Understanding the Effects of a Finite-Length Reaction Zone

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

igh explosives—explosives such as that in a match head. Ordinary sure detonation wave streaks through with very high energy density combustion is a coupled physico- the material at supersonic speeds, H—are used to drive the implo- chemical process in which the inter- turning the material into high-pres- sion of the primary in a nuclear face separating fresh from burnt ener- sure, high-temperature gaseous prod- weapon. That circumstance demands getic material travels as a wave ucts that can do mechanical work at precision in the action of the high through the sample. Exothermic an awesome rate. Figure 1 shows the explosive. To predict with high accura- chemical reactions begin on the sur- initiation of a detonation wave from cy the course of energy release under face of the match head and burn the shock compression through the forma- various conditions is therefore an outer layer of material. The heat tion of a self-sustaining Zeldovich important problem that we face in cer- released is transferred through thermal –von Neumann–Doring (ZND) deto- tifying the safety, reliability, and per- conduction to an adjacent unreacted nation reaction zone behind the shock. formance of nuclear weapons in the layer until that second layer ignites, The power delivered by an explosive stockpile. Here we survey our progress and this layer-by-layer process contin- depends on its energy density and its on the problem of explosives perform- ues until the entire sample is con- ance: predicting the outcome of inten- sumed. The speed of the combustion The pressure plot on this opening page tional detonation of high explosives in wave is relatively low, depending on shows a steady-state detonation wave complex three-dimensional (3-D) both the rate of energy transport from propagating through a cylindrical explo- geometries. The problems of safety one layer to the next and the rate of sive (gray) confined by a low-density (accidental initiation) and reliability the local exothermic chemical reac- inert material (yellow). Red is the high- (reproducible response to a prescribed tions in each layer. est pressure; purple, the lowest. The stimulus) are also under investigation Explosives, in contrast, support reaction starts along the shock (red curve) and ends along the sonic surface but will be only briefly mentioned here. very high speed combustion known as (white curve). A large pressure drop at Explosives belong to the class of detonation. Like an ordinary combus- the edge of the explosive leads to a sig- combustibles known as energetic tion wave, a detonation wave derives nificant lengthening of the chemical materials, which means that they con- its energy from the chemical reactions reaction zone near the edges of the deto- tain both fuel and oxidizer premixed in the material, but the energy trans- nating explosive, a reduction in the on a molecular level. Such materials port occurs not by thermal conduction speed of the detonation wave, and the can support a whole range of combus- but rather by a high-speed compres- development of a curved detonation tion, including ordinary combustion sion, or shock, wave. The high-pres- shock front.

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(a) t0 t1 t2 t3

L1 L2

Distance

(b) 0.3

0.2 ZND reaction 0.1 zone Pressure (Mbar) 0 –5 0 5 10 Distance (mm) (c) 0.4 PZND = 0.36 Mbar

0.3 PCJ = 0.27 Mbar Figure 1. Initiation and Propagation of a ZND Detonation Wave (a) A schematic 1-D (planar) experiment is shown at different times. In the 0.2 experiment, 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 = 0.1 Pressure (Mbar) t2) to a steady-state detonation (t = t3) as the shock sweeps through the explosive and causes chemical energy to be released to the flow at a finite 0 rate. (b) The corresponding pressure-vs-distance snapshots show the evolu- 0.5 1.0 1.5 2.0 tion of an essentially inert shock wave at t = t growing into a classical 1-D 1 Time (µs) ZND detonation structure at t = t3, namely, a shock, or pressure, discontinuity at the ZND point followed by decreasing pressure through the reaction zone, 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 pressure (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 detonation 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.

detonation wave speed. Solid high the United States! This very rapid rate of the initiating shock strength and explosives, like those used in nuclear of energy liberation is what makes depend on only certain properties of weapons, have a detonation speed of solid explosives unique and useful. the explosive before and after passage about 8000 meters per second (m/s), The legacy weapons codes have of the detonation front, namely, the or three times the speed of sound in long used the simple Chapman- initial density of the unreacted materi- the explosive, a high liberated energy Jouguet (CJ) model to compute the al, the liberated energy density of the density of about 5 megajoules per performance of high explosives. In explosive, and the pressure–volume kilogram (MJ/kg), and an initial mate- this classical, one-dimensional (1-D) (P-v) response function of the reacted rial density of about 2000 kilograms model of detonation, it is assumed material (called the mechanical equa- per cubic meter (kg/m3). The product that the chemical reaction rate is infi- tion of state, or EOS). In this CJ limit, of these three quantities yields the nite (and therefore the length of the the explosive performance problem is enormous power density of reaction zone is zero rather than finite, reduced to providing an accurate 80,000,000 MJ/m2/s or 8 × 109 watts as in the opening figure and Figure 1). mechanical EOS for the gaseous prod- 2 per centimeter squared (W/cm ). By That assumption leads to the predic- ucts of detonation, Eg (P,v)—see comparison, a detonation with a sur- tion that the detonation speed is con- Figure 2. face area of 100 centimeters squared stant. Moreover, the values of the det- In this article, we focus on another operates at a power level equal to the onation speed, DCJ, as well as the det- aspect of the performance problem: total electric generating capacity of onation pressure, PCJ,are independent creating accurate 3-D detonation mod-

