Fragmentación de explosivos líquidos en geometría esférica

Introduction

Detonation of a central high explosive charge, surrounded by a shell of material is a common practical geometry in many application areas, where the physics of the interaction of the detonation with the surrounding material continues to be studied. These areas include blast mitigation, fuel-air explosions, metalized explosives, and reactive casings. In this paper, we consider a shell of liquid surrounding an explosive charge in spherical geometry.

The behaviour following the detonation of a high explosive charge underwater has been studied in detail for many years (see for example, Cole [1]). A detonation in deep water can be considered as the case of a shell of water of infinite thickness, and in these conditions, one observes the growth and oscillation of a stable gas bubble. There have been many examples of this phenomenon and we illustrate this behaviour in Fig. 1.

In Fig. 1, one can see the detonation followed by an initial expansion, where some surface structure is observed, which may correspond to charge case effects, but at later times, one can observe the smooth and stable gas globe which is the characteristic of this phase of an underwater explosion.

If one considers a shell of water with a finite thickness, the introduction of an outer boundary changes the stability of the system. In this case, the expansion is no longer stable and the shell forms jet-like structures, suggesting strongly that wave interactions from the outer boundary play a key role in the observed break-up process.

Milne et al. [2, 3] have noted that during the dispersal of both liquid and particle layers, experimental evidence from a wide range of sources shows that the expanding cloud of explosively disseminated material is comprised of “particles” or fragments which have significantly different dimensions from those associated with the original material. Photographic evidence shows characteristic jets or fingers behind these spherically diverging fragments. Examples of the formation of these jet structures are given, for example, in Frost et al. [4] and Ripley and Zhang [5], or in Frost and Zhang [6] in the context of jet formation from heterogeneous blast explosives. Zhang et al. [7] have recently revisited old data to study particle jetting instabilities and hypothesise that charge casing may play an important role. This is an important study area in its own right but is beyond the scope of the current paper, where we deliberately use thin-walled glass cases. The explosive shock wave causes the thin-walled glass casing to fracture into very fine fragments which are carried out by the gas flow behind the shock. Previous work [8] has shown that these glass fragments are visible on flash X-ray radiographs taken during the explosive dispersal of solid particles. While the fracture of the glass into fine fragments could generate fine-scale perturbations at the liquid interface, the size of the glass fragments appears to be much less than the spacing of the subsequent liquid jets.

Further X-ray analysis of these systems has allowed Milne et al. [3] to show that the fragmentation process occurs early in the expansion, around the time that the release wave from the outer boundary reaches the inner boundary of the surrounding material. It is natural to consider the possibility that the breakup into jets is a result of classical hydrodynamic interchange instabilities (see Chandrasekhar [9] for discussion of and introduction to well-known examples, such as Rayleigh–Taylor, Richtmyer–Meshkov, or Kelvin–Helmholtz), and there is a large body of continuing research in this field (see, for example, Mankbadi and Balachandar [10] and references therein). Milne et al. [3] reported that direct numerical simulation of the experimental systems they studied showed that the growth rate of these instabilities was too slow to explain the growth observed by X-ray analysis, so other potential mechanisms were investigated. They showed that, for a given experiment, the numbers of fragments observed in the early-time radiographs and in the late-time photographic evidence are in good agreement. A simple model to predict the number of jets from powders was proposed, where the typical size of the fragment is determined by the thickness of the compressed layer at the time that the release wave from the outer boundary reaches the inner boundary of the surrounding material.

Explosive acceleration of a liquid shell differs from the corresponding situation with solid particles due to the presence of voids in the unperturbed particle bed. In the latter case, the shock propagating through the particles causes considerable compaction, and depending on the ductility of the particles, the compacted bed may approach a fully densified state and have considerable strength after compaction. In the case of shock loading of a liquid, the reflection of an expansion wave at the liquid surface and subsequent tension applied to the bulk liquid will lead to the cavitation of void growth and subsequent breakup of the bubbly liquid into a droplet aerosol [11].

