# Modeling of the Free-Surface Vortex-Driven Bubble Entrainment into Water

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model Description

#### 2.1. The GENTOP Concept

#### 2.2. The New Entrainment Model

#### 2.3. The Turbulence Model

## 3. CFD Set-Up

^{−4}was used to control the mass transfer from cg to G1.

^{−4}were used in this step. The investigation of the influence of the entrainment coefficient was also carried out by performing simulations with two different ${C}_{ent}$. The mass transfer from cg to G1 with the scale of curvature correction ${C}_{scale}$ = 1 was set in the simulations. Finally, the influence of the mesh size was also investigated and presented in the last section.

## 4. Results and Discussion

#### 4.1. The Influence of the Turbulence Model

^{−4}are presented in Figure 4a,b, respectively. In the case of ${C}_{scale}$ = 0, the free surface is almost flat. At this condition, almost no bubble entrainment can be observed. Increasing ${C}_{scale}$ to 0.5 leads to dimple formation. A spot of the dispersed gas fraction is observed, indicating that the entrainment model is activated. However, no bubble entrainment (with ${\alpha}_{dg}$ equal to or more than 0.02) into the connection pipe can be observed. Increasing ${C}_{scale}$ to 1 leads to the formation of the gas core. At this condition, the bubble entrainment into the connection pipe is observed, confirming that the entrainment model successfully takes into account the bubble entrainment phenomenon.

#### 4.2. The Influence of Entrained Bubble Size Distribution

#### 4.3. The Influence of the Entrainment Coefficient

^{−4}.

^{−4}is around 1.7 times that obtained for ${C}_{ent}$ = 1 × 10

^{−4}. This shows that this coefficient can control the entrainment rate. The average cg and dg entrainment values show that dg entrainment is much higher than cg entrainment for both simulations (see Figure 10d). The result of the simulation using ${C}_{ent}$ = 2 × 10

^{−4}, as shown in Figure 10d, is the same as in Figure 8e, so is in the best agreement with the experiment.

#### 4.4. The Influence of the Computational Cell Size

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Latin symbols | |

${A}_{D}$ | interfacial area density (m^{−1}) |

${C}_{D}$ | drag coefficient (dimensionless) |

${C}_{ent}$ | entrainment coefficient (m s^{−1}) |

${C}_{L}$ | lift coefficient (dimensionless) |

${C}_{scale}$ | scaling coefficient of curvature correction (dimensionless) |

${C}_{VM}$ | virtual mass coefficient (dimensionless) |

${C}_{W}$ | wall force coefficient (dimensionless) |

${C}_{\mu}$ | shear-induced turbulence coefficient (dimensionless) |

D | pipe diameter (m) |

${d}_{B}$ | bubble diameter (m) |

${d}_{\perp}$ | maximum horizontal dimension of a bubble (m) |

$Eo$ | Eötvös number (dimensionless) |

$E{o}_{\perp}$ | modified Eötvös number (dimensionless) |

$F$ | force (N m^{−3}) |

$g$ | gravitational acceleration (m^{2} s^{−2}) |

$M$ | momentum transfer term (kg m^{−2} s^{−2}) |

$p$ | pressure (Pa) |

$Re$ | Reynolds number (dimensionless) |

${S}_{M}$ | momentum source due to external body forces (kg m^{−2} s^{−2}) |

$W$ | mean velocity magnitude (m s^{−1}) |

$u$ | velocity vector (m s^{−1}) |

$t$ | time (s) |

y | distance to the wall (m) |

$z$ | axial distance (m) |

Greek symbols | |

$\alpha $ | volume fraction (dimensionless) |

$\Delta x$ | characteristic cell length scale (m) |

$\mu $ | dynamic viscosity (Pa s) |

$\rho $ | density (kg m^{−3}) |

$\sigma $ | surface tension (N m^{−1}) |

${\sigma}_{TD}$ | turbulent Schmidt number (dimensionless) |

$\phi $ | blending function (dimensionless) |

Subscripts and superscripts | |

$fs$ | free surface |

$j$ | phase index |

$G$ | gas |

$L$ | liquid |

$VM$ | virtual mass |

$W$ | wall |

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**Figure 1.**Free-surface vortex observed in Akkats hydropower station, Sweden. The image is taken from [10].

