# Computational Modeling of Bubbles Growth Using the Coupled Level Set—Volume of Fluid Method

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## Abstract

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## 1. Introduction

## 2. Computational Model

#### 2.1. Simple Combined Level Set and Volume of Fluid Method

#### 2.2. Transport of Species Based on One-Fluid Approach

## 3. Bubble Hydrodynamic Verification

#### 3.1. Grid Convergence Study

#### 3.2. Volume Conservation of Rising Bubbles

#### 3.3. Bubble Rising in Stagnant Liquid

## 4. Impact of Transport of Species

#### 4.1. Bubble Growth Rate

#### 4.2. Velocity of a Rising Bubble in a Reactive Flow

#### 4.3. Shape of a Rising Bubble in a Reactive Flow

#### 4.4. Transport of Species Impact on Concentration Field

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Computational setup of a bubble rising in an initially stagnant liquid. The initial liquid volume fraction is presented with the bubble (red) and liquid (blue).

**Figure 2.**Mesh convergence study for a spherical and skirted rising bubble using the simple coupled level set (LS) and volume of fluid (VOF) (s-CLSVOF) solver in a non-reactive medium. The main figure presents the percent difference in the bubble terminal velocity with respect to the terminal velocity at $\Delta x/{R}_{o}=0.0125$ for a spherical (yellow) and skirted (green) bubble. The two insets at the bottom right present the bubble interface for different mesh resolution: Coarse mesh ($\Delta x/{R}_{o}=0.12$, blue), medium mesh ($\Delta x/{R}_{o}=0.04$, red), and fine mesh ($\Delta x/{R}_{o}=0.016$, black).

**Figure 3.**The bubble volume as a function of time normalized with respect to the final bubble volume. The normalized volume of the two (green) and three (black) dimensional spherical bubbles and the two (blue) and three (red) dimensional skirted bubbles are presented as a function of the normalized time. The shapes of the two (top) and three-dimensional (bottom) bubbles are presented as insets with the color of the interface corresponding to the normalized volume profile.

**Figure 4.**Bubble rising characteristic map based on the dimensionless numbers: Reynolds, Bond, and Morton numbers. The corresponding bubble shapes are depicted to the right. The red dots reference the simulations performed in the terminal velocity and shape verification study. This characterization map is reproduced with permission from Bhaga and Weber, 1981 [44].

**Figure 5.**Bubble terminal shape and velocity in a non-reactive system: The experimental results are obtained by Bhaga and Weber, 1981 study [44], the front tracking results (red) are computed by Hua et al., 2007 using three-dimensional simulations [47], and the s-CLSVOF results (blue) are computed using two-dimensional simulations. The error percentage below the bubble figures are the absolute percent error between the experimentally measured and computed terminal velocity based on the $Re$.

**Figure 6.**Simulation of bubble growth as a function of time with a liquid concentration ${c}_{\infty}$ of 1 (blue), 10 (red), 50 (green), and 150 mol m${}^{-3}$ (yellow). The obtained bubble radius as a function of time, $R\left(t\right)\phantom{\rule{3.33333pt}{0ex}}{t}^{\beta}$, is normalized by the initial bubble radius. A power law fitting for $R\left(t\right)\sim {t}^{0.5}$ (black dashed) is presented for reference.

**Figure 7.**Comparison of the bubble velocity at the top of the domain ${U}_{f}$ normalized by the non-reactive terminal velocity ${U}_{\infty}$ for simulations with one volume conserving species (green dot), one non-volume conserving species (blue cross), and both a volume and non-volume conserving species (red x). The steady-state bubble shape from the non-reactive simulation is presented for each case with the proper aspect ratio but not at the same scale.

**Figure 8.**Evolution of an initially small rising bubble in (

**a**) a non-reactive flow evolving into an ellipsoidal bubble and (

**b**) the same initial bubble in a reactive flow growing into a skirted bubble. A larger rising bubble in (

**c**) a non-reactive flow that becomes a skirted bubble and (

**d**) the same initial bubble in a reactive flow shrinking to form a spherical bubble. Four time instances are shown for each bubble, and the dimensions of all four panels are based on the initial radius ${R}_{s}$ of the larger bubble. An inset is included in (

**d**) to enlarge the final state of the rising bubble. (

**e**) The time evolution of the $Re$ and $Bo$ of the rising small bubble in non-reactive flow (black), small growing bubble (blue), large bubble in non-reactive flow (green), and large shrinking bubble (red).

**Figure 9.**Evolution of four bubbles rising through a concentration layer. (

**a**) The initial condition for all four simulations with a bubble containing $c=0$ surrounded by liquid with $c=0$ below a layer with $c=1$. (

**b**,

**c**) Spherical and (

**d**,

**e**) skirted bubbles are allowed to rise through the concentration layer. Results for $He=0.01$ are presented in (

**b**,

**d**) while $He=33$ is presented in (

**c**,

**e**). Four different time instances with vertical ranges with the bottom of the range being the black dashed line and the top of each image corresponding to the blue, orange, yellow, and purple dashed lines in (

**a**).

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**MDPI and ACS Style**

Taqieddin, A.; Liu, Y.; Alshawabkeh, A.N.; Allshouse, M.R.
Computational Modeling of Bubbles Growth Using the Coupled Level Set—Volume of Fluid Method. *Fluids* **2020**, *5*, 120.
https://doi.org/10.3390/fluids5030120

**AMA Style**

Taqieddin A, Liu Y, Alshawabkeh AN, Allshouse MR.
Computational Modeling of Bubbles Growth Using the Coupled Level Set—Volume of Fluid Method. *Fluids*. 2020; 5(3):120.
https://doi.org/10.3390/fluids5030120

**Chicago/Turabian Style**

Taqieddin, Amir, Yuxuan Liu, Akram N. Alshawabkeh, and Michael R. Allshouse.
2020. "Computational Modeling of Bubbles Growth Using the Coupled Level Set—Volume of Fluid Method" *Fluids* 5, no. 3: 120.
https://doi.org/10.3390/fluids5030120