Numerical investigation of particle–blast interaction during explosive dispersal of liquids and granular materials

Abstract

Experiments show that when a high-explosive charge with embedded particles or a charge surrounded by a layer of liquid or granular material is detonated, the flow generated is perturbed by the motion of the particles and the blast wave profile differs from that of an ideal Friedlander form. Initially, the blast wave overpressure is reduced due to the energy dissipation resulting from compaction, fragmentation, and heating of the particle bed, and acceleration of the material. However, as the blast wave propagates, particle–flow interactions collectively serve to reduce the rate of decay of the peak blast wave overpressure. Computations carried out with a multiphase hydrocode reproduce the general trends observed experimentally and highlight the transition between the particle acceleration/deceleration phases, which is not accessible experimentally, since the particles are obscured by the detonation products. The dependence of the particle–blast interaction and the blast mitigation effectiveness on the mitigant to explosive mass ratio, the particle size, and the initial solid volume fraction is investigated systematically. The reduction in peak blast overpressure is, as in experiments, primarily dependent on the mass ratio of material to explosive, with the particle size, density, and initial porosity of the particle bed playing secondary roles. In the near field, the blast overpressure decreases sharply with distance as the particles are accelerated by the flow. When the particles decelerate due to drag, energy is returned to the flow and the peak blast overpressure recovers and reaches values similar to that of a bare explosive charge for low mass ratios. Time–distance trajectory plots of the particle and blast wave motion with the pressure field superimposed, illustrate the weak pressure waves generated by the motion of the particle layer which travel upstream and perturb the blast wave motion. Computation of the particle and gas momentum flux in the multiphase flow generated during explosive particle dispersal indicates that the particle momentum flux is the dominant term in the near field. Both the gas and particle loading must be taken into account when determining the damage to nearby structures following the detonation of a high-explosive charge surrounded by a material layer.

1 Introduction

Surrounding a high-explosive charge with a layer of liquid or porous granular material is a common method to reduce the blast wave signature in the immediate vicinity of the charge. The mitigation of the blast wave is attributed to the dissipation of energy during compaction of a porous material and the transfer of thermal and kinetic energy to the mitigant material [1]. While a number of investigators have experimentally studied the mitigation effectiveness of a variety of materials in the near field (see [2] for a review of past work in this area), relatively little attention has been devoted to examining the interaction of the dispersed material with the blast wave and nearby structures. In some cases, despite a reduction in the blast strength, due to the momentum imparted to the material, the total loading on structures may actually be increased in comparison with a bare explosive charge [3].

In a recent experimental study of the explosive dispersal of liquids and granular materials, the terminal velocity attained by the material [4] and the decay of the blast wave overpressure [2] were determined for a wide range of materials. Loiseau et al. [4] showed that for liquids, the terminal velocity agrees well with the predictions of the classical Gurney model and depends primarily on the liquid to explosive mass ratio, M/C. For granular materials, the Gurney model over-predicts the terminal velocity by 25–60%, with the velocity deficit attributed to energy dissipation during collapse of the porous particle bed and possible particle deformation. The velocity deficit is insensitive to the presence of particle jetting as well as the properties of the granular material. Pontalier et al. [2] found that while the addition of a layer of particles around the explosive charge reduced the peak blast overpressure in the near field, further from the charge (i.e., for $Z \ge 2\, \text {m/kg}^{1/3}$) the overpressure recovered nearly to the value corresponding to a bare charge, for $M/C < 10$. They hypothesized that the particles collectively act effectively as a porous piston with the pressure disturbances from individual particles, or clusters of particles, eventually reaching the blast wave and reducing the rate of decay of the peak blast overpressure. Depending on the size and density of the particles, the explosively dispersed particles may also reach and penetrate the blast wave and perturb the quiescent gas upstream of the blast wave front [5, 6].

A model for the complete process of explosive dispersal of a particle layer must consider first the shock propagation through the particle layer, compaction of the layer, transmission of the shock as a blast wave into the surrounding air, expansion of the dense gas-particle medium, and finally, the dilute gas-particle regime after the particle cloud has expanded over a sufficient volume. Some authors have used the model problem of the sudden release of a pressurized gas-particle mixture to investigate the subsequent multiphase flow, while not considering the initial compaction phase (e.g., [7]). Other researchers have carried out computations to explore some aspects of the overall problem, such as shock interactions with particles and droplets [8], detailed particle–particle interactions [9], or later-time impulse effects from particles [10]. A further complication is that the transient multiphase flow that is generated during explosive dispersal of particles (or liquids) is typically characterized by a non-uniform particle distribution taking the form of clusters of particles or particle jets [11,12,13,14]. A mesoscale model for the multiple particle–particle and particle–flow interactions and formation of particle jets is computationally demanding and beyond the scope of the present investigation.

Experimentally, it is observed that the maximum material velocity is relatively insensitive to the degree of particle jetting and material properties [4], while the blast wave overpressure does exhibit some dependence on particle material properties [2]. It is difficult to isolate the influence of particle jetting on the blast overpressure. However, since pure iron and steel shot have similar densities and mitigation performances, yet iron forms jets and steel shot does not, we expect that material properties have a stronger influence than the degree of particle jetting on blast wave overpressure. This suggests that a simple 1D multiphase model, which does not account for multidimensional jet formation, may be used to gain some insight into the interaction of the dispersed particles with the flow and blast wave, and to see whether the general experimental trends are qualitatively reproduced. The model may then be used to determine the influence of experimental parameters, such as the mass ratio of inert material to explosive M/C, particle size and density, initial bed porosity, as well as the difference between dispersal of a layer of particles and liquid.

In the present paper, the basis of the computational model will first be described, followed by results of simulations to explore the interaction between the dispersed particles and flow behind the blast wave. To facilitate a comparison with experimental results, calculations are carried out with two types of granular powders (glass and steel) and one liquid (water). Emphasis is placed on exhibiting the dependence of various parameters on the decay of the blast wave peak overpressure and impulse, compared to an explosive-generated flow without particles. The results are interpreted with the use of time–distance trajectory plots that illustrate the relative motion of the particles and blast wave.

2 Computational model

The calculations in the present paper are carried out with the EDEN multiphase hydrocode developed by scientists at Fluid Gravity Engineering Ltd (UK). This code has been extensively validated with experimental results for the explosive dispersal of powders and liquids. For example, Milne et al. [13] obtained good agreement with particle radius-time data obtained from radiographic images of the explosive compaction and dispersal of a spherical bed of glass beads. To match the experimental data it was found that a key assumption was necessary, in particular, that the compaction energy was not recovered upon expansion of the particle bed. This has the consequence that a spall layer is not formed during the compaction and expansion of the porous particle bed, which is consistent with the radiographic results. The basic equations that form the basis of the model are given in an Appendix in a previous publication [11].

