# Baseline Model for Bubbly Flows: Simulation of Monodisperse Flow in Pipes of Different Diameters

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Summary of the Experimental Data

#### 2.1. Tests of Liu [36]

_{B}>, liquid mass flux J

_{L}and gas mass flux J

_{G}, were varied in the investigation. Radial profiles of gas fraction α

_{G}, bubble-size d

_{B}, axial liquid velocity u

_{L}and axial liquid turbulence intensity u

_{L}’ were measured at an axial position L/D = 60, so that fully-developed flow conditions are realized. The average bubble diameter was computed in that work from measurements of the chord length using a dual needle resistivity probe based on the assumption of a spherical shape of the bubble. A change in the gas fraction profile from wall to core peak with increasing bubble size was observed, as well as turbulence suppression in the pipe center for combinations of high liquid and low gas mass flux, which correspond to the smallest bubble sizes. Lacking a precise specification, pressure and temperature presumably are at ambient conditions. An overview of the major characteristics of the test cases selected for the present work is given in Table 1, where the letter “L” denotes the tests of Liu [36].

#### 2.2. Tests of Shawkat et al. [33]

#### 2.3. Tests of Hosokawa and Tomiyama [34]

## 3. Description of the Models

#### 3.1. Two-Fluid Model Equations

**T**includes the viscous, as well as the Reynolds stresses. The volumetric interfacial force density ${\mathit{F}}_{G}^{inter}$ is modelled as a sum of different contributions:

#### 3.2. Bubble Forces

_{D}being computed from the correlation given by Ishii and Zuber [43].

_{L}. A correlation proposed by Tomiyama et al. [44] is used to estimate the lift coefficient C

_{L}as a function of the Eötvös number.

_{W}the correlation given by Hosokawa et al. [45]. Due to the small Morton number of the air-water system, log(Mo) = −10.6, we only use the Eötvös-dependent part given in [45].

_{TD}and the turbulent dispersion force coefficient C

_{TD}. A value of σ

_{TD}= 0.9 is used, and the turbulent dispersion force coefficient is set to C

_{TD}= 2.0.

#### 3.3. Turbulence Modelling

#### 3.4. Geometry and Boundary Conditions

_{t}:

_{in}and the turbulent length scale:

#### 3.5. Other Model Aspects

## 4. Simulation Results

#### 4.1. Pipe with D ≈ 5 cm: Liu [36]

#### 4.2. Pipe with D = 20 cm: Shawkat et al. [33]

#### 4.2.1. Single-Phase Flow

^{+}of the midpoints of the near-wall cells in the measurement position range between 1.5 and 50 for case S20 (J

_{L}= 0.45 m/s) and between two and 70 for case S30 (J

_{L}= 0.68 m/s).

#### 4.2.2. Two-Phase Flow

#### 4.3. Pipe with D = 2.5 cm: Hosokawa and Tomiyama [34]

#### 4.3.1. Single-Phase Flow

#### 4.3.2. Two-Phase Flow

_{L}= 1.0 m/s (cases H21 and H22), which are reasonably well reproduced by the simulations. Furthermore, in the bulk of the flow, the gas fraction is reasonably well reproduced by the simulations. For the lower liquid fluxes of J

_{L}= 0.5 m/s (cases H11 and H12), a different picture emerges.

_{L}= 1.0 m/s, which results in a steeper decrease near the wall and a lower velocity in the center of the pipe.

## 5. Models and Data for Drag Force and Rise Velocity

#### 5.1. Critical Assessment of the Drag Force Modelling

- the “degree of contamination” of the water in air-water flows.
- the influence of pipe walls and shear rate.
- the influence of background turbulence.
- the influence of higher gas fractions (swarm effects).

#### 5.2. Review of Further Measurements of Bubble Rise Velocity

_{G}= 0.027 m/s and J

_{G}= 0.347 m/s and the liquid flux between J

_{L}= 0.376 m/s and J

_{L}= 1.391 m/s. The measured bubble size was in the range of 2 mm to 4 mm. The relative velocity was found to vary between 0.1 and 0.4 m/s, depending on the radial position and the gas and liquid superficial velocities.

_{L}= 1.44 m/s and gas superficial velocities of J

_{G}= 0.076 m/s and J

_{G}= 0.16 m/s, he found relative velocities in the range of 0.15 to 0.25 m/s. Under the assumption of a spherical shape, he computed a mean bubble diameter in the range d

_{B}= 3 mm to d

_{B}= 4 mm from the chord lengths that were measured.

