# SU2-NEMO: An Open-Source Framework for High-Mach Nonequilibrium Multi-Species Flows

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Code Architecture and Design

#### 2.1. Mutation++

#### 2.2. SU2 Native Thermochemical Library

_{2}, and Argon. CSU2TCLib demonstrates similar robustness and convergence behavior to CMutationTCLib, and permits easier integration with other SU2 capabilities, but has been shown to be less computationally efficient than CMutationTCLib on benchmark test cases.

## 3. Physical Modeling

**Q**), transport properties, and mixture energies introduced in the next section are provided by the CFluidModel class, either through the CMutationTCLib class linked to the Mutation++ library or CSU2TCLib class containing the native SU2 library.

#### 3.1. Two-Temperature Model

#### Vibrational–Electronic Relaxation

#### 3.2. Finite-Rate Chemical Kinetics

#### 3.3. Transport Properties

#### 3.4. Turbulence Modeling

#### 3.5. Modeling of the Slip Regime

## 4. Numerical Implementation

#### 4.1. Spatial Integration

#### 4.1.1. Convective Flux

- Modified Steger-Warming

- Advection Upstream Splitting Method

^{+}up2 [41] schemes, which contain improvements on the stability characteristics and accuracy of the original AUSM scheme. The AUSM family of schemes offer superior shock-capturing and avoid the presence of carbuncles observed in stagnation regions around blunt bodies. SU2-NEMO, like the base SU2 code, has slope-limiters for higher-order accuracy solutions.

#### 4.1.2. Viscous Flux

#### 4.2. Boundary Conditions

#### 4.3. Time Integration

## 5. Results and Discussion

#### 5.1. Zero-Dimensional Thermal Bath

_{2}and Air-5 gas models within both the SU2 native library and the Mutation++ library. In the cases run using Mutation++, both preferential and non-preferential dissociation models were employed. The Air-5 gas results are compared to results generated using the LeMANS code [45]. The thermal bath is simulated using a 5 × 5 structured mesh with symmetry planes on each side, and an unsteady explicit time-stepping method is used for these simulations.

#### 5.1.1. N_{2} Gas Model

_{2}gas mixture is simulated with an initial concentration of pure diatomic nitrogen. The initial conditions are shown below in Table 1. The comparison of the native (CSU2TCLib) and Mutation++ (CMutationTCLib) CFluidModel classes for temperature relaxation and dissociation are shown in Figure 4a,b, respectively.

^{−6}s. The SU2-Native data, which uses a non-preferential model, is bounded by the SU2-Mutation++ preferential and non-preferential data, with ${T}^{ve}$ reaching a peak value of 11,870 K compared to 11,690 K and 12,100 K, respectively. Comparing the non-preferential data, the peak vibrational–electronic temperature varies by 2.8%. This difference is attributed to minor differences in the characteristic values chosen within the respective models. The non-preferential data exhibit a larger overshoot in both temperature relaxation time-evolutions compared to the preferential data. As a result, the non-preferential simulations have marginally higher rates of dissociation, as seen in Figure 4b. This behavior is caused by the non-preferential model driving the creation and destruction of species at the average ${T}^{ve}$ in the cell, rather than the maximum ${T}^{ve}$; the preferential model will lead to higher dissociation at higher ${T}^{ve}$. In the context of the N

_{2}thermal bath, the impact of the preferential and non-preferential models on dissociation is minimal.

#### 5.1.2. Air-5 Gas Model

^{−7}s, as shown in Figure 5a. Unlike the N

_{2}data, the differences between the non-preferential and preferential models are more significant. The non-preferential models predict greater peak ${T}^{ve}$ values relative to the preferential models: 11.3% for SU2-Mutation++, and 11.8% for LeMANS. Furthermore, both the SU2-Mutation++ models predict higher peak temperatures than the LeMANS models. For example, an increase of 3.5% in the peak non-preferential model vibrational temperature is observed. The differences directly correlate to the dissociation in the Air-5 models; the impact of the preferential dissociation is evident in Figure 5b. In general, the preferential models begin dissociating at a slower rate, due to the preference for higher ${T}^{ve}$. This is most notable in atomic oxygen, around 10

^{−8}s. The percentage of oxygen in the flow is tabulated in Table 3.