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0.3 state of the art for modeling 3-D detonation flows, our models predict deto- CJ detonation state nation propagation only in homoge- 0.25 neous explosives under standard conditions. That is, they do not fully account for the effects that the hetero- 0.20 geneity of the real explosives we use today have on detonation. We there- 0.15 fore conclude this article with our vision for the future of detonation CJ detonation process (Rayleigh line) propagation modeling—one that Pressure (Mbar) 0.10 accounts for that heterogeneity yet EOS of detonation products remains practical for weapons per- (PSCJ(v) isentrope) formance studies. 0.05 Fully expanded state Initial of detonation products state The Detonation Process 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Volume (v/v0) How does a detonation wave reach and then maintain such enormous Figure 2. Maximum Work Obtainable from a CJ Detonation The E (P, v) mechanical equation of state (EOS) is required to model explosive per- power levels as it sweeps through the g explosive? The enormous pressures (a formance. Most often, it is measured along a restricted curve—an isentrope, PS(v), few hundred thousand atmospheres, or or a shock Hugoniot curve, PH(v)—in the state space defined by the thermodynamic variables Eg, P, and v.To characterize the maximum work that a detonation can per- a few hundred kilobars) and tempera- form (shaded area above), we need determine only the principal or CJ expansion tures (2000 to 4000 kelvins) behind isentrope of the detonation products, PSCJ(v), given that we know DCJ, and PCJ. The the detonation front originate from the two curves shown above are the detonation Rayleigh line, shown in red (detonation very rapid release of chemical energy. process), and the detonation products expansion isentrope, PSCJ(v). The area under Reactions are 90 percent complete in the isentrope (to some cut-off pressure) minus the area under the Rayleigh line less than a millionth of a second. As a (work done by the shock in compressing the explosive) is the maximum mechanical work that can be obtained from the explosive. For our high-performance, mono- result of this rapid release, the reac- molecular explosives, such as HMX, this work compared with the available explo- tion zone is very short. But how are sive energy can be very high (more than 90%). We perform experiments to measure the pressures sustained? this isentrope and then construct the Eg(P, v) mechanical EOS for the products of As shown in Figure 3, the reaction detonation, which is an essential ingredient in every model of how detonations do zone is bounded by two surfaces that work on their surroundings. We are working on both better theoretical (Shaw 2002) isolate it from the regions ahead and and experimental (Hill 2002) methods for determining the Eg(P, v) EOS. behind it and thereby maintain its extreme pressure. First is the shock surface, which initiates the reaction. els that account for the effects of waves to go around corners, say, near Because it travels at supersonic speed finite chemical reaction rates (and a small detonator. The models we relative to the unreacted material, it therefore a finite reaction-zone have been developing are specifically prevents any leakage of pressure length behind the detonation front). designed for adaptation to the legacy ahead of the shock. Second is the The finite length of the reaction zone codes and to the Advanced Simulation sonic surface (labeled choked-flow has many effects. For example, it can and Computing (ASCI) high-fidelity state), which moves at the local affect the detonation speed and codes used to study weapons perform- speed of sound in the frame of the therefore the power level at which a ance. Known as detonation shock moving shock front. To explain the detonation engine operates on inert dynamics (DSD), these are subscale effect of this surface, we consider an materials. It also places limits on the (or subgrid) models that capture the observer riding with the shock and minimum size of the explosive and physics of the reaction zone without looking back. The observer sees an the minimum input pressure that will explicitly modeling that zone and, increasing amount of energy release lead to detonation, especially in therefore, without requiring enormous back into the reaction zone as a func- geometries that cause detonation computing time. Although they are tion of distance. This energy release

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Chemical sitivity of explosives such as nitro- Chemical energy release induction glycerine is legendary in this regard. zone zone Because the reaction rate and reac- U tion-zone length of practical explo- U sives depend significantly on pressure U = –D U C CJ and temperature, a sample subjected = – CJ Supersonic inflow to a weak initial shock will experience transients during initiation of detona- Choked-flow state tion. If the sample is a slab of finite (CJ) thickness (L1 in Figure 1) but infinite →∞ Shock state lateral extent (L2 ), the shock can (ZND) pass through the slab in a short time η rz compared with the duration of the transient, and no detonation occurs. Figure 3. The Finite-Length, Self-Sustaining Reaction Zone Conversely, to initiate detonation in a In many respects, the self-sustaining detonation reaction zone operates like a sample of finite thickness with finite rocket motor. The reaction zone is bounded by the shock surface at the detonation reaction-zone length, the shock must front and the choked flow-state surface some distance behind. Those two surfaces have a finite strength. That decrease in isolate the reaction zone from the regions in front of it and behind it and thereby sensitivity caused by a finite reaction- maintain its extreme pressure. Looking backward in the frame that moves with the zone length is what makes practical detonation shock front, one observes that an increasing amount of heat added to explosives safe enough to handle. the flow with increasing distance into the reaction zone acts like a nozzle in a The fact that real explosive sam- rocket, accelerating the flow to sonic speeds, CCJ. ples have a finite lateral extent (L2 in Figure 1) also contributes to reducing serves to accelerate the flow away Real vs Idealized Explosives sensitivity. Some of the energy from the shock front and reduce the released in the reaction zone leaks out pressure, in much the same way as a If an explosive is to be useful in of the sides of the explosive trans- rocket nozzle accelerates the gas engineering applications—be they verse to the direction of detonation ejected from a rocket and thereby mining, nuclear weapons, or modern propagation and thereby reduces the propels the rocket forward. As the “smart” munitions—its chemical reac- support for the forward motion of the reaction is completed, the flow speed tion rate must be essentially zero at shock. If that energy loss is too great, at the end of the reaction zone the ambient state and must become detonation dies out. Thus, the longer becomes equal to the local speed of extremely fast once passage of a the reaction zone in practical explo- sound in the frame moving with the shock wave substantially increases the sives, the more difficult they are to shock, CCJ. As a result, the flow pressure and temperature in the mate- detonate—or, in other words, the becomes choked and thereby stops rial. As mentioned above, in the clas- more insensitive (and safer) they are. any further pressure decrease in the sical CJ model, the chemical reaction One way to control sensitivity is reaction zone. Collectively, these two rate after the shock front has passed is to control the “effective,” or global, effects are referred to as inertial con- infinite, the reaction-zone length goes reaction rate as opposed to the local finement. to zero, and the detonation wave trav- reaction rates. Alfred Nobel used Another way to understand inertial els through the material at a constant this technique to turn the liquid confinement at the sonic surface is to speed and pressure. In reality, the explosive nitroglycerine into dyna- note that the postreaction-zone flow explosives we use in practical applica- mite. Nitroglycerine is an extremely (left of the sonic surface) in the ref- tions do not behave like the ideal CJ sensitive explosive because its high erence frame of the shock is super- model but have finite reaction rates. viscosity allows it to form bubbles sonic. Consequently, the reaction This situation is indeed fortunate. If easily. When these bubbles collapse, zone is essentially isolated from dis- the reaction rate were infinite and the they generate localized high pres- turbances originating in the flow reaction zone of length zero, then sub- sures and temperatures called behind it. Insulated from its surround- jecting even a tiny region of the hotspots. The hotspots serve as initi- ings, detonation is self-propagating, explosive to a high pressure or high ation sites for localized, rapid reac- depending only on what is happening temperature would initiate detonation tion, leading to the establishment of in the reaction zone. of the entire sample. The extreme sen- localized detonation that spreads

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(a) (c) products under the extreme condi- Unreacted Reacted Reaction “solid” “gas” sheet “fire” tions of detonation (about 0.5 megabar in pressure at temperatures of 3000 kelvins). Only recently did we acquire appropriate techniques to address those questions. In particular, Ts Ps we can now generate and characterize Pg ρ s Ug planar shocks using a combination of T Us g ρ g ultrafast lasers and interferometers. In the future, we hope to use ultrafast 50 µm laser spectroscopy to observe, in real time, the chemistry behind those (b) laser-generated shocks (McGrane et 800° al. 2003).