_{L}= 0.175 m/s and J

_{G}= 0.0366 m/s using a four-point optical probe and laser Doppler anemometry. The mean bubble diameter was also in the range of 3 to 4 mm, and the relative velocity varied between 0.21 and 0.27 m/s along the radius of the pipe.

_{L}= 0.22 m/s and two different air superficial velocities of J

_{G}= 0.0024 m/s and J

_{G}= 0.0049 m/s. The average bubble diameter was 2 mm, and the slip velocity between the phases was in the range of 0.19 m/s to 0.25 m/s.

## 6. Discussion and Conclusions

- The baseline model reproduces the experimental data reasonably well independent of the pipe diameter.
- Further improvements of the baseline model are desirable, in particular for the near-wall modelling and the closure for bubble-induced turbulence.

- Knowledge of the full bubble size distribution is essential to explain the observed behavior in bubbly flows. Hence, data including measurements of bubble size distributions are necessary. In addition, such data would also allow a validation of models for bubble coalescence and breakup, as described, e.g., in [77].
- Generally, validation of CFD models for bubbly flow requires more reliable measurement techniques. In particular, reliable measurements of the bubble and liquid velocity are surprisingly scarce at present.
- It is highly desirable that experimental databases for CFD validation provide a comprehensive set of measurements, including both gas and liquid phase properties. At least bubble diameter, liquid and gas velocity profiles and profiles for the liquid turbulent kinetic energy should be given. Otherwise, important features may be missed by the validation.
- Considering the sensitivity of the air-water system to even the smallest contaminations, which are hard to avoid and even harder to quantify, measurement data of other less sensitive systems of gas and liquid should be acquired.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Nomenclature

Notation | Unit | Denomination |

C_{D} | - | drag coefficient |

C_{L} | - | lift coefficient |

C_{TD} | - | turbulent dispersion coefficient |

C_{W} | - | wall force coefficient |

d_{B} | m | bulk volume equivalent sphere bubble diameter |

D | m | pipe diameter |

F | N·m^{−3} | volumetric force density |

g | m·s^{−2} | acceleration of gravity |

I_{t} | - | turbulence intensity |

J | m·s^{−1} | superficial velocity = volumetric flux |

k | m^{2}·s^{−2} | turbulent kinetic energy |

L_{t} | m | turbulent length scale |

L | m | pipe length |

Mo | - | Morton number |

p | Pa | pressure |

r | m | radial coordinate |

t | s | time |

T | N·m^{−2} | stress tensor |

u | m·s^{−1} | velocity |

u’ | m·s^{−1} | fluctuation velocity |

α | - | volume fraction |

ε | m^{2}·s^{−3} | turbulent dissipation rate |

µ | kg·m^{−1}·s^{−1} | dynamic viscosity |

ρ | kg·m^{−3} | density |

σ | N·m^{−1} | surface tension |

ω | s^{−1} | characteristic eddy frequency |

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**Figure 1.**Investigated geometry of all three cases. As indicated by the gray shaded area, axisymmetrical simulations are performed, and only a thin wedge of the pipe has been modelled.

**Figure 2.**Comparison of the radial profiles of the gas fraction obtained using the baseline model with the experimental data of Liu [36]. The solid line is the simulation result and the squares the experimental data (EXP). (

**a**) L11A; (

**b**) L21B; (

**c**) L22A; (

**d**) L21C.

**Figure 3.**Comparison of the radial profiles of the liquid velocity obtained using the baseline model with the experimental data of Liu [36]. The solid line is the simulation result and the squares the experimental data (EXP). (

**a**) L11A; (

**b**) L21B; (

**c**) L22A; (

**d**) L21C.

**Figure 4.**Comparison of the radial profiles of the turbulent kinetic energy obtained using the baseline model in OpenFOAM with the experimental data of Liu [36]. The solid line is the simulation result and the squares the experimental data (EXP). Note that only the axial liquid velocity fluctuations have been measured in the experiments; therefore, the error bars are given to denote the upper and lower limit of the turbulent kinetic energy based on the assumption of isotropic and unidirectional turbulence, respectively. (

**a**) L11A; (

**b**) L21B; (

**c**) L22A; (

**d**) L21C.

**Figure 5.**Radial profiles of the relative velocity obtained using the baseline model. The solid line is the simulation result and the squares the experimental data (EXP). (