#### 5.2. HEG Cylinder

#### 5.3. RAM-C II Test Vehicle

_{2}, O

_{2}, N, O, NO, NO

^{+}, and e

^{−}, with viscous effects absent. A mesh comprised of 316,910 nodes and 267,241 elements is used, with 160 elements normal to the vehicle body. The free-stream conditions for the 61 km flight can be seen in Table 5.

^{+}production, due to dissociative recombination of monatomic nitrogen and oxygen, is the primary mechanism for ion formation at these flight conditions. Because the plasma density is low and the primary focus is on prediction of flow properties, magnetohydrodynamic effects are neglected in this simulation. Results can be seen in Figure 8 and Figure 9.

#### 5.4. Axisymmetric Shock-Wave Boundary Layer Interaction

#### 5.5. Slip Flow over a Cylinder

_{2}gas model is used to simulate this case with free-stream flow conditions illustrated in Table 7.

#### 5.6. NASA X-43a (Hyper-X)

^{+}are negligibly small, confirming that thermochemical effects had little impact on the flow features under the simulated conditions. Select Mach number contours are shown in Figure 19 and Figure 20. Attached oblique shocks are formed by the leading edge of the vehicle. A second ramp on the underside of the vehicle creates a secondary shock, compressing the air further at the inlet of the scramjet. Without the proper amount of compression, there is a significant risk of a scramjet unstart and vehicle failure. The flow behind the scramjet is accelerated, leading to a high-Mach region on the underside of the vehicle and resulting in an expansion jet visible emanating from the rear surface. This region is flanked by lower Mach regions generated by a system of shock-waves that emanate from the vertical and horizontal tails, shown in Figure 20.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AUSM | Advection Upstream Splitting Method |

CFD | Computational Fluid Dynamics |

CFL | Courant–Friedrichs–Lewy |

DSMC | Direct Simulation Monte Carlo |

Kn | Knudsen Number |

V&V | Verification and Validation |

MSW | Modified Steger-Warming |

RANS | Reynolds-Averaged Navier–Stokes |

WSL | Weighted Least-Squares |

## Appendix A. Thermochemical Parameters

Reaction | C_{r} ((cm^{3})/(mol-s)) | η_{r} | ${\mathit{epsilon}}_{\mathit{r}}^{\mathit{A}}$ (K) |
---|---|---|---|

N_{2} + e^{−} ⇌ N + N + e^{−} | 3.0 × 10^{24} | −1.6 | 113,200 |

N_{2} + N_{2} ⇌ 2N + O_{2} | 7.0 × 10^{21} | −1.60 | 113,200 |

N_{2} + O_{2} ⇌ 2N + O_{2} | 7.0 × 10^{21} | −1.60 | 113,200 |

N_{2} + NO ⇌ 2N + NO | 7.0 × 10^{21} | −1.60 | 113,200 |

N_{2} + N ⇌ 2N + NO | 3.0 × 10^{22} | −1.60 | 113,200 |

N_{2} + O ⇌ 2N + NO | 3.0 × 10^{22} | −1.60 | 113,200 |

O_{2} + N_{2} ⇌ 2O + N_{2} | 2.0 × 10^{21} | −1.50 | 59,500 |

O_{2} + O_{2} ⇌ 2O + O_{2} | 2.0 × 10^{21} | −1.50 | 59,500 |

O_{2} + NO ⇌ 2O + NO | 2.0 × 10^{21} | −1.50 | 59,500 |

O_{2} + N ⇌ 2O + N | 1.0 × 10^{22} | −1.50 | 595,200 |

O_{2} + O ⇌ 2O + O | 1.0 × 10^{22} | −1.50 | 595,200 |

NO + N_{2} ⇌ N + O + N_{2} | 5.0 × 10^{15} | 0.00 | 75,500 |

NO + O_{2} ⇌ N + O + O_{2} | 5.0 × 10^{15} | 0.00 | 75,500 |

NO + NO ⇌ N + O + NO | 5.0 × 10^{15} | 0.00 | 75,500 |

NO + N ⇌ N + O + N | 1.1 × 10^{17} | 0.00 | 75,500 |

NO + O ⇌ N + O + O | 1.1 × 10^{17} | 0.00 | 75,500 |

N_{2} + O ⇌ NO + N | 6.4 × 10^{17} | −1.00 | 38,400 |

NO + O ⇌ O_{2} + N | 8.4 × 10^{17} | 0.00 | 19,450 |

N + O ⇌ NO^{+} + e^{−} | 5.3 × 10^{12} | 0.0 | 31,900 |

Reaction | ${\mathbf{a}}_{\mathit{r}}^{\mathit{f}}$ | ${\mathbf{b}}_{\mathit{r}}^{\mathit{f}}$ | ${\mathbf{a}}_{\mathit{r}}^{\mathit{b}}$ | ${\mathbf{b}}_{\mathit{r}}^{\mathit{b}}$ |
---|---|---|---|---|