700° We also have little understanding 1 of how the fine-scale substructures Piston 600° and hotspots in the detonation reac- 500° tion zone affect detonation initiation Width (mm) 400° and propagation. Figure 4(a) shows a photomicrograph of the granular sub- 0 300° 0 1 2 3 4 5 K structure of PBX 9501. Research on Length (mm) the complex, micromechanical, hydro- Figure 4. Substructure of Heterogeneous High Explosives dynamic interactions that develop (a) The photomicrograph shows the granular substructure of PBX 9501 (Skidmore et when such a material is subjected to a al. 1998). (b) A numerical simulation shows the temperature distribution that devel- shock wave is still in its infancy—see ops in a heterogeneous material subjected to rapid, compressive loading (Menikoff Figures 4(b) and 4(c). and Kober 1999). (c) The drawing shows a detailed view of the hotspots that develop when explosives such as PBX 9501 are subjected to a shock wave. We are not yet able to accurately model such complex, micromechanical hydrodynamic interactions. Measuring Reaction-Zone Effects through the otherwise cool material increase the effective, average global and consumes it all. By adding a reaction rate, we formulate a mixture Because the length of the effec- highly porous silica to nitroglycerine, of small, heterogeneous HMX gran- tive reaction zone affects the sensi- Nobel turned the material into a ules and polymeric binder and then tivity to initiation as well as the det- paste, thereby suppressing small bub- press the mixture to a density onation speed and extinction rates, ble formation and dramatically approaching that of pure, crystalline we would like to predict its size. reducing its sensitivity. HMX. By controlling the size of the Since we cannot predict the reaction At Los Alamos, we have followed granules and the final pressed density, scale ab initio, we have taken a more the reverse path. We start from a very we can vary the sensitivity of the phenomenological approach. That is, insensitive explosive and increase its explosive. The granular HMX explo- we have performed macroscale con- sensitivity by using it in the form of sive PBX 9501 requires only tens of tinuum experiments to measure the small granules that serve as centers for kilobars of pressure to initiate detona- effects of the reaction-zone length, initiation of chemical reaction hotspots tion within a fraction of a centimeter. and we have developed continuum and subsequent detonation. A typical Despite our control over the manu- theories and models that, when example of these insensitive, high- facturing and thus the reproducibility forced to match those measurements, mass, high-energy-density solid explo- of explosive detonation, we cannot allow us to infer the global reaction- sives is HMX. To detonate a single predict the effective reaction rates in zone lengths and reaction rates. crystal of this material, a few centime- such heterogeneous HMX explosives The experiments are done on sam- ters on a side and free of most physi- from first principles. One reason is ples whose dimensions run from a few cal defects, requires an input shock that we have been unable to measure to many centimeters. Some experi- pressure of hundreds of kilobars the chemical route by which the solid ments subject the explosive sample to (Campbell and Travis 1985). To explosives decompose to gaseous 1-D detonation hydrodynamic flows

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(a) (planar impacts in simple explosive Target geometries), whereas others generate plate Explosive Magnetic field (B = 750–1200 G) and measure fully 3-D flows (result- sample ing mostly from complex explosive geometries). Although the reaction- zone length can be quite short— about 0.01 millimeter for some of the Impactor packet sensitive explosives—its effects can Lexan projectile be detected because detonation Voltage = BLup hydrodynamics tends to amplify any changes in initial or boundary condi- (b) tions. (This property can be seen in Gun barrel Figure 1, where the transients occur over a distance of many reaction- Figure 5. Magnetic Gauge Measurement of 1-D Detonation Flows zone lengths and later in Figure 10, (a) A nested set of 10 magnetic gauges made of thin conducting wires is embedded where the overall displacement of the obliquely relative to the faces of an explosive sample, and the sample is placed in a multidimensional detonation shock is magnetic field of strength B. (b) The impact of a planar projectile initiates detonation. measured in a number of reaction- When the active elements in the gauge package, of length L and shown in red, are zone lengths.) Still, the experiments moved by the explosive flow, they cut the magnetic flux lines, thereby producing a volt- must be capable of nanosecond time 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. The resolution in order to characterize the gauge package has a thickness of 60 µm and is capable of a time resolution of 20 ns hydrodynamic response of these (Sheffield et al. 1999). explosives. In the high-resolution 1-D experi- 2.5 ment shown in Figure 5, a nested set of C 10 magnetic velocity gauges made of thin conducting wires is embedded 2.0 B obliquely, relative to the faces of an explosive sample, and the sample is placed in a magnetic field (Sheffield et s) A µ 1.5 al. 1999). A planar projectile impacts

(mm/ one of its faces as shown, and the qui- p u escent sample begins to react and ulti- 1.0 mately detonates. The active elements in the gauge package, shown in red, maintain their shape as they move with icle velocity, icle velocity,

rt 0.5 the explosive flow in the direction of Pa the detonation front. As they move, they cut through the magnetic flux 0 lines, producing a voltage directly proportional to the velocity of the flow at each gauge location. Thus, the 10 –0.5 0 0.4 0.8 1.2 1.6 active elements track the history of 10 Time (µs) different particles in the explosive. The gauge package has a thickness of 60 Figure 6. Magnetic Gauge Particle Histories for 1-D Detonation Flow micrometers and is capable of a time Shown here are the results from a magnetic gauge experiment on the insensitive resolution of 20 nanoseconds. Note high explosive PBX 9502. The input pressure was 0.135 Mbar. The experimental that, because the multiple-gauge pack- traces follow the transformation of a planar shock wave into a detonation. (Point A is the input state, point B indicates an interior velocity maximum, and point C is the age is mounted obliquely to the princi- ZND point. The shape change in the particle velocity profile (from an interior velocity pal flow direction, the gauges farther maximum to a maximum at the shock) coincides with the first appearance of a upstream are not perturbed by those choked flow condition, or sonic condition (Sheffield et al. 1998). downstream.

