Science and Technology of Nuclear Installations

Volume 2019, Article ID 2045751, 13 pages

https://doi.org/10.1155/2019/2045751

## Numerical Study of Bubble Rising and Coalescence Characteristics under Flow Pulsation Based on Particle Method

^{1}School of Nuclear Science and Technology, Xi’an Jiaotong University, No. 28, Xianning West road, Xi’an 710049, China^{2}CNNC Key Laboratory on Nuclear Reactor Thermal Hydraulics Technology, Nuclear Power Institute of China, No. 328, Section 1, Changshun Avenue, Chengdu 610213, China

Correspondence should be addressed to Ronghua Chen; nc.ude.utjx.liam@nehchr

Received 17 December 2018; Accepted 7 March 2019; Published 1 April 2019

Academic Editor: Manmohan Pandey

Copyright © 2019 Ronghua Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Two-phase flow instability may occur in nuclear reactor systems, which is often accompanied by periodic fluctuation in fluid flow rate. In this study, bubble rising and coalescence characteristics under inlet flow pulsation condition are analyzed based on the MPS-MAFL method. To begin with, the single bubble rising behavior under flow pulsation condition was simulated. The simulation results show that the bubble shape and rising velocity fluctuate periodically as same as the inlet flow rate. Additionally, the bubble pairs’ coalescence behavior under flow pulsation condition was simulated and compared with static condition results. It is found that the coalescence time of bubble pairs slightly increased under the pulsation condition, and then the bubbles will continue to pulsate with almost the same period as the inlet flow rate after coalescence. In view of these facts, this study could offer theory support and method basis to a better understanding of the two-phase flow configuration under flow pulsation condition.

#### 1. Introduction

Two-phase flow phenomena could occur in nuclear reactor systems (steam generators, boiling water reactor cores, condensers, etc.), conventional power plants, refrigeration, and chemical and aerospace. In a two-phase system, when mass flow density, void fraction, and pressure drop are coupled, the static flow instability (the flow is drifted by a slight disturbance) and the kinetic instability (constant/variable amplitude flow oscillations at a particular frequency) are collectively referred to as flow instability [1]. These flow instability phenomena are usually accompanied by periodic fluctuation in flow rate. As is well known, two-phase flow instability can degrade the performance of the equipment, which will cause many safety problems; for example, the mechanical forces caused by the flow oscillations will make the components undergo harmful forced mechanical vibrations, and continuous mechanical vibration could lead to fatigue failure of components. In addition, bubble dynamics is an indispensable part of the two-phase flow research and the intrinsic driving force of gas-liquid two-phase flow evolution. Therefore, it is necessary to study the bubble behavior under flow pulsation condition.

In experimental research, a great deal of achievements on bubble dynamics have been made until now. In the bubble coalescence visualization experiment of Sanada et al. [2], the high-speed camera was used to capture the rising and coalescence process of bubbles in the stationary liquid phase, and the microscopic characteristics such as the trajectory, velocity, and interface configuration evolution of the bubble coalescence process were obtained. Liu and Zheng [3] used the method of particle image velocimetry to study the motion behavior of bubbles in a stationary liquid with a rectangular column, and the diversity of bubble rising shapes in different viscosity liquids was finally observed. Sathe et al. [4] used the discrete wavelet transform algorithm to reconstruct the PIV flow field distribution around the bubble, which reduced the interference signal during the PIV flow field acquisition process and further obtained the flow field characteristics like phase drift velocity. Basarova et al. [5] experimentally investigated the ascending motion behavior of spherical bubbles in different concentrations of water ethanol and water propanol and obtained characteristic values such as bubble rising velocity and bubble size, which proved the effect of the concentration of the two solutions on the liquid properties and the gas-liquid interface. Maldonado et al. [6] experimentally studied a single bubble of 2.5 mm diameter rises in different liquids. They established a method for calculating the aspect ratio and the rising velocity of bubble in experiment. Meanwhile, they found that the bubble aspect ratio has a unique relationship with the bubble rise velocity under different conditions. Liu et al. [7] performed experiments to study the rise of bubbles in stationary water and glycerol aqueous solution. Specifically, the experiment detected that the bubble shape has a strong connection and interaction with the terminal velocity. At the same time, the inertial force and the surface tension mainly affect the rising shape of the bubble in water, but in the aqueous glycerin solution, the influence of the viscous force needs to be considered. Raymond and Rosant [8] performed experiments on the movement of moderate deformation bubbles in a viscous liquid mixed with glycerin and water, where the dimensionless Reynolds number and the Weber number range are [,100] and [,5], respectively. The bubble aspect ratio and the bubble terminal velocity, which describe the bubble motion characteristics, were obtained by these experiments and the experimental results were also compared with numerical simulation. Sugrue et al. [9] investigated the influences of channel orientation on bubble dynamics. Besides, the bubble dynamics experiments under different fluid conditions [10–13] have been further conducted.

