A numerical model for the study of the difference frequency generated from nonlinear mixing of standing ultrasonic waves in bubbly liquids.

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.