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Figure 6 shows 10 particle-velocity of those shortcomings but also is com- solid, one assumes that the internal histories, one from each magnetic putationally efficient and suitable for energy and density of the mixture are gauge, for the insensitive high explo- use in the ASCI codes. given by sive PBX 9502 subjected to a planar In the standard models, a detonating λλ=− impact with an initial pressure of explosive is described as a continuous EP()by,, vv ()() 1 E ss P , (4) 0.135 megabar (Sheffield et al. 1998). medium that obeys the conservation of + λ EPgg() ,v , The series of particle histories (from mass, momentum, and energy for an left to right) reflects the transforma- Euler fluid: and tion of a planar shock wave supported ∂ρ ρλλ=−()+ by the energy from the projectile 1)+⋅∇ ()ρu = 0 , (1) 11vvsg , (5) impact into detonation supported by ∂t the energy release in the explosive. ∂ρu where λ is the mass fraction of reac- +⋅∇ ()ρuu + IP = 0 , The gauges nearest the impact sur- ∂t (2) tion product gases. face (leftmost trace) record the Closure is brought to this system of progress of what is essentially an equations, namely, Equations (1)–(5), ∂ρe inert shock wave passing through the +⋅∇ []()ρeP +u = 0 , (3) with two additional assumptions. explosive, whereas the later gauges ∂t First, by extending the mechanical show what, at least at first glance, equations of state to include a simple resembles a classical ZND detonation where I is the identity matrix, u is the temperature dependence and then structure (a shock followed by particle velocity in the laboratory assuming that the temperatures of the ρ –1 decreasing particle velocity through frame, P is the pressure, = v is the two phases are equal, T = Ts = Tg, one ⋅ the reaction zone, as in Figure 1). density, e = E + u u/2, and E is the can relate vs and vg and thereby elimi- The shock speed, also recorded in specific internal energy of the reacting nate these intermediate variables from these experiments, shows an initial explosive as a function of density and the problem. Second, one assumes constant-velocity shock that then pressure. The energy E as a function that the rate of conversion of solid to accelerates rapidly to a new, higher of pressure and specific volume, E(P, gas in the reaction zone is given by an speed (approaching the detonation v), is the particular constitutive law (a average, effective global heat-release speed). The point of maximum accel- law determined solely by the intrinsic rate law of the form eration, called the point of detonation properties of the material) that we ∂λ formation, coincides with the shape introduced earlier as the mechanical +⋅u ∇λλ= Rv()P,, , (6) change in the particle-velocity profile EOS, and it must be provided as input ∂t and the first appearance of the condi- to the fluid equations. tion of choked flow (sonic condition). Because the much used Chapman- where the heat-release rate function Jouguet theory requires as input a R(P, v, λ) is constituted so that the mechanical EOS of the form, Eg(Pg, gauge data in Figure 6 and other rate- Modeling the Detonation vg), where the subscript g denotes det- dependent data are reproduced. Reaction Zone onation product gas, some realizations The Lee-Tarver Ignition and Growth of Eg(Pg, vg) are available. To obtain model (Tarver and McGuire 2002) is an To infer more specific information an analogous expression for unreacted example of the standard modeling para- on the global heat-release rate and the solid explosive, Es(Ps, vs), one can digm. It uses an empirical EOS for detonation reaction-zone length from start from a simple Mie-Gruneisen each of the components and takes these and other hydrodynamic meas- EOS and calibrate it to replicate the appropriate account of the detonation urements, we must model the detona- measured jump-off (shock state) value energy in the unreacted explosive. It tion process. We first discuss the stan- of the particle velocity seen with the also uses an empirical form for the dard modeling paradigm. By compar- magnetic gauges (as in Figure 6) and global heat-release rate function in ing its predictions with experiment, the measured shock velocity. To con- Equation (6). The sets of constants for we show that it can model 1-D flows struct a mechanical EOS for the react- the equations of state of each explosive fairly well but has serious shortcom- ing mixture of solid and gas, one typi- have been calibrated to a suite of ings when applied to 3-D flows. cally assumes pressure equilibrium hydrodynamic experiments performed Finally, we show how we have altered between the solid and the gas, P = Ps on each explosive (see the box on the the standard paradigm to create the = Pg. Then, to interpolate between the opposite page). DSD model that not only solves some equations of state for the gas and the The authors and Ashwani Kapila of

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2.5 Rensselaer Politechnic Institute (private communication—2003), used the Lee- Experiment—dashed lines Tarver Ignition and Growth model to 2.0 Calculation—solid lines predict the results of the magnetic gauge experiment for PBX 9502 shown in Figure 6. To solve Equations (1)–(6), s) µ 1.5 these authors and others have developed

(mm/ solution algorithms and adaptive mesh p u refinement codes (Aslam 2003, Fedkiw 1.0 et al. 1999,Quirk 1998, Henshaw and Schwendeman 2003). Here, we used a minimum of 1000 computational zones ticle velocity, ticle velocity, r 0.5 to model the reaction zone. Figure 7 Pa compares the simulation results with the experimental data for PBX 9502. The 0 wave profile that develops far from the initiating piston surface (that is, to the far right) clearly shows a nearly steady- –0.5 0 0.5 1.0 1.5 2.0 state reaction zone. We see that this phe- Time (µs) nomenological model—a simple homogeneous fluid model with a global reac- Figure 7. Comparison of Model with Experiment for 1-D Detonation tion rate—mimics a 1-D experiment Predictions of the standard modeling paradigm are compared with the measured reasonably well. However, it does not particle histories for PBX 9502 shown in Figure 6. The calculations were done using describe the complicated interaction the Lee-Tarver EOS and a recalibrated rate law. The computing mesh had a mini- between hydrodynamic hotspots and mum of 1000 computational zones in the reaction zone. The agreement is reason- fundamental chemical processes, an ably good, but there are noticeable discrepancies. interaction that produces the heat release rate in effective, global, hetero- EOS and Rate Law for the Ignition and Growth Model geneous explosives.

The empirical Jones-Wilkins-Lee forms are used for both the solid (i = s) and the gas (i = g) Application of Standard Models to Multidimensional V EOS T =−−i {PAexp() RVV−− B exp() R } , Flows ω iiiii1i 2 iiCV ρ ρ ω where Vi = 0/ i and i, CVi, Ai, Bi, R1i,and R2i are calibration parameters. The class of models just described has been applied to problems with The internal energy is solved for using the thermodynamic constraint fully 3-D geometries, but our studies show that the solutions contain fea-  ∂  ∂   ∂  ∂  tures that are unsatisfactory for use  − ρ22 eT  + ρ e  T  = P     T . in real performance codes. As an  ∂ρρ ∂P ρρ∂P  ∂  PP example, we consider the propaga- The heat-release rate law is given by tion of a detonation wave in a stack ∗ of right-circular cylinders of explo- Rate Law λλλλ=−− −− 7 RP() ,v , IH()1 ()11()V1a sive—see Figure 8(a). The object is to predict the progress of detonation +−λλ∗∗()− λλ0.111 +−λλ()− λλ3 GH12()11 P GH23() P , as it diffracts from a smaller to a larger coaxial cylinder. We simulated ≤ λ ≤ where 0 1 describes the progress of the global heat-release reaction this experiment with a model similar λ 0 λ 1 λ* λ ( = , is unreacted, and = is fully reacted), H( i – ) is the unit step to that described above but with a λ* function, and I, a, G1, G2, and i are parameters. simpler EOS and a simpler rate law for the reaction zone. The EOS for

