detonation diffraction, dead zones and the ignition-and-growth model
DETONATION DIFFRACTION, DEAD ZONES AND THE IGNITION-AND-GROWTH MODEL G. DeOliveira, A. K. Kapila and D. W. Schwendeman Department of Mathematical Sciences Rensselaer Polytechnic Institute, Troy, NY J. B. Bdzil Los Alamos National Laboratory, Los Alamos, NM W. D. Henshaw and C. M. Tarver Lawrence Livermore National Laboratory, Livermore, CA
Abstract. Of the various macro-scale continuum models of reactive flow in high-energy explosives, the ignition-and-growth model has been the most widely used. Well-resolved computational experiments with the model on corner turning in rigidly confined LX-17 demonstrate that experimentally observed dead zones (sustained regions of chemical inactivity) are not reproduced in the computations. This deficiency is attributed to the fact that the model does not explicitly account for desensitization caused by exposure to shocks of low strength. A modification of the reaction rate that takes account of the desensitization process is proposed. Computations of corner turning with the augmented model show the appearance of dead zones downstream of the corner.
INTRODUCTION Macro-scale modeling of detonation in heterogeneous, high-energy explosives continues to be a challenge. Our knowledge of the thermomechanical behavior of these materials over the broad range of states encountered in a detonation is incomplete, and the same is true of the complex set of reactions that are responsible for the liberation of energy. The morphological complexity, the dearth of information, and the multi-scale nature of material response remain serious obstructions in the path of ab-initio modeling of detonation phenomena at the present time. It is understood that when the heterogeneous explosive is subjected to an initiating shock, meso-scale processes such as friction, pore collapse, shear banding and local plastic deformations generate hot spots, where burning commences and then spreads to consume the entire bulk. With this picture in mind, efforts have been directed at constructing phenomenological, macro-scale, continuum-type models. These in-
clude the Forest-fire model [1], the JTF model [2] and the HVRB model [3], but it is the ignition-and-growth model that has proven to be a workhorse in the field. Originally derived by Lee and Tarver [4], it has been refined and extensively utilized by Tarver and colleagues [5]. Ignition-and-growth treats the explosive as a homogeneous mixture of two distinct constituents, the unreacted explosive and the products of reaction. To each constituent is assigned an equation of state, and a single reaction-rate law is postulated for the conversion of the explosive to products. It is assumed that the two constituents are always in pressure and temperature equilibrium, and in the usual fashion, extensive variables such as energy and volume of the mixture are sums of the corresponding quantities for the individual constituents, weighted by the reaction progress variable λ. The model contains a number of parameters which are calibrated to the explosive of interest so as to reproduce certain sets of experimental data. The model has had considerable success in having provided a framework
within which different classes of experiments can be simulated and studied, although some tuning of the parameters is required. For example, experiments in which the initiation process is the focus of interest require a different parameter set than experiments on established detonation [5], to account for the fact that reaction rates during detonation initiation and detonation propagation are controlled by different phenomena at the meso scale. Not surprisingly, the phenomenology exhibited by the model can depend upon the resolution of the computations as well, and not all simulations reported in the literature are finely resolved [6,7,8]. In this paper we employ the model for a study of detonation diffraction, especially the turning of detonation around an abrupt corner in the explosive. Experiments suggest that a sustained pocket of unreacted material, dubbed a dead zone, may appear downstream of the corner [7,8]. This is a situation in which an established detonation experiences a substantial transient, and our intent is to determine whether the ignition-and-growth model is capable of predicting the observed behavior. We proceed by selecting a single set of parameters for the explosive LX-17, this set having been reported as being appropriate for detonation propagation rather than detonation initiation [5]. One set of calculations is carried out with the standard ignition-and-growth model, and the other with a proposed augmented model which takes into account the desensitization of the explosive subsequent to exposure to a weak shock. The computations employ adaptive mesh refinement and are highly resolved; for example, 36 cells cover the 0 < λ < 8 portion of the steady reaction zone before the detonation turns the corner. (Details of the numerical procedure can be found in [9].) Results are presented for the rigidly confined, axisymmetric, hockey-puck configuration [5,8]. THE MODEL We begin with a brief description of the standard ignition-and-growth model. The mixture variables satisfy the reactive Euler equations which, in a 2D geometry, are Ut + f (U )x + g(U )y = h(U ). The state vector U, the flux vectors f and g, and
the source vector h appearing above are defined as U =
g=
ρ ρu ρw ρE ρλ
f =
ρu ρu2 + p ρuw u(ρE + p) ρuλ
,
ρw ρwu ρw2 + p w(ρE + p) ρwλ
,
h=
0 0 0 0 ρR
,
.
Here ρ is the mixture density, (u, w) the (x, y) components of the velocity, p the mixture pressure, and R the reaction rate. The total energy of the mixture per unit mass is denoted by E, i.e., 1 E = e + (u2 + w2 ), 2 where e is the specific energy of the mixture. The mixture quantities are defined in terms of the corresponding quantities for the individual constituents by the expressions v = e =
(1 − λ)vs + λvg , (1 − λ)es + λeg .
