During hypersonic reentry, a spacecraft experiences several different fluid flow regimes, which usually require the application of different software frameworks to simulate the respective regimes. This study aims to evaluate the hyStrath library for predicting aerodynamic lift and drag coefficients of complex three-dimensional (3D) geometries during hypersonic Mars reentry, using flight data of the Viking 1 mission as reference. A range of altitudes ( $ h = 30 \u2212 140 \u2009 km$) and Mach numbers ( $ M = 13 \u2212 24$) where flight data is available is considered, covering the rarefied, transitional, and continuum fluid flow regimes. The hyStrath library contains a set of modified solvers and state-of-the-art thermophysical and chemistry models within the framework of OpenFOAM, dedicated to modeling high-enthalpy hypersonic flow problems. Depending on the flow regime, the computational fluid dynamics solver *hy2Foam* or direct-simulation Monte Carlo solver *dsmcFoam+* are employed in the study. Because hyStrath is based on OpenFOAM, it allows the use of an unstructured adaptive mesh refinement approach for arbitrary geometries. We obtain excellent results throughout all investigated flow regimes and Mach numbers with an average deviation of 1.5% and 2% from the measured lift and drag coefficients, respectively. The applicability of the framework for accurately modeling both rarefied and continuum Mars reentry problems of complex 3D geometries such as the Viking capsule is demonstrated.

## I. INTRODUCTION

Accurately predicting aerothermal loads on reentering spacecraft will be critical for the design of future Mars mission vehicles and to ensure a safe descent during atmospheric reentry. The steadily increasing interest of (eventually manned) Mars missions such as SpaceX's *Starship* require ever more complicated designs and reentry approaches, making reliable modeling indispensable.

Compared to the Earth, the significantly thinner atmosphere of Mars results in the spacecraft experiencing rarefied flow until lower altitudes. However, during reentry, the majority of deceleration and aerothermal loads take place in the high-enthalpy hypersonic continuum flow regime, invoking several physical phenomena that occur around the reentering vehicle. These include the formation of a bow shock followed by a zone of high temperature and pressure in front of the heat shield. Here, the fluid is characterized by strong thermal and chemical non-equilibrium, yielding dissociation of molecular species and excitation of electronic and vibrational energy modes. Experimental testing of subscale models in wind tunnels with these conditions requires huge amounts of effort. For this reason, a large proportion of research in the hypersonic area has been devoted to numerical methodology in order to reliably predict the flow physics of reentering spacecraft.

The most widely used approach to modeling rarefied fluid flow ( $ Kn \u221e > 1$) is the DSMC method proposed by Bird.^{1,2} Significant extensions and improvements have since been achieved in terms of modeling chemistry,^{3} multiple energy modes,^{4,5} and their relaxation^{6,7} as well as surface-gas interaction models.^{8} Commonly employed DSMC codes are SPARTA,^{9} DAC,^{10} MONACO,^{11} DS3V,^{12} SMILE,^{13} and *dsmcFoam.*^{14} Aerothermal loads during rarefied hypersonic Mars reentry of the MSL mission were extensively investigated by Borner *et al.*^{15} Similarily, DSMC simulations concerned with both transitional and rarefied aerodynamic analysis were carried out by Moss *et al.*^{16,17} for the Pathfinder and by Blanchard *et al.*^{18} for the Viking missions.

The gradual increase in density through loss of altitude results in the fluid flow eventually becoming continuous ( $ Kn \u221e < 0.01$). Compared to the scope of research published in the area of rarefied Mars reentry, more research has been devoted to modeling the continuum flow regime. NASA's compressible code LAURA^{19} was used to study the aerodynamics of the Viking capsule^{20} as well as its backshell heating.^{21} Similar analysis using the DLR TAU code^{22} focused on radiative and convective Viking afterbody heating.^{23} In a publication by Egorov and Pugach,^{24} the HSFlow solver was used to model the flow physics of the ExoMars vehicle traveling at Mach 29 and 69 km altitude. Another study employing the LAURA and DPLR^{25} codes was dedicated to the investigation of brownout and blackout analysis during Mars reentry of the Mars 2020 mission.^{26} Using a customized version of the commercially available code Ansys Fluent, non-equilibrium flowfield analysis of a manned braking system (MBS) lifting body for manned Mars exploration missions was investigated by Viviani *et al.*^{27,28} Detailed studies of continuum surface-gas reactions during Mars reentry were reported from Refs. 29–31 and a technical report summarizing and evaluating different models for modeling Mars reentry using computational fluid dynamics (CFD) including recommendations for computing chemistry, transport properties, and surface–gas interaction were concluded by Noeding.^{32}

In a recent development by Zeng *et al.*,^{33} nonlinear coupled constitutive relations (NCCR) were coupled with the chemical models from Ref. 34 and the two-temperature model from^{7} for modeling hypersonic Earth reentry in the continuum and transitional flow regime. Compared with existing Navier–Stokes and DSMC results, they were able to obtain improved results for characteristic quantities such as the electron number density or heat flux coefficient based on flight data of Earth reentry experiments such as the RAM C-II test.^{35} They attribute the better agreement with the improved capturing of extreme (rarefied) non-equilibrium effects when using the NCCR approach. In a follow-up study,^{36} they investigated jet interaction with high Mach number freestream flow including real gas effects.

