Acoustic signals of underwater explosions near surfaces - Dynaflow, Inc.

Acoustic signals of underwater explosions near surfaces John R. Krieger and Georges L. Chahine Dynaflow, Inc., 10621-J Iron Bridge Road, Jessup, Maryland 20794

共Received 12 November 2004; revised 13 July 2005; accepted 4 August 2005兲 Underwater explosions are conventionally identified and characterized by their seismic and/or acoustic signature based on spherical models of explosion bubbles. These models can be misleading in cases where the bubble is distorted by proximity to the free surface, the bottom, or to a solid object. An experimental and numerical study of the effects of various nearby surfaces on the bubble’s acoustic signature is presented. Measurements from high-speed movie visualizations and acoustic signals are presented which show that the effect of proximity to a rigid surface is to increase the first period, weaken the first bubble pulse, and affect significantly the second period, resulting in a peak value at standoffs of the same order as the maximum bubble radius. These results are compared to results under a free surface, over a bed of sand, and over a cavity in a rigid surface. In all cases, the first period is increasingly lengthened or shortened as the motion of fluid around the bubble is increasingly or decreasingly hindered. The effect of bubble distortion is to weaken the first bubble pulse and increase the bubble size and the duration of the second cycle. © 2005 Acoustical Society of America. 关DOI: 10.1121/1.2047147兴 PACS number共s兲: 43.30.⫺k, 43.30.Lz 关WMC兴

Pages: 2961–2974

I. INTRODUCTION

Underwater explosions produce bubbles of gas and vapor that alternately expand and collapse, generating a train of acoustic pulses that propagates acoustically through the water, and seismically through the seafloor. The timing of these pulses can be used to determine the yield of the explosion, provided its depth can be estimated. In some cases, reverberations from the sea surface and ocean bottom provide information about the explosion depth 共Baumgardt and Der, 1998兲. If the explosion is far from the free surface and the bottom and is otherwise unconstrained, the classic formula, as presented for example by Cole 共1948兲, serves to define the “bubble pulse” period 共T in seconds兲 in terms of the explosive yield 共W in pounds兲, its depth 共Z in feet兲, and a parameter 共K = 4.25 for TNT兲 characterizing the explosive T=K

W1/3 . 共Z + 33兲5/6

共1兲

The ability to assess the character of an underwater explosion 共UNDEX兲 and estimate its yield and depth is important for a variety of reasons. For example, the use of explosives as underwater acoustic sources has a long and rich history 共Weston, 1960兲. A common method of surveying ocean geoacoustic properties is to detonate calibrated charges and measure the propagated signal from acoustic sensors or arrays at various ranges 共Potty et al., 2004, 2003; Spiersberger, 2003; Godin et al., 1999; Popov and Simakina, 1998兲. In addition, the bubble pulse signature is also vital in the context of nuclear test-ban verification 共Weinstein, 1968兲, where a key issue is to distinguish an UNDEX from an underwater earthquake 共Baumgardt, 1999兲 and in tracking dynamite fishing practices 共Woodman et al., 2003兲. Exploiting dynamite fishing explosions as sound sources of opportunity provides another motivation to this study 共Lin et al., 2004兲. In all cases, erroneous conclusions result from the use of Eq. 共1兲 if the explosion is near an obstacle, the bottom, or if J. Acoust. Soc. Am. 118 共5兲, November 2005

it is partly contained. In particular, the bubble generated by an UNDEX can be severely distorted from the presumed spherical geometry by a nearby object or surface. This could result in “bubble pulse periods” that correspond to a yield substantially different from that predicted by Eq. 共1兲. Considerable efforts have been devoted to experimental and theoretical understanding of the physics of underwater explosions as sound sources. These attempts have typically used ad hoc models of bubble migration 共Buratti et al., 1999兲 or have been based on tests with actual full-scale explosives 共Chapman, 1985, 1988; Hannay and Chapman, 1999兲. This paper presents controlled scaled laboratory tests to define the multicycle motion and geometry of the explosion bubble, and compares with a sophisticated numerical simulation based on the dynamics of bubbles near boundaries. Quantitative results 共in particular, depths and time of acoustic radiations兲 that are most important for defining impulse point sources of acoustic propagation are presented. Section II describes the basic physics, Sec. III describes the experimental and numerical conditions of the study, Sec. IV presents the results of the study, and Sec. V provides a discussion of these results in terms of characterization of explosions.

II. BACKGROUND

The detonation of an explosive charge underwater results in the emission of a shock wave and the creation of a high-pressure gas and vapor bubble. The shock wave constitutes the first of a series of acoustic pulses detectable at long ranges, and the remaining bubble pulses are generated by the subsequent motion of the bubble. The pressure of the bubble initially causes it to expand, but inertia of the liquid carries the bubble past its equilibrium point, causing it to overexpand, and then collapse again. The final stages of the collapse are quite violent, and return the bubble nearly to its original state, generating large pressures and another acoustic

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© 2005 Acoustical Society of America

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pulse. The bubble then begins the cycle again and continues until it has broken up and its energy has been dissipated. The shock and subsequent bubble pulses radiate sound as point- or compact impulsive sources at different depths and times. While boundaries affect the sound propagation from a point source in well-understood ways, the problem of how the boundaries affect the multicycle motion of the bubble acoustic source is the focus of this paper.

