Underwater Gas Research Articles

Finite- and Small-Amplitude Underwater Gas Bubble Oscillations Compared

Underwater gas bubble expansion as a function of the ambient-pressure to bubble-pressure ratio p′, in the interval 0⩽p′⩽1, is investigated. At the endpoints, a simple analytic solution to the nonlinear equations of motion in closed form may be found; but, for intermediate values of p′, an approximation is required. In the noncompressive case, representations suitable for large bubble radii, near p′=0, and small-amplitude motion, near p′=1, are derived. A result of this analysis is that the Willis bubble pulse formula, correct in the limit p′ → 0, in agreement with experiment, appears to be valid over much of the interval, but it starts to break down as p′ → 1. The power law governing finite- and small-amplitude periods is different; the actual period has no simple power law dependence, but may be deduced for all p′ by fitting a curve to the asymptotically correct values at p′=0 and p′=1, respectively. Owing to radiation and other loss mechanisms, bubble starting out as finite-amplitude pulsation, with a cusplike shape at the minimum, is gradually transformed into a small-amplitude damped sine wave; concurrently T1 → T̄. (Analysis in this paper refers to case γ = 43.)

Wave field of a directed explosion

We consider an underwater gas explosion in a cylindrical tank which is open at the bottom. The pressure field in the water is a toroidal compression wave with gradually increasing pressure behind the wave front. In the linear approximation the pressure shock occurs only under the cylinder with the gas. Filling of the lower portion of the cylinder with water alters the compression wave profile.

Finite and Small Amplitude Underwater Gas Bubble Oscillations Compared

Bubble oscillation is well described by the nonlinear Keller-Kolodner equation. Although compressibility effects, which strongly influence the motion, are considered, analysis starts with incompressible approximation. The key parameter is p1/p0, the ambient/bubble pressure ratio. This ranges from zero to unity; near the lower limit finite amplitude pulsations occur; near the upper, simple harmonic motion ensues. The equation determining the maximum and minimum bubble radii, with p1/p0 as parameter, is graphed; these endpoints coalesce at a level where p1/p0 = 1. Further, taking compressibility into account, at each depth the end-points also converge to an equilibrium radius. T̄ has a simple power law dependence on p1/p0; this is generally not true for the finite amplitude period T1. The curves for T̄ and T1, as calculated from the Willis formula, cross; hence the actual T1 vs depth may be inferred. The shape of the bubble radius vs time curve changes from pulsation to simple oscillation with depth; this is asymptotically correct at any level, since T1 → T̄.

Expansion of an Underwater Cylinderical Gas Bubble

The theory of the expansion of an underwater spherical gas bubble is rather well known; in this paper the theory for cylindrical gas bubbles is developed. It is customary to treat the motion of the fluid in the incompressible approximation, but this procedure leads to difficulties in the cylindrical case owing to the potential's becoming infinite as r approaches infinity. Therefore, the fluid is treated as compressible, and equations for the bubble radius and the radiated pressure as functions of time are derived. Then these equations are examined in the limit of large speed of sound. Approximate equations obtained by employing simplifying assumptions are analyzed, and the properties of the motion discussed.

A theoretical treatment of combustion in a spherical underwater gas bubble

The combustion equations and the equation of motion of the surface of the bubble are formulated on the assumptions that the gases are ignited at their centre and that the flame front travels outward in the form of a sphere. The equations are integrated to yield gas pressure and bubble radius as functions of time. Provided that radial displacements are small, the flame speed can be assumed constant. The formulation is then simple and the equations can be integrated in closed form to give general non-dimensional solutions. These show that pressure and radius increase at an increasing rate until combustion is complete. Radial oscillations ensue whose amplitude in proportion to the initial radius decreases with increase in the ratio of combustion time to oscillation period. For large radial displacements, the formulation is less simple as finite expansion of the burnt gases at the expense of the unburnt gases is involved, which enhances the flame speed. The effect has already been considered for com­bustion in a closed spherical vessel (Flamm & Mache 1917), and an important approximate relation between the fraction of gas burned and the pressure rise has been derived: n = (p — P 0 )/(P — P 0 ), where n is the fraction of gas burned, p is the corresponding pressure an d P an d P 0 are respectively the final and initial pressures. The analysis of Flamm & Mache, which is in any case not entirely satisfactory, cannot be used if the total gas volume varies, as in the present instance. An energy method has, however, been found to give the required relation (and incidentally to afford a simpler, more rigorous proof of the closed vessel relation and am easure of its accuracy). The equations require numerical integration and some typical results are given in graphical and tabular form . These show two features which are absent when displacements are small: as in closed vessels the combustion time is shorter than that for constant flame speed (typically, by a factor of 3); and, during combustion, pressure fluctuations develop which arise from the interaction of combustion and expansion each of which leads to an acceleration of the other.