipative.Thus, by adopting the scaled variablesof Equation (8), high-frequency fea-tures such as kinks on the front willnot form.

Thus, on the scale of the explosivepiece, the detonation reaction zonelooks like a discontinuity separatingfresh and burnt explosive. Moreover,this discontinuity occurs along a sur-face that propagates according to thedynamics given by Equation (11),which can be viewed as an intrinsicdetonation propagation law for anexplosive. It is important to recognizethat the forms of the coefficient func-tions F (D ), A (D ), and B (D )depend on the material description ofthe explosive (the EOS and the globalheat-release rate).

In addition to specifying a propa-gation law such as Equation (11), weneed to prescribe a boundary condi-tion on the front, where the frontmeets the edge of the explosive piece(see the right portion of Figure 10).Just as we constructed a subscalemodel to mimic the effects of thereaction zone on the front motion, weconstruct a subscale model to mimicthe effect of confinement by adjacentinert materials on the detonationspeed. Roughly speaking, the morecompliant the inert materials, thegreater the deflection of explosivestreamlines and shock angle φ and the

greater the pressure drop. A boundarylayer analysis performed within a dis-tance of one reaction-zone length oneither side of the explosive–inertmaterial interface reveals (see cross-hatched region) that this interactionestablishes a unique shock-edge angle,φc, which is a function of the explo-sive and inert material pair consid-ered. That angle serves as a boundarycondition for Equation (11). Theweaker the confinement, the greaterthe value of φc, up to the point wherethe lateral expansion of the detonationproducts becomes choked (the devel-opment of a sonic state behind theshock, as shown in Figure 10), haltingany further drop in pressure at theshock. This phenomenon occurs at avalue of φ called the sonic angle, φs,which depends solely on the proper-ties of the explosive and is about 45°for our explosives (Aslam and Bdzil2002, Bdzil 1981).

DSD Calibration and Propagation

of Detonation Front

To validate the DSD approach, weused it to compute the detonationfront shapes for the rate law ofEquation (7) and compared the DSDresults with numerical results from thestandard paradigm, Equations (1)–(3),(6), and (7). The good agreement vali-dates the DSD approach, at least for n<2.1675, for which the standardapproach is fairly accurate (Aslam etal. 1998). Although we could havederived a detonation propagation lawdirectly from a calibrated shock-initia-tion model, such as the Lee-TarverIgnition and Growth model, we chose,instead, to derive Equation (11) moregenerically and calibrate it fromexperimental data on multidimension-al detonation. In that way, webypassed artifacts of the homoge-neous-fluid model paradigm and builtin features faithful to real, heteroge-

neous explosives but not easily mod-eled with the standard paradigm.

Calibration data are often obtainedfrom measurements of the detonationspeed and front curvature in explosivecylinders of various sizes (see Figure11). For explosives such as PBX9502, those data can be fit reasonablywell with just the leading term in thepropagation law of Equation (11):

(13)

which specifies a simple relationshipbetween the detonation speed and thedetonation shock-front curvature. Thepropagation law so obtained for PBX9502 predicts that the shock-normaldetonation speed decreases substan-tially with increasing shock-front cur-vature (see Figure 12). Figure 13(a)shows a DSD prediction for the deto-nation front shape at initiation andtwo later times as the detonation prop-agates through an arc of PBX 9502.Figure 13(b), a top-down view, showsthe DSD solution lagging behind thesimple constant-velocity CJ solution.The significant differences betweenthese two argue for the importance ofincluding reaction-zone effects. TheDSD shapes and velocities are in very

κ = ( )F D ,

ω = − ( )± − ( )

ik

k k

2

2 4 2

2

2

B A

A B A



Figure 11. Detonation FrontMeasurement in a Cylinder ofExplosiveThis image, taken by a streak camera,shows the detonation front arrival timevs radius at the planar face of theexplosive cylinder (Hill 1998).

HE edge

Detonationfront trace

good agreement with experiment To obtain the DSD solution shown

in Figure 13, we start from the frontpropagation law and edge boundaryconditions and use level-set methodsto compute the propagation of thefront. These methods work by embed-ding the detonation front in a level-setfunction, Ψ, and then evolving thisfunction according to

(14)

By our definition, the level surface ψ =0 corresponds to the detonation frontinitially and at any subsequent time. Wecompute the actual detonation front byfinding the ψ = 0 contour. The level-setmethodology offers significant compu-tational advantages because it enableseasy handling of complex explosivetopologies and detonation interactions.

