#
A Two-Temperature Open-Source CFD Model for Hypersonic Reacting Flows, Part One: Zero-Dimensional Analysis^{ †}

^{1}

^{2}

^{3}

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^{†}

^{‡}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Governing Equations

#### 2.1.1. Two-Temperature Model

_{2}and O

_{2}, and $3.0\times {10}^{-22}\phantom{\rule{0.222222em}{0ex}}{m}^{2}$ for NO [24].

#### 2.1.2. Non-Equilibrium Navier-Stokes-Fourier Equations

#### 2.2. Chemistry Source Terms

#### 2.2.1. Generalities

#### 2.2.2. Chemistry-Vibration Coupling: The Park TTv Model

#### 2.2.3. Chemistry-Vibration Coupling: The Coupled Vibration-Dissociation-Vibration Model

#### 2.3. Implementation in OpenFOAM 2.3.0

_{2}+ H

_{2}⇌ 2 HI. This reaction takes place at a constant temperature of 700 K and at an initial pressure of 0.528 atm. The species concentration versus time can be easily derived [41] to produce the analytical solution shown in Figure 2. hyFoam results are in very satisfactory agreement thus validating the first stage of the new solver implementation.

## 3. Results and Discussion

#### 3.1. Vibrational-Translational Relaxation of a Single-Species Gas

#### 3.1.1. Case without Electronic Energy

#### 3.1.2. Case with Electronic Energy

#### 3.2. Vibrational-Translational Relaxation of a Non-Reacting Multi-Species Gas

_{2}and N in equal proportions. The initial temperatures remain unchanged with regard to the previous paragraph and number densities are equal to $5.0\times {10}^{22}\phantom{\rule{0.222222em}{0ex}}{\mathrm{m}}^{-3}$ for both species. In Figure 5, the equilibrium temperatures specified on the right-hand side are once more consistent with energy equipartition and the same values are recovered using dsmcFoam and LeMANS. The disaccord between the hy2Foam and dsmcFoam solutions appear to be slightly greater that in the case considered without atomic nitrogen, nonetheless, they do remain satisfactory. The results using LeMANS are not in agreement with hy2Foam; however, a different convention is adopted for Equation (15) in LeMANS where the number density ${n}_{m,s}$ represents the mixture number density and ${\sigma}_{m}$ equals $1\times {10}^{-20}\phantom{\rule{0.222222em}{0ex}}{\mathrm{m}}^{2}$. For clarity, these modifications have been temporarily implemented in hy2Foam and are shown by the red solid line in Figure 5. The temperature profiles are now shown to be nearly superimposed, thus verifying the hy2Foam implementation for a mixture.

_{2}configuration shown in Figure 4, the increase in equilibrium temperature due to the loss of half of the mixture vibrational energy is less pronounced with the inclusion of the electronic mode. This can be explained by the relative importance of the electronic mode of N that brings 1.39 degrees of freedom to the mixture at ${T}^{eq}$, and thus compensates part of the vibrational energy loss.

#### 3.3. Vibrational-Translational and Vibrational-Vibrational Relaxations

_{2}and O

_{2}vibrational energy pools remain distinct throughout the calculation. The solver LeMANS has only one vibrational temperature. Logically, this unique vibrational temperature profile should be somewhere between the ${T}_{v,{\mathrm{N}}_{2}}\left(t\right)$ and ${T}_{v,{\mathrm{O}}_{2}}\left(t\right)$ profiles when the V–V transfer is enabled and this is precisely what is observed in Figure 6.

#### 3.4. Relaxation of a Chemically-Reacting Mixture

_{2}−N

_{2}and N

_{2}−N interactions [47].

#### 3.5. Chemically-Reacting Air

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

Species s | ${\mathcal{M}}_{\mathit{s}}$ (g·m${}^{-3}$) | ${\mathit{h}}_{\mathit{s}}^{\xb0}$ (J·kg${}^{-1}$) | ${\mathit{\theta}}_{\mathit{v},\mathit{s}}$ (K) | ${\mathit{D}}_{\mathit{s}}$ (J·kg${}^{-1}$) |
---|---|---|---|---|

N_{2} | 28.0134 | 0 | 3,371 | 3.36 × 10${}^{7}$ |

O_{2} | 31.9988 | 0 | 2,256 | 1.54 × 10${}^{7}$ |

NO | 30.0061 | 3.04 × 10${}^{6}$ | 2,719 | 2.09 × 10${}^{7}$ |

N | 14.0067 | 3.37 × 10${}^{7}$ | - | - |

O | 15.9994 | 1.56 × 10${}^{7}$ | - | - |

Level i | ${\mathit{g}}_{\mathit{i}}$ | ${\mathit{\theta}}_{\mathbf{el},\mathit{i}}$ (K) |
---|---|---|

