# Numerical Study on Bubble Rising in Complex Channels Saturated with Liquid Using a Phase-Field Lattice-Boltzmann Method

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{Gr}), and Eötvös number (Eo).

## 1. Introduction

_{Gr}), and Eötvös number (Eo). In the following, the specific numerical algorithm of phase-field LB model and its implementation details are presented in Section 2. Several verification examples used to test the phase-field LB model are listed in Section 3. Then, detailed numerical results and the discussions are given in Section 4. Finally, a summary and conclusions for the whole article are given in Section 5.

## 2. Numerical Method

#### 2.1. Phase-Field LB Model

#### 2.1.1. Macroscopic Governing Equations

**u**is the macroscopic velocity vector, M is the mobility, ξ is the interface thickness, and $\widehat{\mathit{n}}$ represents the unit vector normal to the gas-liquid interface, whose direction is away from the liquid side and points to the gas bubble side, that is

**F**

_{b}is the body force, and

**F**

_{s}is the surface tension force due to the presence of interface, which can be calculated by [27]:

_{ϕ}is the chemical potential of the binary-fluid system, which is defined by the derivative of the volumetric free energy E

_{f}with respect to the phase field ϕ:

#### 2.1.2. LBE for Interface Tracking

_{α}is the phase-field distribution function, τ

_{ϕ}is the phase-field relaxation time,

**e**

_{α}represents the lattice-related mesoscopic velocity set. For the lattice of D2Q9 model,

**e**

_{α}is given by [30]:

_{s}is the lattice sound speed and ${c}_{s}=c/\sqrt{3}$, and w

_{α}represents the lattice-related weight coefficient set [30], where w

_{0}= 4/9, w

_{1–4}= 1/9, w

_{5–8}= 1/36. Mobility M is positively correlated with the phase-field relaxation time τ

_{ϕ}as:

_{g}and ρ

_{l}represent the densities of the gas and liquid, respectively.

#### 2.1.3. LBE for Hydrodynamics

_{α}is the forcing term, which is calculated by [20]:

_{α}is the collision operator, here the collision operator with a multiple-relaxation-time (MRT) model [31] is employed because its performance is more stable in the implementation than the Bhatnagar-Gross-Krook (BGK) model:

**M**is an orthogonal transformation matrix to transform the distribution functions from physical space into moment space [31] and

**M**

^{−1}is its inverse matrix, and $\widehat{S}$ is a diagonal relaxation matrix. For the D2Q9 lattice, $\widehat{S}$ can be selected as:

_{g}and μ

_{l}represent the dynamic viscosities of the gas and liquid, respectively.

**u**, so the update sequence of velocity must precede the pressure.

#### 2.2. Numerical Implementation

#### 2.2.1. Discretization

#### 2.2.2. Curved Boundary Treatment

**x**

_{w}is the position of a point on the solid boundary, ${\widehat{\mathit{n}}}_{\mathrm{w}}$ is a unit vector normal to the solid boundary, with its direction pointing away from the solid wall, and ϕ

_{w}is the phase field value of the point on the solid boundary. Θ is related to the equilibrium contact angle θ and is written as:

_{m}is the phase field value of the interpolated point in the fluid (blue dots), and $h=\left|{\mathit{x}}_{\mathrm{m}}-{\mathit{x}}_{\mathrm{w}}\right|$ is the distance from the interpolated point to the solid boundary. ϕ

_{w}is estimated by interpolation:

_{w}with Equation (32):

_{i,j}= ϕ

_{m}. To find ϕ

_{i,j}and impose the contact angle conditions on the solid boundary, ϕ

_{m}is the only unknown parameter, which can be obtained by the bidirectional interpolation scheme by utilizing the phase field values of four nearby nodes, and the coordinate positions of nearby nodes are depicted in Figure 1. If the unknown ϕ

_{i,j}is required for interpolation, it is replaced by the ϕ

_{i,j}from the previous time step.