Here v = 1/ρ is the specific volume and the subscripts s and g refer to the unreacted solid and the gaseous product respectively. The mechanical and thermal equations of state for the two constituents are of the JWL form, es vs0 eg vs0
= =
ps vs − Fs (vs /vs0 ) + Fs (1), ωs pg vg − Fg (vg /vs0 ) − Q, ωg
and pi =
ωi [Ci Ti + Gi (vi /vs0 ) + Fi (vi /vs0 )], vi
i = s, g. Here vs0 is the specific volume of the unreacted solid at the ambient state, Q is the energy of detonation per unit volume of the unreacted solid at the ambient state, Ti is the temperature and Ci the specific heat of constituent i, and
Fi and Gi are the JWL functions given by 1 V − exp(−R1i V ) Fi (V ) = Ai ωi R1i V 1 + Bi − exp(−R2i V ), ωi R2i Ai Gi (V ) = exp(−R1i V ) R1i Bi exp(−R2i V ), + R2i i = s, g. The constants Ai , Bi , ωi , R1i and R2i characterize the explosive. It is assumed that the reactant and the product are in pressure and temperature equilibrium, i.e., ps = pg = p,
and Ts = Tg = T.
Table I. EOS data for LX-17. JWL parameters A (Pa) B (Pa) R1 R2 ω Q (Pa-cc/cc) C (Pa/K) vs0 (m3 /kg)
Unreacted 778.1e11 -0.05031e11 11.3 1.13 0.8938 2.487e6 1/1905
Products 14.8105e11 0.6379e11 6.2 2.2 0.50 0.069e11 1.0e6
The EOS constants used in this study pertain to LX-17 [5] and are listed in table I. Finally, the reaction rate is a pressure-driven rule consisting of three distinct terms, R = RI + RG1 + RG2 ,
RI
x
= I(1 − λ) (vs0 /v − 1 − a) × H(vs0 /v − 1 − a)H(λigmax − λ)
is the ignition term and RG1 RG2
Table II. Rate data for LX-17. Parameter Value I, (s−1 ) 4.0e12 b 0.667 a 0.22 x 7.0 G1 , (1011 P a)−y s−1 ) 4500e6 c 0.667 d 1 y 3 G2 , (1011 P a)−z s−1 ) 30e6 e 0.667 g 0.667 z 1 λigmax 0.02 λG1 max 0.8 λG2 min 0.8 Justification for the form of the rate, and the phenomenological connection of the three terms to the meso-scale processes of hot-spot creation, growth, and flame spread, have been discussed extensively in the literature [5]. For our purposes, suffice it to say that 1 + a is an ignition threshold above which the density must rise for reaction to be initiated, and that once initiated, the reaction must go to eventual completion since the rate expression does not include an explicit extinction mechanism. THE AUGMENTED MODEL
where b
specific to the explosive, limit the contributions of the three terms in the rate law to specified intervals of reaction progress. The rate-law parameters appropriate for LX-17 are listed in table II [5].
= G1 (1 − λ)c λd py H(λG1max − λ), = G2 (1 − λ)e λg pz H(λ − λG2min )
are the growth terms. Also, H is the Heaviside function and I, G1 , G2 , a, b, c, d, e, g, x, y and z are material-specific constants. The switching constants λigmax , λG1max and λG2min , also
It is well known that passage of a weak shock desensitizes the explosive by compacting it, thereby decreasing the fractional volume available for hot-spot formation. As a result the precompressed material requires a stronger stimulus for initiation compared to that required for initiation of the pristine, unshocked material. Many attempts to include desensitization in a reactive model have been reported in the literature. The JTF model [2] does so by restricting creation of hot spots only to the first shock, and allowing a subsequent shock to provide only adiabatic heating. Whitworth and Maw [10] propose modifying
the ignition-and-growth reaction rate by replacing the pressure by the smaller of p and pS , where pS is the pressure of the first shock. Souers et al [4] cut the reaction off when the sum of the acoustic and particle speeds fall below a prescribed minimum. In a more recent study, Souers et al [8] propose a rate law with different rate expressions in different pressure regimes. Any enhancement of the ignition-and-growth model that seeks to capture this phenomenon must contain the following ingredients: (i) A measure of the degree of consolidation of the explosive caused by the shock. (ii) A dependence of the initiation threshold on the degree of consolidation. (iii) A reaction extinction mechanism driven by the degree of consolidation. (iv) A negligible effect of desensitization on the propensity to initiate and on the reactionzone structure when the initiating stimulus is sufficiently strong. Staying within the spirit of the standard ignition-and-growth model, we introduce a desensitization parameter φ and propose that it obey the evolution equation (ρφ)t + (ρuφ)x + (ρwφ)y = ρS. The desensitization rate S is defined as S = Ap(1 − φ)(φ + ), where the rate constant A and the switch are positive constitutive coefficients. In the pristine material, φ = 0 while in the fully desensitized material, φ = 1. The source is zero at the upstream state of zero pressure but is switched on to the value Ap as soon as pressure rises above zero. For any compression p > 0 the source attempts to drive φ to 1. We also propose modifying the reaction rate R in two respects. First, the density threshold in the ignition term RI is prescribed to be a function of φ, i.e., a(φ) = a0 (1 − φ) + a1 (φ),
where a0 and a1 are positive constants, 1 + a0 being the density threshold for ignition of the pristine material and 1 + a1 the same for the fully desensitized material. Second, the first growth term RG1 is now switched on only when λ exceeds an activation minimum λG1min (φ) given by the linear function λG1min (φ) = λc φ, where λc is positive. Thus, RG1 is modified to the desensitized version RG1,D , defined as RG1,D
= G1 (1 − λ)c λd py × H(λ − λG1min (φ))H(λG1max − λ).