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(a) Experimental Setup the unreacted and reacted explosive is ω Igniter Donor explosive obtained by setting Ai = Bi = 0 and s ω Acceptor = g = 2 and by simplifying the rate law to explosive

µ RP()v λλ=− kPn () ,,1 , (7) (b) Release Rate Exponent n = 2 Fine grid Medium where k is a constant and µ is set to µ = 1/2. All constants were selected to mimic the condensed-phase explosive PBX 9502. In the experiment, the explosive is embedded in a low-density plastic. The plastic does not affect the flow in the reaction zone; rather the High resolution explosive behaves as it would if it were Detonation front at time 1 Detonation totally unconfined (floating in free front at space). We simulated the experiment time 2 using the Amrita (Quirk 1998) computational environment, which provides adaptive mesh refinement, simulation scheduling, and documentation of the Fine Low resolution Low results. We also used the Ghost Fluid Medium Coarse interface-tracking algorithm (Fedkiw et al. 1999) to keep a sharp interface Interface at time 2 Interface at time 1 between the explosive and the confin- (heavy white lines) (heavy black lines) ing inert plastic and a Lax-Friedrichs (c) Release Rate Exponent n = 3 solver to update the flow. Fine Medium Figures 8(b) and 8(c) are compos- ites. Each shows two solutions for the pressure—one before and one after the detonation passes into the wider (acceptor) section of the explosive. These two figures differ in that High resolution Detonation front at time 1 they show solutions for two different Detonation energy-release rate functions R— front at Equation (7)—proportional to the time 2 square of the pressure, n = 2, and the cube of the pressure, n = 3, respec- Fine tively. The top and bottom halves of Medium

each figure show results for different resolution Low Coarse resolutions in the steady-state ZND reaction zone, 72 and 18 points, respectively, for the n = 2 solution, and 18 and 9 points, respectively, for Figure 8. Standard Modeling of Detonation in 3-D Geometries the n = 3 solution. For both rate In (a) we show the setup for detonation wave propagation in a stack of right-circular laws, the location of the detonation cylinders, and in (b) and (c) we show the results from the standard modeling para- front depends significantly on numer- digm with different energy-release rate laws, n = 2 and n = 3 with µ = 1/2 in Equation ical resolution. Also, for the n = 3 (7). In each case, top and bottom figures display the results for different resolutions: rate law, the detonation is highly 72 and 18 points in the reaction zone for n = 2 and 18 and 9 points in the reaction zone for n = 3, respectively. In both cases, the results change markedly with increas- unstable: There are very large pres- ing resolution. Also, with the more pressure-sensitive rate law, n = 3, the reaction sure and high-frequency structures in zone shows high-frequency structure that is an artifact of this modeling paradigm the acceptor explosive (wider sec- for heterogeneous explosives.

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tion), and the details of the instability 6 DSD, a Subscale Model are very much resolution dependent. R = kP n(1 – λ)µ of Detonation Short et al. (2003) have analyzed 5 the stability of the classical, steady- We have championed an approach 4 Unstable state ZND reaction-zone structure to to the performance problem that is n small, multi-dimensional disturbances µ = 0.5 philosophically different from the stan- 3 and shown that it is unstable to even µ = 0.67 dard paradigm just described (Aslam et small perturbations whenever n is al. 1996). In the DSD approach, we 2 greater than 2.1675 for the model that µ = 0.33 Stable exploit the fact that the explosive we have described here. Figure 9 pieces of engineering interest are large shows the results of this stability 023451 compared with the reaction-zone analysis. Also, the addition of a noz- Transverse wave number length and substitute a subscale model zling term to Equation (3) (that term for the detailed model of the reaction mimics the energy loss from the reac- Figure 9. Stability Analysis zone. To construct this subscale model, tion zone because of multidimensional Results for the Standard we consider how the detonation reac- flow effects) leads to a further destabi- Detonation Modeling Paradigm tion zone is influenced by weak curva- lization of the reaction zone to 1-D The vertical axis is the pressure expo- ture of the shock front and derive a disturbances. (Note: The high-resolu- nent of the heat-release rate R, and constraint equation relating the speed tion n = 2 simulation also shows some the horizontal axis is the wave number of the detonation front to the shape of signs of instability.) The root of this of the perturbing transverse distur- that front. We then derive a boundary instability can be understood with the bance. For any state above the curve, condition on that equation that relates the ZND detonation is unstable. For µ following argument. Detonation in this edge effects to the detonation wave = 0.5, a ZND detonation is unstable to homogeneous-fluid model is a balance two-dimensional disturbances when- shape. Thus, on the scale of the explo- between shock-initiated energy-releas- ever n > 2.1675. sive, the reaction zone becomes a front, ing reactions and the acoustic transport a discontinuity, separating fresh from of that energy to support the shock. engineering scale problems, where burnt explosive. In this way, DSD When a pressure perturbation in the the reaction-zone length is very much focuses on the two primary goals of reaction zone affects the reaction rate shorter than the dimension of the the performance problem: accurate pre- much more than it does the sound explosive piece. At any instant, about diction of (1) the local detonation speed, then small pressure fluctuations 1010 relatively small computational speed and detonation arrival times in a can disrupt the balance between ener- nodes would be needed in the reac- weapons simulation and (2) the P-v gy liberation and transport, and insta- tion zone, and the computational time (pressure-volume) work that the high- bility can be the result. on a large parallel processing com- pressure detonation products can per- puter would be about 100 days. form on the inert materials with which Second, the high-frequency, acousti- the explosive is in contact. As we will Problems with the Reaction- cally based transverse wave structure see, this approach also filters out the Zone Modeling Paradigm is an artifact of this simple homoge- high-frequency features and vastly neous-fluid model and is not reduces the computational require- Both the dependence on numerical observed in our heterogeneous explo- ments by comparison with the standard resolution and the appearance of the sives. The substructure associated modeling paradigm described above. high-frequency structure in the accep- with the granular structure of real het- Figure 10 shows a detailed view of tor region of the explosive represent a erogeneous explosives derives from the reaction zone and shock front, significant problem for this modeling the material particles, not acoustic with unburnt explosive above and paradigm. In independent studies of waves. In fact, the granularity inhibits burnt explosive below. The reaction- this simple model for the n = 2 case, the formation of large transverse zone length is short compared with we have shown that to predict the det- acoustic waves. Thus, although the the explosive charge dimension, L. In onation speed in the donor section to simple homogeneous-fluid model place of Cartesian coordinates, we use within 10 m/s requires 50 or more reproduces reasonably well the lead- coordinates that are attached to the points in the ZND reaction zone. ing-order features of detonation, such shock surface (see the upper inset in (Aslam et al. 1998). This number as the ZND structure, it fails to Figure 10). The constant η is the dis- translates into a very computationally describe higher-order features of real tance through the reaction zone nor- intensive problem for typical 3-D heterogeneous explosives. mal to the shock surface, and ξ is the