**a**) L11A; (

**b**) L21B; (

**c**) L22A; (

**d**) L21C.

**Figure 6.**Comparison of the results obtained for single-phase flow on different grids with experimental data given by Shawkat [33]. S20: J

_{L}= 0.45 m/s, S30: J

_{L}= 0.68 m/s. The different lines correspond to the y

^{+}values given in Table 3. The squares (EXP) are the experimental data. (

**a**) S20 liquid velocity; (

**b**) S30 liquid velocity; (

**c**) S20 turbulent kinetic energy; (

**d**) S30 turbulent kinetic energy.

**Figure 7.**Comparison of the radial profiles of the gas fraction obtained using the baseline model with the experimental data of Shawkat et al. [33]. The solid line is the simulation result and the squares the experimental data (EXP). (

**a**) S21; (

**b**) S23; (

**c**) S31; (

**d**) S33.

**Figure 8.**Comparison of the radial profiles of the liquid velocity obtained using the baseline model with the experimental data of Shawkat et al. [33]. The solid line is the simulation result and the squares the experimental data (EXP). (

**a**) S21; (

**b**) S23; (

**c**) S31; (

**d**) S33.

**Figure 9.**Comparison of the radial profiles of the turbulent kinetic energy obtained using the baseline with the experimental data of Shawkat et al. [33]. The solid line is the simulation result and the squares the experimental data (EXP). (

**a**) S21; (

**b**) S23; (

**c**) S31; (

**d**) S33.

**Figure 10.**Radial profiles of the relative velocity obtained using the baseline with the experimental data of Shawkat et al. [33]. The solid line is the simulation result and the squares the experimental data (EXP). (

**a**) S21; (

**b**) S23; (

**c**) S31; (

**d**) S33.

**Figure 11.**Comparison of results obtained for single-phase flow on different grids with experimental data given by Hosokawa and Tomiyama [34]. The solid lines are simulation results and the squares denote experimental data (EXP). H10: J

_{L}= 0.5 m/s, H20: J

_{L}= 1.0 m/s. (

**a**) H10 liquid velocity; (

**b**) H20 liquid velocity; (

**c**) H10 turbulent kinetic energy; (

**d**) H20 turbulent kinetic energy.

**Figure 12.**Comparison of the radial profiles of the gas fraction obtained using the baseline model with the experimental data of Hosokawa and Tomiyama [34]. The solid line is the simulation result and the squares the experimental data (EXP). (

**a**) H11; (

**b**) H12; (

**c**) H21; (

**d**) H22.

**Figure 13.**Comparison of the radial profiles of the liquid velocity obtained using the baseline model with the experimental data of Hosokawa and Tomiyama [34]. The solid line is the simulation result and the squares the experimental data (EXP). (

**a**) H11; (

**b**) H12; (

**c**) H21; (

**d**) H22.

**Figure 14.**Comparison of the radial profiles of the turbulent kinetic energy obtained using the baseline model with the experimental data of Hosokawa and Tomiyama [34]. The solid line is the simulation result and the squares the experimental data (EXP). (

**a**) H11; (

**b**) H12; (

**c**) H21; (

**d**) H22.

**Figure 15.**Radial profiles of the relative velocity obtained using the baseline model with the experimental data of Hosokawa and Tomiyama [34]. The solid line is the simulation result and the squares the experimental data (EXP). (

**a**) H11; (

**b**) H12; (

**c**) H21; (

**d**) H22.

**Figure 16.**Terminal rise velocity u (

**a**) and drag coefficient C

_{D}(

**b**) of single air bubbles rising in quiescent water with different “degrees” of contamination. See the text in Section 5 for a detailed explanation of the sources for the different data (symbols) and correlations (lines).

**Table 1.**Overview of the major characteristics of the selected test cases, where the letter “H” denotes the experimental data provided by Hosokawa and Tomiyama [34], “L” the cases of Liu [36], and “S” refers to the data of Shawkat et al. [33]. Nominal values are as reported in the quoted references, and adjusted values are obtained as described in Section 3.5.