This past year, we have developeda first-order accurate, 3-D computa-tional algorithm that uses the level-set method and that runs on parallelcomputing platforms. Results fromthat method are shown in Figure 14,which displays a detonation initiatedsimultaneously at the ends of twosymmetrical legs and propagatingthrough a piece of PBX 9502 withcomplex geometry. The detonationfronts are shown at four different

times. The shape of the explosive,two protruding legs on one side and acylindrical hole on the other, forcesthe detonation fronts to merge (t = t3)and then bifurcate (t = t4). Mergingand bifurcation of different detona-tion fronts are automatically treatedwith the level-set-based DSDapproach. For comparison, the CJwave is also shown (as ahead) in thelast snapshot (t = t4).

New Modeling Paradigms forDetonation Reaction Zones

The methods just described repre-sent the state of the art in detonationmodeling for engineering applications.In closing, we outline our vision forthe future.

The modeling paradigm representedby Equations (1)–(6) has serious short-comings. Principally, such a continuum

∂ψ ∂ ψt Dn+ ∇ = 0 .



Figure 13. DSD Detonation through an Arc of PBX 9502(a) The 3-D composite image shows the progress of the DSD front through an explo-sive arc. Each detonation front is colored by the local instantaneous normal detona-tion speed.The slowing of the detonation near the edges is apparent by the change incolor from red to green. Shown in the inset are plots of the DSD (green) and experi-mental (black) times of arrival of the detonation front along the midline of the planaredge of the arc (measured in units of the cylinder radius).The resulting curves showthe similarity between the DSD and measured wave-front shapes.The DSD and experi-mental plots are set off to display results. (b) This top-down view shows the intersec-tion of the DSD and CJ fronts with a plane passing through the middle of the arc.TheDSD detonation speed slows down by 10% relative to its initial value, whereas the CJdetonation speed is constant. Consequently, there is a growing separation of DSD andCJ fronts.The phase velocities of the DSD wave along the inner and outer surfaces ofthe arc agree well with the experimental values.

7.8

Dn

5.8

x

y z

y x

z

2.0

1.5

1.0

0.5

0

60

(b)

t = 0 µs

t = 10

DSDCJ

t = 19

(a)

70 80 90

Experiment

DSD

t (µs

)

r (mm)

DDSD = 9.961 mm/µsDEXP = 9.953

DDSD = 7.198DEXP = 7.188

DCJ = 7.8

9

8

7

6

5

40 0.2 0.4 0.6

Dn

(mm

/µs)

κ (mm–1)0.8 1.0 1.2

Figure 12. DSD Propagation Lawfor PBX 9502 at 25oCThe DSD propagation law predicts thatdetonation speed normal to the shockDn decreases substantially with increas-ing shock-front curvature κ.

model does not include any phenome-non that might be important for hetero-geneous materials, such as scattering ordispersion of acoustic waves and dissi-pation of energy from the reaction-zonescale to the subreaction-zone scales ofthe hotspots. Current models are effec-tive for homogeneous explosives. Aswe better understand the details of thehotspots and reaction chemistry underdetonation conditions, we will need todevelop better continuum models. Ifsubnano-scale measurements supportour current view that the chemical reac-tions important in detonation areextremely state sensitive, then very

short scale regions where the hotspottemperatures are most extreme willhave a disproportionate effect towardaccelerating the reaction chemistry(Bdzil et al. 1999, Menikoff and Kober1999). These would be subgrain scales,related more to details of the grainshape than the grain volume (Figure 4shows how complex this substructurecan be). The notion of doing numeri-cally resolved meso-scale simulationsof detonation in granular explosivesand then somehow averaging thoseresults to develop appropriate continu-um-level engineering models seems tobe many years away.