^{−}denotes the incoming distribution functions, such that ${\mathit{e}}_{{\alpha}^{-}}=-{\mathit{e}}_{\alpha}$, and the superscript asterisk denotes the pre-streaming or post-collision state of the distribution functions.

## 3. Numerical Validation

#### 3.1. Laplace Law

_{b}is the radius of the bubble. The computational domain of this verification is a square domain with 201 × 201 grids, and periodic boundary conditions are applied at its four boundaries. At the initial moment of the simulation, the center of the single bubble is located at (101,101), and the bubble is immersed in the liquid phase without any external force. The physical properties of the liquid and bubble are fixed as ρ

_{l}/ρ

_{g}= 1000 and μ

_{l}/μ

_{g}= 100. Zheng et al. [16] found that the simulation results were in better agreement with the analytical solutions and the spurious current was smaller when the interface thickness was greater than 4.5 lattice units (lu). Thus, the interface thickness is fixed to 6 lu in this Laplace-law test, and the bubble radius and surface tension are adjusted in the range of 10–50 lu and 0.01–0.1, respectively.

**x**

_{0}is the location of the bubble interface. By comparison, it can be seen that the numerical results are almost the same as the analytical solutions.

#### 3.2. Bubble Deformation

_{b}= N

_{x}/5 is initially placed at (N

_{x}/2, N

_{y}/4) in a rectangular computational domain discretized with N

_{x}× N

_{y}= 512 × 1536 grid cells. A volumetric buoyancy force ${\mathit{F}}_{\mathrm{b}}=\left(\rho -{\rho}_{\mathrm{l}}\right){G}_{y}\widehat{\mathit{y}}$, where G

_{y}is the magnitude of the gravitational acceleration and $\widehat{\mathit{y}}$ is a unit vector with a vertical downward direction, is continuously imposed on the bubble. The periodic boundary conditions are used at the left and right boundaries of the computational domain, while the bounce-back boundary schemes are applied at the top and bottom boundaries. The density and viscosity ratios of the liquid and bubble are fixed as ρ

_{l}/ρ

_{g}= 1000 and μ

_{l}/μ

_{g}= 100. Besides, there are several dimensionless parameters for characterizing bubble dynamics:

- (1)
- The gravity Reynolds number (Re
_{Gr}),$$R{e}_{\mathrm{Gr}}=\frac{\sqrt{{G}_{y}{\rho}_{\mathrm{l}}\left({\rho}_{\mathrm{l}}-{\rho}_{\mathrm{g}}\right){D}_{\mathrm{b}}^{3}}}{{\mu}_{\mathrm{l}}}$$ - (2)
- The Eötvös number (Eo),$$Eo=\frac{{G}_{y}\left({\rho}_{\mathrm{l}}-{\rho}_{\mathrm{g}}\right){D}_{\mathrm{b}}^{2}}{\sigma}$$
- (3)
- The Morton number (Mo),$$Mo=\frac{{G}_{y}\left({\rho}_{\mathrm{l}}-{\rho}_{\mathrm{g}}\right){\mu}_{\mathrm{l}}^{4}}{{\sigma}^{3}{\rho}_{\mathrm{l}}^{2}}$$
- (4)
- The above three are not independent, since $Mo=E{o}^{3}/R{e}_{\mathrm{Gr}}^{4}$.

_{Gr}and Eo obtained by present LBM and other experimental or numerical methods [33,34,35,36,37]. By comparison, the terminal bubble shapes in our simulations can be observed to be highly similar to other research results, which achieves numerical stability under large liquid-to-gas density ratio conditions and demonstrates the qualitative accuracy of the current LB model.

## 4. Numerical Results and Discussion

#### 4.1. Channel Construction and Numerical Initialization

_{b}and D

_{p}marked in the figures are the bubble and particle diameters, the horizontal spacing L of the particles refers to the shortest distance between two horizontally adjacent particles and similarly the vertical spacing H is the shortest distance between two vertically adjacent particles, and S represents the shortest distance between two diagonally adjacent particles. Hereinafter we use the particle spacing normalized by the bubble diameter to characterize the channel width.

