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Boundary element simulations of acoustic self-organization in bubbly liquids

Name
Galia
Surname
Gilmanova
Scientific organization
Center for Micro- and Nanoscale Dynamics of Dispersed Systems, Bashkir State University, Ufa, Russia
Academic degree
Master's degree
Position
Researcher
Scientific discipline
Mathematics & Mechanics
Topic
Boundary element simulations of acoustic self-organization in bubbly liquids
Abstract
The BEM code was developed and example simulations of formation and propagation of the waves of the acoustically induced transparency in three dimensions were carried out. The results were compared with results obtained earlier other methods and a satisfactory agreement was found.
Keywords
3D Helmholtz equation, boundary element method, bubble dynamics, acoustics of bubbly liquids
Summary

Boundary element simulations of acoustic self-organization in bubbly liquids

Galia I. Gilmanova1, Nail A. Gumerov1,2, Iskander S. Akhatov3

 

1Center for Micro- and Nanoscale Dynamics of Dispersed Systems,

Bashkir State University, 32 Zaki Validi Street, Ufa Russia

2Institute for Advanced Computer Studies, University of Maryland, USA

3Skolkovo Institute of Science and Technology, 3 Nobel Street, Moscow, Russia

 

Self-organization of bubbles in acoustic fields, or self-action of the acoustic waves in bubbly liquids is a strongly nonlinear phenomenon due to two-way interaction of the bubbles and the acoustic field. A mathematical model of the phenomenon and preliminary computations predict the existence of the waves of self-induced acoustic transparency, which is also confirmed experimentally [1]. Three dimensional particle-in-cell (PIC) simulations tracking all bubbles in the system were performed by the authors earlier [2], where also a good agreement of the theory and experiment was found [1-2].

The PIC simulations however have some deficiency related to the averaging procedure, when the number of particles in the cell is small enough. Such a procedure is required to match the continuum and discrete models of the bubbly liquid each time step. Moreover, in the current version of the PIC code [2] a finite-difference (FD) solver for the Helmholtz equation is used which performance should be improved. In the present study these two problems were addressed by 1) solving the multiple scattering problem for N bubbles (in the limit of a point monopole acoustic source), so there is no need in the continuum model and spatial averaging, and 2) using the boundary element method (BEM) as an alternative to the FD solver. The BEM is modified to include not only boundaries of the domain into the formulation, but also N sources arbitrarily distributed in the computational domain, so both problems are solved at a given time step by a single method. Solution of the problem enables computation of acoustic radiation (Bjerknes) forces on each bubble and, therefore, simulation of the dynamics of the N-body system. Another advantage of the BEM is it capability to solve the problem in domains of complex shape, which has a practical value for design of experimental setups and more realistic simulations.

 In the present study a pilot version of the BEM code was developed and example simulations of formation and propagation of the waves of the acoustically induced transparency in three dimensions were carried out. The results were compared with results of [2] and a satisfactory agreement was found. Implementation of the fast multipole method (FMM) for acceleration of the BEM (see [3]) is considered as a future work, which should enable the present method for computations of large systems evolving millions of bubbles.

This study is supported in part by Grant of Ministry of Education and Science of the Russian Federation (11.G34.31.0040), Skoltech Partnership Program

[1] N.A. Gumerov, I.S. Akhatov, C.-D. Ohl, S.P. Sametov, M.V. Khazimulin, S.R. Gonzalez-Avila, Robust acoustic manipulation of bubbly liquids. Appl. Phys. Lett., 108, 134102, 2016.

[2] N.A. Gumerov, I.S. Akhatov, C.-D. Ohl, S.P. Sametov, M.V. Khazimulin, G.I. Gilmanova, Waves of acoustically induced transparency in bubbly liquids: theoretical prediction and experimental validation. Proceedings of ASME 2013 International Mechanical Engineering Congress & Exposition, Paper No. 63200, San Diego, CA, USA, 2013.

[3] N.A. Gumerov and R. Duraiswami, A broadband fast multipole accelerated boundary element method for the 3D Helmholtz equation. J. Acoust. Soc. Am. 125(1), 191-205, 2009.