A. Numerical model

Following an underwater explosion detonation, and after a very short period of time where compressible effects are important, the characteristic liquid velocities become small compared to the velocity of sound, enabling one to neglect the liquid compressibility. Similarly, viscous effects can be neglected and one can consider the water flow field as potential, with u = 䉮⌽, where u is the liquid velocity vector and ⌽ is the velocity potential. ⌽ satisfies Laplace’s equation in the fluid domain, D ⵜ2⌽共x,t兲 = 0,

x 苸 D,

共2兲

where x is the spatial variable and t is time 共Chahine and Perdue, 1989兲. On the bubble surface Sb, of local normal, n, ⌽ satisfies the kinematic boundary condition, which expresses equality between the fluid and the free-surface normal velocities dx/dt · n = ⵜ⌽ · n,

x 苸 Sb ,

共3兲

共4兲

where P⬁ is the hydrostatic pressure at the point of initiation of the explosion, Pb共x , t兲 is the local pressure in the liquid at the bubble interface, and z is the vertical coordinate at point x. The pressure inside the bubble is assumed spatially homogeneous, and the bubble content is assumed to be composed of noncondensable gas arising from the explosion products, which follow a polytropic compression law of constant k. These assumptions have been shown, for a wide range of free-field bubbles, to lead to an excellent fit between experimental data and numerical results 共Chahine et al., 1995兲. This leads to the following form of the normal stress boundary condition at the bubble interface: Pb共x,t兲 = Pg0

冋册 V共t兲 V0

k

+ Pv − ␴C,

共5兲

where Pg0 is the initial pressure of the noncondensable gas inside the bubble and Pv is the water vapor pressure. V0 is the initial volume of the bubble, ␴ is the surface tension coefficient, and C共xs , t兲 is twice the local mean curvature at xs given by C = ⵜ · n. The local normal to the surface, n, is defined as 2962

J. Acoust. Soc. Am., Vol. 118, No. 5, November 2005

共6兲

ⵜf , 兩ⵜf兩

共7兲

where f is the bubble surface equation. The appropriate sign is chosen so that the normals point towards the liquid. In addition to conditions 共3兲 and 共4兲, the boundary condition on any nearby solid body St is given by

⳵⌽ = u n, ⳵n

x 苸 St ,

共8兲

where un is the local normal velocity of the solid body in response to the bubble loading. This includes body motion and deformation 共Kalumuck et al., 1995; Chahine and Kalumuck, 1998兲. At infinity, the fluid velocities due to the bubble dynamics vanish, and the boundary condition is 兩lim兩x→⬁兩ⵜ⌽兩 = 0.

共9兲

To complete the description of the problem, appropriate initial conditions are used. At the initiation of the computation, and only at that instant, the bubble dynamics is assumed to be that due to a spherical explosion bubble in a free field. The Rayleigh–Plesset equation 共Plesset, 1948兲 governing spherical bubble dynamics is used to obtain these initial conditions 共i.e., a relationship between the initial bubble radius and the initial bubble wall velocity or gas pressure兲. This results in the following relationship between the initial bubble radius, R0, and the initial gas pressure inside the bubble, Pg0, for R0 = 0: Pg0 =

and the dynamic boundary condition 1 P⬁ − Pb ⳵⌽ = − 兩ⵜ⌽兩2 + − 兩gz兩x苸Sb , 2 ⳵t ␳

n= ±

=

冋

册

3共1 − k兲 p⬁ − pv ␴ 共1 − ␧−2 共1 − ␧−3 0 兲+ 0 兲 ; 3k−3 3 R0 1 − ␧0

␧0

R0 . Rmax

共10兲

Thus, a closed system of equations is obtained to solve for ⌽ using Green’s identity ⍀⌽共x兲 =

冕冕

ny · ⵜy⌽共y兲G共x,y兲dsy

S

−

ny · ⵜyG共x,y兲⌽共y兲dsy ,

共11兲

S

where x is selected on the boundary, y is an integration variable on the surface S , ny is the normal to S, and G共x,y兲 =

1 兩x − y兩

共12兲

is the Green’s function. ⍀ is the solid angle subtended in the fluid at the point x. Equation 共11兲 is an integral equation which relates the potential at any point x to the values of ⌽ and its normal derivatives on the boundary of the liquid domain, S. We can use Eq. 共11兲 to determine at a given computation time the unknown quantities: ⌽ on solid surfaces and normal derivative of the potential, ⳵⌽ / ⳵n, on the free surface, knowing the normal velocities on the solid surfaces and the velocity potential on the free surface. To start the computations, we J. R. Krieger and G. L. Chahine: Explosion signals near surfaces

know the potential on the initial bubble surface through 共⌽ = R0R˙0兲, and we assume a zero body velocity. Once the flow variables are known at a given time step, values at subsequent steps can be obtained by integrating Eqs. 共3兲 and 共4兲, and using an appropriate time-stepping technique. The material derivative, D⌽ / Dt, is then computed using the Bernoulli equation 共4兲, and noting that u = ⵜ⌽

⳵⌽ D⌽ ⳵⌽ = + ⵜ⌽ · ⵜ⌽ = + 兩u兩2 . Dt ⳵t ⳵t

共13兲

This time stepping proceeds throughout the bubble growth and collapse, resulting at each time step in the knowledge of all flow-field quantities and the shape of the bubble. This is the basis of the numerical method used in this paper and which has been extensively used and validated for both axisymmetric 共Zhang et al., 1993; Choi and Chahine, 2003兲 and three-dimensional interactions between UNDEX bubbles and nearby structures 关e.g., Chahine and Perdue 共1989兲; Chahine 共1996兲; Chahine, Kalumuck, and Hsiao, 共2003兲兴.

FIG. 1. Spark-generated bubbles over a rigid surface. Horizontally, four different times are shown—the time of the first volume maximum, just prior to the first collapse, the time of the first collapse, and the time of the second volume maximum—and vertically three different standoff distances are shown, with standoff distance increasing from bottom to top. The ambient pressure was approximately 60 mbar, and Rmax was approximately 15 mm.

