A simple model of ultrasound propagation in a cavitating liquid. Part I: Theory, nonlinear attenuation and traveling wave generation

TL;DR

This paper develops a nonlinear model of ultrasound propagation in cavitating liquids, showing significant attenuation due to inertial bubbles and demonstrating the emergence of traveling waves in such media.

Contribution

It introduces a simplified nonlinear Helmholtz equation accounting for bubble-induced attenuation, extending classical linear theory to high-amplitude regimes with inertial cavitation.

Findings

Attenuation is over 3 orders of magnitude larger than linear predictions at high driving amplitudes.

Viscous dissipation dominates energy loss in small inertial bubbles.

Traveling waves can form in cavitating liquids due to nonlinear attenuation effects.

Abstract

The bubbles involved in sonochemistry and other applications of cavitation oscillate inertially. A correct estimation of the wave attenuation in such bubbly media requires a realistic estimation of the power dissipated by the oscillation of each bubble, by thermal diffusion in the gas and viscous friction in the liquid. Both quantities and calculated numerically for a single inertial bubble driven at 20 kHz, and are found to be several orders of magnitude larger than the linear prediction. Viscous dissipation is found to be the predominant cause of energy loss for bubbles small enough. Then, the classical nonlinear Caflish equations describing the propagation of acoustic waves in a bubbly liquid are recast and simplified conveniently. The main harmonic part of the sound field is found to fulfill a nonlinear Helmholtz equation, where the imaginary part of the squared wave number is directly correlated with the energy lost by a single bubble. For low acoustic driving, linear theory is recovered, but for larger drivings, namely above the Blake threshold, the attenuation coefficient is found to be more than 3 orders of magnitude larger then the linear prediction. A huge attenuation of the wave is thus expected in regions where inertial bubbles are present, which is confirmed by numerical simulations of the nonlinear Helmholtz equation in a 1D standing wave configuration. The expected strong attenuation is not only observed but furthermore, the examination of the phase between the pressure field and its gradient clearly demonstrates that a traveling wave appears in the medium.