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European Journal of Applied Sciences – Vol. 10, No. 6

Publication Date: December 25, 2022

DOI:10.14738/aivp.106.13447. Partom, Y. (2022). Overdriven Detonation Velocity Dependence on Front Curvature. European Journal of Applied Sciences, 10(6).

156-160.

Services for Science and Education – United Kingdom

Overdriven Detonation Velocity Dependence on Front Curvature

Yehuda Partom

Retired from RAFAEL, P.O. Box 2250, Haifa, Israel

ABSTRACT

Overdriven detonation is a forced detonation where the shock state is stronger than

the CJ state of the explosive. Overdriven detonations are usually not steady but

decaying towards the CJ state. It is however possible to have a steady overdriven

detonation wave when the wave forcing agent is steady. For sub-CJ curved and

steady detonations it is known that detonation velocity (D) is a decreasing function

of front curvature (k). This D(k) relation is often used in an approximate and an

efficient procedure, known as Detonation Shock Dynamics (DSD), to calculate the

propagation of a quasi-steady curved detonation front, without having to solve the

entire flow field behind it. In real situations, a divergent front going around an

obstacle may become convergent (negative curvature). To tackle this part of the

front propagation, people using DSD extrapolate the D(k) curve into the negative

curvature region. Matignon et al. [2] performed a test in which they created a steady

overdriven detonation with a negative curvature in an explosive rod. The forcing

agent was a stronger explosive cylinder wrapped around the tested rod. In this way

they were able to measure a point on the D(k) plane above the CJ velocity, to which

they extrapolated the usual D(k) curve. Here we raise the question of the

uniqueness of this extrapolated curve. It is hard to answer this question

experimentally, as candidates to replace the outer explosive are hard to find. But as

we show here, it is quite easy to answer this question computationally. We use our

reactant temperature dependent reactive flow model TDRR [3-5], which we have

calibrated and validated from many test results. As a forcing agent we put a

travelling pressure boundary condition on the test rod. We change independently

the pressure P and the travelling wave speed D, and in this way, we are able to

obtain points on the D(k) plane for negative curvatures and above the CJ state. We

show that these points do not fall on a single D(k) curve, and conclude that there is

no unique D(k) relation for overdriven steady detonations.

INTRODUCTION

For an overdriven detonation, values of the variables at the end of the reaction zone are above

the corresponding values at the CJ state. Overdriven detonations may be realized in the

following situations: 1. for a converging detonation initiated by a very high velocity impact; 2.

when two detonation waves interact at an angle and a detonation Mach stem is formed; or 3.

when a very fast travelling pressure wave is applied at the radial boundary of an explosive rod.

As the reaction rate of an overdriven detonation is very high, it would seem to be of minor

influence, when trying to predict an overdriven detonation configuration. What seems to count

is a reliable products equation of state for the supra compression range. The products equation

of state is usually measured from a cylinder test that samples only the descending part of the

main isentrope, and the common practice is to extrapolate this isentrope to the supra

compression range. In [1] they measured the Hugoniot curves of two standard explosives in

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Partom, Y. (2022). Overdriven Detonation Velocity Dependence on Front Curvature. European Journal of Applied Sciences, 10(6). 156-160.

URL: http://dx.doi.org/10.14738/aivp.106.13447

the supra compression range and found that the experimental curves are substantially above

the curves extrapolated from cylinder test data. Also, when an overdriven detonation is time

dependent, or even when it is steady in one direction and varying in another direction, its

reaction rate may still be of importance. One such case is the steady overdriven detonation in a

rod of explosive.

For a steady detonation in a rod with a free boundary there is the well-known diameter effect

(or size effect), where detonation velocity is lower for a smaller rod diameter. The diameter

effect can be translated into a velocity versus front curvature relation D(k), which depends on

the reaction rate of the explosive. For sensitive (high reaction rate) explosives, the detonation

velocity D changes substantially with the curvature k, and for insensitive explosives the D(k)

curve is almost flat. The D(k) relation can be used by a very efficient procedure to calculate the

propagation of quasi steady diverging fronts, without taking into account the flow field behind

the front. This procedure is called Detonation Shock Dynamics. When using detonation

shockdynamics, a diverging front may sometimes become converging (negative curvature), like

when the front goes around an obstacle. In such a case it is customary to extrapolate the D(k)

relation, determined for positive curvature, into the negative curvature range.

Recently, Matignon et al. [2] devised a test configuration by which they are able to measure a

point on the D(k) curve of an explosive in the negative k range. In this test an overdriven

detonation is forced into an insensitive explosive rod by a stronger (faster) explosive wrapped

around it. The detonation in the inner explosive reaches a steady front with a negative

curvature and with the velocity of the outer explosive. In this way it is possible to determine a

point on the D(k) curve of the inner explosive in the negative curvature range.