It is this modification that introduces a competition between desensitization and growth, and thus provides an extinction mechanism. The modification requires calibration of the four new parameters A, , a1 and λc . The first two define the rate of desensitization and are selected in accord with the information that a 1 GPa shock desensitizes LX-17 within 1 to 2 µs and a 5 GPa shock within 0.2 to 0.3 µs. The choices A = 1000,
= 0.001,
yield desensitization times, i.e., times for φ to rise from 0 to 0.98 behind the shock, of 1.29 µs for a 1 GPa shock and 0.26 µs for a 5 GPa shock. The selection of a1 and λc would presumably require calibration with other desensitization experiments. Here, for demonstration purposes, we have made the choices a1 = 0.50,
λc = 0.01.
NUMERICAL RESULTS: PLANAR CONFIGURATION Initiation with weak shock Figure 1 displays initiation by a 12.54 GPa shock as computed with the standard model. Profiles of reaction progress λ show that initiation is successful, and that after a short transient a steady detonation results. Figure 2 shows the corresponding result for the augmented model, where extinction has occurred subsequent to a short initial burst of reactivity. Profiles of the scaled density vs0 /v and λ are shown, along with those of 1 + a(φ), the initiation threshold density, and λG1min (φ), the activation switch for the
Initiation with strong shock 40 35 30 pressure (GPa)
first growth term. We see that immediately behind the shock the scaled density exceeds the initiation threshold, so that the initiation term is activated. As soon as λ rises above zero the first growth term is activated as well, since the initial value of the growth switch λG1min is zero. Desensitization causes both 1 + a(φ) and λG1min (φ) to rise, and the initiation reaction is deactivated when 1+a(φ) exceeds vs0 /v, while the first growth reaction is cut off when λG1min exceeds λ. The ultimate outcome is that following a negligibly small degree of reaction progress, desensitization has resulted in a cessation of reaction.
25 20 15 10 5 0
1
0
1
2
non−desensitized
3 x (mm)
4
5
6
0.9
FIGURE 3. Successful strong-shock initiation with standard model (dashed line), augmented model (full line).
0.8 0.7 0.6
λ 0.5 0.4
0.03
1.5 0.3 0.2 0.1 0
0.025
1+a( φ )
1.4
vs0 / v
0.02
λ
0.015
λ
1.3 0
1
2
3
4
5
x (mm)
1.2
λ G1MIN 0.01
FIGURE 1. Successful weak-shock initiation with standard model.
1.1
1 0.2
1.5
0.01 1 + a( φ )
1.45 1.4
0.009 0.008
Ignition term deactivated
1.35 1.3
0.007
v s0 / v
0.006
λ G1MIN
1.2
0.004 First Growth term deactivated
λ
1.1
0.003 0.002
1.05 1 3.4
0 0.4
0.6 x (mm)
0.8
1
FIGURE 4. Strong-shock initiation with augmented model: profiles at early time t = 0.2. Desensitization cuts off ignition at λ slightly below 0.02. The growth term is not deactivated as the profile of λ stays above that of λG1min .
0.005 λ
1.25
1.15
0.005
0.001 3.6
3.8
4
4.2 4.4 x (mm)
4.6
4.8
5
0
FIGURE 2. Unsuccessful weak-shock initiation with augmented model: details of failure due to desensitization.
Application of a stronger, 14.97 GPa initiating shock, results in successful initiation for both the standard and the augmented models. Figure 3 shows that once detonation is fully established, both the models produce similar pressure profiles. Detonation in the augmented model lags very slightly behind that in the standard model; a vestige of significant differences in evolution at early times. The profiles in Figure 4, plotted at the early time t = 0.2, exhibit the way in which the
desensitization process manifests itself. We note that the profiles of λG1min (φ) and λ do not cross, and as a result, the growth term is never deactivated and extinction occurs no longer. However, desensitization does affect the size of the ignition term and the duration of λ over which the ignition term is active (through the rise in the density threshold). That, in turn, influences the profile of λ, and hence the size of the growth terms. Collectively these effects are responsible for the small disagreements seen in the pressure profiles of figure 3.
In the numerical setup the boundary of the well is treated as rigid, while outflow conditions are applied at the remaining boundaries. The computation begins at t=0, with the initial condition corresponding to a high-pressure, ambient density slug of reaction products (p = 31.46 GPa, v = vs0 , λ = 1 and φ = 1 ) occupying a hemisphere of radius 7.68 mm centered at the flat surface of the well and embedded within the solid explosive; see figure 5. At t > 0 a shock propagates into the explosive and by the time it has reached the corner, a detonation is well-established.
NUMERICAL RESULTS: THE HOCKEY PUCK GEOMETRY We now consider detonations negotiating a hockey-puck configuration. It is an oft-used experimental arrangement for studying dead zones [5,7,8], consisting of a cylindrical sample of explosive from which a shorter, coaxial cylinder has been carved out to form a well. Figure 5 shows one-half of an axial section with typical experimental dimensions. A hemispherical detonation, centered at the flat face of the well, propagates into the explosive and diffracts around the rightangled corner. Experiments suggest the appearance of a dead zone downstream of the corner. While experimentally measured breakout times are captured well by the standard model [5], we shall see that the augmented model is also able to capture the experimentally observed dead zones.
y 0
axis of symmetry
- 19.05
- 44.45 - 15.0
0
40.0
x
FIGURE 5. The hockey-puck configuration. Dimensions are in mm.