_{b}and a volumetric buoyancy force ${\mathit{F}}_{\mathrm{b}}=\left(\rho -{\rho}_{\mathrm{l}}\right){G}_{y}\widehat{\mathit{y}}$. The phase field in the solid particles is fixed to 0, and in other computational domains the phase field is initialized by the following hyperbolic tangent profile:

_{0}is the initial bubble radius and represents the position of the gas-liquid interface in the radial direction of the bubble, and R(

**x**) is the distance from any position

**x**in the flow domains to the center of the bubble. Unless otherwise specified, in the following simulations, the initial bubble diameter and particle diameter are set as D

_{b}= D

_{p}= 51 lu. The density and viscosity ratios of the liquid and gas are fixed to ρ

_{l}/ρ

_{g}= 1000 and μ

_{l}/μ

_{g}= 100, respectively. Taking into account the balance of numerical stability and accuracy at large density ratios [22], the mobility and interface thickness are taken as M = 0.03 and ξ = 5 lu in the present simulations, respectively. As for the wetting properties, the contact angle of gas-liquid interface on particle surfaces is specified as 40°.

_{D}) is constantly monitored:

_{g}is the average velocity of bubbles rising, which is obtained by averaging the instantaneous y-direction velocities over all gas-phase nodes ($\rho <\left({\rho}_{\mathrm{l}}+{\rho}_{\mathrm{g}}\right)/2$).

#### 4.2. Grid Independence

_{Gr}= 100 and Eo = 20 is selected for testing the grid independence. Five different grids with N

_{x}× N

_{y}= 96 × 1536, 128 × 1536, 176 × 1536, 256 × 1536 and 512 × 1536 grid cells are applied in the bubble rising problem, respectively. The number of grid cells in the y-direction is sufficiently large and constant because its influence on the bubble steady velocity is negligible. The rising velocity variation of the bubble with D

_{b}= N

_{x}/5 over the dimensionless time is present in Figure 5, and the gravity-based dimensionless time is defined as $t*=t\sqrt{{G}_{y}/{D}_{\mathrm{b}}}$. It can be observed that when the grid cells are less than 176 × 1536, the bubble terminal velocity is unstable, and the calculation deviation is unacceptable. Thus, the minimum number of grid cells used in the following simulations is set to 176 × 1536.

#### 4.3. Mass Conservation

_{b}= 0.719 and H/D

_{b}= 0.875 at Re

_{Gr}= 100 and Eo = 20. It is observed that the variation of the normalized total system mass

**M**/

**M**is much less than 10

_{0}^{−6}over a long period of time, indicating that the mass of the gas-liquid two-phase system is conserved well using current phase-field LB model.

#### 4.4. Channel Width Effect

_{Gr}= 100 and Eo = 20. In the wavy vertical channel, the bubble is prone to a stabilized state with a periodically fluctuating terminal velocity, along with C

_{D}fluctuating within the certain ranges. In Figure 7, compared with the channel without particles, the rising velocity of the bubble is lower, and the drag coefficient is larger in the channel when the particles are arranged, which indicates that the presence of particles has a significant hindrance to the movement of the bubble.

_{g}and C

_{D}under different channel widths are given in Figure 8, which are calculated by averaging the regularly fluctuating u

_{g}and C

_{D}over a certain period of time in Figure 7. C

_{D}is found to be lower as the channel width becomes wider, and the bubble rises faster. The same results are observed by Patel et al. [38] in the studies on the effects of the amplitude of the sinusoidal channel walls on bubble dynamics. In addition, the horizontal spacing between particles has a more severe impact on the bubble movement than the vertical spacing of the particles. Because the horizontal spacing of the particles directly affects the interaction between the bubble and the particles, and determines the difficulty for the bubble to pass through each gap, while the vertical spacing affects the fluctuation frequency of the bubble drag force, which can only indirectly act on the drag force. In view of this, unless otherwise stated, the following numerical results are obtained from the simulations in the channels with H/D

_{b}= 0.875.

