Measuring and modeling the bubble population produced by an underwater explosion
Diagram showing the relationship between the shock and pressure waves, and the size and position of the gas-globe during the first moments after the underwater explosion. (a) Initial shock wave from the explosion, and subsequent bubble pulses due to the oscillation of the gas-globe. (b) Pulsations and migration of the gas-globe. (After Ref. 8.)
(Color online) Image of the surface of the quarry immediately prior to the underwater explosion (upper). (a) indicates a floating dock extending approximately 30 m into the quarry; (b) marks a network of 10 cm diameter polyvinyl chloride piping from which pressure sensor transducers are suspended; (c) marks the location of several white buoys suspending a metal cylinder near the charge depth. The lower image was taken ≈2.5 s after detonation.
(Color online) Plan view (upper) and side view (lower) of the acoustic measurement geometry used in the May 2008 quarry test. Forward-loss hydrophone configuration is shown in the solid box. Numbers enclosed in boxes denote hydrophone channel number.
(Color online) Measured attenuation through the bubble cloud for each of the forward loss hydrophones during Event 1 (a), and Event 2 (b). Each plot shows attenuation level in dB vs frequency (in kHz) and time (in s), using the same color scale. Panels are arranged according to the physical position of hydrophones in the water (shown in the diagram in upper right).
(Color online) Comparison of the bubble populations estimated by using the three inversion methods discussed in the text applied to the attenuation measurement at phone 7, approximately 5.25 s after detonation for Event 2.
(Color online) Comparison of attenuation computed using Eq. (1) with the three bubble population estimates discussed in the text (and shown in Fig. 5). Attenuation measured in the quarry Aq (f) is also shown (for Event 2, hydrophone 7, 5.25 s after detonation).
(Color online) Comparison of the output of the three inversion methods with a power-law trendline (ψ pl) and two published bubble population models.
(Color online) The coefficient of the power-law model, m(t,p i ) in Eq. (9) for several forward-loss hydrophones (7-pt low pass filter applied).
(Color online) Number of 100 μm bubbles (ψ(a = 100 μm, t, p i )) vs time estimated for the forward loss vertical array of hydrophones (smoothed using a 3-pt moving average). Each hydrophone plotted separately. Exponential fit shown as a solid black line.
(Color online) Number of bubbles of three sizes (averaged over 6 pings following detonation, excluding first 5 s) vs depth at the charge location. Exponential fits are shown by black dashed lines.
(Color online) Comparison of attenuation measured through the bubble cloud during Event 2 (left) with that predicted using the UNDEX bubble population model (right). Each panel shows attenuation in dB vs frequency (in kHz) and time (in s). All panels use the same color scale. Panels are arranged according to hydrophone position in the water (see Fig. 3).
(Color online) Comparison of the rms measured attenuation with error bars showing ± 1 standard deviation shortly after detonation with attenuation predicted using the UNDEX bubble population model (no error bars) for Event 1 (a), and Event 2 (b). The measured values were averaged over a period of time prior to the onset of exponential-like decay.
Frequency dependent source level of F27 projector estimated from received level the 15.2 m deep hydrophone during Event 2.
The rms error and standard deviation for the three inversion methods applied to the measured attenuation, Aq (f), as discussed in Figs. 5 and 6.
e-folding depths, Dc, for bubbles with radii of 100, 200, and 400 μm. These e-folding depths correspond to the exponential fits (dashed lines) in Fig. 10. R is the correlation coefficient of the fit.
Number of bubbles of various sizes estimated to be in a 1 m2 cylindrical cross-section with the projector at one end and hydrophone 9, 10, or 16 at the other, ψ(a, p i ), at a depth of 13.4 m at the explosion location.
Standard deviation [σg (a = 100 μm, t = 5.25s, z = 13.4 m)] and magnitude scalar [ψ g(a = 100 μm, t = 5.25 s, z = 13.4 m)] for the Gaussian function used for horizontal spatial dependence in the bubble population model.
Percent error between the a = 100 μm bubble population given in Table IV and the values calculated using Eq. (13) with a = 100 μm, t = 5.25 s, z = 13.4 m, and the values for ψg and σg given in Table V.
Values for constants appearing in the UNDEX bubble population model [Eq. (15)].
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