Pulsation behavior of a bubble generated by a deep underwater explosion

The first studies of underwater explosion (UNDEX) were conducted to support naval combat.¹ When an explosive is detonated in deep water, it generates an outward-propagating shock wave and high-pressure, high-temperature gaseous detonation products, the latter of which form an underwater bubble. The bubble expands quickly initially because of its high internal pressure and then keeps expanding because of the inertia of the water until the internal pressure drops below the ambient pressure, whereupon the bubble begins to contract; in the same way, the bubble keeps contracting until the internal pressure exceeds the ambient pressure, whereupon the bubble has completed its first pulsation. Such bubble pulsations may take place dozens of times under suitable conditions, and if the bubble is near a rigid wall or shell they can even result in a high-speed water jet. The energy in the bubble accounts for around 40% of the total energy of the explosive, therefore studying the bubble pulsation behavior is helpful for exploring the energy released by the explosive.

The first theoretical work on bubble behavior was that published in 1917 by Rayleigh.² Subsequently, Plesset³ derived an equation governing the dynamics of a spherical bubble in an infinite body of incompressible fluid, and through subsequent research that equation was improved to include factors such as surface tension and fluid viscosity.^4,5 The governing equation is now known as the Rayleigh–Plesset (RP) equation, and it has been used extensively to study of the behavior of various types of bubble.

UNDEX research received extensive attention during World War II, and in 1948 Cole¹ published the classic textbook Underwater Explosions, containing masses of experimental data and theoretical results. The book detailed the basic aspects of UNDEXs, including research methods, simple theoretical modeling, and empirical pulsation laws. Since then, various scholars^6–10 have done much work on the engineering applications of UNDEX bubbles. Chahine¹¹ and Lauterborn¹² used high-speed cameras to capture the dynamics of spark-generated and laser-generated bubbles in transparent fluids; those results were used to study the pulsation and jetting of UNDEX bubbles and to compare their behaviors. Klaseboer et al.,¹³ Hung et al,¹⁴ Cui et al.,¹⁵ and Zhang et al.¹⁶ conducted UNDEX experiments in pools and tanks and used high-speed cameras to obtain many images of spherical bubbles underdoing pulsation; most of those results were used to verify numerical simulation methods. For the limited depth of the pool, bubble research was mostly based on UNDEX experiments in shallow water.

Because of military security, phenomena related to deep-UNDEX have been little reported, and Cole listed UNDEX bubble results for depths of no more than 200 m. The present paper reports on experiments conducted on deep UNDEX involving bubble pulsation behavior at depths up to 350 m. Meanwhile, the bubble pulsation laws are discussed in relation to experimental data, solutions of the RP equation, and semi-empirical predictions.

A sketch of the experimental setup used in the present study is shown in Fig. 1. The dimensions of the closed steel container are approximately Φ1.5m × 2.1m, and the deep underwater environment is formed by an air compressor. In the container, three quarters of the volume is filled with water and the other quarter is filled with air. Two Φ0.1m toughened-glass observation windows are installed on each side of the container, and a parallel-light source and a high-speed camera are placed at each observation window, respectively.

The bubble is generated by detonating 2g of TNT explosive (trinitrotoluene, C₆H₂CH₃(C₆H₂CH₃(NO₂)₃), ρ=1.63g/cm³). The cylinder explosive package is Φ1.5cm × 0.7cm as shown in Fig. 2(a). The explosive package is located at the center of the light path, approximately 0.8m from the water surface and 0.7m from the bottom of the container. The high-speed camera is placed at the opposite side of the lamp to capture shadow images of the bubble only. The camera frame rate is set as 7,500fps and the exposure time is set as 1/40,000s. Once the container is pressurized, the camera and the explosive are both triggered by the detonator at the same time. Fig. 2(b) shows a snap-shot of the initial moment, and Fig. 2(c) shows an image of the expansion phase of the bubble at a depth of 0.8m.

