# Numerical Simulation of Turbulent Flow in Eccentric Co-Rotating Heat Transfer

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## Abstract

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## 1. Introduction

## 2. Methodology

#### 2.1. Geometry and Mesh Details

#### 2.2. Mathematical Model

#### 2.2.1. Turbulent Flow of an Incompressible Ideal Fluid

- The STD $k-\omega $ model and the transformed $k-\u03f5$ model are multiplied by a blending function and added together. The blending function is designed to take a value of one in the near-wall region, which activates the STD $k-\omega $ model, and zero away from the surface, which activates the transformed $k-\u03f5$ model.
- A damped cross-diffusion derivative term is included in the BSL model equation.
- The model includes a new source term in order to capture the natural convection turbulent heat transfer.
- The modeling constants are different.

#### 2.2.2. Initial and Boundary Conditions

## 3. Validation Study

## 4. Results

#### 4.1. The Effect of the Inner Counterclockwise Rotating Velocity

#### 4.2. The Effect of Outer Counterclockwise Rotating Velocity

#### 4.3. The Effect of Inner–Outer Counterclockwise Rotating Velocity

#### 4.4. The Effect of Inner Clockwise, Outer Counterclockwise Velocity

#### 4.5. Temperature Mixing Performance

#### 4.6. Velocity Distribution

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**A three-dimensional eccentric cylindrical tube: (

**a**) Computational grid and (

**b**) Boundary conditions.

**Figure 4.**Numerical solutions for the SST $k-\omega $ with the buoyancy effect, compared with measurements from previous experiments: (

**a**) Static temperature contour; (

**b**) $\theta ={0}^{\xb0}$; (

**c**) $\theta ={30}^{\xb0}$; (

**d**) $\theta ={60}^{\xb0}$; (

**e**) $\theta ={90}^{\xb0}$; (

**f**) $\theta ={120}^{\xb0}$; (

**g**) $\theta ={150}^{\xb0}$; and (

**h**) $\theta ={180}^{\xb0}$.

**Figure 5.**Numerical solutions for the inner-rotating cylinders: (

**a**) 2 rad/s; (

**b**) 5 rad/s; (

**c**) 10 rad/s; (

**d**) 15 rad/s; (

**e**) 20 rad/s; and (

**f**) 25 rad/s.

**Figure 6.**Numerical solutions for outer-counterclockwise cylinders: (

**a**) 2 rad/s; (

**b**) 5 rad/s; (

**c**) 10 rad/s; (

**d**) 15 rad/s; (

**e**) 20 rad/s; and (

**f**) 25 rad/s.

**Figure 7.**Numerical solutions for the inner–outer counterclockwise rotation with the same angular velocity: (

**a**) 2 rad/s; (

**b**) 5 rad/s; (

**c**) 10 rad/s; (

**d**) 15 rad/s; (

**e**) 20 rad/s; and (

**f**) 25 rad/s.

**Figure 8.**Numerical solutions for the inner clockwise, outer counterclockwise rotations with the same angular velocity: (

**a**) 2 rad/s; (

**b**) 5 rad/s; (

**c**) 10 rad/s; (

**d**) 15 rad/s; (

**e**) 20 rad/s; and (

**f**) 25 rad/s.

**Figure 9.**Variations in mixing temperature with respect to wall rotation velocity. Black, inner wall rotation; red, outer wall rotation; blue, inner–outer clockwise rotation; yellow, inner–outer counterclockwise rotation.

**Figure 10.**Velocity distribution: (

**a**) Inner 5 rad/s; (

**b**) inner 25 rad/s; (

**c**) outer 5 rad/s; (

**d**) outer 25 rad/s; (

**e**) outer 5 rad/s, inner 5 rad/s; (

**f**) inner 25 rad/s, outer 25 rad/s; (

**g**) inner 5 rad/s, outer −5 rad/s; and (

**h**) inner 25 rad/s, outer −25 rad/s.

Solution Methods | Setting |
---|---|

Pressure–Velocity Coupling | Coupled |

Gradient | Least Squares Cell Based |

Pressure | Body Force Weighted |

Density | Second Order Upwind |

Momentum | Second Order Upwind |

Turbulent Kinetic Energy | Second Order Upwind |

Specific Dissipation Rate | Second Order Upwind |

Energy | Second Order Upwind |

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**MDPI and ACS Style**

Kaewbumrung, M.; Charoenloedmongkhon, A. Numerical Simulation of Turbulent Flow in Eccentric Co-Rotating Heat Transfer. *Fluids* **2022**, *7*, 131.
https://doi.org/10.3390/fluids7040131

**AMA Style**

Kaewbumrung M, Charoenloedmongkhon A. Numerical Simulation of Turbulent Flow in Eccentric Co-Rotating Heat Transfer. *Fluids*. 2022; 7(4):131.
https://doi.org/10.3390/fluids7040131

**Chicago/Turabian Style**

Kaewbumrung, Mongkol, and Akapak Charoenloedmongkhon. 2022. "Numerical Simulation of Turbulent Flow in Eccentric Co-Rotating Heat Transfer" *Fluids* 7, no. 4: 131.
https://doi.org/10.3390/fluids7040131