Explosive dispersal of a liquid progresses through several regimes. In the first regime, which has been termed the ejection regime [12], the forces associated with the propagation of the shock wave through the liquid and rarefaction that returns from the liquid surface dominate the motion of the fluid. This is followed by a transition regime in which the explosive forces are comparable to aerodynamic forces on the liquid. In the final regime, the shock forces have decayed, and the motion of liquid droplets is governed by aerodynamic drag and droplet breakup is due to the competition between inertial and surface tension forces.

The present numerical work is concerned with the near-field ejection regime which occurs on a timescale of hundreds of microseconds for charges with mass of the order of 1 kg, where shock forces dominate. The present experimental work measures the jet structure in the late-time regime. Of particular interest is determining the dependence of the scale of the jet-like structures on the physical parameters of the system, including the fluid properties (e.g., density, viscosity, and surface tension) and the ratio of the mass of the liquid to that of the explosive. The number of liquid jets that are formed and the rate of droplet shedding and subsequent evaporation of the liquid will determine the size of the liquid–vapour–air cloud that is ultimately formed, which is of interest if the liquid is combustible.

In the present paper, the dynamics of the explosive dispersal process of a liquid is investigated both experimentally and with a multi-material hydrocode. The computations are used to track the features of the process, including the wave dynamics, and motion of the spall and accretion layers. A working hypothesis for dynamic liquid breakup is proposed in which the maximum thickness of the accretion layer determines the number of jets. This hypothesis is examined in light of experimental results with spherical charges containing a variety of different fluids to determine the influence of fluid properties. Water was chosen as a baseline fluid and other liquids were chosen that have significant variations in fluid properties, including glycerol (similar density and surface tension to water, but much higher viscosity), vegetable oil (similar density to water, but higher viscosity and lower surface tension), and ethanol (similar density and viscosity to water, but lower surface tension). Prior to the presentation of the experimental and computational results, existing theoretical models describing the dynamic fragmentation of liquids will first be considered.

for the glycerol data. These numbers imply primary fragment sizes of order 7 mm in diameter.

We can use the scan data to measure typical widths of the structures and then plot a probability density distribution function for the number of features in a given bin size of widths. Figure 5 shows (in blue) a measured fragment number distribution function from our video data for water. As noted above, this does not have the features of a Mott distribution function (shown in red). We do not observe any small scale features. This may in part be due to the lack of resolution in our basic data, but we see similar distributions for different resolutions. We would definitely expect to see the tail of a Mott distribution clearly, but we do not. Since [3] noted that the number of features seen in early X-ray data was also seen in late-time video data, we include a probability density function obtained by measuring the widths of features in scans of X-ray data for a sand experiment (shown in green). The two measured functions from different measuring systems and at different times are remarkably similar and are more akin to a monomodal than a Mott distribution. We are interested to know what determines this typical size of feature (and associated number of jets).

We can now return to the Grady model in light of the experimental results. For surface-tension-dominated liquid breakup, (1) gives a prediction of the fragment size. For water, one typically has ambient values of γ=0.07J/m2

It is useful at this stage to plot all the modelling results and experimental data in one figure. In Fig. 13, regardless of material, one sees the same basic behaviour for all liquids, both experimentally and numerically, but the underlying curves differ between experiment and theory.

For large F/B(>15)

, experiment suggests a very flat line for liquids at F / B, so that the number of jets is independent of both liquid and F / B. The model predictions are similar in number at high F / B, but show a pronounced decrease in features as F / B increases over the whole range. All the liquids show the opposite trend at the lower end, i.e., the number of jets decreases as the F / B ratio decreases. In Fig. 13, we also show some data from experiments performed using water fills at AWE and reported in [2, 3]. The data lies on the same curve, even though size and the confining case material are very different.

Using (5), we convert the measured number of features along a radial line to the total number of jets in the sphere. When one plots this number against F / B (Fig. 14), one can see two regimes. At low F / B, the number of jets appears to be linearly dependent on F / B. At high F / B, the number tends to a constant value independent of F / B. These results are consistent with hydrodynamic scaling of the break-up mechanism and the fact that strain rate is an important parameter. All the liquids considered in this paper lie on the same lines.

In [3], we noted that the radiographs for powder fills suggested that the primary break-up mechanism showed the features of brittle fracture (with cracks forming and defining the fragment edges). Numerical modelling showed qualitative agreement with this mechanism.