FIGURE 6. Numerical schlieren images (zoomed) at t = 1.6. The standard model (upper) and the augmented model (lower) produce virtually identical results. Figures 6-10 display the computational results. These are presented in the form of numerically-generated schlieren images, some supplemented by pressure contours, for both the standard and the augmented models, arranged to allow a side-by-side comparison. Figure 6 simply shows that at t=1.6 the detonation front has reached the corner. There is no discernible difference in the results of the two models at this time. Figure 7 shows the snapshots at three different time levels, t=2.2, 2.6 and 2.8, with results of the standard model in the upper panel and those of the augmented model in the lower panel. As the detonation turns the corner, the shock in the
near-corner region suffers expansion and weakens, causing the reaction rate behind the shock to drop, and resulting in a progressive retreat of the reaction zone away from the shock. Away from the corner the detonation continues to propagate as a spherical wave, virtually undisturbed. A hint of a kink, clearly visible at t=2.6, develops at the location where the weakened and the undisturbed segments of the wavehead meet. The results for the two models continue to exhibit essentially identical behavior. Figure 8 shows the results from t=2.9 to 3.3. The first distinction between the results of the two models is now apparent at t=2.9, especially in the pressure contours. In the vicinity of the kink the reaction in the standard model has strengthened, thereby creating a ridge of high pressure in the form of a hook curving away from the lead shock. At t=3.1 this hook has grown into a transverse detonation which, at t=3.3, is seen to propagate upwards into the hitherto unreacted region. During
t = 2.2
this time interval, desensitization has clearly come to the fore in the augmented model. Near the kink the pressure is low enough, around 12 GPa at t=2.1, so that the situation is reminiscent of that in figure 2. There is no strengthening of chemical reaction, and hence no transverse wave that would consume the chemically dormant region. Figure 9 shows the results from 3.4 to 3.8. For the standard model the transverse wave continues to propagate upwards, consuming all but a small pocket of unreacted material. The reaction rate within this pocket is large enough so that it continues to react on its own, without needing any help from the reflected shock seen in the snapshot at t=3.8. For the augmented model the detonation continues to propagate but only into the unshocked material, while the dead zone created by the desensitization of the previously shocked material persists. Even at t=4.8, when the detonation breaks out of the left boundary, figure 10 shows that the dead zone remains.
t = 2.4
t = 2.6
FIGURE 7. Numerical schlieren images (zoomed) for the standard model (upper panel) and the augmented model (lower panel). Subsequent to corner turning the reaction zone lags behind the shock. Discernible differences between the two models are yet to appear.
CONCLUSIONS A modification of the ignition-and-growth model of reactive-flow is proposed to explicitly include a mechanism for desensitization when the
explosive is exposed to a weak shock. This is done by introducing a pressure-driven desensitization parameter that is convected with the flow. It is assumed that the compression threshold of the initiation step of the reaction, and a delay in
t = 2.9
t = 3.1
t = 3.3
FIGURE 8. Numerical schlieren images and pressure contours (zoomed) for the standard model (two upper panels) and the augmented model (two lower panels); t from 2.9 to 3.3.
t = 3.4
t = 3.6
t = 3.8
FIGURE 9. Numerical schlieren images and pressure contours (zoomed) for the standard model (two upper panels) and the augmented model (two lower panels); t from 3.4 to 3.8.
the first growth reaction, are functions of the degree of desensitization. Results for the standard and augmented models are compared for detonation diffraction in the LX-17 hockey-puck geometry. While the predictions of the two models away from the corner are virtually identical, the augmented model is also able to capture a sustained dead zone in the vicinity of the corner.
2. J. N. Johnson, P. K.Tang and C. A. Forest. Shock wave initiation of heterogeneous reactive solids, J. Applied Phys. 57 (1985), 4323. 3. J. Starkenberg. Modeling detonation propagation and failure using explosive initiation models in a conventional hydrocode, in The Twelfth Symposium (International) on Detonation (2002), 1001-1007. 4. E. L. Lee and C. M. Tarver. Phenomenological model of shock initiation in heterogeneous explosives, Phys. Fluids 23 (1980), 2362-2372. 5. C. M. Tarver. Ignition and growth modeling of LX-17 hockey puck experiments, Propellants, Explosives, Pyrotechnics 30 (2005), 109-117. 6. P. C.Souers, R. Garza and P. Vitello. Ignition and growth and JWL++ detonation models in coarse zones, Propellants, Explosives, Pyrotechnics 27 (2002), 62-71. 7. P. C. Souers, H. G. Andreski, C. F. Cook III, R. Garza, R. Pastrone, D. Phillips, F. Roeske, P. Vitello and J. D. Molitoris. LX17 corner turning, Propellants, Explosives and Pyrotechnics 29 (2004), 359-367.
FIGURE 10. Numerical schlieren images and pressure contours for the augmented model; t = 4.8. Acknowledgements. The authors acknowledge the support of the Los Alamos National Laboratory, the Lawrence Livermore National Laboratory, and the National Science Foundation. REFERENCES 1. C. L. Mader and C. A. Forest. Twodimensional homogeneous and heterogeneous wave propagation, Tech. Rep. LA6259, Los Alamos Scientific Laboratory, (1976).
8. P. C. Souers, H. G.Andreski, J. Batteux, B. Bratton, C. Cabacungan, C. F. Cook III, S. Fletcher, R. Garza, D. Grimsley, J. Handley, A. Hernandez, P. McMaster, J. D. Molitoris, R. Palmer, J. Prindiville, J. Rodriguez, D. Schneberk, B. Wong and P. Vitello. Dead zones in LX-17 and PBX 9502, Propellants, Explosives, Pyrotechnics, 31 (2006), 89-97. 9. A. K. Kapila, D. W. Schwendeman, J. B. Bdzil and W. D. Henshaw. A study of detonation diffraction in the ignition-andgrowth model. Submitted for publication (2006). 10. N. J. Whitworth and J. R. Maw. Modelling shock desensitization of heterogeneous explosives. AIP Conference Proceedings 370 (1996), 425-428.