Due to the uncertainty of the experiment and the limitations of the experimental conditions, numerical simulation and mechanism studies play an increasingly important role in study of bubble dynamics. Tomiyama et al. [14] investigated the bubble terminal velocity in an unlimited large static liquid dominated by surface tension and proposed a theoretical model applicable to predict the terminal velocity of a twisted spherical bubble with a high nondimensional Reynolds number. Based on the balance of all forces acting on the bubble, Wang et al. [15] analyzed the related forces of bubble and illustrated the bubble interface and the shape of the bubble. Moreover, by theoretical analysis, a conclusion was drawn that the mass and heat transfer in the gas-liquid phase interface provides the possibility of continuous motion of the bubble. Gumulya et al. [16] performed a numerical analysis on the rising behavior of bubbles in viscous liquids. On the basis of the three-dimensional VOF method, the correlation between bubble aspect ratio and several dimensionless numbers in different bubble shapes was emphatically studied. Additionally, they also paid attention to the formation and expansion features of bubble wakes. Yu et al. [17] used the new kind of lattice Boltzmann method to simulate a pair of bubbles with different initial positions and initial velocities, and the interactions between bubbles have been successfully obtained, such as attraction and repulsion. Sanada et al. [18] principally studied the effects of viscosity on coalescence of a bubble upon impact with a free surface; it was concluded that both the liquid film and the bubble wake have a significant impact on the bubble coalescence or bounce.

In the 1990s, a meshless method called the Moving Particle Semi-implicit (MPS) method was proposed by Koshizuka and Oka [19]. It is applicable to the simulation of incompressible fluid, especially those with free surface. In order to solve the problem of particle tracking in the presence of inlet and outlet flows in fluid mechanics, Yoon et al. [20] came up with a hybrid method called MPS-MAFL. At this stage, two-phase simulation based on the MPS-MAFL method has been widely used [21–25] because of its outstanding advantages in capturing phase interface.

Under the condition of inlet flow pulsation, the gas-liquid phase interface changes rapidly and dramatically due to the fluctuation of flow velocity. When using the traditional grid method to simulate the bubble motion characteristics under the condition of velocity pulsation, the mesh quality and the stability of the calculation near the phase interface are very challenging. Nowadays, most of the researches on bubble dynamics were carried out under the static condition, and the study of bubble behavior under flow pulsation condition is insufficient. Therefore, it is reasonable and feasible to study the bubble rising and bubble pairs coalescence characteristics under flow pulsation condition by using the predominance of MPS-MAFL in tracking phase interface. Furthermore, this research will also provide a strong theoretical support for understanding the two-phase flow configuration.

#### 2. Numerical Method and Program Validation

##### 2.1. Governing Equations

In MPS-MAFL method, the continuity equation and the Navier-Stokes equation for the incompressible viscous fluid are as follows:where** u**, , ,** n**, , and are velocity, surface tension coefficient, steam liquid interface curvature, unite vector of phase interface, viscosity, and the motion of a computing point that is adaptively configured during the calculation, respectively. Due to the permission of arbitrary calculation between fully Lagrangian () and Eulerian () calculations, the sharp fluid front is calculated accurately by moving the computing points in Lagrangian coordinates, while the computing point is fixed at the boundaries that are described with Eulerian coordinates [20].

##### 2.2. Particle Interaction Models

In MPS method, the governing equations are solved by particle interaction models such as gradient model, Laplacian model, and divergence model, where a particle interacts with nearby particles covered with a weight function , where is the distance between two particles and is the radius of interaction area. In this study, the kernel function is adopted as weight function, and its expression is as follows:

The gradient and Laplacian of physical quantity on particle are expressed by the summation of physical quantities over its neighboring particles according to the weight function.

The gradient model and Laplacian model are as follows, respectively:where represents spatial dimension and is diffusion coefficient and can be determined as

In the MPS-MAFL method, the divergence model is shown in (7). The velocity divergence between two particles is defined by and the velocity divergence at one particle is calculated from the weighted average of the individual velocity divergences.

##### 2.3. Computational Domain and Boundary Condition

In this study, the calculation domain of the single bubble rising process under flow pulsation condition is a 0.08m wide and 0.47m high two-dimensional region, as shown in Figure 1(a). The initial radius of the bubble is 0.01m and the bubble is located in the center of the channel. The distance between the center of the initial bubble and the inlet of the flow channel is 0.037m. This geometric setting ensures that the influence of the wall can be neglected during bubble motion. The vapor phase property is given based on ideal gas state equation and the liquid area is described by using discretized particles with nonuniform scheme. And the particle number density decreases as the distance from the phase interface increases in the liquid area.