Name | Pipe Diameter | J_{L} (nom) | J_{G} (nom) | J_{L} (adj) | J_{G} (adj) | <d_{B}> | <α_{G}> |
---|---|---|---|---|---|---|---|

mm | m/s | m/s | m/s | m/s | mm | % | |

H11 | 25.0 | 0.5 | 0.018 | 0.5 | 0.018 | 3.2 | 2.5 |

H12 | 25.0 | 0.5 | 0.025 | 0.5 | 0.031 | 4.3 | 4.1 |

H21 | 25.0 | 1.0 | 0.020 | 1.0 | 0.035 | 3.5 | 2.8 |

H22 | 25.0 | 1.0 | 0.036 | 1.0 | 0.042 | 3.7 | 3.2 |

L11A | 57.2 | 0.5 | 0.1 | 0.5 | 0.12 | 2.9 | 15.2 |

L21B | 57.2 | 1.0 | 0.1 | 1.0 | 0.14 | 3.0 | 10.6 |

L21C | 57.2 | 1.0 | 0.1 | 1.0 | 0.13 | 4.2 | 9.6 |

L22A | 57.2 | 1.0 | 0.2 | 1.0 | 0.22 | 3.9 | 15.7 |

S21 | 200 | 0.45 | 0.015 | 0.41 | 0.019 | 4.1 | 2.4 |

S23 | 200 | 0.45 | 0.1 | 0.5 | 0.108 | 5.0 | 10.7 |

S31 | 200 | 0.68 | 0.015 | 0.67 | 0.018 | 3.2 | 1.7 |

S33 | 200 | 0.68 | 0.1 | 0.71 | 0.12 | 4.7 | 10.1 |

**Table 2.**Material properties for the air water system at atmospheric pressure and 25 °C temperature.

ρ_{L} | 997.0 | kg·m^{−3} |

µ_{L} | 8.899 × 10 ^{−4} | kg m^{−1}·s^{−1} |

ρ_{G} | 1.185 | kg·m^{−3} |

µ_{G} | 1.831 × 10 ^{−5} | kg·m^{−1}·s^{−1} |

σ | 0.072 | N·m^{−1} |

**Table 3.**Different grids used in the simulations of the single-phase pipe flow measured by Shawkat et al. [33].

Grid | n_{z} | n_{r} | $\mathit{\psi}\mathbf{=}\frac{\Delta {x}_{\mathit{s}\mathit{m}\mathit{a}\mathit{l}\mathit{l}}}{\Delta {x}_{\mathit{l}\mathit{a}\mathit{r}\mathit{g}\mathit{e}}}$ | $\Delta {x}_{small}\left(mm\right)$ | y^{+} | |
---|---|---|---|---|---|---|

S20 | S30 | |||||

u24 | 224 | 24 | 1.0 | 4.167 | 48.9 | 70.3 |

nu48 | 448 | 48 | 0.5 | 1.443 | 18.1 | 25.4 |

u96 | 896 | 96 | 1.0 | 1.042 | 13.4 | 18.9 |

nu48 | 448 | 48 | 0.1 | 0.528 | 6.3 | 9.5 |

nu192 | 448 | 192 | 0.1 | 0.133 | 1.5 | 2.2 |

**Table 4.**Different grids used in the simulations of the single-phase pipe flow measured by Hosokawa and Tomiyama [34].

Grid | n_{z} | n_{r} | $\mathit{\psi}\mathbf{=}\frac{\Delta {x}_{\mathit{s}\mathit{m}\mathit{a}\mathit{l}\mathit{l}}}{\Delta {x}_{\mathit{l}\mathit{a}\mathit{r}\mathit{g}\mathit{e}}}$ | $\Delta {x}_{small}\left(mm\right)$ | y^{+} | |
---|---|---|---|---|---|---|

H10 | H20 | |||||

nu16 | 399 | 16 | 0.75 | 0.674 | 12.5 | xx |

nu25 | 399 | 25 | 0.5 | 0.346 | 5.9 | 11.6 |

nu35 | 399 | 35 | 0.5 | 0.247 | 4.2 | 7.8 |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license ( http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Kriebitzsch, S.; Rzehak, R.
Baseline Model for Bubbly Flows: Simulation of Monodisperse Flow in Pipes of Different Diameters. *Fluids* **2016**, *1*, 29.
https://doi.org/10.3390/fluids1030029

**AMA Style**

Kriebitzsch S, Rzehak R.
Baseline Model for Bubbly Flows: Simulation of Monodisperse Flow in Pipes of Different Diameters. *Fluids*. 2016; 1(3):29.
https://doi.org/10.3390/fluids1030029

**Chicago/Turabian Style**

Kriebitzsch, Sebastian, and Roland Rzehak.
2016. "Baseline Model for Bubbly Flows: Simulation of Monodisperse Flow in Pipes of Different Diameters" *Fluids* 1, no. 3: 29.
https://doi.org/10.3390/fluids1030029