When one looks at the radiographs for water as reported in [2], one observes (e.g., in Fig. 16) that the primary fragments appear to be liquid ligaments surrounding low-density regions. This suggests that a cavitation as opposed to a cracking mechanism is at play. We will use numerical modelling to further investigate this possibility by testing a potential hypothesis based on the nucleation of cavitation regions. If we build on the observed hydrodynamic scaling and further assume that the number of nucleation sites for cavitation is constant per-unit volume, we can explore whether these two properties can explain the two straight line regions of Fig. 14.

Figure 15 shows that, at low F / B, the number of ligaments between cavitation cells is proportional to the number of cavitation regions and thus will grow with F / B in the same manner as the number of nucleation sites for cavitation. The assumption of a constant number of sites per-unit volume leads to a straight line region at low F / B. At high F / B, there can be more than one cavitation region in the radial direction. The cavitation zones might be expected to join up, so that the number of ligaments or liquid regions between them is no longer proportional to the number of nucleation sites, but might be expected to saturate at the number of cavitation zones in the inner layer. This will lead to an approximately constant number of liquid regions which we take to be the origins of the later jets. Going one step further allows one to note that the intersection of the two straight lines shown in Fig. 14 identifies the case where the average distance between nucleation points is equal to the thickness of the shell at the break-up time.

As in [3], we consider a three-dimensional model of the detonation and expansion. In this case, we randomly seed the initial conditions with nucleation points (here, small volumes of partially cavitated liquid) separated by an average distance determined by the intersection of the curves in Fig. 14.

We have reported here an experimental and numerical study aimed at investigating the mechanisms which determine the number of jets seen in explosive dissemination of liquids. When there is no outer boundary (e.g., an underwater explosion at depth), one observes an extremely stable gas bubble with no jetting. When one introduces an outer boundary, one observes jetting, so we clearly no longer have a stable expansion of the system, suggesting strongly that wave interactions from the outer boundary play a key role in the observed break-up process. We thus sought to investigate in more detail the causes of the observed jetting.

We observed that the distribution of sizes of jets or primary fragments did not fit with a Mott distribution in that no large fragment tail or small scale features were observed. We thus considered the theoretical models for breakup of bulk liquids due to Grady. His early work noted that there was a lack of experimental data to test against and this provided one motivation for our work. The independent variables in these models were density, viscosity, and surface tension (as well as strain rate) which are very amenable to variation in liquids. Our results show trends which do not match the Grady model predictions, suggesting that some other mechanisms dominate in our problem.

We have previously studied explosive dissemination of powders (where the independent variables for Grady models are harder to prescribe experimentally) and have found good empirical correlations with primary fragment size (and thus number of jets) and predicted shell thickness. The shell structure in liquids is more complex due to the presence of a spall layer, so we considered an amended empirical model here (using the maximum thickness of the spall layer as our measure). This measure predicts more jets in liquids than in powders (as observed) and gives reasonable estimates for the number of jets at high (>15) values of F / B, but the underlying shape of the curve of n vs. F / B does not match experiments which (unlike the case of powders) shows fewer jets at low F / B.

If one plots all of the data from our chosen range of liquids with differing initial surface tensions, viscosities, and densities on one graph of measured number of features along a radius against F / B, one sees a common curve. Since experiments were performed at a range of length scales, this shows that hydrodynamic scaling applies. This also suggests that the properties varied in the experiments are not dominant in determining primary fragment size. We note that experiments with very different case materials (glass and plastic) also lie on this curve, so we have no evidence that case effects are dominating. Using spherical symmetry to convert this curve into a plot of total number of jets vs. F / B gives two straight line regions. We hypothesise that these two curves are consistent with a cavitation mechanism, where the number of nucleation sites per-unit volume is a constant. When we perform three-dimensional calculations based on this number of nucleation sites, we get good qualitative agreement with the shape and number of the features observed in experimental radiographs. Prediction (as opposed to post-shot simulation) of the experimental behaviour will require more knowledge of the causes of the initial nucleation sites for cavitation and more work is planned along these lines.