B. Discussion and definitions for analysis

The acoustic bubble pulses, generated by the bubble volume and shape changes, may be treated theoretically primarily as a monopole radiation in the far field with strength dependent on the volume acceleration of the bubble, plus a dipole with strength dependent on the bubble migration, and higher-order poles dependent on its deformation. In the absence of gravity effects or nearby objects, the bubble remains spherical in shape, and the period of its volume oscillations can be predicted analytically. Rayleigh 共1917兲 analyzed the collapse of empty spherical cavities and found the time of collapse to be Tcollapse = 0.915Rmax

冑

␳ , P⬁

共14兲

where Rmax is the initial 共maximum兲 radius of the cavity, and ␳ and P⬁ are the density and pressure of the ambient fluid, respectively. This dependence of the period on the ambient pressure and the bubble size is another formulation of Eq. 共1兲. The effects of vapor pressure, noncondensable gas, surface tension, viscosity, and a time-varying ambient pressure could also be included, to produce the Rayleigh–Plesset equation 共Plesset, 1948兲, the equation of wall motion for a collapsing and rebounding spherical bubble. For given conditions, the period of the bubble oscillation can be predicted from the solutions of this equation. If the explosion occurs near an object or at shallow depths where the effects of gravity are strong, the bubble collapses asymmetrically, resulting in reentrant jet and toroidal bubble formation, or if both effects are strong and in opposite directions, a pear shape and bubble splitting 共Chahine, 1996, 1997兲, and the bubble does not produce the same series of pulses as it would in deep submergence and away from boundaries. The effect of a nearby surface is quantified in terms of the normalized standoff distance J. Acoust. Soc. Am., Vol. 118, No. 5, November 2005

X=

x Rmax

,

共15兲

where x is the standoff distance, i.e., the shortest distance from the initial explosion center to the surface, and Rmax is the maximum radius the bubble would achieve if the explosive were detonated in an infinite medium at the same depth. For large values of the standoff parameter, the bubble is essentially spherical and unaffected by the boundary. As the standoff distance is reduced, the bubble becomes more and more affected by the boundary. At values near unity, the bubble nearly touches the surface on its first expansion, and at values near zero, the first bubble pulsation is quasihemispherical. Examples of this behavior for small-scale explosions generated by a spark in a water tank 共Chahine et al., 1995兲 are shown in Fig. 1. The effect of buoyancy is quantified in terms of the Froude number 共Chahine, 1997兲 F=

P⬁ − Pv . 2␳gRmax

共16兲

This parameter is the ratio of the difference between the ambient pressure at the charge location and the vapor pressure to the difference in pressure between the top and bottom of the bubble. It expresses the relative importance of the compressional force tending to collapse the bubble and the gravity force tending to deform and migrate the bubble upward. For large values of the Froude number, the effect of gravity is relatively weak, and the bubble remains spherical. For small Froude numbers, upward motion and bubble distortion due to gravity are important. This parameter is strongly responsible for differences between UNDEX bubble behaviors at different geometric scales. The Froude number 共or the gravity acceleration term兲 does not appear explicitly in the equations, but the gravity term can be seen in the boundary condition 共4兲 on the bubble and free-surface walls. J. R. Krieger and G. L. Chahine: Explosion signals near surfaces

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FIG. 2. The experimental arrangement for recording bubble dynamics near a rigid surface.

For problems of interaction between an UNDEX bubble and the ocean bottom, or an obstacle located below the bubble, gravity acts in an opposite direction than the bottom, i.e., it tends to force the bubble to have a reentrant jet moving upward, while the bottom induces a downward motion. This results, at small Froude numbers, in large differences in the bubble behavior between two different Froude number conditions for otherwise similar UNDEX conditions. III. EXPERIMENTAL AND NUMERICAL ANALYSIS A. Experimental methods

Bubbles generated by the underwater discharge of a high-voltage charge between two coaxial electrodes, exposed to each other at their ends and contained within a transparent vacuum cell, were used as laboratory-scale models of underwater explosions 共Chahine et al., 1995兲. The cell is 3 ⫻ 3 ⫻ 3 ft.3 made of 1-in.-thick Plexiglas wall, which enables it to withstand reduced pressure down to 0.5 psi. By reducing the pressure in the cell, relatively large and slow-collapsing bubbles were generated, and the transparent walls allowed recording of the bubble dynamics by a high-speed digital camera 共Redlake Imaging model PCI8000S兲. The acoustic signals of the bubbles were measured with a piezoelectric transducer having a 1-␮s rise time 共PCB Piezotronics model 102A03兲. A photograph of the experimental arrangement is shown in Fig. 2. Concerning the physics of the problem, the arrangement electrodes/nearby boundary is quite simple and is further illustrated in the bubble pictures in this paper, such as in Fig. 1. Note that the acoustic properties of the acrylic platform are close to those of water, making the reflection coefficient near zero over the relevant range of reflection angles. Furthermore, reflections from walls and other distant objects resulted in path lengths much longer than the straight-line path from bubble to transducer. Because of this, only the pulses propagating directly from the bubble to the transducer were strong enough to be detectable. The energy of the bubble was controlled by adjusting the voltage supplied to the capacitor of the spark generator. The effects of the tank pressure and of the variations in the discharge output were removed from the data by measuring Rmax and normalizing periods using a characteristic time 2964


共equal to the Rayleigh collapse time omitting the factor 0.915兲 as in Eq. 共15兲. The bubble maximum radius is used to normalize the standoff distance, and is obtained by direct measurement from the video pictures corresponding to each signal. When the bubble was significantly distant from the bottom, it retained a spherical shape at maximum volume and a radius could easily be measured. In cases where the bubble was distorted by the presence of the nearby boundary, an equivalent radius, equal to the radius of a sphere of equal volume, was used. Simulations of identical initial bubbles at various distances show that the bubble achieves the same volume, within ±1%, at the time of its maximum growth regardless of the standoff distance, so that the equivalent Rmax is a consistent measure of bubble size. The volumes of distorted experimental bubbles were calculated from the observed bubble outline at the time of maximum growth, as the volume of a solid of revolution around its vertical axis. The validity of the assumption of axisymmetry is supported by top views, which show circular profiles, and by the mirror symmetry seen in side views. The ambient pressure was calculated based on the absolute pressure in the vacuum cell and the depth of the electrodes, and the vapor pressure was calculated based on the measured ambient water temperature. All times, t, and distances, r, were normalized using the quantities ␳, Rmax, and Pamb − Pv ¯t ⬅ Rmax

¯r ⬅

冑

r Rmax

t

␳ Pamb − Pv

.

