Simulations of Low-Frequency Bubble Pulsations Generated by Impacting Cylindrical Water Jets

W.G. Szymczak and S.L. Means
Acoustics Division

Introduction: The impact of a water jet onto a water surface can entrain air bubbles whose pulsations provide acoustic sources. Such impacts can occur during the breaking of a wave or, on a smaller scale, when a raindrop strikes a puddle of water. A better understanding of this phenomenon can lead to improved characterizations of ambient noise and acoustic detection algorithms for Navy sonar systems. NRL has developed a method for simulating the impact of water jets by using a generalized hydrodynamics model. These simulations provide details of not only the initial formation of the air-entrained bubble during the cavity collapse, but also the bubble's subsequent pulsations as it rises to the surface. Validations are provided by comparisons to experiments of impacting cylindrical water jets.

Generalized Hydrodynamics Model: The simulations rely on a computational method based on a constrained system of conservation laws. The primary constraint is on the density, namely ρ ≤ ρ₀, where ρ₀ is the intrinsic density of the incompressible liquid. The density is used as a fluid of volume variable that delineates the liquid region (where
ρ = ρ₀) from the nonliquid regions (where 0 ≤ ρ < ρ₀). The distinguishing feature of the model is the imposition of this constraint using a density and momentum redistribution algorithm. This algorithm was derived from an approximation to the solution of a Stefan-Boltzmann equation.¹ This algorithm not only enforces the density constraint but also conserves mass and momentum, is spatially invariant, and ensures that the energy is nonincreasing. This feature allows for a physically rational treatment of liquid collisions while providing stability.

Another important feature of the model is that it allows for regions of "spray" where 0 ≤ ρ < ρ₀. Such regions will form when free surfaces become unstable, in particular when Rayleigh-Taylor instabilities occur. These instabilities are, in general, unavoidable in most violent free surface flows, such as the breaking of a wave on the beach or the oscillations of an underwater bubble. Therefore, these spray regions are not a priori suppressed through the use of surface tracking or level set approaches.

The model treats bubbles, defined as connected subsets of the region where 0 ≤ ρ < ρ₀, as uniform pressure regions. For air-entrained bubbles, the pressure is initialized as the ambient air pressure. During the flow field dynamics, the pressure is assumed to behave adiabatically, PV^γ = C, where V is the volume of the bubble that changes due to the fluid motion, is a constant determined at the time the bubble formed, γ = 1.3 is the adiabatic exponent, and P represents the time-dependent bubble pressure.

Computational Results: The generalized hydrodynamics model has been implemented into computer codes denoted BUB2D, for the solution of two-dimensional and axially symmetric problems, and BUB3D for three-dimensional problems. To study details of jet impacts and air entrainment of bubbles, consider an idealized experiment of a falling cylinder of water onto a still water surface. Figure 7 shows the setup of experiments conducted by Kolaini et al.² The case when the water cylinder has radius R = 0.054 m, length L = 0.45 m, and height above the surface H = 0.15 m is considered here. Figure 8 shows images of the experiment from a video of this experiment. The formation of a cavity after the cylinder impacts the air-water surface and the pinching off of a bubble are clearly displayed in these images. For comparison, Fig. 9 shows the results of an axially symmetric computation using BUB2D. In this figure, an annular water jet ejected upward can be seen surrounding the cylinder at time t = 0.2 s, shortly after the impact. The cavity enlarges as the cylinder falls into the water surface at times t = 0.3 s and t = 0.4 s. The cavity has just closed at time t = 0.47 s. Shortly after the cavity closes, two water jets form: one moving downward, piercing the bottom of the bubble and the second, moving upward above the surface as seen at time t = 0.50 s. These water jets have also been observed in high speed photographs of the experiment.² After the formation of the bubble, it begins to pulsate according to the adiabatic pressure assumption. At time t = 0.53 s, the bubble has pulsated twice, each time exhibiting Rayleigh-Taylor instability during the time it is near its minimum volume (maximum pressure). These instabilities are exhibited in the profile of the bubble and the amount of spray seen inside.

Figure 10 shows a comparison of the computed and measured pressure time series at a location 0.4 m from the radial axis and 0.2 m below the original airwater interface. The label "(1)" in the graph of the measured values indicates the time of the initial impact of the cylinder on the quiescent surface and corresponds to the first "spike" in the computed series at time t = 0.175 s. The measured fundamental frequency of 43 Hz is accurately reproduced by the computed frequency of 43.88 Hz. The decay of the amplitude of the pressure pulsations was 10.12% per period for the first 10 oscillations. This value is slightly greater than shown in the experimental data, where the decay is approximately 8% per oscillation. Computationally, this decay is expected from both energy losses due to liquid collisions, and numerical dissipation.

Conclusions: A computational code based on a generalized hydrodynamics model has been shown to be capable of not only predicting bubble formation as the result of liquid impacts, but also the acoustic sources that the bubbles produce as they undergo low-frequency pulsations. This represents an important first step for predicting and characterizing air-entrained bubbles caused by plunging breaking waves, which in turn can be used to enhance acoustic detection and classification algorithms.

Acknowledgments: The authors thank Ali Kolaini for introducing us to this interesting and important benchmark problem and for providing the video tape from which the images of Fig. 8 were extracted.

[Sponsored by ONR]