N_{2} + M ⇌ 2N + O_{2} | 0.5 | 0.5 | 1.0 | 0.0 |

N_{2} + M ⇌ 2O + O_{2} | 0.5 | 0.5 | 1.0 | 0.0 |

NO + M ⇌ N + O + NO | 0.5 | 0.5 | 1.0 | 0.0 |

N_{2} + O ⇌ NO + N | 1.0 | 0.0 | 1.0 | 0.0 |

NO + O ⇌ O_{2} + N | 1.0 | 0.0 | 1.0 | 0.0 |

N + O ⇌ NO^{+} + e^{−} | 1.0 | 0.0 | 1.0 | 0.0 |

Reaction | N (1/cm^{3}) | A_{0} | A_{1} | A_{2} | A_{3} | A_{4} |
---|---|---|---|---|---|---|

N_{2} + M ⇌ 2N + M | 1 × 10^{14} | 3.4907 | 0.83133 | 4.0978 | −12.728 | 0.7487 |

1 × 10^{15} | 2.0723 | 1.38970 | 2.0617 | −11.828 | 0.015105 | |

1 × 10^{16} | 1.6060 | 1.57320 | 1.3923 | −11.533 | −0.004543 | |

1 × 10^{17} | 1.5351 | 1.60610 | 1.2993 | −11.494 | −0.00698 | |

1 × 10^{18} | 1.4766 | 1.62910 | 1.2153 | −11.457 | −0.00944 | |

1 × 10^{19} | 1.4766 | 1.62910 | 1.2153 | −11.457 | −0.00944 | |

O_{2} + M ⇌ 2O + M | 1 × 10^{14} | 1.8103 | 1.9607 | 3.5716 | −7.3623 | 0.083861 |

1 × 10^{15} | 0.91354 | 2.3160 | 2.2885 | −6.7969 | 0.046338 | |

1 × 10^{16} | 0.64183 | 2.4253 | 1.9026 | −6.6277 | 0.035151 | |

1 × 10^{17} | 0.55388 | 2.4600 | 1.7763 | −6.5720 | 0.031445 | |

1 × 10^{18} | 0.52455 | 2.4715 | 1.7342 | −6.55534 | 0.030209 | |

1 × 10^{19} | 0.50989 | 2.4773 | 1.7132 | −6.5441 | 0.29591 | |

NO + M ⇌ N + O + M | 1 × 10^{14} | 2.1649 | 0.078577 | 2.8508 | −8.5422 | 0.053043 |

1 × 10^{15} | 1.0072 | 0.53545 | 1.1911 | −7.8098 | 0.004394 | |

1 × 10^{16} | 0.63817 | 0.68189 | 0.66336 | −7.5773 | −0.011025 | |

1 × 10^{17} | 0.55889 | 0.71558 | 0.55396 | −7.5304 | −0.014089 | |

1 × 10^{18} | 0.5150 | 0.73286 | 0.49096 | −7.5025 | −0.015938 | |

1 × 10^{19} | 0.50765 | 0.73575 | 0.48042 | −7.4979 | −0.016247 | |

N_{2} + O ⇌ N + O + M | 1 × 10^{14} | 1.3261 | 0.75268 | 1.2474 | −4.1857 | 0.02184 |

1 × 10^{15} | 1.0653 | 0.85417 | 0.87093 | −4.0188 | 0.010721 | |

1 × 10^{16} | 0.96794 | 0.89131 | 0.7291 | −3.9555 | 0.006488 | |

1 × 10^{17} | 0.97646 | 0.89043 | 0.74572 | −3.9642 | 0.007123 | |

1 × 10^{18} | 0.96188 | 0.89617 | 0.72479 | −3.955 | 0.006509 | |

1 × 10^{19} | 0.96921 | 0.89329 | 0.73531 | −3.9596 | 0.006818 | |

NO + O ⇌ O_{2} + N | 1 × 10^{14} | 0.35438 | −1.8821 | −0.72111 | −1.1797 | −0.30831 |

1 × 10^{15} | 0.093613 | −1.7806 | −1.0975 | −1.0128 | −0.41949 | |

1 × 10^{16} | −0.003732 | −1.7434 | −1.2394 | −0.94952 | −0.046182 | |

1 × 10^{17} | 0.004815 | −1.7443 | −1.2227 | −0.95824 | −0.45545 | |