The charge configuration used consists of a central spherical ball of C-4 explosive directly surrounded with a spherical layer of particles or liquid. This is contrary to the experiments with liquids [2], where the C-4 is contained within a plastic ball; conversely, for experiments with powders, the particles are in direct contact with the explosive charge. Detonation of the C-4 charge proceeds with a programme burn function, and the detonation products are modelled with the classical Jones–Wilkins–Lee (JWL) equation of state [15]. To model the compaction of the porous particle bed, the Herrmann P-$\alpha$ porous solid equation of state is used, with the Murnaghan equation of state used for the solid, given by (1):

$$\begin{aligned} \left\{ \begin{array}{llll} K = \rho _{\mathrm {s0}} C^2_{\mathrm {s}} \\ n = 4S - 1 \\ P_{\mathrm {s}} = \frac{K}{n} \left[ \left( \frac{\rho _{\mathrm {s}}}{\rho _{\mathrm {s0}}} \right) ^{n} - 1.0 \right] + P_{\mathrm {s0}} \\ \end{array} \right. \end{aligned}$$

(1)

where $\rho _{\mathrm {s0}}$ is the initial solid material density, $C_{\mathrm {s}}$ corresponds to the solid sound speed, S is the slope of the shock/particle velocity curve, and $P_{\mathrm {s0}}$ is the initial solid pressure.

The particle configuration pressure, $\beta$, is assumed to be a function of the solid volume fraction $\phi _{\mathrm {s}}$ and takes the following form (2):

$$\begin{aligned} \beta = a\left[ \frac{(\phi _{\mathrm {s1}} - \phi _{\mathrm {s0}})^2}{(\phi _{\mathrm {s1}} - \phi _{\mathrm {s}})} - (\phi _{\mathrm {s1}} - \phi _{\mathrm {s0}}) \right] \end{aligned}$$

(2)

where $P_{\mathrm {s}}$ and $\rho _{\mathrm {s}}$ are, respectively, the pressure in the solid medium and the density of the solid particles. $\phi _{\mathrm {s0}}$ is the initial value of the particle volume fraction for a packed particle bed ($\phi _{\mathrm {s0}} = \rho _{\mathrm {bulk}}/\rho _{\mathrm {s0}}$), and $\phi _{\mathrm {s1}}$ is the maximum particle volume fraction attained during compaction. The dependence of particle configuration pressure on solid volume fraction $\phi _{\mathrm {s}}$ describes how the inter-particle forces within the particle bed evolve as a function of $\phi _{\mathrm {s}}$. The steepness of this dependence is given by the coefficient a. Values for the parameters used in the above equations (1 and 2) for glass and steel particles are given in Table 1. The values fall in the range of experimental values, i.e., #13 Ballotini glass impact beads (mean particle size of 68 ± 22 $\upmu \mathrm{m}$, and $\rho _{\mathrm {bulk}} = 1500 \pm 80$ $\text {kg}/\text {m}^{3}$) and S-110 hardened steel shot (mean particle size of 280 ± 160 $\upmu \mathrm{m}$, and $\rho _{\mathrm {bulk}} = 4460 \pm 330$ $\text {kg}/\text {m}^{3}$), for the glass and steel beads, respectively. The particles are also considered to be spherical and maintain the same initial size throughout the explosive dispersal event, without fracturing or deforming.

Table 1 Parameters used in the numerical simulations for glass, steel particles, and water

Water is modelled using a Tait equation of state, i.e., (3):

$$\begin{aligned} P_{\mathrm {l}} = K^{\prime }\left[ \left( \frac{\rho _{\mathrm {l}}}{\rho _\mathrm {l0}}\right) ^{n^{\prime }} - 1 \right] \end{aligned}$$

(3)

where $P_{\mathrm {l}}$, $\rho _{\mathrm {l}}$ are, respectively, the pressure in the liquid medium and the density of the liquid droplets. $\rho _\mathrm {l0}$ corresponds to the initial density of water. The water is assumed to fracture upon pressure release into droplets with an imposed size of 100 $\upmu \mathrm{m}$, which is on the order of magnitude of droplet sizes observed in past work [11]. The initial liquid volume fraction $\phi _\mathrm {l0}$ is set close to unity (0.99) to ensure stability of the calculations, and the value of the liquid volume fraction $\phi _{\mathrm {l}}$ evolves when the water is dispersed. The values of the coefficients in (3) are reported in Table 1, in MKS units.

3 Results and discussion

In this section, numerical results are presented for the three different materials considered. Glass particles, which tend to form prominent particle jets when explosively dispersed, were chosen to compare with the extensive experimental data available. Steel particles, which have a higher density and toughness than glass particles, tend not to form jets when dispersed, but expand in a diffuse layer. Water was chosen for a liquid dispersal case due to the availability of experimental data. Note that since the development of multi-dimensional effects cannot be captured with a 1D model, the effect of particle jetting on blast overpressure cannot be critically examined with the current computational investigation. However, as noted earlier, experimental results suggest that material properties have a greater influence on blast wave mitigation than the degree of particle jetting.

Fig. 1

Comparison between Kinney and Graham [16] database (dark full lines) for a conventional chemical explosive charge with energy release equivalent to 1 kg of TNT and calculated peak overpressures (left) and impulses (right) as a function of scaled distance Z, for a 100-g bare C-4 charge shown as red full lines with dots (equivalent to 139 g of TNT for peak overpressures and 115 g of TNT for positive-phase impulses)

In the following sections, peak blast wave overpressures and impulses for an unmitigated C-4 charge determined numerically are first compared with tabulated experimental values (Sect. 3.1). Then, mitigation efficiencies, i.e., the reduction in blast overpressure compared to a bare charge, are compared with experimental data for the three different materials (Sect. 3.2). A systematic study of the influence of initial bed properties on blast peak overpressure and impulse is presented in Sect. 3.3. Finally, the particle–blast interaction is explored through examination of blast overpressure profiles (Sect. 3.4), time of arrival (Sect. 3.5), momentum fluxes, and spatially integrated momenta of both particles and the gas (Sects. 3.6 and 3.7).

Fig. 2

Comparison between numerical calculations (full purple lines with dots) of peak overpressure (left) and impulse (right) for the explosive dispersal of 300 $\upmu \mathrm{m}$ steel particles ($M/C = 55.06$) with experimental data extracted either by videographic tracking of blast wave (dashed purple line) or direct pressure gauge measurements (square purple symbols). For comparison, the numerical calculations for a baseline, bare C-4 charge (solid red lines) are also shown

3.1 Bare C-4 charge

Prior to investigating the influence of different types of materials on the blast wave, it is of interest to compare the EDEN numerical predictions for a bare C-4 charge with the empirically derived tabulated data from Figure 6–4 of Kinney and Graham [16]. This figure shows peak overpressures at 1 atm and 15 $^{\circ }$C computed for a spherical charge of conventional chemical explosive with energy release equivalent to one kilogram of TNT, as a baseline test. Figure 1 shows the comparison between the calculated and referenced normalized peak overpressure $\Delta P_{\mathrm {max}}/P_\mathrm {0}$, with $\Delta P_{\mathrm {max}} = P_{\mathrm {max}} - P_\mathrm {0}$, $P_{\mathrm {max}}$ being the maximum local pressure and scaled impulses $I_{\mathrm {s}}^{+}$. These parameters are plotted as a function of scaled distance Z defined as follows (4):

$$\begin{aligned} Z = \frac{R}{W^{1/3}} \end{aligned}$$

(4)

where Z = scaled distance ($\text {m/kg}^{1/3}$), R = dimensional distance (m), W = (charge mass in equivalent TNT) = mass of C-4 (kg), $\times \, \varepsilon$ (effectiveness factor), with the same effectiveness factor $\varepsilon = 1.39$ as for experimental data [2] for a C-4 charge mass of 100 g.

