Kerry W. Commander

Linear pressure waves in bubbly liquids: Comparison between theory and experiments

Recent work has rendered possible the formulation of a rigorous model for the propagation of pressure waves in bubbly liquids. The derivation of this model is reviewed heuristically, and the predictions for the small‐amplitude case are compared with the data sets of several investigators. The data concern the phase speed, attenuation, and transmission coefficient through a layer of bubbly liquid. It is found that the model works very well up to volume fractions of 1%–2% provided that bubble resonances play a negligible role. Such is the case in a mixture of many bubble sizes or, when only one or a few sizes are present, away from the resonant frequency regions for these sizes. In the presence of resonance effects, the accuracy of the model is severely impaired. Possible reasons for the failure of the model in this case are discussed.

Nonlinear bubble dynamics

The standard approach to the analysis of the pulsations of a driven gas bubble is to assume that the pressure within the bubble follows a polytropic relation of the form p=p0(R0/R)3κ, where p is the pressure within the bubble, R is the radius, κ is the polytropic exponent, and the subscript zero indicates equilibrium values. For nonlinear oscillations of the gas bubble, however, this approximation has several limitations and needs to be reconsidered. A new formulation of the dynamics of bubble oscillations is presented in which the internal pressure is obtained numerically and the polytropic approximation is no longer required. Several comparisons are given of the two formulations, which describe in some detail the limitations of the polytropic approximation.

Finite‐element solution of the inverse problem in bubble swarm acoustics

The bubble population near the ocean surface is of considerable interest. This population affects surface scattering strength, propagation near the surface, and the exchange of gases between the atmosphere and the sea. Both optical and acoustical means have been used to measure the bubble population with varying degrees of success. The acoustic method requires measurements at multiple frequencies and their subsequent conversion to bubble densities through either the resonance theory approximation or numerical solution of the resulting integral equation. In this paper, a numerical solution to the integral equation is obtained using the method of weighted residuals with linear B splines as basis functions. A regularization technique is employed to stabilize the solution. A number of plausible bubble distribution functions are generated along with their acoustic properties to test the robustness of the technique. This approach is shown to yield very accurate bubble distributions from high‐quality attenuation...

Off-resonance contributions to acoustical bubble spectra

Comparisons of experimental optical and acoustical bubble size spectra disagree in size distribution at the low end of the spectrum. Previous methods of obtaining acoustic bubble size information relied strictly on resonant acoustical scattering and absorption theory. For some plausible distributions, these traditional methods greatly overpredict the number of bubbles present in a volume of fluid for bubble radii of 50 μ or less. Two cases are investigated that show the magnitude of departure from a priori bubble size distributions and are used to benchmark traditional acoustical scattering and absorption‐based methods for obtaining spectra. A third possible bubble distribution is presented that is consistent with resonant approximation theory; however, the acoustic properties of this distribution are inconsistent with measurements from naturally occurring bubble populations. It is argued that off‐resonance contributions to acoustical bubble spectra determinations need to be included.

Solving the inverse scattering problem to obtain acoustic bubble spectra

Determination of acoustic bubble spectra near the surface of the ocean, using resonance scattering approximations, leads to considerable overestimation of the numbers of bubbles with radii less than 50 microns [J. Acoust. Soc. Am. Suppl. 1 82, S109 (1988)]. Several numerical methods for solving the exact inverse scattering problem were examined and results for each are presented. In these studies, attenuation and backscattering strengths, as a function of frequency, were used as input in each of the techniques, and the computer bubble distribution was compared with known (a priori) distributions. Sensitivity of the numerical techniques to dependence on the attenuation and backscattering functions was explored for each method as well. Improvements over resonance theory approximations are shown through a systematic study of each technique using known bubble distributions. [Work supported by ONT.]

High‐frequency surface scattering phenomena in high sea states.

