# Numerical Simulation of High-Density Ratio Bubble Motion with interIsoFoam

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## Abstract

**:**

## 1. Introduction

## 2. Code Description and Methods

## 3. Results

#### 3.1. interIsoFoam Validation

#### 3.1.1. 2D Stationary Drop

`symmetry`BC has been set on the sides of the square, while the

`empty`BC, which is the standard condition in OpenFOAM for boundaries positioned in the non-considered direction of a 2D model, has been used on the faces. Through the utility

`setAlphaField`, we have imposed ${r}_{\alpha}=1$ in the CVs inside the circumference and ${r}_{\alpha}=0$ in the external CVs, initializing the drop.

`pointCellsLeastSquares`scheme for the gradient discretization, the

`limitedLinearV`scheme with a blending factor equal to 1 for the velocity advection term discretization and the

`vanLeer`scheme has been employed for the volume fraction advection discretization. Three PISO cycles and three external PIMPLE cycles have been performed, using the solver

`smoothSolver`with the

`symGaussSeidel`smoother for the velocity resolution and the GAMG solver with the DIC smoother for the pressure. The isoAdvector method with 2 plic-RDF iterations in the reconstruction-step of the interface has been used. As indicated in [24], the maximum CFL number was imposed equal to 0.2 to guarantee a second order convergence for the interface advection step.

^{−7}(about 125 s of transient) in the range $1\xf74.5$ h.

#### 3.1.2. 2D Rising Bubble

`noSlip`condition for the bottom and top boundaries and the

`slip`condition, which imposes a zero shear stress, on the vertical boundaries. For pressure, the

`zeroGradient`BC has been employed, whereas for the faces normal to the non-considered direction the

`empty`condition has been employed. The light phase constituting the bubble (${\alpha}_{2}$) is initialized as described in Section 3.1.1, as well as the numerical set up, with the only difference in the value of the gravity acceleration which, in this case, has a finite value (Table 2). The mesh used is triangular and unstructured, shown in Figure 3b.

#### 3.1.3. 3D Rising Bubble

`noSlip`condition for the velocity and

`zeroGradient`for the pressure on all faces. The mesh used, shown in Figure 8b, is prismatic and unstructured. The mesh was generated from a 2D mesh extruded along the cylinder axis direction for a s number of steps. The resolution of the grid is greater around the axis of symmetry of the cylinder, where a uniform grid size h has been imposed, to maximize the resolution of the bubble. The mesh grows towards the edge of the cylinder, where the elements reach their maximum size. The same numerical setup described in Section 3.1.1 was used.

#### 3.1.4. Co-Axial Coalescence of Two Bubbles

`setAlphaField`utility, has the same diameter d as the first bubble shown Figure 8a and is positioned aligned with it with respect to the y-direction at a distance from its centroid equal to $1.5d$. The characteristic parameters for this case are: $Eo=16,Mo=2\times {10}^{-4},{\eta}_{\rho}=100$ and ${\eta}_{\mu}=100$.

#### 3.2. Rising He Bubble in PbLi

#### 3.2.1. Axisymmetric and Skirted Regimes

#### 3.2.2. Oscillatory Regime

#### 3.2.3. Peripheral and Central Breakup Regimes

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Computational domain, BCs, ICs (

**a**) and the detail of the mesh at the interface for the 2D static drop (

**b**). Red colour corresponds to ${r}_{\alpha}=1$ whereas the blue one to ${r}_{\alpha}=0$.

**Figure 2.**Pressure profile on the horizontal axis passing through the centre of the model (

**a**) and velocity vectors on the upper-right quarter, representing the spurious velocities, for the M3 mesh (${\eta}_{\rho}=1$) (

**b**).

**Figure 3.**Computational domain, BCs, ICs (

**a**) and mesh detail on the bubble at $t=0$ (

**b**) for the 2D rising bubble test. The black line in (

**b**) is the ${r}_{\alpha}=0.5$ contour.

**Figure 4.**Evolution of ${u}_{c}$ (

**a**) and $\xi $ (

**b**) for the C1 case provided by plicRDF on M4 mesh compared with Ref. [10].

**Figure 5.**2D bubble shape evolution for the C1 (

**a**) and C2 (

**b**) scenario represented through contour ${r}_{\alpha}=0.5$.

**Figure 6.**Evolution of ${u}_{c}$ (

**a**) and $\xi $ (

**b**) for the 2D C2 case provided by plicRDF compared with FreeLIFE code in Ref. [29].

**Figure 7.**Bubble shape at the end of the transient for cases C1 (

**a**), C2 (

**b**) and C3 (

**c**) represented through the half contour ${r}_{\alpha}=0.5$.

**Figure 8.**Half computational domain, BCs, ICs (

**a**) and mesh detail on the xz plane that cuts the bubble in half at $t=0$ (

**b**) for the 3D rising bubble test. The bubble, coloured in cyan, is represented through the isovolume $0\le \alpha \le 0.5$.