The scaled blast impulse and arrival time (analysed in Sect. 3.5) are defined, respectively, by (5) and (6):

$$\begin{aligned} I^{+}_{\mathrm {s}} = \frac{I^{+}}{W^{1/3}} \end{aligned}$$

(5)

and

$$\begin{aligned} t_{\mathrm {s}} = \frac{t_\mathrm {a}}{W^{1/3}} \end{aligned}$$

(6)

where $I^{+}_{\mathrm {s}}$ = blast wave scaled impulse (bar$\cdot$ms/kg$^{1/3}$), $I^{+}$ = blast wave positive-phase impulse (bar$\cdot$ms), $t_{\mathrm {s}}$ = scaled blast arrival time ($\hbox {ms/kg}^{1/3}$), $t_\mathrm {a}$ = actual blast arrival time (ms), W is the charge mass in equivalent TNT defined previously, with $\varepsilon = 1.15$ for impulses and $\varepsilon = 1.39$ for arrival times, in conformity with [2]. Overall, good agreement for peak overpressures is found, with a maximal overpressure discrepancy of 30% in the region 1 $\text {m/kg}^{1/3}\,{<}\,Z\,{<}\,$ 3 $\text {m/kg}^{1/3}$ where the computed values fall below the tabulated values. Scaling laws have been also verified with different charge masses (75 g, 28 g, and 1 kg). Changing the mass of C4 will just shift the curve to the left (higher explosive mass) or to the right (lower explosive mass). In comparison, calculated positive-phase impulses fall above the tabulated experimental values in the range with a maximal discrepancy of 68% at $Z = 0.7\, \text {m/kg}^{1/3}$, but slightly below in the region 1.5 $\text {m/kg}^{1/3}\,{<}\,Z\,{<}\,$ 2 $\text {m/kg}^{1/3}$. Given the discrepancies between the calculations and experimental data, a relative study will be carried out in this paper. For the remainder of the presentation of results, the numerically calculated peak overpressures and impulses for the bare C-4 charge will define the baseline case.

Fig. 3

Comparison between numerical calculations of peak overpressure (left) and impulse (right) with experimental data extracted by either videography or pressure gauges for the explosive dispersal of 50 $\upmu \mathrm{m}$ glass particles with mass ratios $M/C = 6.5$ and $M/C = 46.85$

Fig. 4

Comparison between numerical calculations of peak overpressure (left) and impulse (right) with experimental data extracted by either videography or pressure gauges for the explosive dispersal of water with $M/C = 32.21$ (droplet sizes of 100 $\upmu \mathrm{m}$)

3.2 Dispersal of granular media and liquids

In this section, computed peak overpressures and impulses generated in the air during explosive dispersal of either granular powders or liquids are reported and compared with the experimental data (reported in [2]). The calculated baseline case for a bare, or unmitigated, C-4 charge is also shown to quantify the attenuation of the pressure and impulse due to the presence of the inert material. In Figs. 2, 3, and 4, blast properties for explosive charges surrounded by three different materials are investigated with various material to charge mass ratios M/C, including steel ($M/C =55.06$), glass ($M/C = 6.50$ and $M/C = 46.85$), and water ($M/C = 32.21$). Properties of the materials (density, bulk density, particle size) and coefficients used in the model are reported earlier in Table 1. The computed peak blast overpressures (shown as full lines with dots in the figures) are compared with experimental results for the overpressure obtained in two different ways: (i) tracking the blast wave trajectory to estimate the shock Mach number and hence inferring the shock overpressure with the Rankine–Hugoniot relation (dashed lines), or (ii) by direct measurement of blast overpressure with side-on piezoelectric pressure gauges (square and diamond symbols) [2].

The calculated peak overpressure values generally agree with the experimental values within the error associated with the experimental results over the range of scaled distance simulated and mass ratios (see [2] for a discussion of the scatter in the experimental results and errors associated with the different measurement techniques). The maximal discrepancies in terms of peak pressure are 6% for steel ($M/C = 55.06$, at $Z = 1.4\, \text {m/kg}^{1/3}$) and 9% for glass ($M/C = 46.85$, at $Z = 1.6\, \text {m/kg}^{1/3}$). Maximal discrepancies for water ($M/C = 32.21$, at $Z = 0.6\, \text {m/kg}^{1/3}$) and glass ($M/C = 6.5$, at $Z = 0.7 \,\text {m/kg}^{1/3}$) are higher than the experimental estimated errors, i.e., respectively, 35 and 55%. Nevertheless, overall, the peak overpressure decays are reproduced by the numerical calculations for each material. Factors that are not accounted for in the model, which may contribute to the differences between the computed and experimental data, include the energy loss during deformation and fracture of the particles during particle compaction, and the relative velocity induced fragmentation and evaporation of water droplets.

Fig. 5

Peak overpressures (left) and impulses (right) as a function of scaled distance for explosive dispersal of glass particles with mass ratios of $M/C = 1, 10$, and 100 ($\phi _{\mathrm {s0}} = 0.6$, particle size $= {50}\,{\upmu \mathrm{m}})$

With respect to the computed overpressure results for water, note the presence of a kink in the overpressure curve at a scaled distance $Z = 0.5\, \text {m/kg}^{1/3}$ (a similar bend appears for the other materials at lower Z values). The physical mechanism responsible for this feature in the overpressure curve will be discussed later (Sect. 3.3.1). The presence of such a kink in the experimental data has not been observed, although experimental measurements of blast overpressure in the very near field, e.g., $Z\,{<}\,0.5\, \text {m/kg}^{1/3}$, are challenging since the blast wave is obscured by the detonation products at early times and pressure gauges are subject to particle impacts very close to the charge, which perturbs the blast pressure signal.

The calculated scaled positive-phase air impulse is also shown in comparison with experimental data from pressure gauges in Figs. 2, 3, and 4. In general, the calculated air impulses exceed significantly the experimental measured values for particle dispersal (51–61% for steel, 36–52% for glass $M/C = 6.5$, 62–105% for glass $M/C = 46.85$), whereas they closely match the experimental results for water (maximum error of 8%). The cause of the overestimation of the positive-phase blast impulse in comparison with the experimental data is not known, but the energy loss due to particle deformation and fracture, as well as the influence of particle clustering and jetting on the particle drag and energy transfer from the particles to the flow, may contribute to this discrepancy. In the case of the explosive dispersal of water, it is also interesting to note that despite surrounding the explosive charge with a mass of water 30 times the mass of the explosive, the positive-phase impulse values are close to that of a bare C-4 charge in the far field ($Z > 4\, \text {m/kg}^{1/3}$).