As listed in Table I, five deep-UNDEX experiments are conducted in the present study, in which the maximum equivalent depth is 350 m. The actual internal pressure is measured by a pressure senor inside the container; the uncertainty in the internal pressure is less than 3%.

Influenced by the shape of the explosive package, the bubble is shaped initially like a capsule with the detonator direction as the rotational axis. Because of the high internal pressure, the bubble expands rapidly to become a sphere. Therefore, we reason that the bubble motion is axisymmetric and that the bubble profile in the shadow images is the same in the other directions. We construct a coordinate system in which to analyze the bubble behavior: the point of contact between the explosive and the detonator is taken as the coordinate origin, the detonator direction is taken as one axis, and the direction perpendicular to the detonator direction is taken as the other axis. The time series of the bubble radius is then obtained from the image proportion. Because the experiments are conducted in a container, the results may be affected by the following two factors. 1) Because of the limited amount of explosive, considerable amounts of detonator powder and debris are generated beside the bubble. Travelling at high speeds initially, these fragments form an irregular boundary at the bubble surface, and this boundary may affect the bubble-radius data that are obtained. 2) The container is made of cast iron and is closed, thereby forming a rigid boundary condition in the experiment. Therefore, according to the theory of shock-wave propagation, most of the generated shock wave is reflected from the container wall. Assuming that the shock wave spreads at the speed of sound in water (i.e., ∼1,400m/s), its round trip takes approximately 1ms. This means that the dynamics of the deep-UNDEX bubble may be disturbed by the reflected shock wave at 1ms after initiation.

The high-speed camera records images of the deep-UNDEX bubble for each of the test cases. The bubble motion is analyzed from these images, and the bubble radius and period are investigated with solutions of the RP equation and by empirical prediction.

Figure 3 shows images of the first pulsation process in each case. These images cover the complete first bubble pulsation, including the initial dilation stage, the maximum bubble size, and the contraction phase. As shown in Fig. 3, because of the shape of the explosive package, the bubble is shaped initially like a capsule in each case, but this shape changes with the dilation. For the 0.8m and 100m depths, the bubble evolves rapidly into a sphere; the shape of the explosive package influences only part of the initial stage of the bubble expansion. This phenomenon is the same as that reported in Ref. 16, which showed that an explosive with a small slenderness ratio could form a spherical bubble easily. And in the 0.8m and 100m depths, the bubble dilated beyond the view of the observation window, thus only the early part of the expansion phase was obtained. However, when the bubble contracted to its minimum size, it was again visible in the observation field, thus the bubble pulsation period was obtained.

For the 200m, 300m, 350m depths, the bubble shape is again influenced by the shape of the explosive package. In the initial expansion stage, the top and bottom of the bubble clearly expand more rapidly than do the left and right sides, making the bubble capsule-shaped again. However, the dilation reverses this trend, and in each case the bubble is shaped like an oblate spheroid at the moment of maximum expansion. In the contraction stage, the top and bottom of the bubble contract more rapidly than do the left and right sides, thus there is no regular spherical bubble during the entire first pulsation. Because of the considerable amounts of surrounding debris and powder, it is difficult to detect the jet phenomenon at the moment of minimum bubble size.

In the bubble motion images, it is found that the shape of the bubble was affected by the initial charge shape at the initial stage of the bubble expansion in each case. However compared with the shallow water cases, the bubble is more difficult to form a spherical bubble in deep water at the all bubble motion process, which indicates that the water depth has a certain influence on the bubble movement.

In addition, the maximum radius and the pulsation period both decrease with the water depth, which is consistent with the predictions of previous research. Furthermore, floating motion of the bubble is observed only at the end of the contraction stage; because the first pulsation process is unaffected by buoyancy, we do not discuss bubble migration herein.