#### 4.5. Surface Tension Effect

_{g}and ascending C

_{D}are observed in Figure 10 when the surface tension is gradually increasing. This is because the bubble resists deformation more strongly when colliding with the particles, which restricts the bubble rising velocity. In contrast, the bubble with smaller surface tension is more likely to pass through the narrow gaps in the channel by deforming or even breaking. This phenomenon is consistent with the numerical results of Patel et al. [38], and similar to the studies on the effect of surface tension on liquid penetration by Shi et al. [39].

#### 4.6. Bubble Diameter Effect

#### 4.7. Driving Force Effect

**F**

_{d}can be added to the right side of the governing Equation (4) to simulate the changes in the pressure difference of the packed bed in the actual industry. Here we use the dimensionless ratio

**F**

_{d}/(G

_{y}·ρ

_{l}) to measure the magnitude of the driving force.

_{g}increases linearly while C

_{D}decreases rapidly. Because the additional driving force pushes the bubble to pass through the channel faster and more smoothly, the rising velocity is significantly increased, so that u

_{g}increases and C

_{D}decreases.

#### 4.8. Bubble Flow Pattern

_{Gr}range of 0–350 and Eo range of 0–250. Figure 15 depicts the divisions of flow patterns while changing Re

_{Gr}and Eo with fixed channel widths L/D

_{b}= 0.719 and H/D

_{b}= 0.875 in the wavy vertical channel and L/D

_{b}= 1.344 and H/D

_{b}= 0.875 in the S-shaped curved channel. Eo is found to have more significant effects on the flow patterns compared to Re

_{Gr}, and lower Eo numbers often correspond to the bubbles with higher integrity because the surface tension that is related to Eo number plays a leading role in bubble deformation. In the S-shaped channel, the flow pattern Ds occupies a wider range than the flow pattern Dw in the wavy channel, indicating that the bubbles are more likely to reach the state of fragmentation and dispersion because of more frequent collisions with particles.

_{Gr}and Eo increase, and this is attributed to the changes in liquid viscosity and bubble surface tension that have significant impacts on the gas-liquid interface. The increase of Re

_{Gr}and Eo leads to the reduction of liquid viscosity and surface tension, respectively. The reduced liquid viscosity causes the weakening of the viscous effects, which results in the increase of bubble wobbling [38], and the lowered surface tension further promotes bubble deformation and breakup.

_{Gr}and Eo, channel width also has an important influence on the bubble flow patterns. The flow pattern divisions according to the channel widths of two types of channels are demonstrated in Figure 16. The horizontal spacing between particles is found to have more obvious impacts on the flow of bubbles compared to the vertical spacing. The horizontal spacing of the particles directly affects the interaction between the bubble and the particles, thereby affecting the bubble shapes and flow patterns.

_{Gr}and Eo. Conversely, in the relatively complex S-shaped channel, the transitions of the flow patterns are more sensitive to the variations of channel width; four different flow patterns are revealed by changing the channel width. Under this Re

_{Gr}and Eo condition, the flow pattern Cs is more conducive to the even distribution of the dispersed phase in practice because too-small channel width is not prone to the disintegration and flow of the dispersed phase, and too large channel width reduces the contact between the dispersed phase and the particles.

## 5. Conclusions

- (1)
- The present LB model is tested through three aspects of Laplace law, bubble deformation, and mass conservation, and it has been proven to have good stability, accuracy, and conservation from both qualitative and quantitative perspectives.
- (2)
- In the simulations of bubble rising in complex channels, the effects of channel width, surface tension, bubble diameter and additional driving force on bubble motion are investigated in detail. The larger channel width and additional driving force as well as smaller bubble diameter and surface tension lead to lower drag coefficients, which are conducive to smooth passage through the channels for the bubble.
- (3)
- Four and five types of bubble flow patterns are divided according to different bubble evolution processes under different Re
_{Gr}, Eo and channel structures conditions in the wavy vertical channel and S-shaped curved channel, respectively. The detailed flow pattern diagrams are drawn for flow pattern recognition. To some extent, this study has some guiding significance for the regulation of bubble flow patterns in the industrial packed beds.