3.3 Parametric study

To gain more insight into the decay of the blast wave overpressure very near the charge, including the presence of the kink in Fig. 4, calculated results will be presented over a larger range of scaled distance Z, up to the location of the initial material layer. The effect of initial material parameters on the blast peak overpressure and positive-phase impulse, including the mass ratio M/C, particle size, and initial solid volume fraction, is also presented in the following sections.

Fig. 6

Peak overpressures (left) and impulses (right) as a function of scaled distance for explosive dispersal of steel particles with mass ratios of $M/C = 1, 10$, and 100 ($\phi _{\mathrm {s0}} = 0.6$, particle size $= {50}\,{\upmu \mathrm{m}})$

3.3.1 Effect of mitigant to charge mass ratio M/C

Three values of M/C, 1, 10, and 100 were chosen for a fixed mass of C-4 ($C = 100$ g), for steel and glass particles. For each value of M/C, the initial volume fraction of the powder is fixed at a nominal value of 0.6 with a particle size of 50 $\upmu \mathrm{m}$. As the solid volume fraction is fixed, for increasing masses of powder, the thickness of the powder layer increases and is indicated by dashed lines in Figs. 5 and 6. The thickness of the granular layer depends on the density of the material, since a denser material occupies less volume for the same mass of powder and particles are all assumed to be spherical.

For scaled distances greater than $Z = 0.4\, \text {m/kg}^{1/3}$, the attenuation of the peak overpressure for both glass and steel increases with increasing M/C as expected with an attenuation of more than one order of magnitude for $M/C = 100$. For $Z < 0.4\, \text {m/kg}^{1/3}$, this observation does not hold since, depending on the M/C ratio, the shock may still be propagating within the granular layer, and the particle acceleration phase is ongoing. At $Z = 1\, \text {m/kg}^{1/3}$, for glass particles, the peak overpressure attenuation relative to that of a bare C4 charge is approximately 19, 69, 96% for, respectively, $M/C = 1, 10, 100$. For steel particles, the attenuation is slightly less than for glass particles, with values of 64 and 94% for $M/C = 10$ and 100, respectively, as observed experimentally [2]. On the contrary, for $M/C= 1$, the attenuation is slightly higher with a value of 25% (c.f., 19% for glass particles). Nevertheless, discrepancies between the peak blast overpressure mitigated by glass and steel particles at a fixed M/C and Z appear less compared to experimental data. For example, at $M/C = 100$ and $Z = 0.8\, \text {m/kg}^{1/3}$, values of numerical peak overpressure are, respectively, 0.3 and 0.45 (difference of 0.15) for glass and steel, compared to 0.37 and 0.64 (difference of 0.27) estimated from Fig. 15 of [2]. However, this analysis is cautionary given the experimental error bars.

For all materials, for scaled distances around $Z = 0.3\, \text {m/kg}^{1/3}$ or $Z = 0.5\, \text {m/kg}^{1/3}$, depending on the mass ratio, a kink appears in the peak overpressure curves, which is most pronounced for the largest M/C values. The kink separates a region of rapid pressure decay, for small scaled distances, from a region where the pressure decay is reduced and the peak overpressures approach values of an unmitigated charge. During the acceleration phase of the particles at early times, energy is extracted from the flow and the blast pressure drops rapidly. However, at some point the particles will reach their maximum velocity, while the ambient flow velocity continues to decrease rapidly as the blast wave and flow expand spherically outwards. Depending on the inertia of the particles, there will be a period of time where the particles are moving faster than the ambient gas and will generate pressure perturbations to the local flow, or bow shocks if the relative flow velocity is supersonic. During this time the pressure disturbances will travel upstream and influence the pressure behind the blast wave, reducing the rate of blast pressure decay. The kink is also the location where the blast peak overpressure is attenuated the most. Due to the pressure recovery, the peak blast overpressure recovers to that of a bare charge, in the far field, after $Z = 3\, \text {m/kg}^{1/3}$, for $M/C = 1$ and 10. When the mass ratio is increased to a value of 100, with the larger particle mass, the timescales for energy transfer to and from the particles increase. Hence, at a distance of $Z = 3\, \text {m/kg}^{1/3}$ the peak blast overpressure is still attenuated relative to that of a bare charge (i.e., a value of 83% for glass particles and 75% for steel particles), and the overpressure approaches the bare charge value at further distances from the charge.

Fig. 7

Peak overpressures (left) and impulses (right) as a function of scaled distance for explosive dispersal of water with mass ratios of $M/C = 1$, 10, and 50 (droplet size $=\,{50}\,{\upmu \mathrm{m}}$)

The rate of decay of the blast overpressure is primarily dependent on the mass ratio M/C. Comparing the results for glass and steel particles at a specific mass ratio indicates that the dependence of blast overpressure on material properties is of secondary order, with more attenuation for glass particles than steel particles in the near field. Presumably, the lower inertia and higher heat capacity of the glass particles contribute to this difference. However, at the lowest mass ratio ($M/C= 1$), in the mid-to-far field ($Z\,{>}\,$1 $\text {m/kg}^{1/3}$), steel particles can potentially reduce slightly more the blast peak overpressure. This depends on the motion of the particles, on the pressure disturbances created and how the particles are subjected to drag.

The same analysis is carried out with water, but the $M/C = 100$ case has been replaced by a value of $M/C = 50$. Droplet sizes are assumed to be 50 $\upmu \mathrm{m}$ to compare with solid particles, but no significant differences of mitigation effectiveness are observed between the liquid and the powders (see Fig. 7). This is not consistent with experimental results which show that, on average, solid powders outperform liquids by almost a factor of two at reducing the peak blast overpressure [2]. However, fragmentation of the water droplets after they are formed due to stripping at the high particle Weber numbers and subsequent evaporation of the fine droplets are not accounted for in the computations, which will influence the predictions for the water case at later times.

