# Large Eddy Simulation of Conjugate Heat Transfer in a Ribbed Channel: Reynolds Number Effect

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Methods and Code Validation

#### 2.1. Numerical Methods

_{i}in Equation (2) are mass source and momentum forcing that satisfy the no slip condition on a solid surface, respectively. Using the IBM, the code that could contain solids was later revised to solve for convective heat transfer as well [36].

_{ij}and q

_{j}are the sub-grid scale (SGS) stress and heat flux, respectively; τ

_{ij}is determined as a dynamic SGS model using scale correlation by framing a test filter around the cell [37,38]; and q

_{j}is also determined dynamically like τ

_{ij}to secure better results in problems where velocity and temperature are dissimilar [39]. The simulation was carried out for 10,000 time steps to reach a fully developed flow. After the initial steps, an additional 10,000 time steps (t U

_{b}/D

_{h}= 5) were performed to obtain the statistics.

#### 2.2. Code Validation

## 3. Results and Discussion

#### 3.1. Effect of Reynolds Number on the Isothermal Ribbed Channel

_{0}, which represents the Nusselt number in the absence of ribs. In the channel wall between the ribs (Figure 4a), the laminar flow (green line in Figure 4a) shows a value less than 1 in the entire section. The data reported in [8] (green triangles in Figure 4a) and the difference occurring near the rib can be considered as a three-dimensional effect based on the observations reported in [8].

#### 3.2. Time-Averaged Thermal Fields in the Conducting Ribbed Channel

#### 3.3. Turbulent Heat Transfer Statistics

#### 3.4. Thermal Performance

_{f}) is the ratio of heat transfer rate between the bare base and the fin, and is defined using the following equation [42]. Usually, it should be 2 or more.

## 4. Conclusions

- In pure convection, when the Reynolds number is lowered from 30,000 to 7000, the heat transfer increases by 5% on the channel wall, but decreases by 20% on the rib.
- When the thermal conductivity ratio is more than 10, the Reynolds number effect is stronger in the rib than in the wall. When Re = 7000, the heat transfer coefficient ratio in the rib is larger than that when Re = 30,000.
- Compared with the turbulent flow, the effect of conduction in the laminar flow is observed at a low thermal conductivity ratio, and the effect of heat transfer promotion is not large in the typical ribbed channel geometry of the gas turbines.
- In the turbulent flow, when K* = 100 or more, the heat transfer promotion effect of the ribbed channel can be expected even at a low Reynolds number. If K* = 10 or less, then the heat transfer promotion performance of the rib becomes worse than that in the laminar flow, and thus, the effect of the rib cannot be expected under these conditions.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

A_{c:b} | cross-sectional area at the base |

A_{rib} | rib surface area |

Bi | Biot number (=hd/k_{s}) |

C* | heat capacity ratio (=(ρ c_{p})_{f}/(ρ c_{p})_{s}) |

d | thickness of the channel wall |

f | frication factor |

f_{i} | momentum forcing |

k | thermal conductivity |

K* | thermal conductivity ratio (=k_{s}/k_{f}) |

ms | mass source/sink |

Nu | Nusselt number (=hD_{h}/k_{f}) |

q” | heat flux |

q | heat transfer rate |

q_{f} | heat transfer rate through a fin |

Re | bulk Reynolds number (=U_{b}D_{h}/ν) |

t | time |

T | temperature |

T_{b} | bulk temperature |

T_{w} | wall temperature |

U_{b} | bulk velocity |

Greek symbols | |

ε_{f} | fin effectiveness |

η_{f} | fin efficiency |

ν | kinematic viscosity |

θ | dimensionless temperature (=(T − T_{b})/(T_{w} − T_{b})) |

Θ | time-averaged dimensionless temperature |

ω | index function between the solid and the fluid |

Subscripts | |

f | fluid or fin |

rms | root-mean-square value |

s | solid |

0 | fully developed value in a smooth pipe |

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**Figure 1.**Computational domain and grid system: (

**a**) schematic of the internal cooling passage; (

**b**) computational domain; and (

**c**) grid system.

**Figure 3.**Time-averaged flow and temperature fields on the isothermal walls for each Reynolds number; (

**a**) time-averaged streamlines; (

**b**) time-averaged temperature field.

**Figure 4.**Effect of Reynolds number on the local Nusselt number along the wall; (

**a**) Nusselt number along the interface between the ribs and (

**b**) Nusselt number on the rib.

**Figure 5.**Time-averaged temperature fields; (

**a**) K* = 566.26, (

**b**) K* = 100.00, (

**c**) K* = 10.00, and (

**d**) K* = 1.00.

**Figure 6.**Effect of the Reynolds number on the local Nusselt number along the channel wall between the ribs: (

**a**) for high thermal conductivity ratios (K* = 566.26 and 100.00) and (

**b**) for low thermal conductivity ratios (K* = 10.00 and 1.00).

**Figure 7.**Effect of Reynolds number on the local convective heat transfer in the rib: (

**a**) for high thermal conductivity ratios (K* = 566.26 and 100.00) and (

**b**) for low thermal conductivity ratios (K* = 10.00 and 1.00).

**Figure 8.**Turbulent heat flux distributions for (

**a**) K* = 566.26, (

**b**) K* = 100.00, (

**c**) K* = 10.00, and (

**d**) K* = 1.00.

**Figure 9.**Temperature fluctuation contours for (

**a**) K* = 566.26, (

**b**) K* = 100.00, (

**c**) K* = 10.00, and (

**d**) K* = 1.00.

**Figure 11.**Heat transfer enhancement of the ribbed channel: (

**a**) total heat transfer rate and (

**b**) thermal performance.

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**MDPI and ACS Style**

Ahn, J.; Song, J.C.; Lee, J.S.
Large Eddy Simulation of Conjugate Heat Transfer in a Ribbed Channel: Reynolds Number Effect. *Processes* **2022**, *10*, 1928.
https://doi.org/10.3390/pr10101928

**AMA Style**

Ahn J, Song JC, Lee JS.
Large Eddy Simulation of Conjugate Heat Transfer in a Ribbed Channel: Reynolds Number Effect. *Processes*. 2022; 10(10):1928.
https://doi.org/10.3390/pr10101928

**Chicago/Turabian Style**

Ahn, Joon, Jeong Chul Song, and Joon Sik Lee.
2022. "Large Eddy Simulation of Conjugate Heat Transfer in a Ribbed Channel: Reynolds Number Effect" *Processes* 10, no. 10: 1928.
https://doi.org/10.3390/pr10101928