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

Symbols | |

c_{s} | Lattice sound speed |

C_{D} | Drag coefficient |

D_{b} | Bubble diameter |

D_{p} | Particle diameter |

e_{α} | Lattice-related mesoscopic velocity set |

E_{f} | Volumetric free energy |

Eo | Eötvös number |

F_{α} | Forcing term of the hydrodynamic LBE |

F_{b} | Body force |

F_{d} | Additional driving force |

F_{s} | Surface tension force |

${g}_{\alpha}^{\mathrm{eq}}$ | Equilibrium hydrodynamic distribution function |

${\overline{g}}_{\alpha}$ | Modified hydrodynamic distribution function |

${\overline{g}}_{\alpha}^{\mathrm{eq}}$ | Modified equilibrium hydrodynamic distribution function |

G_{y} | Gravitational acceleration |

${h}_{\alpha}$ | Phase-field distribution function |

${h}_{\alpha}^{\mathrm{eq}}$ | Equilibrium phase-field distribution function |

H | Vertical spacing between two vertically adjacent particles |

L | Horizontal spacing between two horizontally adjacent particles |

M | Mobility |

M | Orthogonal transformation matrix |

M | Total mass of the gas-liquid system |

M_{0} | Initial total mass of the gas-liquid system |

Mo | Morton number |

$\widehat{\mathit{n}}$ | Unit vector normal to the gas-liquid interface |

${\widehat{\mathit{n}}}_{\mathrm{w}}$ | Unit vector normal to the solid boundary |