The behaviour of the scaled positive-phase impulse shown in the right plots of Figs. 5, 6, and 7 also indicates that particle material properties have minimal influence on the impulse. The maximal impulse attenuation, for all the materials, takes place in the near field ($Z < 1\, \text {m/kg}^{1/3}$), in the same region with the kink in the peak overpressure decay mentioned earlier. In this region, the positive-phase impulse mitigation, relative to that of a bare charge, for M/C going, respectively, from 1 to 100, is 32–69% (glass), 35–66% (steel), and 44–66% (water). After $Z = 2\, \text {m/kg}^{1/3}$, impulses for water at each M/C value are superimposed with the curve of an unmitigated charge (at $M/C = 100$, water impulses are even higher), as observed experimentally [2]. For glass particles, in the same region, a similar behaviour is observed for $M/C = 1$ and 10. In the case of the explosive dispersal of steel particles, virtually no differences are observed with an unmitigated charge at $M/C = 1$. However, at $M/C = 10$, the impulse recovery takes place later than for glass particles, at around $Z = 4\, \text {m/kg}^{1/3}$. At the same distance, at $M/C = 100$, values of impulses are still reduced by around 23% for both types of solid particles compared to a bare charge. Differences in attenuation and recovery of impulses between the two types of solid particles are attributed to differences in heat capacity and density. Conversely, in the near field ($Z < 1\, \text {m/kg}^{1/3}$), vaporization is likely to contribute to the differences in mitigation performance. Specifically, the liquid may attenuate the blast impulse more, as compared to solid particles at $M/C = 1$. In the far field ($Z > 2\, \text {m/kg}^{1/3}$), it is hypothesized that the motion of the vapour produced behind the blast front, during the explosive dispersal of water, will contribute to the total impulse recovery.

Fig. 8

Peak overpressures (left) and impulses (right) as a function of scaled distance for explosive dispersal of glass particles with particle sizes of 50 $\upmu \mathrm{m}$, 0.5 mm, and 5 mm ($M/C = 10$, $\phi _{\mathrm {s0}} =0.6$)

Fig. 9

Peak overpressures (left) and impulses (right) as a function of scaled distance for explosive dispersal of steel particles with particle sizes of 50 $\upmu \mathrm{m}$, 0.5 mm, and 5 mm ($M/C = 10$, $\phi _{\mathrm {s0}} =0.6$)

Fig. 10

Peak overpressures (left) and impulses (right) as a function of scaled distance for explosive dispersal of water with droplet sizes of 0.5 and 5 mm ($M/C = 10$)

Fig. 11

Peak overpressures (left) and impulses (right) as a function of scaled distance for explosive dispersal of glass particles solid mass fractions of $\phi _{\mathrm {s0}} = 0.5$, 0.6, and 0.85 ($M/C = 10$, particle size $= {50}\,\upmu \mathrm{m}$)

3.3.2 Effect of particle size

The influence of particle size on the decay of the peak blast wave overpressure is shown for glass (Fig. 8) and for steel (Fig. 9) for values of $M/C = 10$ and $\phi _{\mathrm {s0}} = 0.6$. In this case, the thickness of the granular layer is constant for the three curves. For glass particles, the particle size has little impact on the peak overpressure decay, whereas for steel particles, the effect of particle size is more significant. In the very near field ($Z = 0.4\, \text {m/kg}^{1/3}$), the shock attenuation is weakest for the largest steel particles (57% for 5-mm particles), compared to the smallest particles (86% for 0.05-mm particles). On the contrary, at the same distance, the differences of peak overpressure attenuation for glass particles are less, i.e., respectively, 85% (largest particles) and 90% (smallest particles). The poor mitigation performance for the largest particles is presumably a result of the large particle inertia and length of time for the particles to accommodate to changes in the local flow velocity. Nevertheless, in the far field ($Z > 2\, \text {m/kg}^{1/3}$), this trend is reversed and the smallest particles attenuate the peak overpressure the least. In the case of water (Fig. 10), differences of mitigation effectiveness are only evident in the mid-to-far field ($Z > 1\, \text {m/kg}^{1/3}$). The mitigation performance varies from 15% (smallest droplets, 0.5 mm) to 59% (biggest droplets, 5 mm), at $Z = 2\, \text {m/kg}^{1/3}$. However, the results in this regime are uncertain due to a lack of a droplet fragmentation model as noted earlier.

Similar trends are observed as in Figs. 5, 6, and 7 for the positive-phase impulses, except that discrepancies in mitigation are more apparent in the far field ($Z > 2\, \text {m/kg}^{1/3}$) for glass particles, contrary to the near field ($Z < 1\, \text {m/kg}^{1/3}$) for steel particles. Indeed, the impulse attenuation at $Z = 0.4\, \text {m/kg}^{1/3}$ for glass particles is only between 25% (5-mm particles) and 35% (0.05-mm particles). In comparison, the attenuation for steel particles is between 0 and 43%, respectively, whereas at $Z = 4\, \text {m/kg}^{1/3}$, the impulse mitigation becomes 47% (c.f. 29% for steel) for the largest particles (the blast impulse is effectively not attenuated by the smallest particles). The effect of droplet size on impulse in the case of the explosive dispersal of water (Fig. 10) is more predominant in the same region as for peak overpressures, i.e., in the mid-to-far field, with an attenuation between 34% (0.5-mm droplets) and 48% (5-mm droplets) at $Z = 2\, \text {m/kg}^{1/3}$.

In general, since smaller particles accommodate to the local flow velocity more rapidly than larger particles, energy is extracted more rapidly from the flow during particle acceleration and conversely given back to the flow more rapidly during particle deceleration. Hence, smaller particles tend to lead to more rapid decay of the blast pressure in the near field and greater pressure recovery in the far field. Since the size distribution of particles generated during explosive particle dispersal, in general, differs from the initial particle size due to particle compaction, deformation, and sintering effects, it is difficult to predict the effect of particle size without also addressing the details of the particle bed compaction and fracture processes. In any case, the effect of particle size appears to be a secondary effect compared with the effect of varying the mass ratio M/C.

3.3.3 Effect of initial solid volume fraction $\phi _{\mathrm {s0}}$

The influence of the initial solid volume fraction for a constant mass ratio of $M/C = 10$ and particle size of 50 $\upmu \mathrm{m}$ on blast overpressure and impulse for glass particles is shown in Fig. 11. Three values of volume fraction, i.e., 0.50, 0.60, and 0.85, are investigated. Note that in this case the thickness of the granular powder layer decreases with increasing solid volume fraction. The dependence of peak overpressure on initial solid volume fraction is weak, with the highest solid fraction leading to slightly less blast pressure attenuation, as would be expected since with a denser layer with a lower void fraction, less energy dissipation occurs upon compaction of the bed. The maximum discrepancy appears at $Z = 1\, \text {m/kg}^{1/3}$, and the blast peak overpressure attenuation, relative to that of a bare charge, is between 57 and 70%. Conversely, the blast impulse attenuation at $Z = 0.7\,\text {m/kg}^{1/3}$ varies from 59 to 63%. In the far field ($Z> 2\,\text {m/kg}^{1/3}$), for both peak overpressure and impulse, all curves are superimposed on top of the unmitigated blast case. No significant changes were observed comparing the plots for glass and steel particles, and hence, the results for steel particles are not shown.