1 × 10^{18} | −0.009758 | −1.7386 | −1.2436 | −0.949 | −0.046159 | |

1 × 10^{19} | −0.002428 | −1.7415 | −1.2331 | −0.95365 | −0.04585 | |

N + O ⇌ NO^{+} + e− | 1 × 10^{14} | −2.1852 | −6.6709 | −4.2968 | −2.2175 | −0.50748 |

1 × 10^{15} | −1.0276 | −7.1278 | −2.637 | −2.95 | −0.0021 | |

1 × 10^{16} | −0.65871 | −7.2742 | −2.1096 | −3.1823 | 0.01331 | |

1 × 10^{17} | −0.57924 | −7.3079 | −1.9999 | −3.2294 | 0.016382 | |

1 × 10^{18} | −0.53538 | −7.3252 | −1.937 | −3.2572 | 0.01823 | |

1 × 10^{19} | −0.52801 | −7.3281 | −1.9264 | −3.2618 | 0.01854 |

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**Figure 2.**Diagrams of extensions of SU2 classes implemented in SU2-NEMO for simulating nonequilibrium flows.

**Figure 3.**SU2 CFluidModel with CNEMOGas, with native and Mutation++ thermochemical libraries as child classes.

**Figure 7.**HEG cylinder surface plots with comparison ro experimental [48] results.

**Figure 10.**Diagram of experimental [53] configuration for axisymmetric shock-wave boundary-layer interaction.

${\mathit{T}}_{\mathbf{\infty}}^{\mathit{tr}}$ (K) | ${\mathit{T}}_{\mathbf{\infty}}^{\mathit{ve}}$ (K) | P_{∞} (atm) | $\mathit{Y}{\left[{\mathit{N}}_{2}\right]}_{\mathbf{\infty}}$ | $\mathit{Y}{\left[\mathit{N}\right]}_{\mathbf{\infty}}$ |
---|---|---|---|---|

20,000 | 300 | 27.25 | 1.0 | 0.0 |

${\mathit{T}}_{\mathbf{\infty}}^{\mathit{tr}}$ (K) | ${\mathit{T}}_{\mathbf{\infty}}^{\mathit{ve}}$ (K) | P_{∞} (atm) | $\mathit{Y}{\left[{\mathit{N}}_{2}\right]}_{\mathbf{\infty}}$ | $\mathit{Y}{\left[{\mathit{O}}_{2}\right]}_{\mathbf{\infty}}$ | $\mathit{Y}{\left[\mathbf{NO}\right]}_{\mathbf{\infty}}$ | $\mathit{Y}{\left[\mathit{N}\right]}_{\mathbf{\infty}}$ | $\mathit{Y}{\left[\mathit{O}\right]}_{\mathbf{\infty}}$ |
---|---|---|---|---|---|---|---|

15,000 | 300 | 20.42 | 0.767 | 0.233 | 0.0 | 0.0 | 0.0 |

Non-Preferential | Preferential | |
---|---|---|

SU2-Mutation++ | 14.91% | 13.27% |

LeMANS | 17.48% | 15.42% |

U_{∞} (m/s) | p_{∞} (Pa) | ρ_{∞} (kg/m^{3}) | T_{∞} (K) | M_{∞} | $\mathit{Y}{\left[{\mathit{N}}_{2}\right]}_{\mathbf{\infty}}$ | $\mathit{Y}{\left[{\mathit{O}}_{2}\right]}_{\mathbf{\infty}}$ | $\mathit{Y}{\left[\mathbf{NO}\right]}_{\mathbf{\infty}}$ | $\mathit{Y}{\left[\mathit{N}\right]}_{\mathbf{\infty}}$ | $\mathit{Y}{\left[\mathit{O}\right]}_{\mathbf{\infty}}$ |
---|---|---|---|---|---|---|---|---|---|