Number 28 2003 Los Alamos Science 105 High-Explosives Performance

κ > 0 D scaled detonation speed is (D /D – n ~ n CJ η φ 1) = D = εD. n^ η= constant To construct an asymptotic solu- s tion—a solution in the limit that ε << C Sh ho o 1 —for the multidimensional detona- ke ck ξ d-f su tion reaction-zone flow, we first intro- low rfa Stre s ce ur η duce into the standard detonation ξ aml fa rz = constantine ce model, Equations (1)–(3) and (6), the s following slowly changing, scaled, φ = φ High explosive (HE) φ Dn c s independent variables: in this L example ~ ~ 12 12~ Inert φ tt==εξεξφεφ , , = ,(8) Shock curve s HE interface Choked-flow curve where φ is the shock normal angle Streamline defined in Figure 10. We then expand T the solution vector Y = (ρ,uη P) as End of reaction () () () YY=+01εε Y +2 Y2 + ... , (9)

32 32 (10) uuξ =+ε ξ ... ,

HE–inert material interface region The leading order term in the solution vector (designated with a superscript 0) represents the 1-D ZND solution, Figure 10. Multidimensional Reaction Zone and the DSD Shock- whereas the terms proportional to Attached Coordinates powers of ε bring in the effects of A multidimensional reaction zone in a cylindrical detonating explosive (gray) is multidimensionality and time depend- weakly confined at its edges by a low-density inert material (yellow). As shown, the ence. A compatibility constraint reaction zone is typically short compared with the dimension of the explosive emerges on the solution that forces a charge, L. The upper inset depicts a segment of the 3-D reaction zone with the relationship between the shock curva- intrinsic shock-attached coordinates used in DSD analysis. In the DSD limit, the ture κ and various derivatives of the ratio of the reaction-zone thickness to the scale of the explosive charge is very, very η ε scaled detonation speed D small, rz/L = O( ) << 1, and a subscale front model takes the place of the detailed reaction-zone model. The crosshatched area, approximately the width of the reaction DD zone and straddling the explosive–inert material interface, defines the region where FD()=+κ AD() an analysis of the boundary region is performed. As explained in the text, that Dt analysis leads to boundary conditions for the subscale model. 2 (11) ∂ D − BD() + . . . , ∂ξ 2 distance along the shock measured DSD assumes that the detonation from the centerline of the explosive. reaction zone departs from its 1-D where F (D ) is a decreasing function Thus, curves of constant η are parallel (planar) steady-state ZND form by a of D with F (0) = 0, A (D ) > 0 and to the shock, and the lines of constant small amount. That small departure is B (D ) > 0. A term-by-term examina- ξ are in the direction of the local nor- determined by both the size of the tion of the right-hand side of this mal to the shock surface. Because the shock curvature κ measured in units equation shows that (1) increasing κ η features of interest are on the scale of of the reaction-zone length scale rz, (shock curvature) slows the detonation η κ ε the explosive, we define a dimension- ( rz = O( ) << 1) and the departure wave, (2) the acceleration term ε η η less scale, = rz/L << 1, where rz of Dn, the detonation speed in the A (D )(DD /Dt) acts like inertia and is the scale of the detonation reaction- direction of the shock normal vector, resists changes in the front speed and zone length. This scale aids in the der- from DCJ, the detonation speed for a shape, and (3) the dissipative term ivation of the subscale model. 1-D steady-state wave. The relevant B (D )(∂2D /∂ξ2) damps high fre-

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quencies and thus the formation of greater the pressure drop. A boundary neous explosives but not easily mod- kinks on the wave front. The disper- layer analysis performed within a dis- eled with the standard paradigm. sion relation for the linearized form of tance of one reaction-zone length on Calibration data are often obtained Equation (11) either side of the explosive–inert from measurements of the detonation material interface reveals (see cross- speed and front curvature in explosive 2 hatched region) that this interaction cylinders of various sizes (see Figure ω =−ik ()BA2 establishes a unique shock-edge angle, 11). For explosives such as PBX 24 2 φ ±−kkABA()2 (12) c,which is a function of the explo- 9502, those data can be fit reasonably sive and inert material pair consid- well with just the leading term in the ered. That angle serves as a boundary propagation law of Equation (11): reveals that, at low transverse frequen- condition for Equation (11). The cies (small values of the wave number weaker the confinement, the greater κ = FD() , (13) φ k), Equation (11) predicts dissipative, the value of c, up to the point where transverse waves moving along the the lateral expansion of the detonation which specifies a simple relationship front. On the other hand, at high products becomes choked (the devel- between the detonation speed and the transverse frequencies (large k), opment of a sonic state behind the detonation shock-front curvature. The Equation (11) is purely dissipative. shock, as shown in Figure 10), halting propagation law so obtained for PBX Thus, by adopting the scaled variables any further drop in pressure at the 9502 predicts that the shock-normal of Equation (8), high-frequency fea- shock. This phenomenon occurs at a detonation speed decreases substan- φ φ tures such as kinks on the front will value of called the sonic angle, s, tially with increasing shock-front cur- not form. which depends solely on the proper- vature (see Figure 12). Figure 13(a) Thus, on the scale of the explosive ties of the explosive and is about 45° shows a DSD prediction for the deto- piece, the detonation reaction zone for our explosives (Aslam and Bdzil nation front shape at initiation and looks like a discontinuity separating 2002, Bdzil 1981). two later times as the detonation prop- fresh and burnt explosive. Moreover, agates through an arc of PBX 9502. this discontinuity occurs along a sur- Figure 13(b), a top-down view, shows face that propagates according to the DSD Calibration the DSD solution lagging behind the dynamics given by Equation (11), and Propagation simple constant-velocity CJ solution. which can be viewed as an intrinsic of Detonation Front The significant differences between detonation propagation law for an these two argue for the importance of explosive. It is important to recognize To validate the DSD approach, we including reaction-zone effects. The that the forms of the coefficient func- used it to compute the detonation DSD shapes and velocities are in very tions F (D ), A (D ), and B (D ) front shapes for the rate law of depend on the material description of Equation (7) and compared the DSD the explosive (the EOS and the global results with numerical results from the heat-release rate). standard paradigm, Equations (1)–(3), In addition to specifying a propa- (6), and (7). The good agreement vali- HE gation law such as Equation (11), we dates the DSD approach, at least for n edge need to prescribe a boundary condi- <2.1675, for which the standard tion on the front, where the front approach is fairly accurate (Aslam et meets the edge of the explosive piece al. 1998). Although we could have (see the right portion of Figure 10). derived a detonation propagation law Just as we constructed a subscale directly from a calibrated shock-initia- Detonation model to mimic the effects of the tion model, such as the Lee-Tarver front trace reaction zone on the front motion, we Ignition and Growth model, we chose, construct a subscale model to mimic instead, to derive Equation (11) more the effect of confinement by adjacent generically and calibrate it from Figure 11. Detonation Front Measurement in a Cylinder of inert materials on the detonation experimental data on multidimension- Explosive speed. Roughly speaking, the more al detonation. In that way, we This image, taken by a streak camera, compliant the inert materials, the bypassed artifacts of the homoge- shows the detonation front arrival time greater the deflection of explosive neous-fluid model paradigm and built vs radius at the planar face of the streamlines and shock angle φ and the in features faithful to real, heteroge- explosive cylinder (Hill 1998).