Fig. 12

Comparison of the blast pressure profile for a bare C-4 charge (dashed black lines) with that of the same charge surrounded with a layer of 50 $\upmu \mathrm{m}$ glass particles ($M/C = 10$, $\phi _{\mathrm {s0}} = 0.6$) at scaled distances of a $Z = 0.33\, \text {m/kg}^{1/3}$, b $Z = 0.65\, \text {m/kg}^{1/3}$, c $Z = 0.82\, \text {m/kg}^{1/3}$, and d $Z = 3.27\, \text {m/kg}^{1/3}$ (red curves). The particle volume fraction ($\phi = \phi _{\mathrm {s}})$ is also shown (dotted blue curves) and appears to influence the blast overpressure (figure courtesy of [17])

3.4 Overpressure profiles

To elucidate the particle–blast interaction in more detail, it is instructive to compute pressure-time histories. Figure 12 presents blast pressure histories for glass particles ($M/C = 10, \phi _{\mathrm {s0}} = 0.6$, particle size 50 $\upmu \mathrm{m}$) at four different scaled distances as well as the results for an unmitigated charge. Solid volume fractions are also plotted to correlate the presence of the particles with the blast wave structure. At every distance, the arrival of the blast wave from a mitigated charge is delayed compared to an unmitigated blast as the blast travels at a lower speed. Moreover, by comparing mitigated overpressures with the baseline of a pure C-4 charge (dashed lines), mitigation efficiencies appear more substantial in the near field.

Fig. 13

Blast wave trajectory plotted on a scaled time–distance plot for a 100-g bare C-4 charge. Also shown on the plot is a grayscale mapping of the logarithm of the normalized pressure

With the addition of particles, the blast overpressure is reduced in the near field and hence the time for the blast wave to arrive at a given location is increased. At the farthest location ($Z = 3.27\, \text {m/kg}^{1/3}$), the peak blast overpressure for the mitigated charge is essentially the same as for the bare C-4 charge, although the arrival time is longer. For the intermediate distances, the motion of the particles clearly has a strong influence on the shape of the pressure histories, particularly at the smallest distance where the particle number density is still relatively high and the pressure trace exhibits a double peak structure at a time that is coincident with the arrival of the particles. As the particle density falls with distance, the perturbation to the blast wave trace decays until the blast wave trace is essentially the same as that of the bare C-4 charge at a distance of $Z = 3.27\, \text {m/kg}^{1/3}$. Notably, a comparison between Figs. 5 and 12 indicates that the particles influence the peak blast overpressure even if the particles do not follow closely behind the blast front. Indeed, from Fig. 12, the arrival time of the particles is delayed compared to the arrival time of the blast front for all the distances, although the peak overpressures are located after the kinks in Fig. 5 (phase of pressure recovery). This evidence supports the hypothesis of the creation of pressure disturbances by the particles that travel upstream in the disturbed flow which ultimately merge with the blast front [2]. At the third distance ($Z = 0.82\, \text {m/kg}^{1/3}$), the blast overpressure exhibits a plateau. This plateau appears when the pressure disturbances created by the particles combined with the overpressure behind the shock are of the order of the peak overpressure value. This phenomenon is possible since the pressure disturbances decay more slowly than the peak overpressure due to the propagation in a hot, shocked medium, contrary to the blast front, which propagates in air at ambient temperature.

3.5 Time–distance trajectories of particles and blast wave

3.5.1 Bare C-4 charge

The blast trajectory, or scaled time of arrival plotted as a function of normalized distance, for the baseline case of a 100-g bare C-4 charge is shown in Fig. 13, superimposed with the logarithm of the air overpressure shown as a greyscale map. The thickness of the high-pressure region between the leading blast wave front and trailing edge of the blast wave structure increases with time as the trailing edge slows down and eventually begins to implode towards the centre at a scaled distance of about $Z = 0.59\, \text {m/kg}^{1/3}$. This imploding wave eventually reflects at the origin generating a secondary outgoing shock, although the secondary shock is not visible due to the overpressure scaling chosen. A faint curved line within the high-pressure region represents the contact surface which reaches a maximum scaled radius of a little less than $Z = 0.8\, \text {m/kg}^{1/3}$. Further details of the blast wave structure for this baseline case can be obtained by consulting the classical work of Brode [18].

Fig. 14

The plot on the left shows trajectories of the blast wave (blue curve) and particles (shown as a grayscale map of solid volume fraction) on a scaled time–distance plot for the explosive dispersal of 50 $\upmu \mathrm{m}$ glass particles, at $M/C = 1$, with initial solid fraction of $\phi _{\mathrm {s0}} = 0.6$. The blast trajectory for a 100-g bare C-4 charge (red curve) is shown for comparison. On the right, the same information is shown, except that now the grayscale map corresponds to the logarithm of the scaled pressure. The location of the particles in time and space is indicated by the coloured region, with the colour mapping corresponding to the logarithm of the scaled pressure of the interstitial gas between the particles

Fig. 15

Similar plots as shown in Fig. 14 but for the explosive dispersal of 50-$\upmu \mathrm{m}$ glass particles with mass ratio $M/C = 10$ ($\phi _{\mathrm {s0}} = 0.6$)

3.5.2 Charges with glass particles

Figure 14 shows a similar plot as in Fig. 13, except for the case of a 100-g C-4 charge surrounded with glass particles with $M/C = 1$ (50-$\upmu \mathrm{m}$ particles, initial solid volume fraction $\phi _{\mathrm {s0}} = 0.6$). The blast wave trajectory for the baseline of a bare 100-g C-4 charge is also shown for comparison. The location of the dispersed layer of glass particles as a function of time is indicated on the plot on the right by the coloured region, with the colour scale representing the logarithm of the interstitial gas pressure between particles, in a similar manner as the grayscale. The plot appearing on the left side of the figure shows only a grayscale map of the solid volume fraction, and not the gas pressure, for clarity. It is apparent that for these relatively small, light, glass particles, the particles catch up to the blast at a distance of about $Z = 0.4\, \text {m/kg}^{1/3}$ and then continue to follow closely behind the blast wave front, for this case of a small mass ratio. Although the solid volume fraction decreases as the particle cloud expands radially outwards, the particles remain within a narrow region in space and time.

If the mass ratio is increased to a value of $M/C = 10$, Fig. 15 shows that the blast wave velocity is reduced and the blast wave trajectory lags considerably behind the corresponding trajectory for an unmitigated, bare C-4 charge. Furthermore, the radial zone over which the particles are contained expands and the particle zone becomes more diffuse, as shown by the coloured region on the right of the figure or the grayscale map of solid volume fraction shown on the left. The leading edge of the particle zone falls behind the blast wave front. The distance between the front of the region containing the glass particles and the blast wave continues to increase as the particles slow due to drag. In the case of large, dense (e.g., steel) particles, the particles may have sufficient inertia to catch and penetrate the slowing blast wave front, as observed experimentally [6]. The air overpressure in the region between the particle front and the blast wave remains relatively high, suggesting that the porous particle front acts effectively as a (porous) piston to generate pressure disturbances that propagate ahead into the shocked air region. These pressure disturbances travel faster than the leading shock front and ultimately merge with the blast wave, reducing the rate of decay of the blast wave overpressure, until the mitigated peak blast overpressure returns to the values of that of a bare charge.

