# Turbulent Bubble-Laden Channel Flow of Power-Law Fluids: A Direct Numerical Simulation Study

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Description of the Power-Law Model

## 3. Numerical Method and Flow Configuration

#### 3.1. Governing Equations and Numerical Framework

_{b}ensures the sharpness of the volume fraction field, and its value on the global grid is computed from the single markers ${f}_{b}$ as $f=max\left({f}_{b}\right)$. In addition, the total surface tension contribution of multiple interface segments present in one computational cell is determined as the sum of the individual surface tension terms [18].

#### 3.2. Case Definition

## 4. Results and Discussion

#### 4.1. Average Wall Shear Stress and Viscosity Profiles

#### 4.2. First- and Second-Order Fluid Statistics

#### 4.3. Dissipation Rates

#### 4.4. Bubble Deformation

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Parameters for the Case Definition

Symbol | Parameter | Value |
---|---|---|

H | Channel half width | 1 |

${\rho}_{l}$ | Density of the liquid phase | 1 |

${\rho}_{g}$ | Density of the gas phase | $0.1$ |

${\mu}_{l}$ | Dynamic viscosity of the liquid phase for $n=1$ | $1/3000$ |

${\mu}_{g}$ | Dynamic viscosity of the gas phase | $1/3000$ |

K | Consistency index | $1/3000$ |

${d}_{b}$ | Bubble diameter | $0.25$ |

g | Gravitational acceleration | $0.1$ |

$\sigma $ | Surface tension coefficient | $0.02$ ($Eo=0.3125$), $1.667\times {10}^{-3}$ ($Eo=3.75$) |

N | Number of bubbles | 72 |

$\dot{V}\phantom{\rule{0.166667em}{0ex}}(\mathrm{and}\phantom{\rule{0.166667em}{0ex}}\dot{V}/{L}_{z})$ | Constant volumetric flow rate | $1.2636$ ($0.8044$) |

${u}_{\tau}^{init}$ | Friction velocity of the initialization setup | $0.0424$ |

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**Figure 1.**Apparent dynamic viscosity ${\mu}_{a}$, normalized with the viscosity of the Newtonian fluid $\mu $, as a function of the shear rate $\dot{\gamma}$.

**Figure 2.**Downflow channel configuration. The gray areas represent the no-slip walls, the x- and z-direction are treated with periodic boundaries.

**Figure 3.**Grace diagram for bubble shapes [34]. Purple point and lines specify case “tph10_loEo”, blue point and lines specify case “tph10_hiEo”.

**Figure 4.**Total instantaneous wall-averaged shear stress ${\tau}_{w}$, normalized by ${\mu}_{l}({u}_{\tau}^{init}/H)$ to yield $R{e}_{\tau}$, as a function of the dimensionless time $t/T$ for the single-phase (

**a**) and the two-phase setups with $Eo=0.3125$ (

**b**) and $Eo=3.75$ (

**c**). Statistics are collected starting at $t/T=4.24$, which is indicated by the vertical dotted line.

**Figure 5.**Average apparent dynamic viscosity of the liquid phase ${\langle {\mu}_{a}\rangle}_{l}$, normalized by the dynamic viscosity of the Newtonian fluid ${\mu}_{l}$, as a function of the wall-normal coordinate for the single-phase (

**a**) and the two-phase setups with $Eo=0.3125$ (

**b**) and $Eo=3.75$ (

**c**).

**Figure 6.**Average gas volume fraction field $\overline{f}$ as a function of the wall-normal coordinate for the two-phase setups with $Eo=0.3125$ (

**a**) and $Eo=3.75$ (

**b**).

**Figure 7.**Average stream-wise velocity, normalized by the friction velocity of the initialization setup to yield $\langle {u}^{+}\rangle $, as a function of the wall-normal coordinate for the single-phase (

**a**) and the two-phase setups with $Eo=0.3125$ (

**b**) and $Eo=3.75$ (

**c**).

**Figure 8.**Semilogarithmic plot of the average stream-wise velocity, normalized by the actual friction velocity to yield $\langle {U}^{+}\rangle $, as a function of the dimensionless distance from the wall for the single-phase (

**a**) and the two-phase setups with $Eo=0.3125$ (

**b**) and $Eo=3.75$ (

**c**).

**Figure 9.**Root mean square of the stream-wise velocity fluctuations ${u}^{\prime}$, normalized by the friction velocity of the initialization setup, as a function of the wall-normal coordinate for the single-phase (

**a**) and the two-phase setups with $Eo=0.3125$ (

**b**) and $Eo=3.75$ (

**c**).

**Figure 10.**Root mean square of the wall-normal velocity fluctuations ${v}^{\prime}$, normalized by the friction velocity of the initialization setup, as a function of the wall-normal coordinate for the single-phase (

**a**) and the two-phase setups with $Eo=0.3125$ (

**b**) and $Eo=3.75$ (

**c**).

**Figure 11.**Root mean square of the span-wise velocity fluctuations ${w}^{\prime}$, normalized by the friction velocity of the initialization setup, as a function of the wall-normal coordinate for the single-phase (

**a**) and the two-phase setups with $Eo=0.3125$ (

**b**) and $Eo=3.75$ (

**c**).

**Figure 12.**Average dissipation rate $\langle \epsilon \rangle $, normalized by ${\epsilon}_{ref}$, as a function of the wall-normal coordinate for the single-phase (