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9 (a) 7.8 z

8 n D

7 y 5.8 x

(mm/µs) 6 n D 5 2.0

4 1.5 00.20.40.60.8 1.0 1.2 s)

κ –1 DSD 1.0 µ

(mm ) ( t 0.5 Figure 12. DSD Propagation Law Experiment for PBX 9502 at 25oC 0 The DSD propagation law predicts that 60 70 80 90 detonation speed normal to the shock r (mm)

Dn decreases substantially with increas- µ ing shock-front curvature κ. (b) DDSD = 9.961 mm/ s DEXP = 9.953 DSD CJ t = 10 good agreement with experiment To obtain the DSD solution shown in Figure 13, we start from the front D = 7.8 t = 19 propagation law and edge boundary CJ D = 7.198 conditions and use level-set methods DSD x DEXP = 7.188 to compute the propagation of the t = 0 µs front. These methods work by embed- y z ding the detonation front in a level-set function, Ψ, and then evolving this function according to 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- ∂ψ ∂+∇= ψ sive arc. Each detonation front is colored by the local instantaneous normal detona- tDn 0 . (14) tion speed. The slowing of the detonation near the edges is apparent by the change in color from red to green. Shown in the inset are plots of the DSD (green) and experi- By our definition, the level surface ψ = mental (black) times of arrival of the detonation front along the midline of the planar 0 corresponds to the detonation front edge of the arc (measured in units of the cylinder radius). The resulting curves show initially and at any subsequent time. We the similarity between the DSD and measured wave-front shapes. The DSD and experi- compute the actual detonation front by mental plots are set off to display results. (b) This top-down view shows the intersec- finding the ψ = 0 contour. The level-set tion of the DSD and CJ fronts with a plane passing through the middle of the arc. The methodology offers significant compu- DSD detonation speed slows down by 10% relative to its initial value, whereas the CJ tational advantages because it enables detonation speed is constant. Consequently, there is a growing separation of DSD and easy handling of complex explosive CJ fronts. The phase velocities of the DSD wave along the inner and outer surfaces of topologies and detonation interactions. the arc agree well with the experimental values. This past year, we have developed a first-order accurate, 3-D computa- times. The shape of the explosive, New Modeling Paradigms for tional algorithm that uses the level- two protruding legs on one side and a Detonation Reaction Zones set method and that runs on parallel cylindrical hole on the other, forces computing platforms. Results from the detonation fronts to merge (t = t3) The methods just described repre- that method are shown in Figure 14, and then bifurcate (t = t4). Merging sent the state of the art in detonation which displays a detonation initiated and bifurcation of different detona- modeling for engineering applications. simultaneously at the ends of two tion fronts are automatically treated In closing, we outline our vision for symmetrical legs and propagating with the level-set-based DSD the future. through a piece of PBX 9502 with approach. For comparison, the CJ The modeling paradigm represented complex geometry. The detonation wave is also shown (as ahead) in the by Equations (1)–(6) has serious short- fronts are shown at four different last snapshot (t = t4). comings. Principally, such a continuum

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Given our lack of detailed informa- t 1 tion about the relationship between the geometry of explosive grains and t the constitutive properties of our 2 t 3 materials (including, for example, heat conductivity) under detonation conditions, a more realistic midterm t 4 CJ goal is to develop subscale models for the reaction zone in which behaviors D n on the grain and subgrain scales are 9.30 parameterized in terms of longer 8.55 wavelength variables. We are work- 7.80 ing to develop rational asymptotic 7.05 models that indicate not only how the explosives’ global heat-release rate 6.30 should be modeled but also how the CJ presence of granularity and hotspots DSD in these materials affects the basic modeling structure—that is, what modifications need to be made to DSD Equations (1)–(6). For example, sig- DSD nificant density variations in a material on a short-wavelength scale will Detonator appear on the long-wavelength con- DSD tinuum scale as dispersion terms added to the basic conservation laws—refer to Equations (1)–(3). The Figure 14. A 3-D DSD Calculation presence of such terms could be This example shows how DSD can handle the merging of separated detonation expected to scatter acoustic waves fronts, the acceleration of the detonation in regions where the fronts converge, and and inhibit the detonation modeling the bifurcation of the detonation wave around obstacles. Detonation starts in the two legs (left side of the figure) and progresses toward the cylindrical hole on the right. instabilities that we have observed in Four snapshots show the progress of the detonation waves through the sample, and our homogeneous-explosive models. the DSD and CJ calculations are compared on the fourth snapshot. The detonation Whatever improved modeling para- front is colored with the local value of the detonation speed. The inset shows an digms are developed for the continu- oblique view. um response of heterogeneous explosives, we expect that, for the foresee- model does not include any phenome- short scale regions where the hotspot able future, models will have to be non that might be important for hetero- temperatures are most extreme will calibrated to experiments if they are geneous materials, such as scattering or have a disproportionate effect toward to make the accurate predictions of dispersion of acoustic waves and dissi- accelerating the reaction chemistry detonation propagation necessary for pation of energy from the reaction-zone (Bdzil et al. 1999, Menikoff and Kober weapons simulations. scale to the subreaction-zone scales of 1999). These would be subgrain scales, the hotspots. Current models are effec- related more to details of the grain tive for homogeneous explosives. As shape than the grain volume (Figure 4 we better understand the details of the shows how complex this substructure hotspots and reaction chemistry under can be). The notion of doing numeri- detonation conditions, we will need to cally resolved meso-scale simulations develop better continuum models. If of detonation in granular explosives subnano-scale measurements support and then somehow averaging those our current view that the chemical reac- results to develop appropriate continu- tions important in detonation are um-level engineering models seems to extremely state sensitive, then very be many years away.