p | Macroscopic pressure |

Δp | Pressure difference between inside and outside the bubble |

R_{b} | Bubble radius |

Re | Reynolds number of the rising bubble |

Re_{Gr} | Gravity Reynolds number |

S | Shortest spacing between two diagonally adjacent particles |

$\widehat{S}$ | Diagonal relaxation matrix |

t | Time |

t* | Gravity-based dimensionless time |

u | Macroscopic velocity vector |

u_{g} | Bubble rising velocity |

w_{α} | Lattice-related weight coefficient set |

x | Coordinates of the lattice nodes |

x_{w} | Position of the point on the solid boundary |

$\widehat{\mathit{y}}$ | Unit vector with a vertical downward direction |

δt | Unit time |

δx | Unit lattice length |

ξ | Interface thickness |

μ | Fluid mixed viscosity |

μ_{g} | Gas viscosity |

μ_{l} | Liquid viscosity |

μ_{ϕ} | Chemical potential |

ϕ | Phase-field variable |

ϕ_{m} | Phase field value of the interpolated point |

ϕ_{w} | Phase field value of the point on the solid boundary |

ρ | Fluid mixed density |

ρ_{g} | Gas density |

ρ_{l} | Liquid density |

σ | Surface tension |

τ | Hydrodynamic relaxation time |

τ_{ϕ} | Phase-field relaxation time |

θ | Contact angle |

Ω_{α} | Collision operator of the hydrodynamic LBE |

Ψ(ϕ) | Bulk free energy |

## References

- Tailleur, R.G.; Hernandez, J.; Rojas, A. Selective hydrogenation of olefins with mass transfer control in a structured packed bed reactor. Fuel
**2008**, 87, 3694–3705. [Google Scholar] [CrossRef] - Yuan, R.; He, Z.; Zhang, Y.; Wang, W.; Chen, C.; Wu, H.; Zhan, Z. Partial oxidation of methane to syngas in a packed bed catalyst membrane reactor. AIChE J.
**2016**, 62, 2170–2176. [Google Scholar] [CrossRef] - Miladinovic, N.; Weatherley, L.R. Intensification of ammonia removal in a combined ion-exchange and nitrification column. Chem. Eng. J.
**2008**, 135, 15–24. [Google Scholar] [CrossRef] - Huggins, T.M.; Haeger, A.; Biffinger, J.C.; Ren, Z.J. Granular biochar compared with activated carbon for wastewater treatment and resource recovery. Water Res.
**2016**, 94, 225–232. [Google Scholar] [CrossRef] [Green Version] - Li, Q.; Luo, K.H.; Kang, Q.J.; He, Y.L.; Chen, Q.; Liu, Q. Lattice Boltzmann methods for multiphase flow and phase-change heat transfer. Prog. Energy Combust. Sci.
**2016**, 52, 62–105. [Google Scholar] [CrossRef] [Green Version] - Gunstensen, A.K.; Rothman, D.H.; Zaleski, S.; Zanetti, G. Lattice Boltzmann model of immiscible fluids. Phys. Rev. A
**1991**, 43, 4320–4327. [Google Scholar] [CrossRef] [PubMed] - Shan, X.; Chen, H. Lattice Boltzmann model for simulating flows with multiple phases and components. Phys. Rev. E
**1993**, 47, 1815–1819. [Google Scholar] [CrossRef] [Green Version] - Swift, M.R.; Osborn, W.R.; Yeomans, J.M. Lattice Boltzmann simulation of nonideal fluids. Phys. Rev. Lett.
**1995**, 75, 830–833. [Google Scholar] [CrossRef] [Green Version] - He, X.; Chen, S.; Zhang, R. A lattice Boltzmann scheme for incompressible multiphase flow and its application in simulation of Rayleigh-Taylor instability. J. Comput. Phys.
**1999**, 152, 642–663. [Google Scholar] [CrossRef] - Inamuro, T.; Ogata, T.; Tajima, S.; Konishi, N. A lattice Boltzmann method for incompressible two-phase flows with large density differences. J. Comput. Phys.
**2004**, 198, 628–644. [Google Scholar] [CrossRef] - Lee, T.; Lin, C.L. A stable discretization of the lattice Boltzmann equation for simulation of incompressible two-phase flows at high density ratio. J. Comput. Phys.
**2005**, 206, 16–47. [Google Scholar] [CrossRef] - Lee, T.; Liu, L. Lattice Boltzmann simulations of micron-scale drop impact on dry surfaces. J. Comput. Phys.
**2010**, 229, 8045–8063. [Google Scholar] [CrossRef] - Cahn, J.W.; Hilliard, J.E. Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys.
**1958**, 28, 258–267. [Google Scholar] - Chiappini, D.; Bella, G.; Succi, S.; Toschi, F.; Ubertini, S. Improved lattice Boltzmann without parasitic currents for Rayleigh-Taylor instability. Commun. Comput. Phys.
**2010**, 7, 423–444. [Google Scholar] [CrossRef] - Guo, Z.; Zheng, C.; Shi, B. Force imbalance in lattice Boltzmann equation for two-phase flows. Phys. Rev. E
**2011**, 83, 036707. [Google Scholar] [CrossRef] - Zheng, H.W.; Shu, C.; Chew, Y.T. A lattice Boltzmann model for multiphase flows with large density ratio. J. Comput. Phys.