As obtained from the experimental images, time series of the bubble radius are plotted in Fig. 4. The error bars for the radius correspond to the difference between the shape of the bubble and that of a regular aspherical bubble. In each case, the rate of increase of the bubble radius decreases after approximately 1.0ms (hereinafter referred to as the switch point). All curves are increasing at a higher rate before the switch point, but after the switch point the increase is slower for the 0.8m and 100m depths and the curves actually decrease for the 200m, 300m, and 350m depths. This corresponds well to the discussion above regarding the effect of the reflect shock wave. Therefore, herein we consider the radius data to be accurate and reliable in the first 1.0ms.

We use the RP equation to compute the dynamics of a free-field spherical bubble, and we use the results by Chahine¹¹ on spark/laser bubbles to verify the solutions. The RP equation is based on the theory of one-dimensional incompressible flows, and in its simplest form it is written as

where R(t) is the bubble radius as a function of time, ρ_$w$ is the density of water (ρ_$w$=1,000kg/m³), P_∞ = P₀ is the ambient pressure infinitely far from the bubble, and P_b is the pressure inside the bubble. The RP equation assumes that the gas inside the bubble behaves polytropically, namely that the detonation products expand isentropically.¹⁷ The polytropic law is written as

where P is the gas pressure, ρ is the gas density, and γ is the polytropic index of the gas. Equation (2) can be rewritten as

where P_g0 and V₀ are the initial internal pressure and volume of the bubble, respectively. Substituting Eq.(3) into Eq.(1) and replacing V with R³ gives

with the bubble boundary conditions (R_t=0 = r, $Ṙt=0=0$⁠) and other parameters, the RP equation can be solved using the Runge–Kutta method.

The reactive gaseous detonations in the bubble are in the very complicated high-temperature/high-pressure phase in the initial stage. According to Deal,¹⁸ the equation of state of the detonation products can be isentropic expansion law of Eq.(2), which was tested by plane-wave generated experiment. However, the temperature and density of the products change greatly during the expansion process, thereby changing the expansion process. The product-expansion process is always divided into either two or three stages,¹⁹ namely the high-pressure stage and that of ordinary pressure (the latter of which can be subdivided into the medium-pressure stage and low-pressure stage) as expressed by

In the high-pressure phase, the value of k is 2.5–3.5 (as obtained experimentally) and the volume of the products is relatively constant. However, when the internal pressure P_cj (the TNT detonation pressure, P_cj=15-20×10⁹Pa) drops to P_k (1.5-2.0×10⁹Pa), the products follow the expansion law for ordinary pressure wherein the value of γ is 1.25–1.4, thereby complying fully with the ideal gas law under ordinary conditions.

For the high-pressure stage, the relationship between pressure and density can be expressed as

in the high-pressure phase, and the relationship between pressure and bubble radius can also be derived. With k equal 3 at the high-pressure phase, the internal pressure of the products decreases from the detonation pressure P_CJ to 1% of P_CJ, the bubble radius is only approximately 1.6 times the original radius of the explosive package. Thus, we can neglect the high-pressure phase of the gaseous detonation products when calculating their motion.

In a previous study by Chahine et al.¹¹ an explosively generated bubble was fitted well by the solution of the RP equation with γ equal to 1.667 and the initial condition P_g0 or P_k equal to 1.75×10⁹Pa. In the present study, P_g0 or P_k equal to 2.0×10⁹Pa is used, and the value of γ is obtained by fitting the experimental results. The bubble pulsation motion according to the RP equation and that according to the experimental data are shown in Fig. 4. The solutions fit well with the undisturbed stage of the experiment, and it is also shown that the influences of the reflect wave can be neglected in the bubble expansion phase for the 300m and 350m depths. For the 0.1m and 100m depths, the solutions of the RP equation fit well with the experimental results when γ equals 1.25. However, as the water depth is increased, the value of γ changes. For the 200m, 300m, and 350m depths, the solutions fit well when γ equals 1.3. This change means that the effect of water depth on the bubble motion includes not only the ambient pressure but also those factors that change the state of the gaseous detonation products. This phenomenon has not been seen in previous studies.

