The aim of this paper is the study of the behavior of nonlinear standing ultrasonic waves in bubbly liquids and the generation of the difference frequency by nonlinear mixing of several signals. To this end we present a new numerical model based on the finite-volume method and the finite-difference method. This model solves the differential system formed by the wave equation and a Rayleigh-Plesset equation coupling the acoustic pressure field with the bubble vibrations. We consider a resonator filled with a bubbly liquid excited by an ultrasonic pressure source. The numerical experiments presented here are performed by modifying the source amplitude and frequency, the void fraction in the liquid, as well as the length of the resonator. The results allow us to observe the physical effects due to the presence of the bubbles in the liquid: nonlinearity, dispersion, attenuation. The nonlinear frequency mixing performed in the resonator is also evidenced. The amplitude of the generated difference frequency is studied as a function of the pressure amplitude and for several primary frequencies. Our results suggest that a better response is obtained for primary frequencies situated below the bubble resonance. They show a very high difference-frequency amplitude response for a cavity resonant at one wavelength of the difference frequency in the bubbly medium. This analyze could be useful for some practical applications.
This paper studies the nonlinear resonance of a cavity filled with a nonlinear biphasic medium made of a liquid and gas bubbles at a frequency generated by nonlinear frequency mixing. The analysis is performed through numerical simulations by mixing two source signals of frequencies well below the bubble resonance. The finite-volume and finite-difference based model developed in the time domain simulates the nonlinear interaction of ultrasound and bubble dynamics via the resolution of a differential system formed by the wave and Rayleigh–Plesset equations. Some numerical results, consistent with the literature, validate our procedure. Other results reveal the existence of a frequency shift of the cavity resonance at the difference-frequency component, which rises with pressure amplitude and evidences the global changes undergone by the bubbly medium under finite amplitudes. Finally, this work shows the enhancement of the amplitude of the difference-frequency component generated by parametric excitation using the nonlinear resonance shift, which is more pronounced when the second primary frequency is constant, the first one is varied to match the nonlinear resonance, and both have the same amplitude.
The objective of this work is to develop versatile numerical models to study the nonlinear distortion of ultrasounds and the generation of low-ultrasonic frequency signals by nonlinear frequency mixing in two and three-dimensional resonators filled with bubbly liquids. The interaction of the acoustic field and the bubble vibrations is modeled through a coupled differential system formed by the multi-dimensional wave equation and a Rayleigh-Plesset equation. The numerical models we develop are based on multi-dimensional finite-volume techniques and a time discretization carried out by finite differences. Numerical experiments are performed for complex modes in many different cavities considering different kinds of boundary conditions and taking advantage of the dispersive character of the bubbly fluid to match specific resonances of the cavities. Results show the distribution of fundamental and harmonics for single frequency excitation and difference-frequency component for two-frequency excitation that are promoted by the strong nonlinearity of the bubbly medium. The numerous simulations analyzed suggest that the new numerical models developed and proposed in this paper are useful to understand the behavior of ultrasounds in bubbly liquids for sonochemical processes and applications of nonlinear frequency mixing.
Several numerical models have been developed in different configurations to simulate the behaviour of finite-amplitude ultrasound when interacting with tiny gas bubbles in a liquid. Since this interaction is highly nonlinear, specific models must be developed to understand the propagation of the waves in this kind of dispersive media for which their nonlinear and attenuation coefficients, as well as the sound speed, are extremely dependent on the ratio of the driven frequency to the bubble resonance. The bubble volume variation is mathematically modelled in the time domain through a Rayleigh-Plesset equation with terms up to the second order, whereas the time-dependent acoustic field relies on the wave equation in one or several dimensions. Both differential equations are coupled and auxiliary conditions are imposed. The differential systems are solved by the developed numerical models. In this paper we study in a three-dimensional resonator with axial symmetry how new harmonics obtained by nonlinear distortion can be enhanced by taking the nonlinear resonance effect into account, and we show that the generation of new frequency components by nonlinear frequency mixing exists. We also analyse the stable cavitation phenomenon in a three-dimensional focused field with axial symmetry by considering a nonlinear dependence of bubble generation in the liquid and the existence of primary Bjerknes forces.
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