Modeling acoustic cavitation in homogeneous mixture framework

doi:10.1016/j.ijmultiphaseflow.2019.103142

International Journal of Multiphase Flow

Volume 122, January 2020, 103142

https://doi.org/10.1016/j.ijmultiphaseflow.2019.103142 Get rights and content

Highlights

•
A homogeneous mixture cavitation model is proposed and validated by acoustic cavitation.
•
The bounded Rayleigh–Plesset equation is modified due to the homogeneous treatment.
•
The compressibility of liquid should be considered but not gas.

Abstract

A homogeneous mixture cavitation model is proposed in this study. The bounded Rayleigh–Plesset equation is modified due to the homogeneous treatment and the compressibility of liquid is considered. The acoustic cavitation in stationary liquid is simulated to validate the improvements in proposed model: firstly, cavitation with up to 3 bubbles is simulated, the predicted bubble dynamics agree well with the predictions by the VOF method; then two experiments of shock-wave-induced cavitation are simulated and good agreements are obtained. More factors need to be considered to extend the application range of the proposed model in future.

Introduction

Cavitation usually needs to be avoided in fluid machineries, while it can be utilized in biomedical field such as ultrasound imaging, stone fragmentation, drug delivery, sonoporation and so on (Brennen 2015). Numerical tools have been widely used to analyze cavitation, but it is still far beyond precise to predict hydrodynamic cavitating flows. Interface capturing methods, such as VOF and level set, are efficient ways to solve gas-liquid flows with or without the consideration of phase change (Lauer et al., 2012; Ding et al., 2017; Singh and Premachandran, 2018). However, there are usually huge number of bubbles growing from tiny small nuclei, which makes interface capturing methods inapplicable for predicting hydrodynamic cavitating flows. Thus homogeneous assumption is introduced to reduce the computational cost, but it also brings many new issues, some of which are hard to handle.

Under homogeneous assumption, there is no interface between liquids and gases, which makes it hard to consider the influences of bubble deformation, coalescence, breakup, turbulence-cavitation interaction, slip velocity, phase change, compressibility and so on (Coutier-Delgosha et al., 2006; Huang et al., 2014; Zhao et al., 2016; Ye and Li, 2016; Ye et al., 2017; Donghua et al., 2018; Chen et al., 2019; Asnaghi et al., 2017). Despite most of these factors, let's consider the simplest situation: a shock wave passes the bubbles in stationary liquid. The phase change usually has little influence on bubble dynamics, since the saturated vapor pressure at normal temperature is extremely small comparing with the pressure amplitude of shock waves. The huge difference of compressibility between liquid and gas makes the sound speed vary in a wide range, which plays an important role in cavitation. The compressibility of liquid can be neglected, while that of gas dominates the propagation of pressure in cavitation regions. The compressibility of gas, sometimes together with that of liquid, are considered more and more frequently (Saito et al., 2007; Gnanaskandan and Mahesh, 2015, 2016; Chen et al., 2016; Messahel et al., 2018).

The Rayleigh–Plesset equation (Plesset and Prosperetti, 1977) solves the dynamics of a single bubble driven by a far-field pressure. But for numerical simulations, the local pressure is known but not the far-field pressure. Thus we think the Rayleigh–Plesset equation for a bubble in a bounded region (Wang, 2017) should be used instead, which predicts the bubble dynamics driven by a known pressure at the boundary of the region. Moreover, when the homogeneous assumption is introduced, the boundary pressure used in the bounded Rayleigh–Plesset equation should be deduced from the local pressure, which will be described in Section 2.

A homogeneous mixture cavitation model should first have the ability to well predict the shock-wave-induced cavitation in stationary liquid, which is the main purpose of this paper. In the proposed model, the 2D and 3D bounded Rayleigh–Plesset equations are modified due to the homogeneous treatment; the compressibility of liquid is considered while the gas density is set to be constant. In the validation part, cavitation with up to 3 bubbles triggered by pressure reduction are simulated and comparisons with the predictions by VOF method are made. Then the multibubble surface cavitation (Bremond et al., 2006) and collapse of cavitation clusters (Arora et al., 2007) caused by shock waves are simulated and comparisons are made with the experimental results.

Section snippets

Three-dimensional equation

The compressibility of liquid can be neglected when calculating the dynamics of a bubble (Wang, 2017). The Navier–Stokes equation for the incompressible flow outside an axisymmetric bubble in the r direction is (Brennen, 1995): $\frac{\partial u_{r}}{\partial t} + u_{r} \frac{\partial u_{r}}{\partial r} = - \frac{1}{ρ_{L}} \frac{\partial p}{\partial r} + \frac{μ_{L}}{ρ_{L}} (\frac{\partial^{2} u_{r}}{\partial r^{2}} + \frac{2}{r} \frac{\partial u_{r}}{\partial r} - \frac{2 u_{r}}{r^{2}})$ where ρ_L and μ_L are respectively the density and dynamic viscosity of liquid. According to mass conservation, the radial velocity outside the bubble is as: $u_{r} = \frac{R^{2} \dot{R}}{r^{2}}$ where R is bubble radius and the over dot denotes the

Validation

All flows in the validation part are considered to be laminar and the gravity is neglected. The gas density is set to 0.1 kg/m³. ${\dot{R}}_{\min}$ in Eq. (21) is set to −30 m/s so as to inhibit the bubble rebound and improve the numerical stability. If the bubble rebound plays an important role, such as the collapse of a single spherical bubble, much lower minimum collapse rate and smaller time-step size are needed.

Conclusions and prospects

A homogeneous mixture cavitation model has been presented. The bounded Rayleigh–Plesset equations for 2D and 3D bubbles are modified, so as to obtain the bubble dynamics of the homogeneous flow as close as possible to the real two-phase flow. According to this study, the compressibility of liquid should be considered but not gas. In the validation part, the cavitation with up to 3 bubbles in stationary liquid was simulated, the bubble volume and pressure field predicted by the proposed model

Declaration of Competing Interest

We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted.

Acknowledgments

Financial support from the National Natural Science Foundation of China (No. 51606169, 51776188), Zhejiang Provincial Natural Science Foundation of China (No. LQ18E050015) and Department of Education of Zhejiang Province (No. Y201636549, Y201941643).

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View full text

Modeling acoustic cavitation in homogeneous mixture framework

Highlights

Abstract

Introduction

Section snippets

Three-dimensional equation

Validation

Conclusions and prospects

Declaration of Competing Interest

Acknowledgments

Int. J. Multiph. Flow

Int. J. Multiph. Flow

Int. J. Heat Mass Transf.

Int. J. Multiph. Flow

Int. J. Heat Mass Transf.

Int. J. Multiph. Flow

Int. J. Multiph. Flow

Comput. Fluids

Comput. Fluids

Int. J. Multiph. Flow

Int. J. Multiph. Flow

Int. J. Multiph. Flow

Int. Commun. Heat Mass Transf.

Int. Commun. Heat Mass Transf.