Abstract. Of the various macro-scale continuum models of reactive flow in high-energy explosives, the ignition-and-growth model has been the most widely used. Well-resolved computational experiments with the model on corner turning in rigidly confined LX-17 demonstrate that experimentally observed dead zones (sustained regions of chemical inactivity) are not reproduced in the computations. This deficiency is attributed to the fact that the model does not explicitly account for desensitization caused by exposure to shocks of low strength. A modification of the reaction rate that takes account of the desensitization process is proposed. Computations of corner turning with the augmented model show the appearance of dead zones downstream of the corner.
INTRODUCTION Macro-scale modeling of detonation in heterogeneous, high-energy explosives continues to be a challenge. Our knowledge of the thermomechanical behavior of these materials over the broad range of states encountered in a detonation is incomplete, and the same is true of the complex set of reactions that are responsible for the liberation of energy. The morphological complexity, the dearth of information, and the multi-scale nature of material response remain serious obstructions in the path of ab-initio modeling of detonation phenomena at the present time. It is understood that when the heterogeneous explosive is subjected to an initiating shock, meso-scale processes such as friction, pore collapse, shear banding and local plastic deformations generate hot spots, where burning commences and then spreads to consume the entire bulk. With this picture in mind, efforts have been directed at constructing phenomenological, macro-scale, continuum-type models. These in-
clude the Forest-fire model [1], the JTF model [2] and the HVRB model [3], but it is the ignition-and-growth model that has proven to be a workhorse in the field. Originally derived by Lee and Tarver [4], it has been refined and extensively utilized by Tarver and colleagues [5]. Ignition-and-growth treats the explosive as a homogeneous mixture of two distinct constituents, the unreacted explosive and the products of reaction. To each constituent is assigned an equation of state, and a single reaction-rate law is postulated for the conversion of the explosive to products. It is assumed that the two constituents are always in pressure and temperature equilibrium, and in the usual fashion, extensive variables such as energy and volume of the mixture are sums of the corresponding quantities for the individual constituents, weighted by the reaction progress variable λ. The model contains a number of parameters which are calibrated to the explosive of interest so as to reproduce certain sets of experimental data. The model has had considerable success in having provided a framework
within which different classes of experiments can be simulated and studied, although some tuning of the parameters is required. For example, experiments in which the initiation process is the focus of interest require a different parameter set than experiments on established detonation [5], to account for the fact that reaction rates during detonation initiation and detonation propagation are controlled by different phenomena at the meso scale. Not surprisingly, the phenomenology exhibited by the model can depend upon the resolution of the computations as well, and not all simulations reported in the literature are finely resolved [6,7,8]. In this paper we employ the model for a study of detonation diffraction, especially the turning of detonation around an abrupt corner in the explosive. Experiments suggest that a sustained pocket of unreacted material, dubbed a dead zone, may appear downstream of the corner [7,8]. This is a situation in which an established detonation experiences a substantial transient, and our intent is to determine whether the ignition-and-growth model is capable of predicting the observed behavior. We proceed by selecting a single set of parameters for the explosive LX-17, this set having been reported as being appropriate for detonation propagation rather than detonation initiation [5]. One set of calculations is carried out with the standard ignition-and-growth model, and the other with a proposed augmented model which takes into account the desensitization of the explosive subsequent to exposure to a weak shock. The computations employ adaptive mesh refinement and are highly resolved; for example, 36 cells cover the 0 < λ < 8 portion of the steady reaction zone before the detonation turns the corner. (Details of the numerical procedure can be found in [9].) Results are presented for the rigidly confined, axisymmetric, hockey-puck configuration [5,8]. THE MODEL We begin with a brief description of the standard ignition-and-growth model. The mixture variables satisfy the reactive Euler equations which, in a 2D geometry, are Ut + f (U )x + g(U )y = h(U ). The state vector U, the flux vectors f and g, and
the source vector h appearing above are defined as U =
g=
ρ ρu ρw ρE ρλ
f =
ρu ρu2 + p ρuw u(ρE + p) ρuλ
,
ρw ρwu ρw2 + p w(ρE + p) ρwλ
,
h=
0 0 0 0 ρR
,
.
Here ρ is the mixture density, (u, w) the (x, y) components of the velocity, p the mixture pressure, and R the reaction rate. The total energy of the mixture per unit mass is denoted by E, i.e., 1 E = e + (u2 + w2 ), 2 where e is the specific energy of the mixture. The mixture quantities are defined in terms of the corresponding quantities for the individual constituents by the expressions v = e =
(1 − λ)vs + λvg , (1 − λ)es + λeg .
Here v = 1/ρ is the specific volume and the subscripts s and g refer to the unreacted solid and the gaseous product respectively. The mechanical and thermal equations of state for the two constituents are of the JWL form, es vs0 eg vs0
= =
ps vs − Fs (vs /vs0 ) + Fs (1), ωs pg vg − Fg (vg /vs0 ) − Q, ωg
and pi =
ωi [Ci Ti + Gi (vi /vs0 ) + Fi (vi /vs0 )], vi
i = s, g. Here vs0 is the specific volume of the unreacted solid at the ambient state, Q is the energy of detonation per unit volume of the unreacted solid at the ambient state, Ti is the temperature and Ci the specific heat of constituent i, and
Fi and Gi are the JWL functions given by 1 V − exp(−R1i V ) Fi (V ) = Ai ωi R1i V 1 + Bi − exp(−R2i V ), ωi R2i Ai Gi (V ) = exp(−R1i V ) R1i Bi exp(−R2i V ), + R2i i = s, g. The constants Ai , Bi , ωi , R1i and R2i characterize the explosive. It is assumed that the reactant and the product are in pressure and temperature equilibrium, i.e., ps = pg = p,
and Ts = Tg = T.