,

共17兲

共18兲

Although spark-generated bubbles are considerably smaller than typical explosion bubbles, surface tension and viscosity effects remain negligible, and the bubble dynamics is an accurate scaled representation of a full-scale case having the same normalized standoff and Froude number 共Chahine et al., 1995兲. Conversion between scales is accomplished using the characteristic length lcharac = Rmax and the characteristic time Tcharac = Rmax共␳ / ⌬P兲1/2, as above. Differences between various scaled results obviously exist if the Froude numbers, i.e., the relative influence of gravity, are not the same. B. Error estimates

One of the principal sources of experimental uncertainties lies in the determination of Rmax for each case, which affects both the normalized standoff distance and the normalized period. Repeated measurements for several different cases give a standard deviation of about 1% of the mean. In cases where an equivalent Rmax was measured for the bubble at its second maximum, the standard deviation was about 4%, since the axisymmetric assumption probably introduces more significant errors in these cases. Vapor pressure is a function of temperature, and uncertainty in the temperature measurement led to errors of about 2% or less. The least significant digit on the pressure gauge is 1 mbar, or about J. R. Krieger and G. L. Chahine: Explosion signals near surfaces

2% for low-pressure tests. For high-pressure tests, in which the bubbles are small, the dominant source of uncertainty lay in the measurement of standoff, which could have been as much as 10%. In the graphs presented here, it was decided to rely upon the scatter among a large number of data points as an indication of repeatability. One series of tests was conducted with each test repeated five times, which produced standard deviations in normalized period and normalized standoff of approximately 5%.

C. Numerical approach

DYNAFLOW’s boundary element method code 共Choi and Chahine, 2003兲 was used to conduct simulations of bubbles under conditions matching those of the spark tests. 2DYNAFS© is based on the boundary element method described in Sec. II for solving potential flows with arbitrary boundaries, and calculates the motion of one or more interfaces in an axisymmetric geometry. Each interface is discretized in a meridional plane, each node is advanced at each time step using the velocities and the unsteady Bernoulli equation, and the new velocity potentials and normal velocities are calculated using a discretized version of Green’s equation 共Zhang et al., 1993兲. In cases where a reentrant jet impacts the opposite wall of the bubble, the dynamic cut relocation algorithm of Best 共1994兲 was used to continue computations with a new toroidal bubble. For laboratory-scale simulations, the initial conditions of the bubble were specified as the same ambient pressure as the corresponding experimental test, and an initially spherical explosion bubble with a radius and pressure such as to produce an Rmax value of 1.3 cm, which is a typical value for the spark tests in the pressure range tested. 2DYNAFS©

IV. RESULTS A. Bubble dynamics near a rigid surface

Figure 3 shows an example series of pressure signals from bubbles generated by spark discharges at varying distances above a flat, rigid plate, with each signal offset vertically in the figure by an amount proportional to its corresponding standoff distance. The time has been normalized as in Eq. 共15兲. Each signal contains initial strong fluctuations composed of a combination of the initial shock wave and electrical noise associated with the spark discharge, followed by two and sometimes three detectable peaks generated by the bubble collapses. Two principal differences among the signals can be observed. First, the timing of the first collapse pulses appears in general delayed for bubbles near the plate. Second, the first bubble pulse decreases in strength dramatically as the bubble approaches the plate. For bubbles at moderate distances from a rigid surface, Chahine and Bovis 共1983兲 have demonstrated, using matched asymptotic expansions for large X, that J. Acoust. Soc. Am., Vol. 118, No. 5, November 2005

FIG. 3. Pressure signals from a spark-generated bubble with varying standoff distance. Signals were measured using a quartz pressure transducer several inches to one side of the bubble. The ambient pressure was 1 atmosphere and the maximum bubble radius about 6 mm. All signals have been offset vertically by an amount proportional to the respective standoff distance.

冉冕

T* ⬵ 1.83 1 +

1 4X

Rmax

R0

冊冉

R共t兲dt ⬇ 1.83 1 +

冊

0.79 , 4X 共19兲

*

where T is the normalized period of the bubble in the presence of the surface. This first-order approximation provides a useful theoretical comparison for moderate values of standoff distance, although it is not valid for small values. In order to investigate the effects of proximity to a rigid boundary over the entire range of standoff distances, several series of spark tests were conducted using approximately constant energy but varying standoff distances, including a concentrated series of measurements for values of X less than 1, and these results were compared to the results from a series of simulations using 2DYNAFS©. The measured and calculated first bubble periods are shown in Fig. 4 along with a curve calculated from Eq. 共19兲, and a correlation calculated from Eq. 共20兲, presented below. Both the measurements and the simulations agree with the predictions of the first-order asymptotic expansion 共19兲

FIG. 4. First bubble period as a function of standoff distance. Experimental data were collected for several cell pressures, and simulations were conducted using an ambient pressure of 50 mbar plus a correction for depth, and an Rmax value of 1.3 cm. J. R. Krieger and G. L. Chahine: Explosion signals near surfaces

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FIG. 5. Normalized second period as a function of standoff. Experimental data are shown from the same set of tests as in Fig. 4, normalized using Eqs. 共17兲 and 共18兲.

for values of X larger than 1, where the period is approximately twice the characteristic time, as expected. The contribution of this work is the description of a nonasymptotic region where larger bubble deformations occur. As the bubble nears the bottom, the period shows a steady gradual increase, up to a value at the bottom approximately 20%– 25% greater than the “free” value. This increase is consistent theoretically with the additional energy due to an image bubble on the opposite side of the surface. If a hemispherical bubble is generated exactly at the surface, it will act with its mirror image as a single spherical bubble having twice the energy as the actual bubble, and according to the period– energy relation will have a period greater than the actual bubble if it were spherical by a factor of 21/3 = 1.26. Thus, the theory predicts a normalized period of 2 ⫻ 0.915= 1.83 for free bubbles, and with the image a period of 2 ⫻ 0.915 ⫻ 21/3 = 2.31 for bubbles on a rigid surface. Both experimental measurements and numerical simulations using 2DYNAFS© agree with these extrema and show a steady progression in between, while, as expected, the asymptotic expansion 共and the classical corrections兲 fails at the lower values of X. A general analytic prediction of the bubble first period over all standoff distances is lacking. However, the empirical expression