N_{2} | ||

ground | 1 | 0 |

1 | 3 | 7.223157 × 10${}^{4}$ |

2 | 6 | 8.577863 × 10${}^{4}$ |

3 | 6 | 8.605027 × 10${}^{4}$ |

4 | 3 | 9.535119 × 10${}^{4}$ |

5 | 1 | 9.805636 × 10${}^{4}$ |

6 | 2 | 9.968268 × 10${}^{4}$ |

7 | 2 | 1.048976 × 10${}^{5}$ |

8 | 5 | 1.116490 × 10${}^{5}$ |

9 | 1 | 1.225836 × 10${}^{5}$ |

10 | 6 | 1.248857 × 10${}^{5}$ |

11 | 6 | 1.282476 × 10${}^{5}$ |

12 | 10 | 1.338061 × 10${}^{5}$ |

13 | 6 | 1.404296 × 10${}^{5}$ |

14 | 6 | 1.504959 × 10${}^{5}$ |

O_{2} | ||

ground | 3 | 0 |

1 | 2 | 1.139156 × 10${}^{4}$ |

2 | 1 | 1.898474 × 10${}^{4}$ |

3 | 1 | 4.755974 × 10${}^{4}$ |

4 | 6 | 4.991242 × 10${}^{4}$ |

5 | 3 | 5.092269 × 10${}^{4}$ |

6 | 3 | 7.189863 × 10${}^{4}$ |

NO | ||

ground | 4 | 0 |

1 | 8 | 5.467346 × 10${}^{4}$ |

2 | 2 | 6.317140 × 10${}^{4}$ |

3 | 4 | 6.599450 × 10${}^{4}$ |

4 | 4 | 6.906121 × 10${}^{4}$ |

5 | 4 | 7.049998 × 10${}^{4}$ |

6 | 4 | 7.491055 × 10${}^{4}$ |

7 | 2 | 7.628875 × 10${}^{4}$ |

8 | 4 | 8.676189 × 10${}^{4}$ |

9 | 2 | 8.714431 × 10${}^{4}$ |

10 | 4 | 8.886077 × 10${}^{4}$ |

11 | 4 | 8.981756 × 10${}^{4}$ |

12 | 2 | 8.988446 × 10${}^{4}$ |

13 | 2 | 9.042702 × 10${}^{4}$ |

14 | 2 | 9.064284 × 10${}^{4}$ |

15 | 4 | 9.111763 × 10${}^{4}$ |

N | ||

ground | 4 | 0 |

1 | 10 | 2.766470 × 10${}^{4}$ |

2 | 6 | 4.149309 × 10${}^{4}$ |

O | ||

ground | 9 | 0 |

3 | 5 | 2.283029 × 10${}^{4}$ |

4 | 1 | 4.861993 × 10${}^{4}$ |

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**Figure 1.**Two visions of the future of air-space transportation. (

**a**) Orion spacecraft. Credit: National Aeronautics and Space Administration (NASA); (

**b**) cFASTT-1 hypersonic airliner.

**Figure 3.**V–T relaxation towards equilibrium of a N

_{2}heat bath. (

**a**) Vibrational heating: ${T}_{tr}^{0}$ = 10,000 K, ${T}_{v,{\mathrm{N}}_{2}}^{0}=1000$ K, and ${p}^{0}=1$ atm; (

**b**) Vibrational cooling: ${T}_{tr}^{0}=3000$ K, ${T}_{v,{\mathrm{N}}_{2}}^{0}$ = 10,000 K, and ${p}^{0}=1$ atm.

**Figure 7.**Influence of the chemistry-vibration model and chemical rate constants on a chemically-reacting N

_{2}−N heat bath. (

**a**)Temperature versus time; (

**b**)Normalised number density versus time.

**Figure 8.**Chemically-reacting N

_{2}−N heat bath in an initial state of thermal equilibrium. (

**a**) Temperature versus time; (

**b**) Normalised number density versus time.

**Figure 9.**Thermo-chemical relaxation of a 0.79 N

_{2}–0.21 O

_{2}heat bath. (

**a**) Temperature versus time; (

**b**) Normalised number densities versus time, where the dsmcFoam symbols are represented as: N

_{2}(+), O

_{2}(×), NO (Δ), N (⊡), and O (⊙).

Reaction Type | Forward, ${\mathit{T}}_{\mathit{c},\mathit{f}}$ | Backward, ${\mathit{T}}_{\mathit{c},\mathit{b}}$ |
---|---|---|

dissociation | ${T}_{P}$ | ${T}_{tr}$ |

exchange | ${T}_{tr}$ | ${T}_{tr}$ |

associative ionisation | ${T}_{tr}$ | ${T}_{ve,m}$ |

electron impact ionisation | ${T}_{ve,\mathrm{ref}}$ | ${T}_{ve,\mathrm{ref}}$ |

charge exchange | ${T}_{tr}$ | ${T}_{tr}$ |

Reaction Rate | Arrhenius Law Constants | ||
---|---|---|---|

A | β | ${\mathit{T}}_{\mathit{a}}$ | |

Park 1993 | $7.0\times {10}^{21}$ | −1.6 | 113,200 |

QK | $2.47\times {10}^{18}$ | −0.62 | 113,500 |

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**MDPI and ACS Style**

Casseau, V.; Palharini, R.C.; Scanlon, T.J.; Brown, R.E.
A Two-Temperature Open-Source CFD Model for Hypersonic Reacting Flows, Part One: Zero-Dimensional Analysis. *Aerospace* **2016**, *3*, 34.
https://doi.org/10.3390/aerospace3040034

**AMA Style**

Casseau V, Palharini RC, Scanlon TJ, Brown RE.
A Two-Temperature Open-Source CFD Model for Hypersonic Reacting Flows, Part One: Zero-Dimensional Analysis. *Aerospace*. 2016; 3(4):34.
https://doi.org/10.3390/aerospace3040034

**Chicago/Turabian Style**

Casseau, Vincent, Rodrigo C. Palharini, Thomas J. Scanlon, and Richard E. Brown.
2016. "A Two-Temperature Open-Source CFD Model for Hypersonic Reacting Flows, Part One: Zero-Dimensional Analysis" *Aerospace* 3, no. 4: 34.
https://doi.org/10.3390/aerospace3040034