**2006**, 218, 353–371. [Google Scholar] [CrossRef] - Fakhari, A.; Rahimian, M.H. Phase-field modeling by the method of lattice Boltzmann equations. Phys. Rev. E
**2010**, 81, 036707. [Google Scholar] [CrossRef] - Fakhari, A.; Lee, T. Finite-difference lattice Boltzmann method with a block-structured adaptive-mesh-refinement technique. Phys. Rev. E
**2014**, 89, 033310. [Google Scholar] [CrossRef] - Fakhari, A.; Geier, M.; Lee, T. A mass-conserving lattice Boltzmann method with dynamic grid refinement for immiscible two-phase flows. J. Comput. Phys.
**2016**, 315, 434–457. [Google Scholar] [CrossRef] [Green Version] - Fakhari, A.; Bolster, D. Diffuse interface modeling of three-phase contact line dynamics on curved boundaries: A lattice Boltzmann model for large density and viscosity ratios. J. Comput. Phys.
**2017**, 334, 620–638. [Google Scholar] [CrossRef] [Green Version] - Fakhari, A.; Mitchell, T.; Leonardi, C.; Bolster, D. Improved locality of the phase-field lattice-Boltzmann model for immiscible fluids at high density ratios. Phys. Rev. E
**2017**, 96, 053301. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Fakhari, A.; Bolster, D.; Luo, L.S. A weighted multiple-relaxation-time lattice Boltzmann method for multiphase flows and its application to partial coalescence cascades. J. Comput. Phys.
**2017**, 341, 22–43. [Google Scholar] [CrossRef] [Green Version] - Fakhari, A.; Li, Y.; Bolster, D.; Christensen, K.T. A phase-field lattice Boltzmann model for simulating multiphase flows in porous media: Application and comparison to experiments of CO
_{2}sequestration at pore scale. Adv. Water Resour.**2018**, 114, 119–134. [Google Scholar] [CrossRef] - Mitchell, T.; Leonardi, C.; Fakhari, A. Development of a three-dimensional phase-field lattice Boltzmann method for the study of immiscible fluids at high density ratios. Int. J. Multiph. Flow
**2018**, 107, 1–15. [Google Scholar] [CrossRef] [Green Version] - Magaletti, F.; Picano, F.; Chinappi, M.; Marino, L.; Casciola, C.M. The sharp-interface limit of the Cahn-Hilliard/Navier-Stokes model for binary fluids. J. Fluid Mech.
**2013**, 714, 95–126. [Google Scholar] [CrossRef] - Chiu, P.H.; Lin, Y.T. A conservative phase field method for solving incompressible two-phase flows. J. Comput. Phys.
**2011**, 230, 185–204. [Google Scholar] [CrossRef] - Jacqmin, D. Calculation of two-phase Navier-Stokes flows using phase-field modeling. J. Comput. Phys.
**1999**, 155, 96–127. [Google Scholar] [CrossRef] - Jacqmin, D. Contact-line dynamics of a diffuse fluid interface. J. Fluid Mech.
**2000**, 402, 57–88. [Google Scholar] [CrossRef] - Geier, M.; Fakhari, A.; Lee, T. Conservative phase-field lattice Boltzmann model for interface tracking equation. Phys. Rev. E
**2015**, 91, 063309. [Google Scholar] [CrossRef] [Green Version] - He, X.; Luo, L.S. Theory of the lattice Boltzmann method: From the Boltzmann equation to the lattice Boltzmann equation. Phys. Rev. E
**1997**, 56, 6811–6817. [Google Scholar] [CrossRef] [Green Version] - Lallemand, P.; Luo, L.S. Theory of the lattice Boltzmann method: Dispersion, dissipation, isotropy, Galilean invariance, and stability. Phys. Rev. E
**2000**, 61, 6546–6562. [Google Scholar] [CrossRef] [Green Version] - Yu, D.; Mei, R.; Shyy, W. A unified boundary treatment in lattice Boltzmann method. AIAA J.
**2003**. [Google Scholar] [CrossRef] - Hua, J.; Lou, J. Numerical simulation of bubble rising in viscous liquid. J. Comput. Phys.
**2007**, 222, 769–795. [Google Scholar] [CrossRef] - Fakhari, A.; Rahimian, M.H. Simulation of an axisymmetric rising bubble by a multiple relaxation time lattice Boltzmann method. Int. J. Mod. Phys. B
**2009**, 23, 4907–4932. [Google Scholar] [CrossRef] - Huang, H.; Huang, J.J.; Lu, X.Y. A mass-conserving axisymmetric multiphase lattice Boltzmann method and its application in simulation of bubble rising. J. Comput. Phys.
**2014**, 269, 386–402. [Google Scholar] [CrossRef] - Liang, H.; Li, Y.; Chen, J.; Xu, J. Axisymmetric lattice Boltzmann model for multiphase flows with large density ratio. Int. J. Heat Mass Transf.
**2019**, 130, 1189–1205. [Google Scholar] [CrossRef] [Green Version] - Bhaga, D.; Weber, M.E. Bubbles in viscous liquids: Shapes, wakes and velocities. J. Fluid Mech.
**1981**, 105, 61–85. [Google Scholar] [CrossRef] [Green Version] - Patel, T.; Patel, D.; Thakkar, N.; Lakdawala, A. A numerical study on bubble dynamics in sinusoidal channels. Phys. Fluids
**2019**, 31, 052103. [Google Scholar] [CrossRef] - Shi, Y.; Tang, G.H.; Lin, H.F.; Zhao, P.X.; Cheng, L.H. Dynamics of droplet and liquid layer penetration in three-dimensional porous media: A lattice Boltzmann study. Phys. Fluids
**2019**, 31, 042106. [Google Scholar] [CrossRef] - Clift, R.; Grace, J.R.; Weber, M.E. Bubbles, Drops, and Particles; Academic Press: New York, NY, USA, 1978. [Google Scholar]

