Numerical models for the study of the nonlinear frequency mixing in two and three-dimensional resonant cavities filled with a bubbly liquid

Abstract

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.

Modeling and Simulation of Parametric Nonlinear Focused Ultrasound in Three-Dimensional Bubbly Liquids with Axial Symmetry by a Finite-Element Model

“…Although the numerical model in the above references did not allow to get the same geometrical shape at the source, this comparison led to a quite good concordance. With the single-frequency source, the maximal pressure amplitude at 2f with the model from [30] was 40% of the fundamental frequency amplitude f, whereas it is 46% here. In the case of the dual-frequency source, the maximal pressure amplitude at f d with the model from [31] was 5% of the pressure amplitude at the primary frequency f 1 , whereas it is 6% with the current model presented here.…”

“…It must be noted here that to verify and validate the numerical model presented in this paper, it was used with the parameters given in Sections 3.1 and 3.2 and compared with another model based on a diferent kind of numerical methods, published earlier in the literature [30,31]. Although the numerical model in the above references did not allow to get the same geometrical shape at the source, this comparison led to a quite good concordance.…”

“…Tis model is thus able to track in time the behavior of ultrasound in the bubbly medium in nonlinear confgurations. It has a strong advantage over other models [14,30,31], since the FEM allows us to modify the geometrical conditions of the physical problem easily. Tis important beneft will be the topic of future works.…”

A Numerical Study on the Penetrability and Directivity of the Difference-Frequency Component Beam in Bubbly Liquids Obtained via Parametric Acoustic Array

“…In this section we consider our problem in a two-dimensional domain of dimensions L x × L y . The following partial differential equations system is considered [2,4,5,28], in which the acoustic pressure is p(x, y, t) and the bubble volume variation is v(x, y, t) = V(x, y, t) − v 0g , where (x, y) are the two-dimensional space coordinates and V(x, y, t) is the instantaneous bubble volume located at position (x, y). Subscripts t, x, and y stand for partial derivatives with respect to t, x, and y:…”

Nonlinear Maximization of the Sum-Frequency Component from Two Ultrasonic Signals in a Bubbly Liquid

“…The control of the numerous parameters which come into play in the set-up of experimental systems to study the behavior of finite-amplitude ultrasound propagating in liquids with a stable population of gas bubbles is a very difficult task. The development of numerical models able to approximate the nonlinear response of such systems is thus necessary to analyze the generation of harmonics, difference and sum frequencies during the propagation of finite-amplitude acoustic waves in bubbly liquids [19][20][21][22][23].…”