冉

T* ⬵ 1.83 1 +

0.1975

冑0.5776 + X2

冊

,

共20兲

does a reasonably good job as a correlation, as shown in Fig. 4. This expression matches the theoretical values for X = 0 and X → ⬁, and agrees with the asymptotic expansion formula for large X. It may be used to predict the period of an explosion bubble of any size near a large, flat, rigid surface, as long as the effects of buoyancy are not too strong, i.e., as long as the Froude number is above approximately 10. Similar expression could be derived for other values of F but were not attempted here. The period of the second cycle, shown in Fig. 5, depends on the history of the bubble through the first cycle, and exhibits more variation with standoff than does the first period. The second period is normalized using the same characteristic time as the first period. At large standoff distances, the second period is nearly half that of the first cycle, which according to the period–energy relation for spherical bubbles, and for negligible migration, corresponds to an en2966


FIG. 6. First bubble pulse strength as a function of standoff. The maximum pressure recorded by the transducer 3 in. from the bubble during the first bubble pulse is shown from the same tests as in Fig. 4 and Fig. 5.

ergy of 共 21 兲 or 13% of the original energy, or an energy loss at the first collapse of about 87%. This differs slightly from period ratios and energy losses for free-explosion bubbles 共Cole, 1948兲, but is consistent with reported values for lasergenerated bubbles 共Vogel et al., 1989; Vogel and Lauterborn, 1988兲. As the standoff distance decreases to a value of X ⬇ 1, the second period increases to a large fraction of its original value, suggesting a reduction in the bubble energy loss during the first collapse. A reduction in energy loss is also suggested by a reduction in emitted acoustic energy, as is indicated in Fig. 6, which shows a pronounced minimum in the peak pulse pressure near X = 1. However, a change in period is also to be expected for these standoff distances simply due to the formation of a vortex ring bubble, whose period is known to be larger than that of a spherical bubble of equal energy 共Chahine and Genoux, 1983兲. As the standoff distance decreases below X ⬇ 1, the period again decreases. This behavior of exhibiting an extremum near X ⬇ 1 is also shown in the measurements of Lindau and Lauterborn 共2003兲, who measured maximum and minimum volumes of laser-generated bubbles near a plate and found that the ratio of maximum to minimum volume near X ⬇ 1 was reduced by roughly 2 orders of magnitude from its value very near and very far from the plate. In other words, the strength of compression of the bubble contents is greatly weakened when the standoff distance is approximately 1 bubble radius from the surface. Note that the value of peak pressure shown in Fig. 6 is probably lower than the actual value due to the extreme brevity of the pulse 共laser-induced bubbles produce pulses with a duration of tens of nanoseconds; see Vogel and Lauterborn, 1988兲, and the relatively large size and slow response 共rise time of 1 ␮s兲 of the quartz transducer. The pressures in all cases were measured several inches to the side, and were normalized to correspond to a 3-in. separation between electrodes and transducer, using an assumed 1 / r dependence of pressure with distance. Since standoff values much less and much greater than the bubble size correspond, respectively, to hemispherical and spherical symmetry, while standoff values nearly equal to the bubble size result in highly nonspherical bubbles and in the development of strong linear and rotational motion, it seems reasonable to explain the extrema in all of these properties in terms of bubble distortion during the first cycle and 3

J. R. Krieger and G. L. Chahine: Explosion signals near surfaces

FIG. 7. Ratio of second maximum radius to first maximum radius as a function of standoff distance. Data are shown from the same set as shown in previous figures.

energy loss during the first collapse. For hemispherically and spherically symmetric bubble collapse, all of the energy of the fluid is directed into compression of the gaseous contents of the bubble, producing small minimum volumes, large maximum pressures, and large energy losses due to acoustic radiation. For values of X ⬇ 1, much of the fluid energy goes into the formation of a jet and a toroidal bubble, and the asymmetric collapse deprives the gas of some of its compression energy, producing larger minimum volumes, smaller pressures, and weaker acoustic emission. Larger second-cycle bubble volumes, as well as larger second periods, are also observed near X = 1. Figure 7 shows approximate measurements of the bubble equivalent radius at the second volume maximum 共Rmax 2兲, as a fraction of the first Rmax value. Similar to the second period, the second maximum radius displays a maximum near X = 1. Furthermore, the similarity between the radius ratio and the ratio of second to first period, shown in Fig. 8, suggests that the period scales approximately as the equivalent maximum radius for nonspherical bubbles as for spherical bubbles, and both are decreased in nearly the same proportion by the loss of energy during the first collapse. Note that Cole 共1948兲 reports a period ratio of approximately 0.77 for freeexplosion bubbles, and a decrease in the ratio as the explosion approaches the bottom, while a period ratio between 0.5 and 0.6 is consistently obtained for free spark-generated

FIG. 9. Ratio of second peak pressure to first peak pressure as a function of standoff distance. Data are shown from the same experimental tests as shown in previous figures.