Fig. 16

Similar plots as shown in Fig. 14 but for the explosive dispersal of 50-$\upmu \mathrm{m}$ glass particles with mass ratio $M/C = 100$ ($\phi _{\mathrm {s0}} = 0.6$)

The trends observed when increasing M/C from 1 to 10 are accentuated when the mass ratio is increased further to a value of 100, as shown in Fig. 16. Here the initial particle layer is considerably thicker due to the higher mass of glass and when the blast wave emerges from the particle layer, it immediately runs ahead of the particle front, which moves more slowly than for smaller mass ratios. With the slower expansion of the particle layer, the pressure in the detonation products takes a longer time to reduce in magnitude. Despite the lower velocity of the particle front, pressure disturbances, visible as gradations in the grayscale region between the particle front and blast wave, propagate ahead of the particle front into the shocked air region behind the blast wave. The region where the accumulation of these pressure disturbances begins to influence the blast wave propagation and reduce the rate of pressure decay occurs at scaled distances $Z > 0.5\, \text {m/kg}^{1/3}$ (see Fig. 8), which is just beyond the region shown in Fig. 16.

Fig. 17

Similar plots as shown in Fig. 14 but for the explosive dispersal of 50-$\upmu \mathrm{m}$ steel particles with a mass ratio of $M/C = 100$ ($\phi _{\mathrm {s0}} = 0.6$)

3.5.3 Charges with steel particles

If the mass ratio is held constant at 100, but glass particles are replaced by steel particles, differences are seen. The change in the behaviour of the trajectories of the particles and blast wave is shown in Fig. 17. Due to the higher sound speed in the steel particles, the shock travels faster through the particles before transmitting a blast wave into the air. As the detonation products expand, they plough up the inner surface of the particle layer towards the outer surface, which is moving at a lower velocity. This has the result that the accumulated layer of particles, or accretion layer, has a noticeably smaller thickness than for glass particles. The gas pressure within the particle layer remains relatively low compared with the pressure in the detonation products. Pressure disturbances are again visible emanating forward from the front surface of the particle layer and travelling at the sound speed in the shocked air. These pressure disturbances eventually catch up to the blast wave front, reducing the rate of decay of the peak blast wave overpressure.

3.5.4 Charges surrounded with a water layer

For the case of dispersal of a layer of water with a mass ratio of 100, the resulting trajectories are shown in Fig. 18. Contrary to the dispersal of solid powders, a spall layer is present which runs ahead of the accretion layer. The dynamics of the dispersal of a water layer is described more fully in the previous publications [11, 19]. When the initial shock propagating through the water layer reaches the surface of the water layer, a blast wave is transmitted outwards and a rarefaction wave is created, which moves back through the shocked water. This expansion wave depressurizes the liquid, causing cavitation, and generating a two-phase bubbly liquid zone between the accretion and spall layers. As the low pressures within the cavitation region cannot be plotted with the log pressure scaling, the pressure is set to zero in this region. Similarly as for solid powders, pressure disturbances are visible emanating from the front of the liquid spall layer into the shocked air region towards the leading blast wave front.

Fig. 18

Similar plots as shown in Fig. 14 but for the explosive dispersal of water with a mass ratio of $M/C = 100$ and droplet size of 50 $\upmu \mathrm{m}$

Fig. 19

Time history of the momentum flux of particles and air plotted as a function of scaled time for the explosive dispersal of 50-$\upmu \mathrm{m}$ glass particles at scaled distances of $Z = 0.5\, \text {m/kg}^{1/3}$, $Z = 1.92\, \text {m/kg}^{1/3}$, and $Z = 2.5\, \text {m/kg}^{1/3}$ ($M/C = 10,\phi _{\mathrm {s0}} = 0.6$)

3.6 Momentum fluxes

Following the analysis given in [3], it is possible to compute the momentum fluxes of particles and the air at different distances from the charge for the dispersal of glass particles (with $M/C= 10$, solid volume fraction $\phi _{\mathrm {s0}} = 0.6$, particle size = 50 $\upmu \mathrm{m}$). This analysis is relevant for assessing the impulse applied to a nearby structure. Momentum fluxes are defined as the product of the density of each phase multiplied by the square of the velocity of the phases ($\rho _\mathrm {p} v_\mathrm {p}^2 = \phi _{\mathrm {s}} \rho _{\mathrm {s0}} v_\mathrm {p}^2$ for particles and $\rho _\mathrm {a} v_\mathrm {a}^2$ for the air). Figure 19 shows the history of the momentum fluxes of the particles and air passing three different scaled distances ($Z = 0.5, 1.92$, and 2.5 $\text {m/kg}^{1/3}$) for glass at $M/C = 10$. For the smallest scaled distance, the momentum flux of the particles has a higher peak value in comparison with the momentum flux of the air (about 28 times higher). The decay in the particle momentum flux with distance is due to the decrease in the particle number density and velocity as the cloud of particles expands radially outwards and slows due to drag.

At $Z = 1.92\, \text {m/kg}^{1/3}$, the peak value of the momentum flux of the particles has decreased substantially and is on the order of the value for that of air. Finally, at $Z = 2.5\, \text {m/kg}^{1/3}$, both peaks have reduced considerably and the integrated particle momentum flux is less than that of air by a factor of 25, due to the reduction in particle velocity from drag. As expected, with increasing distance, the time separation between the arrival of the blast and the arrival of the particles passing the same location increases.

To find the total impulse applied to a structure from a blast-driven multiphase flow, it is necessary to integrate the force applied to the structure from three components, i.e., (i) the force applied from the blast wave as it diffracts around the structure, (ii) the drag force from the gas flow behind the blast wave as it moves around the structure, which is proportional to the drag coefficient and gas momentum flux, and (iii) the force from particle impacts, which depends on the particle momentum flux and the coefficient of restitution associated with the particle collisions (see [3] for a discussion of the various contributions to impulse in a multiphase flow). The relative contribution to the total impulse from the particles and air depends on the location from the charge. From Fig. 19, it is apparent that the momentum flux of the particles dominates the corresponding value for the gas in the near field ($Z \lesssim 1\, \text {m/kg}^{1/3}$), whereas the opposite holds true further away from the charge. Hence, the impulse associated with the particle collisions dominates the impulse from the gas motion in the near field, and vice versa after.

Fig. 20

Spatially integrated momenta of the air, detonation products, and glass particles, as a function of time, for a 100-g bare C-4 charge (dashed lines) and for a charge surrounded by 50-$\upmu \mathrm{m}$ glass particles ($\phi _{\mathrm {s0}} = 0.6$) with mass ratios (from left to right) of $M/C = 1$, $M/C = 10$, and $M/C = 100$ (solid lines)

Fig. 21

Early time behaviour of the same momenta values shown in Fig. 20

3.7 Spatially integrated momenta

The dynamics of the interaction between the particles and the flow of the shocked gas behind the blast wave can be illustrated by considering the variation of the total momenta (integrated in space) of the particles, shocked air, and C-4 detonation products as a function of time. Figure 20 shows the history of the total momenta of the particles, air, and detonation products for the dispersal of glass particles for M/C values of 1, 10, and 100. Figure 21 shows expanded views of the early time of the same plots. Also shown in Figs. 20 and 21 is the history of the total air and product gas momenta for the baseline case of a bare C-4 charge.