**Figure 2.**Phase field distribution contour (

**a**) of the stable bubble and phase field distribution comparison in the radial direction (

**b**) through the bubble.

**Figure 5.**Variations of the bubble rising velocity with dimensionless time using five different grids.

**Figure 7.**Variations of u

_{g}(

**a**) and C

_{D}(

**b**) with dimensionless time at Re

_{Gr}= 100 and Eo = 20.

**Figure 12.**Variations of u

_{g}(

**a**) and C

_{D}(

**b**) with additional driving force at Re

_{Gr}= 100 and Eo = 20.

**Figure 13.**Bubble evolution diagrams of flow patterns Aw (

**a**), Bw (

**b**), Cw (

**c**), and Dw (

**d**) in the wavy vertical channel with L/D

_{b}= 0.719.

**Figure 14.**Bubble evolution diagrams of flow patterns As (

**a**), Bs (

**b**), Cs (

**c**), Ds (

**d**), and Es (

**e**) in the S-shaped curved channel.

**Figure 15.**Flow pattern divisions according to Re

_{Gr}and Eo in the wavy vertical channel (

**a**) and S-shaped curved channel (

**b**).

**Figure 16.**Flow pattern divisions according to the channel widths of the wavy vertical channel (

**a**) and S-shaped curved channel (

**b**) at Re

_{Gr}= 100 and Eo = 20.

**Table 1.**Comparison of terminal bubble shapes observed in experiments and predicted by front tracking method, previous and present LBMs.

Case | Re_{Gr} | Eo | Experiments (Bhaga and Weber, 1981) [37] | Front Tracking Method (Hua and Lou, 2007) [33] | LBM (Liang et al., 2019) [36] | Present LBM |
---|---|---|---|---|---|---|

A1 | 1.67 | 17.7 | ||||

A2 | 79.88 | 32.2 | ||||

A3 | 134.63 | 115 | ||||

A4 | 30.83 | 339 | ||||

A5 | 49.72 | 641 |

Case | Re of Experiments [37] | Re of Present LBM | Relative Error (%) |
---|---|---|---|

A1 | 0.232 | 0.211 | 9.05 |

A2 | 55.3 | 47.8 | 13.56 |

A3 | 94.0 | 87.5 | 6.91 |

A4 | 18.3 | 16.4 | 10.38 |

A5 | 30.3 | 27.5 | 9.24 |

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**MDPI and ACS Style**

Yu, K.; Yong, Y.; Yang, C.
Numerical Study on Bubble Rising in Complex Channels Saturated with Liquid Using a Phase-Field Lattice-Boltzmann Method. *Processes* **2020**, *8*, 1608.
https://doi.org/10.3390/pr8121608

**AMA Style**

Yu K, Yong Y, Yang C.
Numerical Study on Bubble Rising in Complex Channels Saturated with Liquid Using a Phase-Field Lattice-Boltzmann Method. *Processes*. 2020; 8(12):1608.
https://doi.org/10.3390/pr8121608

**Chicago/Turabian Style**

Yu, Kang, Yumei Yong, and Chao Yang.
2020. "Numerical Study on Bubble Rising in Complex Channels Saturated with Liquid Using a Phase-Field Lattice-Boltzmann Method" *Processes* 8, no. 12: 1608.
https://doi.org/10.3390/pr8121608