# Prediction of Wall Heat Fluxes in a Rocket Engine with Conjugate Heat Transfer Based on Large-Eddy Simulation

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## Abstract

**:**

## 1. Introduction

## 2. The JAXA Chamber Configurations

## 3. Computational Setup

#### 3.1. Numerical Approach to Solve the Reactive Flow Field

#### 3.1.1. Numerical Scheme

#### 3.1.2. Liquid Injection Model

#### 3.1.3. Kinetics for the Combustion Model

#### 3.2. Numerical Approach to Solve the Heat Equation in the Solid

#### 3.3. Conjugate Heat Transfer Coupling Approach

#### 3.4. Meshes and Boundary Conditions

## 4. Results

#### 4.1. Description of the Two-Phase Flow

#### 4.2. Flame Structure

#### 4.3. Comparison between Ribbed and Smooth Chambers

#### 4.3.1. Stratification in the Inter-Rib Region

#### 4.3.2. Heat Transfer

#### 4.4. Heat Transfer: Comparison with Experiment

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 5.**Fluid mesh for JAXA calculation. Prism aspect ratio is kept under five. There are only two stacks of prisms to limit cell deformation.

**Figure 7.**Multiple iso-surfaces of temperature (from 3000 to 3600 K); transparency depends on temperature. The ${30}^{\circ}$ sector simulated is duplicated to represent three injectors flames.

**Figure 9.**Ribbed JAXA chamber: cuts of mean axial velocity u at ${0}^{\circ}$ (

**bottom**) and ${15}^{\circ}$ (

**top**). Recirculation zones are evidenced by the dashed line circles.

**Figure 10.**Ribbed JAXA chamber: cuts of radial RMS velocity ${u}_{r}^{RMS}$ with superimposed field of liquid volume fraction ${\alpha}_{l}$ at ${0}^{\circ}$ (

**bottom**) and ${15}^{\circ}$ (

**top**).

**Figure 11.**Ribbed chamber: instantaneous cuts of heat release HR with superimposed iso-lines of temperature.

**Figure 12.**Ribbed JAXA chamber instantaneous cut of the outer injector. (

**Top**): oxygen mass fraction ${Y}_{{O}_{2}}$; (

**bottom**): Takeno index, [−1]: non-premixed flame, and [1] premixed flame.

**Figure 13.**Scatter plots of heat release, temperature, and species mass fraction, plotted along the mixture fraction z colored by Takeno index, compared to 1D pure diffusion flame (NP-NP) and mixed Non Premixed-Premixed (NP-P) flame ($a=5000$ s${}^{-1}$).

**Figure 14.**Ribbed JAXA chamber strain rate distribution taken along the stoichiometric iso-surface (for $z={z}_{st}$).

**Figure 15.**Time-average temperature of the inner wall of the solid liner for JAXA smooth (

**top**) and ribbed (

**bottom**) burners extracted from AVTP averaged solutions.

**Figure 16.**Time-averaged temperature on cross-section cuts in fluid and solid domains for the smooth and ribbed configurations.

**Figure 17.**Time-averaged $OH$ mass fraction on cross-section cuts for the smooth and ribbed configurations.

**Figure 18.**Time-averaged ${H}_{2}$ and ${H}_{2}{O}_{2}$ mass fractions on cross-sections cuts for the smooth and ribbed configurations.

**Figure 19.**Time-averaged fields of radial and axial velocity ${u}_{r}$ and ${u}_{x}$ on cross-section cuts on the ribbed combustion chamber.

**Figure 20.**Time-averaged wall heat flux of the ${30}^{\circ}$ sector simulation of smooth (

**top**) and ribbed (

**bottom**) burners.

**Figure 21.**Time-averaged of wall heat flux circumferential for the ribbed and smooth configurations at three axial positions. The distance from the axis of the chamber to the wall position R is given to identify ribs and inter-rib regions.

**Figure 22.**Time-averaged wall heat flux of the ribbed and smooth configuration obtained by the present CHT-LES confronted to the experiment. RANS calculation published by JAXA [11] are also represented.

**Table 1.**Geometrical characteristics of the two JAXA calorimeter chambers [11].

Ribbed Case | Smooth Case | |
---|---|---|

chamber length [mm] | 153 | 117 |

chamber diameter [mm] | 65/67 | 66 |

number of injectors | 18 | 18 |

**Table 2.**Operating conditions of the two JAXA calorimeter chambers [11].

Ribbed | Smooth | |
---|---|---|

${O}_{2}/{H}_{2}$ mass ratio | $5.6$ | $5.3$ |

chamber pressure [bar] | 35 | 36 |

total mass flow (${H}_{2}+{O}_{2}$) [kg·m${}^{-3}$] | $0.616$ | $0.626$ |

${O}_{2}$ injection temperature [K] | 95 | 95 |

${H}_{2}$ injection temperature [K] | 275 | 275 |

Ribbed | Smooth | |
---|---|---|

$We$ | 14,485 | 14,500 |

$R{e}_{l}$ | 92,000 | 90,700 |

J | 1.8 | 1.85 |

cone length L [mm] | 10.6 | 10.85 |

injection diameter $\mu $ [m] | 41.2 | 40.7 |

${\theta}_{lip}$ | ${38.62}^{\circ}$ | ${38.54}^{\circ}$ |

${\alpha}_{l}$ | 0.118 | 0.12 |

$St$ | 0.76 | 0.11 |

Copper Alloy | Inconel600 | |
---|---|---|

density ${\rho}_{s}$ | 8814 kg/m${}^{3}$ | 8470 kg/m${}^{3}$ |

heat capacity ${C}_{s}$ at 300 K | 377 J/kg/K | 444 J/kg/K |

conductivity ${\lambda}_{s}$ at 300 K | 322 W/m/K | 14.9 W/m/K |

Ribbed Chamber | Smooth Chamber | |
---|---|---|

Fluid mesh | $30.6\phantom{\rule{3.33333pt}{0ex}}M.\phantom{\rule{3.33333pt}{0ex}}cells$ | $31.5\phantom{\rule{3.33333pt}{0ex}}M.\phantom{\rule{3.33333pt}{0ex}}cells$ |

Solid mesh | $25\phantom{\rule{3.33333pt}{0ex}}M.\phantom{\rule{3.33333pt}{0ex}}cells$ | $8\phantom{\rule{3.33333pt}{0ex}}M.\phantom{\rule{3.33333pt}{0ex}}cells$ |

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**MDPI and ACS Style**

Potier, L.; Duchaine, F.; Cuenot, B.; Saucereau, D.; Pichillou, J.
Prediction of Wall Heat Fluxes in a Rocket Engine with Conjugate Heat Transfer Based on Large-Eddy Simulation. *Entropy* **2022**, *24*, 256.
https://doi.org/10.3390/e24020256

**AMA Style**

Potier L, Duchaine F, Cuenot B, Saucereau D, Pichillou J.
Prediction of Wall Heat Fluxes in a Rocket Engine with Conjugate Heat Transfer Based on Large-Eddy Simulation. *Entropy*. 2022; 24(2):256.
https://doi.org/10.3390/e24020256

**Chicago/Turabian Style**

Potier, Luc, Florent Duchaine, Bénédicte Cuenot, Didier Saucereau, and Julien Pichillou.
2022. "Prediction of Wall Heat Fluxes in a Rocket Engine with Conjugate Heat Transfer Based on Large-Eddy Simulation" *Entropy* 24, no. 2: 256.
https://doi.org/10.3390/e24020256