Given our lack of detailed informa-tion about the relationship betweenthe geometry of explosive grains andthe constitutive properties of ourmaterials (including, for example,heat conductivity) under detonationconditions, a more realistic midtermgoal is to develop subscale models forthe reaction zone in which behaviorson the grain and subgrain scales areparameterized in terms of longerwavelength variables. We are work-ing to develop rational asymptoticmodels that indicate not only how theexplosives’ global heat-release rateshould be modeled but also how thepresence of granularity and hotspotsin these materials affects the basicmodeling structure—that is, whatmodifications need to be made toEquations (1)–(6). For example, sig-nificant density variations in a mate-rial on a short-wavelength scale willappear on the long-wavelength con-tinuum scale as dispersion termsadded to the basic conservationlaws—refer to Equations (1)–(3). Thepresence of such terms could beexpected to scatter acoustic wavesand inhibit the detonation modelinginstabilities that we have observed inour homogeneous-explosive models.Whatever improved modeling para-digms are developed for the continu-um response of heterogeneous explo-sives, we expect that, for the foresee-able future, models will have to becalibrated to experiments if they areto make the accurate predictions ofdetonation propagation necessary forweapons simulations. �



Dn

9.30

8.55

7.80

7.05

6.30

t1

t2 t3

t4CJ

CJ

DSD

DSDDSD

DSDDetonator

Figure 14. A 3-D DSD CalculationThis example shows how DSD can handle the merging of separated detonationfronts, the acceleration of the detonation in regions where the fronts converge, andthe bifurcation of the detonation wave around obstacles. Detonation starts in the twolegs (left side of the figure) and progresses toward the cylindrical hole on the right.Four snapshots show the progress of the detonation waves through the sample, andthe DSD and CJ calculations are compared on the fourth snapshot. The detonationfront is colored with the local value of the detonation speed. The inset shows anoblique view.

Further Reading

Aslam, T. D. 2003. A Level Set Algorithm forTracking Discontinuities in HyperbolicConservation Laws II: Systems of Equations(to be published in J. Sci. Comput. 19 (1–3),December 2003).

Aslam, T. D., and J. B. Bdzil. 2002. “Numericaland Theoretical Investigations on Detonation-Inert Confinement Interaction.” TwelfthSymposium (Int.) on Detonation, San Diego,California, August 11–16, 2002. Los AlamosNational Laboratory document LA-UR-01-4664 (to be published in Proceedings—Twelfth International DetonationSymposium). [Online]:http://www.sainc.com/onr/detsymp/technicalProgram.htm

Aslam, T. D., J. B. Bdzil, and L. G. Hill. 2000.“Extensions to DSD Theory: Analysis ofPBX 9502 Rate Stick Data.” InProceedings—Eleventh InternationalDetonation Symposium, ONR 33300-5, pp.21–29. Arlington, Virginia: Office of NavalResearch.

Aslam, T. D., J. B. Bdzil, and D. S. Stewart.1996. Level Set Methods Applied toModeling Detonation Shock Dynamics. J.Comput. Phys. 126 (2): 390.

Bdzil, J. B. 1981. Steady-State Two-Dimensional Detonation. J. Fluid Mech.108: 195.

Bdzil, J. B., R. Menikoff, S. F. Son, A. K.Kapila, and D. S. Stewart. 1999. Two-PhaseModeling of Deflagration-to-DetonationTransition in Granular Materials: A CriticalExamination of Modeling Issues. Phys.Fluids 11 (2): 378.

Campbell, A. W., and J. R. Travis. 1985.“Shock Desensitization of PBX-9404 andComposition B-3.” In Proceedings—EighthInternational Detonation Symposium, J. M.Short, Ed., pp. 1057–1068. Naval SurfaceWeapons Center, NSWC MP 86-194.

Fedkiw, R. P., T. Aslam, B. Merriman, and S.Osher. 1999. A Non-Oscillatory EulerianApproach to Interfaces in MultimaterialFlows (the Ghost Fluid Method). J. Comput.Phys. 152 (2): 457.

Henshaw, W. D., and D. W. Schwendeman.2003. “An Adaptive Numerical Scheme forHigh-Speed Reactive Flow on OverlappingGrids.” Lawrence Livermore NationalLaboratory preprint UCRL-JC-151574 (sub-mitted to J. Comput. Phys.).

Hill, L. G. 2002. “Development of the LANLSandwich Test.” In Shock Compression ofCondensed Matter—2001: Proceedings ofthe Conference of the American PhysicalSociety, Topical Group on ShockCompression of Condensed Matter, M. D.Furnish, Y. Horie, and N. N. Thadhani, Eds.Vol. 620 (1), pp. 149–152. Secaucus, NewJersey: Springer-Verlag.