Here we question the uniqueness of D(k) in the negative curvature range. We suspect that

because of the nature of overdriven detonations, they cannot be independent of the boundary

conditions or the properties of the forcing agent. It is difficult to explore this uniqueness

question experimentally, as an arsenal of different forcing agents is practically nonexistent. But

it is rather easy to explore this uniqueness question computationally. All one needs for that is a

hydrocode that employs a valid reactive flow model.

In what follows we use our reactant temperature dependent reactive flow model (TDRR) [3-5],

which has been calibrated and validated for many shock- initiation and detonation situations.

Even if someone raises doubts as to the validity of TDRR for a specific explosive, this is of no

consequence regarding the question of uniqueness explored here. Our TDRR reactive flow

model with the parameters that we use describes the behavior of a possible explosive. We show

in what follows that D(k) of this explosive is not unique in the negative curvature range, and

this may put in doubt the uniqueness of D(k) there for any explosive.

COMPUTATION RESULTS

We use our TDRR [3-5] reactive flow model, which is employed in the commercial 2DPISCES

hydrocode, to calculate an overdriven detonation in an insensitive high explosive rod. We force

the overdriven detonation into the rod by applying a travelling wave pressure boundary

condition on its radial boundary. The travelling wave boundary condition has two independent

parameters: 1) the travelling wave velocity (D>DCJ); and 2) the initial boundary pressure

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European Journal of Applied Sciences (EJAS) Vol. 10, Issue 6, December-2022

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(P0>PCJ). At each point on the boundary the pressure decays by the same rate. From the

computations we obtain for each pair of (P0,D) a steady overdriven detonation front of negative

curvature. We denote the curvature on the axis by k0, and for each pair of (D,P0) we get a pair

of (D,k0), which we plot as a point on a (D,k) plane.

We use an insensitive high explosive for which TDRR has been calibrated. The rod diameter is

20mm and its length is 120mm. The left boundary is held rigid, and after 100mm of propagation

into the rod the detonation front becomes steady. We use three values of D: 8.0, 8.5 and 9.0

km/s, all higher than DCJ=7.71 km/s, and three values of P0: 30, 35 and 40 GPa, all higher than

PCJ=28.2 GPa. The time constant for the exponential pressure decay on the boundary is 1.2μs.

In Fig. 1 we show our computed detonation fronts every 2 μs, from the run with D=8.5 km/s

and P0=35 GPa.

Figure 1. Detonation fronts every 2μs, for D=8.5km/s and P0=35GPa. (The vertical lines are

code artifacts with no physical meaning).

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Partom, Y. (2022). Overdriven Detonation Velocity Dependence on Front Curvature. European Journal of Applied Sciences, 10(6). 156-160.

URL: http://dx.doi.org/10.14738/aivp.106.13447

We see from Fig. 1 that the front is curved at the central section of the rod and is flat (conical)

at the outer section. This is similar to Matignon et al. [2] experimental results.

In Fig. 2 we show an enlarged plot of two front shapes from Fig. 1, at 80 and 100 mm along the

rod. The second front shape is cutoff at the point where its straight part begins. We fit this front

shape by:

(1)

From which we have:

(2)

In Fig. 3 we show the results of all our 9 runs.

Figure 2. Front shapes at 80 and 100mm into the rod, from the run of Fig. 1. The curvature on

the axis is k0=-0.01546/mm.

Figure 3. D(k0) results of overdriven detonation in a rod, from 9 runs with different values of D

and P0, and for P0=30,35,40 GPa respectively.

2 4 z Ar Br = +

"

0 k z (0) 2A = - = -

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European Journal of Applied Sciences (EJAS) Vol. 10, Issue 6, December-2022

Services for Science and Education – United Kingdom

We see from Fig. 3, that for different values of the travelling pressure waves on the boundary,

we get different D(k0) curves. This result would put in doubt the uniqueness of D(k) in the

negative curvature range.

SUMMARY

We question here the uniqueness of the D(k) relation for overdriven detonations (or for

negative curvature). To this end we run a 2D hydrocode with a reactive flow model, for

overdriven detonations in a rod. We force the overdriven detonation by applying a travelling

wave boundary condition on the radial boundary. We change separately the travelling wave

velocity (D), which is also the steady detonation velocity in the rod, and the initial boundary

pressure P0. From each run we obtain the value of the negative curvature on the axis, and we

plot the pairs (D,k0) on the (D,k) plane. From the results we conclude that there is no unique

D(k) relation in the negative curvature range.

References

P.K. Tang et al., A study of the overdriven behaviors of PBX-9501 and PBX-9502, 11th symposium on detonation,

1058-1064 (1998).

C. Matignon et al., Detonation propagation of converging front in IHE: comparison of direct numerical simulation

and detonation shock dynamics experimental data, 14th symposium on detonation, 1182-1190 (2010).

Y. Partom, A Void collapse model for shock initiation, 7th symposium on detonation, 506 (1981).

Y. Partom, Characteristics code for shock initiation, LANL report, LA-10773 (1986).

Y. Partom, Hydro-Reactive computations with a temperature dependent reaction rate, shock compression in

condensed matter conference, Atlanta GA (2001).