**Figure 9.**Evolution of Ga (

**a**) and $\mathsf{\Phi}$ (

**b**) for the 3D C1 case provided by plicRDF (M4 mesh) compared with Ref. [10].

**Figure 10.**Evolution of Ga (

**a**) and $\mathsf{\Phi}$ (

**b**) for the 3D C2 case provided by plicRDF compared with Ref. [10].

**Figure 11.**Evolution of ${u}^{*}$ (

**a**) and $\mathsf{\Phi}$ (

**b**) for the 3D C3 case provided by plicRDF compared with Ref. [10].

**Figure 12.**Bubble shapes for the coalescence test represented through the half contour ${r}_{\alpha}=0.5$, with velocity vectors, at four different times: ${t}^{*}=0.42$ (

**a**), ${t}^{*}=0.60$ (

**b**), ${t}^{*}=0.63$ (

**c**) and ${t}^{*}=0.97$ (

**d**).

**Figure 13.**Evolution of Ga for the coalescence case provided by plicRDF compared with Ref. [10].

**Figure 14.**Water-air phase plot showing different regions of bubble behaviours, adapted from Ref. [38].

**Table 1.**Mesh data and results for the mesh sensitivity study and for the ${\eta}_{\rho}=1000$ case, simulated with the M2 mesh.

Grid Size h (m) | Number of Elements | ${\mathit{L}}_{1}\left(\mathit{u}\right)$ | $\mathit{E}(\mathbf{\Delta}\mathit{p})$ | |
---|---|---|---|---|

M1 | 0.04 | 1400 | 2.68 × 10^{−3} | 0.125 |

M2 | 0.02 | 5678 | 1.84 × 10^{−3} | 0.068 |

M3 | 0.01 | 22,902 | 1.30 × 10^{−3} | 0.146 |

${\eta}_{\rho}=1000$ | 0.02 | 5678 | 4.1 × 10^{−1} | 0.075 |

${\mathit{\eta}}_{\mathit{\rho}}$ | ${\mathit{\eta}}_{\mathit{\mu}}$ | g (m s${}^{-2}$) | $\mathit{\gamma}$ (N m${}^{-1}$) | Eo | Ga | |
---|---|---|---|---|---|---|

C1 | 10 | 10 | 0.98 | 24.5 | 10 | 35 |

C2 | 1000 | 100 | 0.98 | 1.96 | 125 | 35 |

Grid Size h (m) | Number of Elements | |
---|---|---|

M1 | 0.01250 | 29,344 |

M2 | 0.00833 | 66,854 |

M3 | 0.00625 | 116,142 |

M4 | 0.00500 | 184,398 |

isoAlpha | plicRDF | |||||
---|---|---|---|---|---|---|

${\mathit{\xi}}_{\mathit{min}}$ | ${\mathit{u}}_{\mathit{c},\mathit{max}}$ (m s${}^{-1}$) | ${\mathit{y}}_{\mathit{c}}$ (t = 3 s) | ${\mathit{\xi}}_{\mathit{min}}$ | ${\mathit{u}}_{\mathit{c},\mathit{max}}$ (m s${}^{-1}$) | ${\mathit{y}}_{\mathit{c}}$ (t = 3 s) | |

M1 | 0.8939 | 0.2414 | 1.0831 | 0.8945 | 0.2413 | 1.0832 |

M2 | 0.8947 | 0.2409 | 1.0833 | 0.8946 | 0.2408 | 1.0834 |

M3 | 0.8952 | 0.2415 | 1.0839 | 0.8953 | 0.2413 | 1.0840 |

M4 | 0.8952 | 0.2417 | 1.0840 | 0.8955 | 0.2414 | 1.0841 |

${\mathit{\xi}}_{\mathit{min}}$ | ${\mathit{t}|}_{\mathit{\xi}={\mathit{\xi}}_{\mathit{min}}}$ | ${\mathit{u}}_{\mathit{c},\mathit{max}}$ (m s${}^{-1}$) | ${\mathit{t}|}_{{\mathit{u}}_{\mathit{c}}={\mathit{u}}_{\mathit{c},\mathit{max}}}$ | ${\mathit{y}}_{\mathit{c}}$ (t = 3 s) | |
---|---|---|---|---|---|

isoAlpha | 0.8952 | 1.8900 | 0.2415 | 0.9260 | 1.0839 |

plicRDF | 0.8953 | 1.9000 | 0.2413 | 0.9280 | 1.0840 |

Ref. [10] | 0.9005 | 1.8934 | 0.2414 | 0.9260 | 1.0809 |

Ref. [29] | 0.9014 | 1.9070 | 0.2419 | 0.9281 | 1.0812 |

**Table 6.**Benchmark results for the C2 case for both isoAlpha and pliRFD method compared with Ref. [29].

${\mathit{\xi}}_{\mathit{min}}$ | ${\mathit{t}|}_{\mathit{\xi}={\mathit{\xi}}_{\mathit{min}}}$ | ${\mathit{u}}_{\mathit{c},\mathit{max}}$ (m s${}^{-1}$) | ${\mathit{t}|}_{{\mathit{u}}_{\mathit{c}}={\mathit{u}}_{\mathit{c},\mathit{max}}}$ | ${\mathit{y}}_{\mathit{c}}$ (t = 3 s) | |
---|---|---|---|---|---|