Hill, L. G., Bdzil, J. B., and T. D. Aslam.2000.“Front Curvature Rate Stick Measurementsand Detonation Shock DynamicsCalibration for PBX9502 over a WideTemperature Range.” In Proceedings—Eleventh International DetonationSymposium, ONR 33300-5, pp. 1029–1037.Arlington, Virginia: Office of NavalResearch.

McGrane, S. D., D. S. Moore, and D. J. Funk.2003. Sub-Picosecond Shock Interferometryof Transparent Thin Films. J. Appl. Phys. 93(9): 5063.

Menikoff, R., and E. M. Kober. 1999.“Compaction Waves in Granular HMX,”Los Alamos National Laboratory report LA-13546-MS.

Quirk, J. J. 1998. AMRITA: “A ComputationalFacility (for CFD Modeling).” In 29thComputational Fluid Dynamics, vonKarman Institute Lecture Series. ISSN0377-8312.

Shaw, M. S. 2002. “Direct Simulation ofDetonation Products Equation of State by aComposite Monte Carlo Method,” Twelfth

Symposium (Int.) on Detonation, San Diego,California, August 11–16, 2002 (to be pub-lished in Proceedings—Twelfth InternationalDetonation Symposium).[Online]:http://www.sainc.com/onr/detsymp/technicalProgram.ht

Sheffield, S. A., R. L. Gustavsen, and R. R.Alcon. 2000. “In Situ Magnetic GaugingTechnique Used at LANL—Method andShock Information Obtained.” In ShockCompression of Condensed Matter—1999,AIP Conference Proceedings, M. D.Furnish, L. C. Chhabildas, and R. S.Hixson, Eds., Vol. 505, p. 1043. Secaucus,New Jersey: Springer-Verlag.

Sheffield, S. A., R. L. Gustavsen, L. G. Hill,and R. R. Alcon. 2000. “ElectromagneticGauge Measurements of Shock InitiatingPBX9501 and PBX9502 Explosives.” InProceedings—Eleventh InternationalDetonation Symposium, ONR 33300-5, pp.451–458. Arlington, Virginia: Office ofNaval Research.

Short, M., J. B. Bdzil, G. J. Sharpe, and I.Anguelova. Detonation Stability for aCondensed Phase Reaction Model with aVariable Reaction Order (submitted toCombust. Theory Modell.).

Skidmore, C. B., D. S. Phillips, and N. B.Crane. 1997. Microscopical Examination ofPlastic-Bonded Explosives. Microscope 45(4): 127.

Tarver, C., and E. McGuire. 2002. “ReactiveFlow Modeling of the Interaction of TATBDetonation Waves with Inert Materials.”Twelfth Symposium (Int.) on Detonation,San Diego, California, August 11–16, 2002(to be published in Proceedings—TwelfthInternational Detonation Symposium).[Online]: http://www.sainc.com/onr/det-symp/technicalProgram.htm

Watt, S. D., and G. J. Sharpe. One-Dimensional Linear Stability of CurvedDetonations (submitted to Proc. R. Soc.Lond., Ser. A).



(Left to right): Rudolph Henninger, Tariq Aslam, John Bdzil,and James Quirk.

John B. Bdzil received his Ph.D. in physical chemistry and physics fromthe University of Minnesota in 1969. John began his career at Los Alamosin 1972 as a staff member in the Dynamic Experimentation Division andbecame a fellow of the Laboratory in 2001. His research at Los Alamos hasbeen in detonation hydrodynamics. Along with Prof. Scott Stewart of theUniversity of Illinois, John is codeveloper of detonation shock dynamics, amodel now widely used throughout the world. John characterizes hisapproach to explosive modeling as unique at the national laboratories as itrelies on mathematical analysis and asymptotic methods rather than ondirect numerical simulation. He collaborates extensively with universityresearchers in his work on explosive modeling, including work on damagedexplosive and accidental initiation of detonation in such materials. John hasserved as a member of the trilaboratory support team for the University ofIllinois ASCI Alliances Center for the past five years and is a member of theLaboratory’s University Alliances Board. For many years, he served as ajudge for ski racing events in northern New Mexico.

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