bubbles in the ambient pressure conditions considered here, and this ratio increases as the bubbles approach the bottom. The energy losses for spark-generated bubbles are consistent, however, with reported energy losses for laser-produced bubbles 共Vogel et al., 1989兲. Also, the influence of gravity in the two cases is very different, since the Froude number for the conditions reported by Cole is approximately half of the value for a typical spark test reported here, and the opposing effects of gravity and a nearby bottom produce significantly different effects from the effects of a surface alone, such as bubble lengthening and splitting 共Chahine, 1996兲. In addition, energy losses between one bubble cycle and the following for free-field bubble have been shown experimentally, over a very large range of explosion conditions, to depend on a parameter equivalent to the Froude number, i.e., Rmax / Z, where Z is the hydrostatic head at the explosion center 共Snay, 1962; Chahine and Harris, 1997兲. The ratio of the peak pressures between the second and the first bubble cycle is also very instructive, and shows a very distinctive peak at a normalized standoff distance of approximately X = 0.9. This is clearly illustrated in Fig. 9. B. Bubble dynamics near a free surface

It is well known that proximity to a free surface reduces the first period of an explosion bubble. An analysis using the image theory can be applied to a free surface, similar to that for a rigid surface but with an opposite sign, resulting in an analogous formula 共Chahine and Bovis, 1983兲

冉

T* ⬇ 1.83 1 −

FIG. 8. Ratio of second period to first period as a function of standoff distance. Data are shown from the same experimental tests as shown in previous figures. J. Acoust. Soc. Am., Vol. 118, No. 5, November 2005

冊

0.79 . 4X

共21兲

In this case, X is the normalized depth, or standoff distance from the free surface. This formula is essentially the same as that given by Cole 共1948兲, and except for depths shallower than 1 bubble radius, agrees quite well with both laboratoryscale experimental data, and with simulations, as shown in Fig. 10. Note that the Froude number for the experimental conditions is approximately 8, indicating a mild influence of buoyancy, while Eq. 共21兲 assumes negligible influence of buoyancy 共F Ⰷ 1兲. Also note that data are unavailable below approximately X = 0.5, as the spark-generated bubbles vent J. R. Krieger and G. L. Chahine: Explosion signals near surfaces

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FIG. 12. Schematic of spark-generated bubbles over a hemispherical cavity in a rigid surface. FIG. 10. Normalized first period versus normalized depth for bubbles near the free surface.

their contents to the atmosphere in this region, lose their identity, and fail to emit bubble pulses. In the range 0.5 ⬍ X ⬍ 1.0, the top of the bubble does rise above the equilibrium free-surface level, but the surface is not breached—it maintains its integrity and exhibits behavior resembling an elastic membrane, and a water layer remains between the bubble and the free surface 共Chahine, 1977兲. As with bubbles near a rigid surface, the first bubble pulse and the second cycle behavior of bubbles near a free surface show more complex variations than the period of the first cycle. The first bubble pulse strength, and the ratios of second to first period and second cycle bubble size to first cycle bubble size, are shown in Fig. 11. When the bubble is close to the surface, the surface has a repulsive effect which forces the bubble downward during and after the collapse. When the bubble is sufficiently deep, the effect of buoyancy dominates over the effect of the free surface, and the bubble is distorted and lifts upward. At some intermediate depth, the two opposing effects cancel each other, and the bubble is observed to remain nearly spherical and to pulsate without rising or descending. Under the laboratory conditions shown here, this occurs at a normalized depth of approximately 2. Unfortunately, when the bubble remains spherical and stationary it collapses at the tip of the electrode, and this tends to make the bubble pulse magnitudes somewhat erratic in this region. Figure 11 shows that, near a normalized depth of 2, the strength of the first bubble pulse is relatively strong, and that the second bubble cycle is relatively short with a small maxi-

FIG. 11. Size and duration of the second bubble cycle, and magnitude of the first bubble pulse for bubbles near a free surface. 2968


mum bubble radius. Thus, just as for bubbles near a rigid surface, when the collapse is symmetric the compression of the bubble contents is relatively strong, resulting in relatively large energy losses and weaker second cycles. When the bubble is distorted during collapse, the compression is weakened, resulting in smaller energy losses and stronger second cycles. C. Bubble dynamics near a nonflat rigid surface

In addition to being nonrigid, surfaces near explosion bubbles are often not perfectly flat. Explosions near the seafloor may result in a crater. Explosions near or on an object may interact with a cavity in the object. As a simple model of a bubble interaction with a nonflat surface, a series of simulations and experimental tests of a bubble near a flat surface containing a hemispherical cavity was conducted. These tests are characterized not only by the normalized standoff distance x / Rmax, but also by the ratio of cavity size to maximum bubble size, Rc / Rmax. The standoff distance for these tests is defined relative to the bottom of the cavity, as shown in Fig. 12 in order to avoid ambiguities in the limit as the wall cavity radius becomes very large. If the explosion bubble is very much larger or very much smaller than the cavity, the behavior of the bubble resembles that near a flat surface, with the bubble interacting with a large region of the surface around the cavity, or with the bottom of the cavity, respectively. If the bubble size is of the same order as the cavity size, however, the behavior is different. If the bubble is initiated at the bottom of the cavity, it splits during collapse, as shown in Fig. 13. The most prominent effect of the cavity on the acoustic signals is the significant increase in first period for bubbles partially enclosed within the cavity. Figure 14 shows the normalized first period as a function of normalized standoff for various ratios of cavity size to bubble size, with Fig. 14共a兲 showing experimental data, and Fig. 14共b兲 showing the results of simulations. In both cases, a curve similar to the normalized period curve for flat surfaces is produced when the cavity to bubble

FIG. 13. Collapse, splitting, and rebound of an explosion bubble initiated at the bottom of a moderately sized hemispherical cavity. J. R. Krieger and G. L. Chahine: Explosion signals near surfaces

FIG. 14. Normalized first period of bubbles over a cavity in a rigid surface.