isoAlpha | 0.4743 | 3.0000 | 0.2491 | 0.7170 | 1.1175 |

plicRDF | 0.4725 | 3.0000 | 0.2491 | 0.7180 | 1.1175 |

TP2D | 0.5869 | 2.4004 | 0.2524 | 0.7332 | 1.1380 |

FreeLIFE | 0.4647 | 3.0000 | 0.2514 | 0.7281 | 1.1249 |

MooNMD | 0.5144 | 3.0000 | 0.2502 | 0.7317 | 1.1376 |

${\mathit{\eta}}_{\mathit{\rho}}$ | ${\mathit{\eta}}_{\mathit{\mu}}$ | g (m s${}^{-2}$) | $\mathit{\gamma}$ (N m${}^{-1}$) | Eo | Mo | |
---|---|---|---|---|---|---|

C1 | 100 | 100 | 9.81 | 8.37 × 10^{−3} | 116 | 41.1 |

C2 | 100 | 100 | 9.81 | 2.86 × 10^{−3} | 339 | 43.1 |

C3 | 714 | 6670 | 9.81 | 2.49 × 10^{−2} | 39.4 | 0.065 |

Grid Size h (cm) | Axial Divisions s | Number of Elements | |
---|---|---|---|

M1 | 0.067 | 120 | 222,720 |

M2 | 0.050 | 160 | 774,400 |

M3 | 0.033 | 240 | 2,881,200 |

M4 | 0.025 | 320 | 7,128,320 |

**Table 9.**Mesh sensitivity study results for 3D rising bubble C1 case. Absolute error E has as reference value that of the M4 mesh.

$\mathbf{\Phi}$ | ${\mathit{E}}_{\mathbf{\Phi}}(\%)$ | Ga | ${\mathit{E}}_{Ga}(\%)$ | |
---|---|---|---|---|

M1 | 0.8341 | 2.807 | 7.0894 | 0.263 |

M2 | 0.8181 | 0.901 | 7.1176 | 0.134 |

M3 | 0.8126 | 0.234 | 7.1032 | 0.068 |

M4 | 0.8107 | - | 7.1080 | - |

Ga | ${\mathit{E}}_{Ga}(\%)$ | $\mathbf{\Phi}$ | ${\mathit{E}}_{\mathbf{\Phi}}(\%)$ | |
---|---|---|---|---|

pliRDF | 7.09 | - | 0.82 | - |

Ref. [10] | 7.02 | 0.98 | 0.81 | 0.71 |

Ref. [31] | 7.16 | 1.00 | - | - |

Ref. [30] | 7.0 | 1.27 | - | - |

${\mathit{u}}^{*}$ | ${\mathit{E}}_{{\mathit{u}}^{*}}(\%)$ | |
---|---|---|

pliRDF | 0.6195 | - |

Ref. [10] | 0.6110 | 1.40 |

Ref. [32] | 0.6226 | 0.49 |

$\mathit{d}/2$ (mm) | ${\mathit{\eta}}_{\mathit{\rho}}$ | ${\mathit{\eta}}_{\mathit{\mu}}$ | Ga | Eo | |
---|---|---|---|---|---|

B1 | 0.2 | 1.24 × 10^{5} | 61 | 44 | 8.44 × 10^{−3} |

B2 | 116 | 5.00 × 10^{4} | 100 | 12 | 4.86 × 10^{3} |

B3 | 1 | 1.24 × 10^{5} | 61 | 497 | 0.21 |

B4 | 10 | 1.24 × 10^{5} | 61 | 1.57 × 10^{4} | 21.1 |

B5 | 100 | 1.24 × 10^{5} | 61 | 4.97 × 10^{5} | 2.11 × 10^{3} |

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**MDPI and ACS Style**

Siriano, S.; Balcázar, N.; Tassone, A.; Rigola, J.; Caruso, G.
Numerical Simulation of High-Density Ratio Bubble Motion with interIsoFoam. *Fluids* **2022**, *7*, 152.
https://doi.org/10.3390/fluids7050152

**AMA Style**

Siriano S, Balcázar N, Tassone A, Rigola J, Caruso G.
Numerical Simulation of High-Density Ratio Bubble Motion with interIsoFoam. *Fluids*. 2022; 7(5):152.
https://doi.org/10.3390/fluids7050152

**Chicago/Turabian Style**

Siriano, Simone, Néstor Balcázar, Alessandro Tassone, Joaquim Rigola, and Gianfranco Caruso.
2022. "Numerical Simulation of High-Density Ratio Bubble Motion with interIsoFoam" *Fluids* 7, no. 5: 152.
https://doi.org/10.3390/fluids7050152