For the smallest mass ratio ($M/C = 1$), the shock transmitted into the packed particle bed rapidly accelerates the particles to a maximum momentum in less than 100 $\upmu$s. The particles then decelerate due to drag, and the momentum of the shocked air increases as energy is transferred from the particles to the air. The deficit in the air momentum, compared with the bare C-4 case, is a result of the momentum transferred to the particles. However, this deficit completely recovers after about 5 ms, at which point the particle momentum has decayed nearly to zero.

When the mass ratio is increased from 1 by a factor of 10 or 100, the time for the particle momentum to reach a maximum increases from 0.1 ms to about 0.2 and 1 ms, for $M/C = 10$ and 100, respectively. The maximum particle momentum attained scales with $({M/C})^{0.47-0.52}$ (intermediate mass ratios M/C of 2, 3, 5, 15, 30, 50 have been simulated to establish this scaling law), consistent with Gurney theory, which predicts a dependence of $({M/C})^{1/2}$ [20]. The combined spatially integrated momentum of both particles and the air is generally higher than or equal to the momentum of an unmitigated charge and stays higher for a longer time with increasing M/C. Despite the increase in maximum particle momentum at the largest M/C value, the particle momentum eventually decreases leading to an increase in the gas momentum until it recovers to the same value as the baseline case at a time of about 7 ms, at which point the particles still have a significant, residual momentum. The time at which the particle and air momenta are exactly equal increases from about 0.15–1–5 ms, for $M/C = 1$, 10, and 100, respectively, which corresponds to a power law dependence on M/C of approximately $({M/C})^{0.76-0.84}$.

4 Conclusions

The explosive dispersal of a layer of granular material or liquid has been investigated computationally with a 1D multiphase hydrocode and compared with recent experimental results [2]. The computational results are consistent with the main experimental finding, i.e., that the mitigation of the blast wave overpressure by the surrounding inert material is primarily dependent on the mitigant to explosive mass ratio M/C, with the particle and bed properties, and the degree of particle jetting, playing a secondary role. Another salient observation from the experiments is that while the blast overpressure is reduced in the near-to-mid field (0.5 $\text {m/kg}^{1/3}\,{<}\,Z\,{<}\,$ 2 $\text {m/kg}^{1/3}$), the rate of decay of the blast overpressure is reduced with distance and the overpressure nearly recovers to the value for a bare charge for small values of the mass ratio ($M/C\,{<}\,10$), in the far field ($Z\,{<}\, 2\, \text {m/kg}^{1/3}$). The hypothesis that the particles act as a porous piston and generate pressure disturbances which reach the blast wave front and slow the rate of pressure decay has been confirmed with the computations. In particular, by plotting the trajectories of the particles and blast wave as well as the gas pressure field on a time–distance plot, the pressure disturbances generated at the front of the expanding particle cloud can be visualized. Since these pressure perturbations move at the sound speed of the hot, shocked air behind the blast wave, they move faster than the blast wave and can eventually overtake the blast wave front.

The phase of pressure recovery follows the earlier phase when the particles accelerate and extract energy from the flow, reducing the blast overpressure in the very near field. The numerical simulations are useful for investigating the near-field phenomena which are not accessible experimentally [2]. The location of this transition, where the peak blast overpressure and positive-phase impulse exhibit the most reduction, as compared to a bare charge, appears in the region 0.3 $\text {m/kg}^{1/3}\,{<}\,Z\,{<}\,$ 0.6 $\text {m/kg}^{1/3}$, with the transition occurring at larger distances with increasing M/C.

While the blast wave decay for a mitigated charge is not strongly dependent on the particle properties, changes to the material properties lead to small systematic changes in the decay rate of the blast overpressure. For example, when the particle size and/or particle packing density is reduced, the particles accommodate more quickly to the ambient flow velocity. Hence, during the particle acceleration phase at early times, energy is transferred more quickly to the particles by the flow and the rate of decay of the peak blast overpressure increases. Conversely, as the particles decelerate and transfer energy back to the flow, smaller particles lead to a more rapid recovery of the blast overpressure. A comparison between steel and glass particles indicates that the latter are more effective at reducing the peak blast overpressure and impulse. However, the discrepancies observed are less than that found experimentally [2].

The computational results for the blast wave overpressures for explosive dispersal of liquids or particles generally agreed with the experimental results, considering the degree of scatter in the experimental results, except in the near field (0.5 $\text {m/kg}^{1/3}\,{<}\,Z\,{<}\,$ 1 $\text {m/kg}^{1/3}$), for a mass ratio below around 30. On the contrary, in the far field ($Z\,{>}\,2\, \text {m/kg}^{1/3}$), impulses for glass and steel powders exceed experimental values. The differences are likely related to energy dissipation during particle deformation and fracture, or liquid droplet fragmentation, which are not captured in the computations. Surrounding an explosive charge with a layer of either particles or a liquid leads to a residual reduction of the peak blast overpressure in the far field ($Z\,{>}\,2\, \text {m/kg}^{1/3}$), especially at high M/C. Nevertheless, the effect on the positive-phase blast impulse differs for the two cases: particles lead to a reduction in blast impulse, whereas a liquid has virtually no effect on the blast impulse, in comparison with a bare explosive charge and that for all M/C. The independence of impulse on the presence of water for the explosive dispersal of water was reproduced by the computations. The impulse is an integrated measure of the blast overpressure and is dependent on the details of the flow field behind the blast wave. Hence, a more accurate prediction of the blast impulse will likely require a consideration of the particle deformation effects noted above, as well as the particle clustering and jetting that is characteristic of explosive particle dispersal.

Computations show that the momentum flux of the particles exceeds that of the gas, particularly in the near field ($Z \le 1\, \text {m/kg}^{1/3}$), and hence, the primary contribution to impulse to a structure in this region is from particle impacts. This observation, together with the tendency of the blast overpressure and impulse to recover in the far field when particles or liquids are explosively dispersed, is an important factor that must be considered when attempting to mitigate the blast effects from a high-explosive charge at a given distance.

To extend the current work, refinements of the model are required to take into account the various mechanisms of energy dissipation (deformation, fracturing, cavitation, etc.) not modelled in the present study, as well as multi-dimensional effects. Investigation of the partition of kinetic and internal energy within the flow is helpful to quantify the energy transfer between the phases. Initial work in this direction is presented in another publication [21]. Another possible extension of the current work could be the investigation of the combined impulse from the particles and the blast applied to a structure. In addition, an analysis of the maximal radial extent of the particle cloud as a function of M/C and material properties is of interest to quantify the hazards associated with the explosive dispersal of particles.