size ratio tends towards very small or very large values, since, in the limit of very small cavities, the bubble should interact with a flat surface with negligible influence of the cavity, and in the limit of very large cavities, the bubble should interact with the bottom of the cavity as if it were a very large flat surface. Note that curves for large values of Rc / Rmax maintain higher values than curves for small values of Rc / Rmax at large standoff distances, since smaller bubbles imply that larger normalized standoff distances are required to emerge from and to escape the effects of the wall cavity. If the wall cavity to bubble size ratio is not extremely large or small, i.e., if the wall cavity and maximum bubble radii are of the same order, then there is a marked increase in the bubble period when the explosion is initiated near or within the wall cavity, demonstrating the effect of increased inertia of the surrounding water due to increasing confinement of the explosion. Figure 15 shows more clearly the effect of the relative cavity and bubble sizes. The normalized first period is shown from experiments and simulations of explosion bubbles initiated at the bottom of the crater, for various values of Rc / Rmax. As expected, the period approaches the theoretical J. Acoust. Soc. Am., Vol. 118, No. 5, November 2005

value for an infinite flat surface for large and small values of Rc / Rmax, and increases by a factor of almost 2 in the range Rc / Rmax ⬃ 2 – 3, indicating a maximum restriction of bubble motion when the bubble size is a large fraction of the size of the containing cavity. Figure 16 and Fig. 17 show the experimental results for the second bubble oscillation period, and the amplitude of the first bubble pressure pulse, for bubbles near a cavity. Overall behavior is similar to that near a flat surface, except that the curves are shifted towards larger standoff values as the bubbles become smaller, due to the fact that smaller bubbles must be further away from the bottom of the cavity in order to escape its effects.

D. Bubble dynamics near a bed of sand

In order to investigate the effect of nonrigid surfaces, a series of experimental tests over a bed of sand was conducted, for comparison to the rigid surface. Commercial playground-quality sand 共grain size approximately 0.1– 1.0 mm兲 was thoroughly rinsed and poured into a bed J. R. Krieger and G. L. Chahine: Explosion signals near surfaces

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FIG. 15. Normalized first period of explosion bubbles initiated at the bottom of a cavity, as a function of the cavity size relative to the bubble size.

2 cm deep. Each bubble growth and collapse disturbed the surface of the sand to a certain extent, so the surface was smoothed flat before each test. When the bubble is within approximately 2 radii from the surface, the bubble tends to lift the sand underneath it into a mound during its collapse, as shown in Fig. 18. The dynamics of the first cycle appears otherwise unaffected compared to behavior over a rigid surface, and shows similar distortions and similar jetting behavior at similar standoff distances. Furthermore, the first period of the bubble over sand is within experimental variability of the first period over a rigid surface, as shown in Fig. 19, indicating that the ability of a solid surface to deform slightly or fluidize within a shallow region is not sufficient to significantly alter the hydrodynamics around a bubble near the surface. Greater elasticity, such as that of a free surface, is required to produce large variations in the bubble behavior. The most pronounced difference between the effect of a sandy bottom and of a rigid bottom lies in the behavior following the impact of the bubble with the sand. For a rigid surface, the bubble and remnants of the bubble may be observed in motion for a long time, but when a bubble strikes the sandy surface, the sand appears to immediately swallow the bubble, with fragments occasionally reemerging only af-

FIG. 16. Normalized second period of the bubble oscillations. 2970


ter a long time. A second bubble pulse is normally detectable, indicating a second collapse, but the strength of this pulse is much diminished for bubbles very near to the sand, as shown in Fig. 20. 共As in Fig. 3, each signal has been offset vertically by an amount proportional to the standoff distance. Note that several signals are overlaid at certain standoff distances, representing multiple tests at those distances.兲 The second period, shown as the ratio of second to first period in Fig. 21, shows a similar trend to the period ratio for bubbles over a rigid surface, except for somewhat diminished values for small standoff distances. The reduced period of the second cycle shown in Fig. 20 and Fig. 21 for standoff distances less than about 1, which is the region where the bubble collapses directly against the sandy surface, indicates a weakened rebound and second cycle, presumably due to dissipation of the bubble’s energy as it rebounds within and moves through the sand. The weakened second bubble pulse is consistent with greater energy losses as the pulse propagates from deeper within the sand bed. E. Spectra of the first bubble pulse

In cases where the explosion occurs near a rigid surface, the bubble pulse is often not a single peak, but has a smaller,

FIG. 17. Amplitude of the first bubble pulse peak pressure. J. R. Krieger and G. L. Chahine: Explosion signals near surfaces

FIG. 18. Growth and collapse of an explosion bubble over a bed of sand.

initial peak preceding the primary collapse peak, as shown in Fig. 22, which shows typical peak profiles for normalized standoff distances of 0.06, 0.30, 0.45, and 0.69. The initial, or precursor, peak is evident for X = 0.06 and quite pronounced at X = 0.3. Also evident is the decrease in peak amplitude for standoff distances near X = 1. A multitude of peaks, such as Fig. 22共c兲, is often observed for standoff values in the neighborhood of X = 1, and sometimes for bubbles strongly influenced by gravity as well, and is probably a result of partial or complete bubble breakup during collapse. The precursor peak is known to occur as a result of impact of the reentrant jet on the surface 共Chahine and Duraiswami, 1994; Thrun et al., 1992兲, while the main peak is produced at the bubble’s volume minimum. This precursor peak, which is most pronounced at a normalized standoff value of approximately 0.3, may also hold some explanation for the minimum in the second bubble period which occurs at this standoff value, due to increased energy loss accompanying the jet impact and pulse emission. Bubble collapse is an unstable process, and any perturbations, such as disruption of flow around an electrode, or any other object present in a nonideal flow field, will promote breakup of the bubble. This breakup has sometimes been observed in overhead views to produce daughter bubble fragments which collapse separately. In order to identify the effect of the changing features of the bubble pulse on spectra of the acoustic signal, a series of spectra of the first bubble pulse has been compared. Figure 23 shows profiles of the first bubble pulse and precursor peak, with the primary peak centered at t = 0, while Fig. 24 shows spectra of the first bubble pulse for standoff distances less than approximately unity. In Fig. 23, the signals from five repeated tests at each standoff distance have been overlaid, as an indication of variability in the signals. As is evident in Fig. 23, the jet impact peak becomes pronounced at X ⬇ 0.3, although with variable delay between it and the primary peak, and both the precursor peak and the collapse peak become more variable before nearly vanishing as the