Table I. EOS data for LX-17. JWL parameters A (Pa) B (Pa) R1 R2 ω Q (Pa-cc/cc) C (Pa/K) vs0 (m3 /kg)
Unreacted 778.1e11 -0.05031e11 11.3 1.13 0.8938 2.487e6 1/1905
Products 14.8105e11 0.6379e11 6.2 2.2 0.50 0.069e11 1.0e6
The EOS constants used in this study pertain to LX-17 [5] and are listed in table I. Finally, the reaction rate is a pressure-driven rule consisting of three distinct terms, R = RI + RG1 + RG2 ,
RI
x
= I(1 − λ) (vs0 /v − 1 − a) × H(vs0 /v − 1 − a)H(λigmax − λ)
is the ignition term and RG1 RG2
Table II. Rate data for LX-17. Parameter Value I, (s−1 ) 4.0e12 b 0.667 a 0.22 x 7.0 G1 , (1011 P a)−y s−1 ) 4500e6 c 0.667 d 1 y 3 G2 , (1011 P a)−z s−1 ) 30e6 e 0.667 g 0.667 z 1 λigmax 0.02 λG1 max 0.8 λG2 min 0.8 Justification for the form of the rate, and the phenomenological connection of the three terms to the meso-scale processes of hot-spot creation, growth, and flame spread, have been discussed extensively in the literature [5]. For our purposes, suffice it to say that 1 + a is an ignition threshold above which the density must rise for reaction to be initiated, and that once initiated, the reaction must go to eventual completion since the rate expression does not include an explicit extinction mechanism. THE AUGMENTED MODEL
where b
specific to the explosive, limit the contributions of the three terms in the rate law to specified intervals of reaction progress. The rate-law parameters appropriate for LX-17 are listed in table II [5].
= G1 (1 − λ)c λd py H(λG1max − λ), = G2 (1 − λ)e λg pz H(λ − λG2min )
are the growth terms. Also, H is the Heaviside function and I, G1 , G2 , a, b, c, d, e, g, x, y and z are material-specific constants. The switching constants λigmax , λG1max and λG2min , also
It is well known that passage of a weak shock desensitizes the explosive by compacting it, thereby decreasing the fractional volume available for hot-spot formation. As a result the precompressed material requires a stronger stimulus for initiation compared to that required for initiation of the pristine, unshocked material. Many attempts to include desensitization in a reactive model have been reported in the literature. The JTF model [2] does so by restricting creation of hot spots only to the first shock, and allowing a subsequent shock to provide only adiabatic heating. Whitworth and Maw [10] propose modifying
the ignition-and-growth reaction rate by replacing the pressure by the smaller of p and pS , where pS is the pressure of the first shock. Souers et al [4] cut the reaction off when the sum of the acoustic and particle speeds fall below a prescribed minimum. In a more recent study, Souers et al [8] propose a rate law with different rate expressions in different pressure regimes. Any enhancement of the ignition-and-growth model that seeks to capture this phenomenon must contain the following ingredients: (i) A measure of the degree of consolidation of the explosive caused by the shock. (ii) A dependence of the initiation threshold on the degree of consolidation. (iii) A reaction extinction mechanism driven by the degree of consolidation. (iv) A negligible effect of desensitization on the propensity to initiate and on the reactionzone structure when the initiating stimulus is sufficiently strong. Staying within the spirit of the standard ignition-and-growth model, we introduce a desensitization parameter φ and propose that it obey the evolution equation (ρφ)t + (ρuφ)x + (ρwφ)y = ρS. The desensitization rate S is defined as S = Ap(1 − φ)(φ + ), where the rate constant A and the switch are positive constitutive coefficients. In the pristine material, φ = 0 while in the fully desensitized material, φ = 1. The source is zero at the upstream state of zero pressure but is switched on to the value Ap as soon as pressure rises above zero. For any compression p > 0 the source attempts to drive φ to 1. We also propose modifying the reaction rate R in two respects. First, the density threshold in the ignition term RI is prescribed to be a function of φ, i.e., a(φ) = a0 (1 − φ) + a1 (φ),
where a0 and a1 are positive constants, 1 + a0 being the density threshold for ignition of the pristine material and 1 + a1 the same for the fully desensitized material. Second, the first growth term RG1 is now switched on only when λ exceeds an activation minimum λG1min (φ) given by the linear function λG1min (φ) = λc φ, where λc is positive. Thus, RG1 is modified to the desensitized version RG1,D , defined as RG1,D
= G1 (1 − λ)c λd py × H(λ − λG1min (φ))H(λG1max − λ).
It is this modification that introduces a competition between desensitization and growth, and thus provides an extinction mechanism. The modification requires calibration of the four new parameters A, , a1 and λc . The first two define the rate of desensitization and are selected in accord with the information that a 1 GPa shock desensitizes LX-17 within 1 to 2 µs and a 5 GPa shock within 0.2 to 0.3 µs. The choices A = 1000,
= 0.001,
yield desensitization times, i.e., times for φ to rise from 0 to 0.98 behind the shock, of 1.29 µs for a 1 GPa shock and 0.26 µs for a 5 GPa shock. The selection of a1 and λc would presumably require calibration with other desensitization experiments. Here, for demonstration purposes, we have made the choices a1 = 0.50,
λc = 0.01.
