Numerical investigation of the translational motion of bubbles: The comparison of capabilities of the time-resolved and the time-averaged methods.

In the present study, the accuracies of two different numerical approaches used to model the translational motion of acoustic cavitational bubble in a standing acoustic field are compared. The less accurate but less computational demanding approach is to decouple the equation of translational motion from the radial oscillation, and solve it by calculating the time-averaged forces exerted on the bubble for one acoustic cycle. The second approach is to solve the coupled ordinary differential equations directly, which provides more accurate results with higher computational effort. The investigations are carried out in the parameter space of the driving frequency, pressure amplitude and equilibrium radius. Results showed that both models are capable to reveal stable equilibrium positions; however, in the case of oscillatory solutions, the difference in terms of translational frequency may be more than three fold, and the amplitude of translational motion obtained by the time-averaged method is roughly 1.5 times higher compared to the time-resolved solution at particular sets of parameters. This observation implies that where the transient behaviour is important, the time-resolved approach is the proper choice for reliable results.

Memory-friendly fixed-point iteration method for nonlinear surface mode oscillations of acoustically driven bubbles: from the perspective of high-performance GPU programming.

A fixed-point iteration technique is presented to handle the implicit nature of the governing equations of nonlinear surface mode oscillations of acoustically excited microbubbles. The model is adopted from the theoretical work of Shaw [1], where the dynamics of the mean bubble radius and the surface modes are bi-directionally coupled via nonlinear terms. The model comprises a set of second-order ordinary differential equations. It extends the classic Keller-Miksis equation and the linearized dynamical equations for each surface mode. Only the implicit parts (containing the second derivatives) are reevaluated during the iteration process. The performance of the technique is tested at various parameter combinations. The majority of the test cases needs only a single reevaluation to achieve 10-9 error. Although the arithmetic operation count is higher than the Gauss elimination, due to its memory-friendly matrix-free nature, it is a viable alternative for high-performance GPU computations of massive parameter studies.

Dataset of exponential growth rate values corresponding non-spherical bubble oscillations under dual-frequency acoustic irradiation.

The dataset described in this paper is related to the paper [1], in which the exponential growth rate values of a spherically oscillating gas bubble modelled by the Keller-Miksis equation are recorded. As the bubble is excited by dual-frequency, the employed parameters are the pressure amplitudes and frequencies corresponding to the first and second harmonic components, respectively, the phase shift between harmonic components, and the equilibrium bubble radius. At each parameter combinations the exponential growth rate values corresponding to mode 2 up to mode 6, and the maximum bubble radius are stored. The huge amount of numerical data are generated that are stored in a public repository [2]. The present paper describes the generated data, the applied numerical model and the implementation details of the program code used to generate the data on graphics processing units (GPUs).

GPU accelerated numerical investigation of the spherical stability of an acoustic cavitation bubble excited by dual-frequency.

The spherical stability of an acoustic cavitation bubble under dual-frequency excitation is investigated numerically. The radial dynamics is described by the Keller-Miksis equation, which is a second-order ordinary differential equation. The surface dynamics is modelled by a set of linear ordinary differential equation according to Hao and Prosperetti (1999), which takes into account the effect of vorticity by boundary layer approximation. Due to the large amount of investigated parameter combinations, the numerical computations were carried out on graphics processing units. The results showed that for bubble size between RE=2µm and 4µm, the combination of a low and a high frequency, and the combination of two close but not equal frequencies are important to prevent the bubble losing its shape stability, while reaching the chemical threshold (Rmax/RE=3) (Kalmár et al., 2020). The phase shift between harmonic components of dual-frequency excitation has no effect on the shape stability.

Relationship between the radial dynamics and the chemical production of a harmonically driven spherical bubble.