Willis²⁰ derived the energy equation for the incompressible radial motion of a free-field explosion gas bubble, namely

where P₀ is the hydrostatic pressure, E(R) is the internal energy of the gas bubble which only stands a relatively small part of the total energy, and Y is the total energy. In this expression, the first term represents the kinetic energy of the flow of the surrounding fluid, and the second term represents the work done by the bubble against the hydrostatic pressure to expand to radius R. The internal energy can be neglected when the bubble expands to its maximum radius. Thus it can be expressed as

where R_max is the maximum bubble radius. This expression simply reflects the relationship between the maximum bubble radius and the total energy Y when dR/dt = 0. With transforming, Eq.(8) can be expressed as:

And using the mass of the explosive W to represent the total energy Y, we obtain R_max as

where K [m^4/3/kg^1/3]is a constant particular to the given type of explosive which can be fixed by experiment, W [kg] is the mass of the explosive, and Z [m] is the depth. On separating the variables and integrating, Eq.(8) gives

In this expression, R₀ is the initial bubble radius, which can be taken as zero given that it is much smaller than the maximum radius. By integrating and transforming Eq.(11) and incorporating Eq.(10), we have

where J[s·m^5/6/kg^1/3] is a constant particular to the given type of explosive. Eq. (10) and (12) are the classical “semi-empirical” expressions for the maximum radius and the first pulsation period, respectively, of an UNDEX bubble.

Figure 5 shows the curves of the bubble pulsation period and maximum radius with depth. Although the experimental data are affected by the reflected wave, the figure shows minimal errors in the bubble pulsation process between the experimental data and the solutions of the RP equation. Regarding the pulsation period, the various sets of results are all reasonably close as shown in Fig. 5(a). Regarding the maximum radius, the experimental data are lower than both the empirical prediction and the RP solutions as shown in Fig. 5(b). The experimental pulsation period shows minimal error against the empirical prediction, but there are clear differences in the maximum radius between the experimental data and the empirical prediction. The influence of water depth on the pulsation period is greater than it is on the maximum radius, which is also shown in Eq.(12) as T∝f(H^5/6) and in Eq.(11) as R_max∝g(H^1/3).

In addition, the total energy of the explosive is converted into shock wave energy and the bubble energy after blast. According to Eq.(9), the kinetic energy of the bubble can be calculated out and the empirical formula of the bubble maximum radius can be derived out also. With the bubble maximum radius obtained in this paper, it is shown that the experimental data is fitted well with empirical predictions at the 0.8m and 100m depths, however the experimental data is smaller than the empirical predictions at the 200m 300m and 350m depths. This phenomenon implies that the bubble may counts lesser explosive energy in the deep water.

Because it is difficult to obtain the shock-wave signal in the present experiment, we cannot confirm how much energy is converted into the shock wave and the bubble motion at the 200m, 300m and 350m depth. However, the water depth certainly influences the bubble pulsation behavior, the state of the gaseous products, and the explosive energy-release process.

In this paper, the pulsation behavior of deep-UNDEX bubbles is studied both experimentally and theoretically, from which we draw the following main conclusions. 1) In the deep-UNDEX experiments, the bubble pulsation motion is revealed by the high-speed camera for equivalent water depths up to 350m. It is found that the shape of the explosive package exerts a considerable influence in the early dilating stage. In the deep-water cases, the bubble during the whole first pulsation is never a regular spherical bubble. 2) Analysis of the deep-UNDEX bubble pulsation shows that the bubble dynamics are described well by the RP equation. Meanwhile, the experimental data indicate that the expansion process of the gaseous detonation products of the deep UNDEX obey a polytropic law. The value of the polytropic index γ in that polytropic law changes from 1.25 to 1.3 as the depth is increased. 3) Regarding the bubble pulsation laws, the experimental pulsation period and that obtained from the RP equation agree with that predicted empirically, whereas the maximum radius is smaller than that predicted empirically. This phenomenon shows that the depth not only creates a high hydrostatic pressure for the bubble but also changes the energy-release process of the deep UNDEX.

This work was supported by the National Key R&D Plan of China (grant no. 2016YFC0801204).