NUMERICAL RESULTS: PLANAR CONFIGURATION Initiation with weak shock Figure 1 displays initiation by a 12.54 GPa shock as computed with the standard model. Profiles of reaction progress λ show that initiation is successful, and that after a short transient a steady detonation results. Figure 2 shows the corresponding result for the augmented model, where extinction has occurred subsequent to a short initial burst of reactivity. Profiles of the scaled density vs0 /v and λ are shown, along with those of 1 + a(φ), the initiation threshold density, and λG1min (φ), the activation switch for the
Initiation with strong shock 40 35 30 pressure (GPa)
first growth term. We see that immediately behind the shock the scaled density exceeds the initiation threshold, so that the initiation term is activated. As soon as λ rises above zero the first growth term is activated as well, since the initial value of the growth switch λG1min is zero. Desensitization causes both 1 + a(φ) and λG1min (φ) to rise, and the initiation reaction is deactivated when 1+a(φ) exceeds vs0 /v, while the first growth reaction is cut off when λG1min exceeds λ. The ultimate outcome is that following a negligibly small degree of reaction progress, desensitization has resulted in a cessation of reaction.
25 20 15 10 5 0
1
0
1
2
non−desensitized
3 x (mm)
4
5
6
0.9
FIGURE 3. Successful strong-shock initiation with standard model (dashed line), augmented model (full line).
0.8 0.7 0.6
λ 0.5 0.4
0.03
1.5 0.3 0.2 0.1 0
0.025
1+a( φ )
1.4
vs0 / v
0.02
λ
0.015
λ
1.3 0
1
2
3
4
5
x (mm)
1.2
λ G1MIN 0.01
FIGURE 1. Successful weak-shock initiation with standard model.
1.1
1 0.2
1.5
0.01 1 + a( φ )
1.45 1.4
0.009 0.008
Ignition term deactivated
1.35 1.3
0.007
v s0 / v
0.006
λ G1MIN
1.2
0.004 First Growth term deactivated
λ
1.1
0.003 0.002
1.05 1 3.4
0 0.4
0.6 x (mm)
0.8
1
FIGURE 4. Strong-shock initiation with augmented model: profiles at early time t = 0.2. Desensitization cuts off ignition at λ slightly below 0.02. The growth term is not deactivated as the profile of λ stays above that of λG1min .
0.005 λ
1.25
1.15
0.005
0.001 3.6
3.8
4
4.2 4.4 x (mm)
4.6
4.8
5
0
FIGURE 2. Unsuccessful weak-shock initiation with augmented model: details of failure due to desensitization.
Application of a stronger, 14.97 GPa initiating shock, results in successful initiation for both the standard and the augmented models. Figure 3 shows that once detonation is fully established, both the models produce similar pressure profiles. Detonation in the augmented model lags very slightly behind that in the standard model; a vestige of significant differences in evolution at early times. The profiles in Figure 4, plotted at the early time t = 0.2, exhibit the way in which the
desensitization process manifests itself. We note that the profiles of λG1min (φ) and λ do not cross, and as a result, the growth term is never deactivated and extinction occurs no longer. However, desensitization does affect the size of the ignition term and the duration of λ over which the ignition term is active (through the rise in the density threshold). That, in turn, influences the profile of λ, and hence the size of the growth terms. Collectively these effects are responsible for the small disagreements seen in the pressure profiles of figure 3.
In the numerical setup the boundary of the well is treated as rigid, while outflow conditions are applied at the remaining boundaries. The computation begins at t=0, with the initial condition corresponding to a high-pressure, ambient density slug of reaction products (p = 31.46 GPa, v = vs0 , λ = 1 and φ = 1 ) occupying a hemisphere of radius 7.68 mm centered at the flat surface of the well and embedded within the solid explosive; see figure 5. At t > 0 a shock propagates into the explosive and by the time it has reached the corner, a detonation is well-established.
NUMERICAL RESULTS: THE HOCKEY PUCK GEOMETRY We now consider detonations negotiating a hockey-puck configuration. It is an oft-used experimental arrangement for studying dead zones [5,7,8], consisting of a cylindrical sample of explosive from which a shorter, coaxial cylinder has been carved out to form a well. Figure 5 shows one-half of an axial section with typical experimental dimensions. A hemispherical detonation, centered at the flat face of the well, propagates into the explosive and diffracts around the rightangled corner. Experiments suggest the appearance of a dead zone downstream of the corner. While experimentally measured breakout times are captured well by the standard model [5], we shall see that the augmented model is also able to capture the experimentally observed dead zones.
y 0
axis of symmetry
- 19.05
- 44.45 - 15.0
0
40.0
x
FIGURE 5. The hockey-puck configuration. Dimensions are in mm.