The sonochemical activity and the radial dynamics of a harmonically excited spherical bubble are investigated numerically. A detailed model is employed capable to calculate the chemical production inside the bubble placed in water that is saturated with oxygen. Parameter studies are performed with the control parameters of the pressure amplitude, the forcing frequency and the bubble size. Three different definitions of collapse strengths (extracted from the radius vs.time curves) are examined and compared with the chemical output of various species. A mathematical formula is established to estimate the chemical output as a function of the collapse strength; thus, the chemical activity can be predicted without taking into account the chemical kinetics into the bubble model. The calculations are carried out by an in-house code exploiting the high processing power of professional graphics cards (GPUs). The results shown that chemical activity can be approximated qualitatively from the values of relative expansion. This could be helpful in order to optimise chemical output of sonochemical reactors either from measurement data or simulations as well.

GPU accelerated study of a dual-frequency driven single bubble in a 6-dimensional parameter space: The active cavitation threshold.

The active cavitation threshold of a dual-frequency driven single spherical gas bubble is studied numerically. This threshold is defined as the minimum intensity required to generate a given relative expansion (Rmax-RE)/RE, where RE is the equilibrium size of the bubble and Rmax is the maximum bubble radius during its oscillation. The model employed is the Keller-Miksis equation that is a second order ordinary differential equation. The parameter space investigated is composed by the pressure amplitudes, excitation frequencies, phase shift between the two harmonic components and by the equilibrium bubble radius (bubble size). Due to the large 6-dimensional parameter space, the number of the parameter combinations investigated is approximately two billion. Therefore, the high performance of graphics processing units is exploited; our in-house code is written in C++ and CUDA C software environments. The results show that for (Rmax-RE)/RE=2, the best choice of the frequency pairs depends on the bubble size. For small bubbles, below 3µm, the best option is to use just a single frequency of a low value in the giant response region. For medium sized bubbles, between 3µm and 6µm, the optimal choice is the mixture of low frequency (giant response) and main resonance frequency. For large bubbles, above 6µm, the main resonance dominates the active cavitation threshold. Increasing the prescribed relative expansion value to (Rmax-RE)/RE=3, the optimal choice is always single frequency driving with the lowest value (20kHz here). Thus, in this case, the giant response always dominates the active cavitation threshold. The phase shift between the harmonic components of the dual-frequency driving (different frequency values) has no effect on the threshold.

Study of non-spherical bubble oscillations under acoustic irradiation in viscous liquid.

The effect of dissipation on the shape stability of a harmonically excited bubble is investigated. The employed liquid is the highly viscous glycerine. The rate of the dissipation is controlled through the alteration of viscosity of the liquid by varying its temperature. The mean radius of the bubble during its radial oscillation is described by the Keller-Miksis equation. Two approaches are used to describe the surface oscillations. The first model solves the surface dynamics equations of each mode together with the transport equation of the vorticity in the liquid domain. The second model approximates the transport equation, which is a partial differential equation, with a boundary layer approximation reducing the required computational resources significantly. The comparison of the surface models shows qualitative agreement at low dissipation rate; however, at high viscosity the application of the full transport equation is mandatory. The results show that an increasing rate of dissipation can significantly extend the shape stable domains in the excitation frequency-pressure amplitude parameter plane. Nevertheless, the collapse strength is decreasing due to the highly damped oscillations. It has been found that an optimal range of dissipation rate in terms of temperature can be defined expressing a good compromise between the collapse strength and surface stability. The computations are carried out by an in-house GPU accelerated initial value problem solver.

The effect of high viscosity on the collapse-like chaotic and regular periodic oscillations of a harmonically excited gas bubble.

In the last decade many industrial applications have emerged based on the rapidly developing ultrasonic technology such as ultrasonic pasteurization, alteration of the viscosity of food systems, and mixing immiscible liquids. The fundamental physical basis of these applications is the prevailing extreme conditions (high temperature, pressure and even shock waves) during the collapse of acoustically excited bubbles. By applying the sophisticated numerical techniques of modern bifurcation theory, the present study intends to reveal the regions in the excitation pressure amplitude-ambient temperature parameter plane where collapse-like motion of an acoustically driven gas bubble in highly viscous glycerine exists. We report evidence that below a threshold temperature the bubble model, the Keller-Miksis equation, becomes an overdamped oscillator suppressing collapse-like behaviour. In addition, we have found periodic windows interspersed with chaotic regions indicating the presence of transient chaos, which is important from application point of view if predictability is required.