**Figure 4.**(

**a**) Average interfacial deformation represented by iso-surface of ${\alpha}_{cg}$ = 0.5 and (

**b**) average bubble entrainment represented by iso-surface of ${\alpha}_{dg}$ = 0.02. The results are obtained from the simulations using ${C}_{ent}$ = 2 × 10

^{−4}.

**Figure 5.**Transient profile of gas entrainment rate of: (

**a**) potentially continuous gas cg, (

**b**) dispersed gas dg, and (

**c**) total gas. (

**d**) Average gas entrainment rate of cg and dg and (

**e**) total. The results are obtained from the simulations using ${C}_{ent}$ = 2 × 10

^{−4}. The experimental gas entrainment rate is based on [71,72].

**Figure 6.**Velocity vectors represent the tangential projection of the average liquid velocity on horizontal planes for cases: (

**a**) ${C}_{scale}$ = 0, (

**b**) ${C}_{scale}$ = 0.5, (

**c**) ${C}_{scale}$ = 1. The background color represents the average liquid vorticity magnitude. The results are obtained from the simulations using ${C}_{ent}$ = 2 × 10

^{−4}. Horizontal planes H1, H2, and H3 are located 10, 20, and 30 mm above the bottom surface of the cylindrical vessel, respectively.

**Figure 7.**The contour of the dispersed gas fraction at t = 6 s plotted on the vertical central plane for the simulations of (

**a**) Sim. A, (

**b**) Sim. B, and (

**c**) Sim. C. The results are obtained from the simulations using ${C}_{ent}$ = 2× 10

^{−4}and ${C}_{scale}$ = 1.

**Figure 8.**Transient profile of gas entrainment rate of: (

**a**) potentially continuous gas, (

**b**) dispersed gas, and (

**c**) total gas. Average gas entrainment rate of: (

**d**) cg and dg and (

**e**) total. The results are obtained from the simulations using ${C}_{ent}$ = 2 × 10

^{−4}. The experimental gas entrainment rate is based on [71,72].

**Figure 9.**(

**a**) Average interfacial deformation represented by isosurface of ${\alpha}_{cg}$ = 0.5 and (

**b**) average bubble entrainment represented by isosurface of ${\alpha}_{dg}$ = 0.02. The results are obtained from the simulations using ${C}_{scale}$ = 1.

**Figure 10.**Transient profile of gas entrainment rate of: (

**a**) potentially continuous gas, (

**b**) dispersed gas and (

**c**) total gas. (

**d**) Average gas entrainment rate of cg and dg and (

**e**) total. The results are obtained from the simulations using ${C}_{scale}$ = 1. The experimental gas entrainment rate is based on [71,72].

**Figure 11.**Velocity vectors represent the tangential projection of the average liquid velocity on horizontal planes for cases (

**a**) ${C}_{ent}$ = 1× 10

^{−4}and (

**b**) ${C}_{ent}$ = 2 × 10

^{−4}. The background color represents the average liquid vorticity magnitude. The results are obtained from the simulations using ${C}_{scale}$ = 1. Horizontal planes H1, H2, and H3 are located 10, 20, and 30 mm above the bottom surface of the cylindrical vessel, respectively.

**Figure 12.**Average interfacial deformation, represented by isosurface, of ${\alpha}_{cg}$ = 0.5 for cases (

**a**) Mesh A and ${C}_{scale}$ = 1, (

**b**) Mesh B and ${C}_{scale}$ = 1 and (

**c**) Mesh B and ${C}_{scale}$ = 0.85.

Force | Formulation | Ref. | No. |
---|---|---|---|

Drag | ${F}_{drag}=-\frac{3}{4{d}_{B}}{C}_{D}{\rho}_{L}{\alpha}_{G}\left|{u}_{G}-{u}_{L}\right|\left({u}_{G}-{u}_{L}\right)$ | [53] | (5) |

${C}_{D,bubb}=max\left({C}_{D,sphere},min\left({C}_{D,ellipse},{C}_{D,cap}\right)\right)$, | (6) | ||

${C}_{D,sphere}=\frac{24}{Re}\left(1+0.1R{e}^{0.75}\right)$, ${C}_{D,ellipse}=\frac{2}{3}\sqrt{Eo}$, ${C}_{D,cap}=\frac{8}{3}$ | |||

Lift | ${F}_{lift}=-{C}_{L}{\rho}_{L}{\alpha}_{G}\left({u}_{G}-{u}_{L}\right)\times \left(\nabla \times {u}_{L}\right)$ | [54] | (7) |