FIGURE 6. Numerical schlieren images (zoomed) at t = 1.6. The standard model (upper) and the augmented model (lower) produce virtually identical results. Figures 6-10 display the computational results. These are presented in the form of numerically-generated schlieren images, some supplemented by pressure contours, for both the standard and the augmented models, arranged to allow a side-by-side comparison. Figure 6 simply shows that at t=1.6 the detonation front has reached the corner. There is no discernible difference in the results of the two models at this time. Figure 7 shows the snapshots at three different time levels, t=2.2, 2.6 and 2.8, with results of the standard model in the upper panel and those of the augmented model in the lower panel. As the detonation turns the corner, the shock in the
near-corner region suffers expansion and weakens, causing the reaction rate behind the shock to drop, and resulting in a progressive retreat of the reaction zone away from the shock. Away from the corner the detonation continues to propagate as a spherical wave, virtually undisturbed. A hint of a kink, clearly visible at t=2.6, develops at the location where the weakened and the undisturbed segments of the wavehead meet. The results for the two models continue to exhibit essentially identical behavior. Figure 8 shows the results from t=2.9 to 3.3. The first distinction between the results of the two models is now apparent at t=2.9, especially in the pressure contours. In the vicinity of the kink the reaction in the standard model has strengthened, thereby creating a ridge of high pressure in the form of a hook curving away from the lead shock. At t=3.1 this hook has grown into a transverse detonation which, at t=3.3, is seen to propagate upwards into the hitherto unreacted region. During
t = 2.2
this time interval, desensitization has clearly come to the fore in the augmented model. Near the kink the pressure is low enough, around 12 GPa at t=2.1, so that the situation is reminiscent of that in figure 2. There is no strengthening of chemical reaction, and hence no transverse wave that would consume the chemically dormant region. Figure 9 shows the results from 3.4 to 3.8. For the standard model the transverse wave continues to propagate upwards, consuming all but a small pocket of unreacted material. The reaction rate within this pocket is large enough so that it continues to react on its own, without needing any help from the reflected shock seen in the snapshot at t=3.8. For the augmented model the detonation continues to propagate but only into the unshocked material, while the dead zone created by the desensitization of the previously shocked material persists. Even at t=4.8, when the detonation breaks out of the left boundary, figure 10 shows that the dead zone remains.
t = 2.4
t = 2.6
FIGURE 7. Numerical schlieren images (zoomed) for the standard model (upper panel) and the augmented model (lower panel). Subsequent to corner turning the reaction zone lags behind the shock. Discernible differences between the two models are yet to appear.
CONCLUSIONS A modification of the ignition-and-growth model of reactive-flow is proposed to explicitly include a mechanism for desensitization when the
explosive is exposed to a weak shock. This is done by introducing a pressure-driven desensitization parameter that is convected with the flow. It is assumed that the compression threshold of the initiation step of the reaction, and a delay in
t = 2.9
t = 3.1
t = 3.3
FIGURE 8. Numerical schlieren images and pressure contours (zoomed) for the standard model (two upper panels) and the augmented model (two lower panels); t from 2.9 to 3.3.
t = 3.4
t = 3.6
t = 3.8
FIGURE 9. Numerical schlieren images and pressure contours (zoomed) for the standard model (two upper panels) and the augmented model (two lower panels); t from 3.4 to 3.8.
the first growth reaction, are functions of the degree of desensitization. Results for the standard and augmented models are compared for detonation diffraction in the LX-17 hockey-puck geometry. While the predictions of the two models away from the corner are virtually identical, the augmented model is also able to capture a sustained dead zone in the vicinity of the corner.
2. J. N. Johnson, P. K.Tang and C. A. Forest. Shock wave initiation of heterogeneous reactive solids, J. Applied Phys. 57 (1985), 4323. 3. J. Starkenberg. Modeling detonation propagation and failure using explosive initiation models in a conventional hydrocode, in The Twelfth Symposium (International) on Detonation (2002), 1001-1007. 4. E. L. Lee and C. M. Tarver. Phenomenological model of shock initiation in heterogeneous explosives, Phys. Fluids 23 (1980), 2362-2372. 5. C. M. Tarver. Ignition and growth modeling of LX-17 hockey puck experiments, Propellants, Explosives, Pyrotechnics 30 (2005), 109-117. 6. P. C.Souers, R. Garza and P. Vitello. Ignition and growth and JWL++ detonation models in coarse zones, Propellants, Explosives, Pyrotechnics 27 (2002), 62-71. 7. P. C. Souers, H. G. Andreski, C. F. Cook III, R. Garza, R. Pastrone, D. Phillips, F. Roeske, P. Vitello and J. D. Molitoris. LX17 corner turning, Propellants, Explosives and Pyrotechnics 29 (2004), 359-367.
FIGURE 10. Numerical schlieren images and pressure contours for the augmented model; t = 4.8. Acknowledgements. The authors acknowledge the support of the Los Alamos National Laboratory, the Lawrence Livermore National Laboratory, and the National Science Foundation. REFERENCES 1. C. L. Mader and C. A. Forest. Twodimensional homogeneous and heterogeneous wave propagation, Tech. Rep. LA6259, Los Alamos Scientific Laboratory, (1976).
8. P. C. Souers, H. G.Andreski, J. Batteux, B. Bratton, C. Cabacungan, C. F. Cook III, S. Fletcher, R. Garza, D. Grimsley, J. Handley, A. Hernandez, P. McMaster, J. D. Molitoris, R. Palmer, J. Prindiville, J. Rodriguez, D. Schneberk, B. Wong and P. Vitello. Dead zones in LX-17 and PBX 9502, Propellants, Explosives, Pyrotechnics, 31 (2006), 89-97. 9. A. K. Kapila, D. W. Schwendeman, J. B. Bdzil and W. D. Henshaw. A study of detonation diffraction in the ignition-andgrowth model. Submitted for publication (2006). 10. N. J. Whitworth and J. R. Maw. Modelling shock desensitization of heterogeneous explosives. AIP Conference Proceedings 370 (1996), 425-428.