5956 | 476 | 1.547 × 10^{−3} | 901 | 8.98 | 0.7543 | 0.00713 | 0.01026 | 6.5 × 10^{−7} | 0.2283 |

M_{∞} | P_{∞} (Pa) | ρ_{∞} (kg/m^{3}) | T_{∞} (K) | $\mathit{Y}{\left[{\mathit{N}}_{2}\right]}_{\mathbf{\infty}}$ | $\mathit{Y}{\left[{\mathit{O}}_{2}\right]}_{\mathbf{\infty}}$ |
---|---|---|---|---|---|

23.9 | 19.7 | 2.7024 × 10^{−4} | 254 | 0.767 | 0.233 |

M_{∞} | P_{∞} (Pa) | T_{∞} (K) | T_{wall} (K) | $\mathit{Y}{\left[{\mathit{N}}_{2}\right]}_{\mathbf{\infty}}$ | $\mathit{Y}{\left[{\mathit{O}}_{2}\right]}_{\mathbf{\infty}}$ | $\mathit{Y}{\left[\mathbf{NO}\right]}_{\mathbf{\infty}}$ | $\mathit{Y}{\left[\mathit{N}\right]}_{\mathbf{\infty}}$ | $\mathit{Y}{\left[\mathit{O}\right]}_{\mathbf{\infty}}$ |
---|---|---|---|---|---|---|---|---|

7.11 | 550.13 | 80.0 | 311.0 | 0.77 | 0.23 | 0.0 | 0.0 | 0.0 |

Kn_{∞} | M_{∞} | U_{∞} (m/s) | ρ_{∞} (kg/m^{3}) | T_{∞} (K) | T_{wall} (K) |
---|---|---|---|---|---|

0.01 | 10 | 2883 | 1.974 × 10^{−5} | 200 | 500 |

0.05 | 10 | 2883 | 3.949 × 10^{−6} | 200 | 500 |

Kn = 0.01 | Kn = 0.05 | |||||
---|---|---|---|---|---|---|

SU2-NEMO | DSMC | Δ | SU2-NEMO | DSMC | Δ | |

Peak Pressure Coefficient | 1.7764 | 1.7634 | 0.74 % | 1.8370 | 1.8201 | 0.93 % |

Peak Heat flux Coefficient | 0.1480 | 0.1503 | −1.53 % | 0.3254 | 0.3099 | 4.97 % |

Peak Skin friction Coefficient | 0.1026 | 0.1045 | −1.81 % | 0.2173 | 0.2049 | 6.05 % |

M_{∞} | P_{∞} (Pa) | T_{∞} (K) | $\mathit{Y}{\left[{\mathit{N}}_{2}\right]}_{\mathbf{\infty}}$ | $\mathit{Y}{\left[{\mathit{O}}_{2}\right]}_{\mathbf{\infty}}$ | $\mathit{Y}{[-]}_{\mathbf{\infty}}$ |
---|---|---|---|---|---|

7 | 190 | 239 | 0.77 | 0.23 | 0.0 |

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## Share and Cite

**MDPI and ACS Style**

Maier, W.T.; Needels, J.T.; Garbacz, C.; Morgado, F.; Alonso, J.J.; Fossati, M.
SU2-NEMO: An Open-Source Framework for High-Mach Nonequilibrium Multi-Species Flows. *Aerospace* **2021**, *8*, 193.
https://doi.org/10.3390/aerospace8070193

**AMA Style**

Maier WT, Needels JT, Garbacz C, Morgado F, Alonso JJ, Fossati M.
SU2-NEMO: An Open-Source Framework for High-Mach Nonequilibrium Multi-Species Flows. *Aerospace*. 2021; 8(7):193.
https://doi.org/10.3390/aerospace8070193

**Chicago/Turabian Style**

Maier, Walter T., Jacob T. Needels, Catarina Garbacz, Fábio Morgado, Juan J. Alonso, and Marco Fossati.
2021. "SU2-NEMO: An Open-Source Framework for High-Mach Nonequilibrium Multi-Species Flows" *Aerospace* 8, no. 7: 193.
https://doi.org/10.3390/aerospace8070193