FIG. 19. First period of a bubble over a bed of sand. J. Acoust. Soc. Am., Vol. 118, No. 5, November 2005

FIG. 20. The first and second bubble pulses of bubbles over a bed of sand. Signals were measured using a quartz pressure transducer several inches to one side of the bubble. The ambient pressure was 1 atmosphere and the maximum bubble radius about 6 mm. All signals have been offset vertically by an amount proportional to the respective standoff distance.

standoff distance approaches X ⬇ 1. Similar behavior may be observed in the spectra. The full width at half-maximum of the bubble pulse is typically approximately 16 ␮s, the logarithm of the reciprocal of which is 4.8. A distinct peak in the average spectra can be discerned at a value of 4.8, at least for the lower standoff distances where the pulse is strong. At standoff distances of approximately 0.5, where the bubble pulses become more erratic, the spectra do as well. At standoff distances of approximately 0.3, where the jet impact peak is most distinct and the interval is approximately 80 ␮s, the spectra show a faint rise at 104.1 Hz= 1 / 80 ␮s. V. CONCLUSIONS

We have presented measurements of the acoustic signals of spark-generated bubbles 共laboratory-scale underwater explosion bubbles兲, as well as simulations for similar conditions, which demonstrate the effects of bubble deformations on the acoustic signals. The influence of a nearby rigid surface increases the first period up to a value of approximately 25% greater than the free-field value when the explosion is right at the wall, and increases the second period significantly near a normalized standoff value of X = 1, due to jet

FIG. 21. Ratio of second period to first period for explosion bubbles over sand. J. R. Krieger and G. L. Chahine: Explosion signals near surfaces

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FIG. 22. Experimentally measured acoustic pressure profiles taken around the time of the first bubble pulse. At the lower standoffs, a precursor pulse can be seen 共noted 1兲, and is attributed to reentrant jet impact.

and toroidal bubble formation during the first collapse. This increase in the second period corresponds to reductions in the first bubble pulse pressure and increases in the second cycle bubble size near X = 1. The presence of a free surface produces similar but opposite effects for moderate distances, causing a decrease in the bubble period for bubbles close to the surface. In addition, the distortion caused by the free surface has similar effects on the bubble pulse and second period as the distortion caused by a rigid surface, namely a weakening of the bubble pulse and a lengthening of the second period of the

FIG. 24. Averaged spectra of the first bubble pulses from bubbles near a rigid surface. Each spectrum has been offset vertically by an amount proportional to the standoff distance.

bubble. The influence of a sandy surface produces similar effects to a rigid surface for moderate distances, but the sand tends to engulf the bubble and dissipate its energy for small standoff distances, greatly reducing the strength of later bubble pulses and the period of later cycles. An explosion bubble near or within a hemispherical cavity in a rigid surface exhibits behaviors similar to a bubble near a flat rigid

FIG. 23. Precursor peaks for bubbles within approximately 1 maximum bubble radius of a rigid surface. Each signal has been offset vertically by an amount proportional to the standoff distance. Five signals from repeated tests have been overlaid over each other for each standoff distance.

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wall if the bubble is very much larger or smaller than the size of the cavity, but if the bubble size is of the same order as that of the cavity, the restriction of motion caused by the cavity surrounding the bubble causes significant increases in the duration of the bubble cycle and the period between pulses. These results are consistent with general rules summarized as follows: Any factors which tend to confine the bubble, i.e., which restrict the motion of the fluid around the bubble, will tend to increase the first period. Any factors which tend to distort the bubble, e.g., large buoyancy forces or the presence of nearby objects, will tend to increase the minimum volume of the bubble during its collapse and thus tend to weaken the compression during collapse, resulting in weaker bubble pulses and longer periods in subsequent cycles. The significance of these results to seismo-acoustic signals can be demonstrated by considering a standard 1.8 -pound signal charge, which at a depth of 56 ft. produces a maximum bubble radius of about 1 m, and a first bubble period of about 0.12 s. If the explosion is within 1 m of the bottom at this depth, the period will be larger than this, and could be as much as 0.15 s, according to the data presented here. The period is weakly dependent on the yield, as indicated by Eq. 共1兲, so that this difference in the periods will produce an even greater difference in estimates of the yield. Given the same depth, a period of 0.15 s would correspond in the inverse problem to an overestimate of the yield of 3.3 pounds, if Eq. 共1兲 is used. This factor of 2 excess could also be predicted from the observation that a hemispherical bubble on the bottom will have a period corresponding to a spherical bubble of twice the energy. The results for explosions over a hemispherical cavity indicate that, if the explosion is near a partial enclosure, the first bubble period could be increased by as much as a factor of 2, depending on the proximity and size of the enclosure. A factor of 2 increase in the period would result in an overprediction of the yield by a factor of 8. Finally, we remark that, given this new ability to make accurate predictions of bubble pulse, depths, and times, one can numerically model the multiple impulsive point sources 共shock and bubble pulses兲 to simulate a pressure time series that defines more realistically the acoustic radiation from an underwater explosion. This capability could be useful for refined geoacoustic measurements using calibrated explosives, since each impulse occurs at a different depth and time. It should also be possible, by inverting bubble pulse measurements to estimate yield and depth of an explosion 共Gumerov and Chahine, 2000兲, to calibrate dynamite fishing and other “explosions of opportunity.” ACKNOWLEDGMENTS

This work was sponsored by the DCI Postdoctoral Fellowship Program and was conducted at DYNAFLOW, INC., 具www.dynaflow-inc.com典. The authors wish to thank many colleagues at DYNAFLOW, who have contributed to one aspect or another of this study, and most particularly Dr. JinKeun Choi, Dr. Chao-Tsung Hsiao, and Mr. Gary Frederick. J. Acoust. Soc. Am., Vol. 118, No. 5, November 2005

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