${C}_{L}\{\begin{array}{c}min\left[0.288tanh\left(0.121Re\right),f\left(E{o}_{\perp}\right)\right]\text{}E{o}_{\perp}4\\ f\left(E{o}_{\perp}\right)\text{}for\text{}4E{o}_{\perp}10\\ -0.27\text{}10E{o}_{\perp \text{}}\end{array}$ | [55] | (8) | |

$f\left(E{o}_{\perp}\right)=0.00105E{o}_{\perp}{}^{3}-0.0159E{o}_{\perp}{}^{2}-0.0204E{o}_{\perp}+0.474$ | (9) | ||

$E{o}_{\perp}=\frac{g\left({\rho}_{L}-{\rho}_{G}\right){d}_{\perp}^{2}}{\sigma}$ | (10) | ||

${d}_{\perp}={d}_{B}\sqrt[3]{1+0.163E{o}^{0.757}}$ | [56] | (11) | |

Wall lubrication | ${F}_{wall}=\frac{2}{{d}_{B}}{C}_{W}{\rho}_{L}{\alpha}_{G}{\left|{u}_{G}-{u}_{L}\right|}^{2}\widehat{y}$ | [57] | (12) |

${C}_{W}\left(y\right)=f\left(Eo\right){\left(\frac{{d}_{B}}{2y}\right)}^{2}$ | [58] | (13) | |

$f\left(Eo\right)=0.0217Eo$ | [59] | (14) | |

Turbulent dispersion | ${F}_{TD}=-\frac{3}{4}{C}_{D}\frac{{\alpha}_{G}}{{d}_{B}}\left|{u}_{G}-{u}_{L}\right|\frac{{\mu}_{L}^{turb}}{{\sigma}_{TD}}\left(\frac{1}{{\alpha}_{L}}+\frac{1}{{\alpha}_{G}}\right)\nabla {\alpha}_{G}$ | [60] | (15) |

${\sigma}_{TD}=0.9$ | [49] | (16) | |

Virtual mass | ${F}_{VM}=-{C}_{VM}{\rho}_{L}{\alpha}_{G}\left(\frac{{D}_{G}{u}_{G}}{{D}_{t}}-\frac{{D}_{L}{u}_{L}}{{D}_{t}}\right)$ | [61,62,63] | (17) |

${C}_{VM}=0.5$ | (18) |

Velocity Groups | dg | cg | |||
---|---|---|---|---|---|

Morphology | Polydispersed | Continuous | |||

Bubble classes | G1 | G2 | G3 | G4 | G5 |

Diameter [mm] | 1 | 3 | 5 | 7 | ≥9 |

Entrainment fraction (Sim. A) | 1 | 0 | 0 | 0 | −1 |

Entrainment fraction (Sim. B) | 0.50 | 0.50 | 0 | 0 | −1 |

Entrainment fraction (Sim. C) | 0.25 | 0.25 | 0.25 | 0.25 | −1 |

Method/Model | ${\mathit{L}}_{\mathit{g}}$ (mm) | Gas Entrainment Rate (m^{3}/s) | $\mathbf{Max}.\text{}\mathit{\omega}$ (1/s) around the Vortex Tip |
---|---|---|---|

Ikoma exp. | 2.0 × 10^{−8} [71,72] | ||

Mesh A, ${\mathrm{C}}_{\mathrm{scale}}$ = 1 | 16 | 2.3 × 10^{−8} | 369 |

Mesh B, ${\mathrm{C}}_{\mathrm{scale}}$ = 1 | 27 | 6.9 × 10^{−8} | 620 |

Mesh B, ${\mathrm{C}}_{\mathrm{scale}}$ = 0.85 | 16 | 2.4 × 10^{−8} | 384 |

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Putra, R.A.; Lucas, D.
Modeling of the Free-Surface Vortex-Driven Bubble Entrainment into Water. *Water* **2020**, *12*, 709.
https://doi.org/10.3390/w12030709

**AMA Style**

Putra RA, Lucas D.
Modeling of the Free-Surface Vortex-Driven Bubble Entrainment into Water. *Water*. 2020; 12(3):709.
https://doi.org/10.3390/w12030709

**Chicago/Turabian Style**

Putra, Ryan Anugrah, and Dirk Lucas.
2020. "Modeling of the Free-Surface Vortex-Driven Bubble Entrainment into Water" *Water* 12, no. 3: 709.
https://doi.org/10.3390/w12030709