**a**) and the two-phase setups with $Eo=0.3125$ (

**b**) and $Eo=3.75$ (

**c**).

**Figure 13.**Average turbulent kinetic energy $\langle k\rangle $, normalized by ${\left({u}_{\tau}^{init}\right)}^{2}$, as a function of the wall-normal coordinate for the single-phase (

**a**) and the two-phase setups with $Eo=0.3125$ (

**b**) and $Eo=3.75$ (

**c**).

**Figure 14.**Average dissipation rate of the liquid phase ${\langle \epsilon \rangle}_{l}$, normalized by ${\epsilon}_{ref}$, as a function of the wall-normal coordinate for the single-phase (

**a**) and the two-phase setups with $Eo=0.3125$ (

**b**) and $Eo=3.75$ (

**c**).

**Figure 15.**Isosurfaces for volume fraction $f=0.5$ (gray) and wall-normal slice of the velocity magnitude on the mid-plane. (

**a**) setup with $n=1$ and $Eo=0.3125$; (

**b**) setup with $n=1$ and $Eo=3.75$. It is worth noting that some bubbles intersect with the periodic boundaries, which should not be confused with a concave surface topology.

**Table 1.**DNS case overview in terms of power-law index n, bubble count ${N}_{b}$, Eötvös number $Eo$, bubble Reynolds number $R{e}_{b}$, and dimensionless distance from the wall ${y}^{+}$.

Case | n | ${\mathit{N}}_{\mathit{b}}$ | $\mathit{Eo}$ | ${\mathit{Re}}_{\mathit{b}}$ | ${\mathit{y}}^{+}$ |
---|---|---|---|---|---|

sph07 | 0.7 | 0 | - | - | 0.9574 |

sph10 | 1 | 0 | - | - | 0.6178 |

sph13 | 1.3 | 0 | - | - | 0.4428 |

tph07_loEo | 0.7 | 72 | 0.3125 | 132.30 | 1.1743 |

tph10_loEo | 1 | 72 | 0.3125 | 123.35 | 0.6687 |

tph13_loEo | 1.3 | 72 | 0.3125 | 117.58 | 0.4559 |

tph07_hiEo | 0.7 | 72 | 3.75 | 109.12 | 1.1860 |

tph10_hiEo | 1 | 72 | 3.75 | 105.85 | 0.6750 |

tph13_hiEo | 1.3 | 72 | 3.75 | 100.51 | 0.4582 |

**Table 2.**Average normalized liquid dynamic viscosities for all nine setups. The viscosity fields have been averaged over all spatial directions and over time, which is indicated by ${\langle \phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\rangle}^{x,y,z,t}$.

${\langle {\mathit{\mu}}_{\mathit{a}}\rangle}_{\mathit{l}}^{\mathit{x},\mathit{y},\mathit{z},\mathit{t}}/{\mathit{\mu}}_{\mathit{l}}$ | |||
---|---|---|---|

Single Phase | $\mathit{Eo}=0.3125$ | $\mathit{Eo}=3.75$ | |

n = 0.7 | 1.23 | 0.97 | 0.97 |

n = 1 | 1 | 1 | 1 |

n = 1.3 | 0.99 | 1.08 | 1.09 |

**Table 3.**Mean $\overline{A}$ and fluctuation ${A}^{\prime}$ of the normalized surface area, averaged over all bubbles.

$\mathit{Eo}=0.3125$ | $\mathit{Eo}=3.75$ | |||||
---|---|---|---|---|---|---|

$\mathit{n}=\mathbf{0.7}$ | $\mathit{n}=\mathbf{1}$ | $\mathit{n}=\mathbf{1.3}$ | $\mathit{n}=\mathbf{0.7}$ | $\mathit{n}=\mathbf{1}$ | $\mathit{n}=\mathbf{1.3}$ | |

$\overline{A}$ | $1.0190$ | $1.0188$ | $1.0187$ | $1.0842$ | $1.0783$ | $1.0705$ |

${A}^{\prime}$ | $6.8385\times {10}^{-4}$ | $4.8121\times {10}^{-4}$ | $3.2722\times {10}^{-4}$ | $2.7642\times {10}^{-2}$ | $2.3220\times {10}^{-2}$ | $1.8235\times {10}^{-2}$ |

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**MDPI and ACS Style**

Bräuer, F.; Trautner, E.; Hasslberger, J.; Cifani, P.; Klein, M.
Turbulent Bubble-Laden Channel Flow of Power-Law Fluids: A Direct Numerical Simulation Study. *Fluids* **2021**, *6*, 40.
https://doi.org/10.3390/fluids6010040

**AMA Style**

Bräuer F, Trautner E, Hasslberger J, Cifani P, Klein M.
Turbulent Bubble-Laden Channel Flow of Power-Law Fluids: A Direct Numerical Simulation Study. *Fluids*. 2021; 6(1):40.
https://doi.org/10.3390/fluids6010040

**Chicago/Turabian Style**

Bräuer, Felix, Elias Trautner, Josef Hasslberger, Paolo Cifani, and Markus Klein.
2021. "Turbulent Bubble-Laden Channel Flow of Power-Law Fluids: A Direct Numerical Simulation Study" *Fluids* 6, no. 1: 40.
https://doi.org/10.3390/fluids6010040