The determination of acoustic scattering from a rough sea surface has received considerable attention in recent years. Knowledge of whitecap characteristics [J. Wu, IEEE J. Oceanic Eng. 17, 150–158 (1992)] has led to a better understanding of surface scattering phenomena [D. F. McCammon and S. T. McDaniel, IEEE J. Oceanic Eng. 15, 95–100 (1990)]. For this study a substantial amount of backscattering data was acquired with autonomous underwater vehicles running between 50 and 100 ft of the surface. These vehicles used a short pulse length sonar operating at 215 kHz. The use of a high‐frequency sonar allows deep penetration of densely populated bubble plumes. Because of the high repetition rate, good spatial variation of backscattering strength from breaking waves has been obtained. Previous measurements by other scientists of bubble distribution functions in breaking waves allows an approximate conversion of scattering strength at the sonar frequency to most frequencies of interest. The database here consi...

A finite element solution of the inverse scattering problem for determining acoustic bubble spectra

The determination of acoustic bubble spectra in the ocean from multi‐frequency attenuation measurements was improved by numerical solution of the inverse scattering problem [K. W. Commander and E. Mortiz, in Proceedings of IEEE Oceans 89, 1181–1185 (1989)]. Although this numerical solution offers an improvement over the resonance approximation (at small radii), it too suffers a loss of accuracy at the small radii end of the bubble spectrum. The numerical solution of this problem has been improved by replacing the Fourier series approximation to the unknown distribution by a piecewise linear polynomial constructed from a series of basis functions. The basis functions used were linear B‐splines or “hat” functions. The resulting coefficient matrices are much less ill‐conditioned than those from the previous method because of the local support of the basis functions. This improvement leads to a more accurate solution of the inverse scattering problem near the end points. Improvements in accuracy due to the f...

Off‐resonance contributions to acoustic backscattering from bubble swarms

Recent discrepancies between measurements of bubble populations near the surface in the ocean by optical and acoustical methods have been discussed in detail by MacIntyre [F, MacIntyre, “On reconciling optical and acoustical bubble spectra in the mixed layer,” in Oceanic Whitecaps, edited by E. C. Monohan and G. MacNiocaill (Reidel, New York, 1986), pp. 75–94.] Another possible explanation for the very large number of small bubbles found by acoustic methods is the potential for overestimates of bubble numbers through the use of resonance scattering theory in the bubble population calculations. In this work it is shown that for some bubble distribution of interest, the acoustic backscattering for a particular frequency is actually dominated by scattering contributions from off‐resonance bubbles. This effect is more prevalent for high frequencies, causing overestimation of the number of small bubbles, which is precisely where the two methods disagree. The more complete acoustic calculations may not complete...

Bubble swarm acoustics

A study of the sensitivity of sound attenuation in bubbly liquids to variations of bubble distribution functions and void fractions was conducted using several theoretical models [A. L. Anderson and L. D. Hampton, J. Acoust. Soc. Am. 67, 1865–1889 (1980); A. Prosperetti, Ultrasonics 22, 115–124 (1984); R. Caflisch, M. J. Miksis, G. C. Papanicolaou, and L. Ting, J. Fluid Mech. 153, 259–273 (1985)]. The distributions studied include uniform, Gaussian, naturally occurring, and others. The frequencies covered ranged from dc to 500 kHz. Differences in predicted attenuations between the theories are discussed as well as limitations of theories vis‐a‐vis empirical data. [Work supported by ONT.]

An exact formulation for the internal pressure of a cavitation bubble

Recent experiments have shown that when the acoustic driving frequency is near one of the bubbles harmonic resonances, the theoretical values predicted by the Rayleigh‐Plesset equation are inconsistent with observed values. This inconsistency lead Prosperetti to consider the internal pressure term in the Rayleigh‐Plesset equation in a more general manner. In the past, the internal pressure of a bubble was assumed to be accurately predicted by a polytropic approximation. The internal pressure is computed from the conservation equations, resulting in a more accurate formulation. The new method also provides additional information about the internal thermodynamics of a bubble, which is explored in some detail. The two models are examined using some of the recent techniques in dynamical systems. “Feigenbaum trees” are shown for the two models of interest. This method for analyzing an equation is shown to be very sensitive to the internal pressure term, and thus it is an appropriate method for comparing diffe...