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Further Reading Fedkiw, R. P., T. Aslam, B. Merriman, and S. Symposium (Int.) on Detonation, San Diego, Osher. 1999. A Non-Oscillatory Eulerian California, August 11–16, 2002 (to be pub- Aslam, T. D. 2003. A Level Set Algorithm for Approach to Interfaces in Multimaterial lished in Proceedings—Twelfth International Tracking Discontinuities in Hyperbolic Flows (the Ghost Fluid Method). J. Comput. Detonation Symposium). Conservation Laws II: Systems of Equations Phys. 152 (2): 457. [Online]:http://www.sainc.com/onr/detsymp/ (to be published in J. Sci. Comput. 19 (1–3), Henshaw, W. D., and D. W. Schwendeman. technicalProgram.ht December 2003). 2003. “An Adaptive Numerical Scheme for Sheffield, S. A., R. L. Gustavsen, and R. R. Aslam, T. D., and J. B. Bdzil. 2002. “Numerical High-Speed Reactive Flow on Overlapping Alcon. 2000. “In Situ Magnetic Gauging and Theoretical Investigations on Detonation- Grids.” Lawrence Livermore National Technique Used at LANL—Method and Inert Confinement Interaction.” Twelfth Laboratory preprint UCRL-JC-151574 (sub- Shock Information Obtained.” In Shock Symposium (Int.) on Detonation, San Diego, mitted to J. Comput. Phys.). Compression of Condensed Matter—1999, California, August 11–16, 2002. Los Alamos Hill, L. G. 2002. “Development of the LANL AIP Conference Proceedings, M. D. National Laboratory document LA-UR-01- Sandwich Test.” In Shock Compression of Furnish, L. C. Chhabildas, and R. S. 4664 (to be published in Proceedings— Condensed Matter—2001: Proceedings of Hixson, Eds., Vol. 505, p. 1043. Secaucus, Twelfth International Detonation the Conference of the American Physical New Jersey: Springer-Verlag. Symposium). [Online]: Society, Topical Group on Shock Sheffield, S. A., R. L. Gustavsen, L. G. Hill, http://www.sainc.com/onr/detsymp/technical Compression of Condensed Matter,M. D. and R. R. Alcon. 2000. “Electromagnetic Program.htm Furnish, Y. Horie, and N. N. Thadhani, Eds. Gauge Measurements of Shock Initiating Aslam, T. D., J. B. Bdzil, and L. G. Hill. 2000. Vol. 620 (1), pp. 149–152. Secaucus, New PBX9501 and PBX9502 Explosives.” In “Extensions to DSD Theory: Analysis of Jersey: Springer-Verlag. Proceedings—Eleventh International PBX 9502 Rate Stick Data.” In Hill, L. G., Bdzil, J. B., and T. D. Aslam.2000. Detonation Symposium, ONR 33300-5, pp. Proceedings—Eleventh International “Front Curvature Rate Stick Measurements 451–458. Arlington, Virginia: Office of Detonation Symposium, ONR 33300-5, pp. and Detonation Shock Dynamics Naval Research. 21–29. Arlington, Virginia: Office of Naval Calibration for PBX9502 over a Wide Short, M., J. B. Bdzil, G. J. Sharpe, and I. Research. Temperature Range.” In Proceedings— Anguelova. Detonation Stability for a Aslam, T. D., J. B. Bdzil, and D. S. Stewart. Eleventh International Detonation Condensed Phase Reaction Model with a 1996. Level Set Methods Applied to Symposium, ONR 33300-5, pp. 1029–1037. Variable Reaction Order (submitted to Modeling Detonation Shock Dynamics. J. Arlington, Virginia: Office of Naval Combust. Theory Modell.). Comput. Phys. 126 (2): 390. Research. Skidmore, C. B., D. S. Phillips, and N. B. Bdzil, J. B. 1981. Steady-State Two- McGrane, S. D., D. S. Moore, and D. J. Funk. Crane. 1997. Microscopical Examination of Dimensional Detonation. J. Fluid Mech. 2003. Sub-Picosecond Shock Interferometry Plastic-Bonded Explosives. Microscope 45 108: 195. of Transparent Thin Films. J. Appl. Phys. 93 (4): 127. Bdzil, J. B., R. Menikoff, S. F. Son, A. K. (9): 5063. Tarver, C., and E. McGuire. 2002. “Reactive Kapila, and D. S. Stewart. 1999. Two-Phase Menikoff, R., and E. M. Kober. 1999. Flow Modeling of the Interaction of TATB Modeling of Deflagration-to-Detonation “Compaction Waves in Granular HMX,” Detonation Waves with Inert Materials.” Transition in Granular Materials: A Critical Los Alamos National Laboratory report LA- Twelfth Symposium (Int.) on Detonation, Examination of Modeling Issues. Phys. 13546-MS. San Diego, California, August 11–16, 2002 Fluids 11 (2): 378. Quirk, J. J. 1998. AMRITA: “A Computational (to be published in Proceedings—Twelfth Campbell, A. W., and J. R. Travis. 1985. Facility (for CFD Modeling).” In 29th International Detonation Symposium). “Shock Desensitization of PBX-9404 and Computational Fluid Dynamics, von [Online]: http://www.sainc.com/onr/det- Composition B-3.” In Proceedings—Eighth Karman Institute Lecture Series. ISSN symp/technicalProgram.htm International Detonation Symposium, J. M. 0377-8312. Watt, S. D., and G. J. Sharpe. One- Short, Ed., pp. 1057–1068. Naval Surface Shaw, M. S. 2002. “Direct Simulation of Dimensional Linear Stability of Curved Weapons Center, NSWC MP 86-194. Detonation Products Equation of State by a Detonations (submitted to Proc. R. Soc. Composite Monte Carlo Method,” Twelfth Lond., Ser. A).

John B. Bdzil received his Ph.D. in physical chemistry and physics from the University of Minnesota in 1969. John began his career at Los Alamos in 1972 as a staff member in the Dynamic Experimentation Division and became a fellow of the Laboratory in 2001. His research at Los Alamos has been in detonation hydrodynamics. Along with Prof. Scott Stewart of the University of Illinois, John is codeveloper of detonation shock dynamics, a model now widely used throughout the world. John characterizes his approach to explosive modeling as unique at the national laboratories as it relies on mathematical analysis and asymptotic methods rather than on direct numerical simulation. He collaborates extensively with university researchers in his work on explosive modeling, including work on damaged explosive and accidental initiation of detonation in such materials. John has served as a member of the trilaboratory support team for the University of Illinois ASCI Alliances Center for the past five years and is a member of the (Left to right): Rudolph Henninger, Tariq Aslam, John Bdzil, Laboratory’s University Alliances Board. For many years, he served as a judge for ski racing events in northern New Mexico. and James Quirk.

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