In the present study, hypersonic reentry of the Viking 1 lander is simulated for altitudes spanning from 140 km (rarefied, $ Ma \u221e = 25$) down to 30 km (continuum, $ Ma \u221e = 13$) using the hyStrath library^{37,38} that contains DSMC and CFD solvers that are built upon existing OpenFOAM^{39} solvers, namely, *rhoCentralFoam* and *dsmcFoam*. They were extended with thermodynamic and chemical non-equilibrium models by Casseau *et al.*^{37,38} so that they are able to better capture the physical phenomena of high-enthalpy hypersonic flows. The goal of the present research is to evaluate the *dsmcFoam+* and *hy2Foam* solvers for predicting aerodynamic coefficients of Mars reentry vehicles traveling in the rarefied and continuum hypersonic flow regime at non-zero angle-of-attack. The Viking lander was selected for this purpose due to the good availability of geometric and aerodynamic data. During reentry, various on-board sensors acquired data of the atmospheric properties and spatial accelerations, from which the lift and drag coefficients were derived.^{40}

## II. VIKING LANDER REENTRY

The Viking 1 lander capsule successfully landed on the surface of Mars on July 20, 1976 after a 10 min reentry journey. The geometry of the capsule and telemetry of the velocity and freestream temperature is shown in Fig. 1 from the point of reentry at 140 km down to 30 km with the corresponding flow regimes denoted on the right side.

In the present study, the investigated range of reentry cases was divided into steps of 10 km in the rarefied and transitional regimes and 5 km in the continuum regime, yielding a total of 14 performed simulations. The freestream properties and telemetry of the capsule for each altitude were taken from analyses carried out by Blanchard and Walberg^{40} and Nier *et al.*^{41} and set accordingly at the numerical boundary conditions, which are illustrated in Fig. 2. A table detailing freestream properties (species mass fraction, pressure and temperature) and telemetry (angle-of-attack and velocity) for each simulated altitude can be found in Appendix A of Table I.

In Eq. (1), *λ* denotes the particle mean free path, *k _{B}* is the Boltzmann constant, $ T \u221e$ is the freestream temperature, $ d \xaf$ is the mean particle kinetic diameter of the mixture, $ p \u221e$ is the freestream pressure, and $L$ is the characteristic length, i.e., the capsule diameter.

## III. METHODOLOGY

### A. Rarefied and transitional regime

Both the transitional and rarefied regime ( $ Kn \u221e > 0.01$) were simulated using the *dsmcFoam+* solver which is an extension of the existing *dsmcFoam* solver by,^{14} based on the DSMC method originally proposed by Bird.^{1} Key features required for modeling hypersonic flow problems were implemented by White *et al.*^{42} including (molecular) electro-vibrational energy modes and chemical reactions. One of the main benefits is the design within the OpenFOAM framework, allowing the utilization of fully unstructured meshes for arbitrary geometries. This aspect played a crucial role in our study since the wake of the spacecraft produces a very-low-density rarefied flow zone that would lead to numerical and unphysical instabilities when employing the continuum solver despite $ Kn \u221e \u2248 0.1$. With an unstructured grid, we could substantially increase the cell volume in the wake of the Viking capsule while decreasing the cell volume in the front zone affected by the bow shock, ensuring both sufficient number of particles per cell and resolution of the mean free path.

DSMC simulations were carried out including 9 species: CO_{2}, N_{2}, Ar, O_{2}, CO, O, N, NO, and C. 16 reactions were considered using the quantum-kinetic (Q-K) chemical reaction model according to Ref. 43 and implemented into *dsmcFoam* by Scanlon *et al.*^{44} A table containing all chemical reactions formulas and types is provided in Appendix B (see Table II).

*w*were set in a manner so that the mean-collision time $ t mc$ to time step ratio and mean free path to cell size ratio remained below 1, respectively

The computational meshes were generated with the *snappyHexMesh* mesher available in the OpenFOAM framework. We implemented a semi-automated procedure to adaptively refine the mesh based on the output files written by the solver since the *dsmcFoam+* implementation does not allow for run-time adaptive mesh refinement (AMR). In our approach, the macro field of the Mach and density gradient were evaluated after quasi-steady-state was achieved and logarithmic 3D contour surfaces exported using the post-processing tool ParaView. The 3 D surfaces were used as refinement regions within *snappyHexMesh* and the DSMC simulation restarted. This algorithm was repeated until the mean-free particle path was adequately resolved in all cells. An overview of the computational meshing approach is shown in Fig. 3, outlining colored boundary conditions and pre/post adaptive mesh refinement section cuts.

Freestream properties were enforced at the farfield and outlet boundaries whereby each crossing particle was deleted. The Variable Soft Sphere (VSS) model introduced by Koura and Matsumoto^{45} was utilized to model binary collision events, using the species-dependent collision parameters from the study of Ref. 46 (for species-wise VSS properties, see Appendix C of Table III). Particle-wall interactions were modeled with the diffuse-specular wall boundary condition provided in *dsmcFoam+*, using a diffuse fraction of 0.9 for rarefied flow cases ( $ Kn \u221e > 1$) and 1.0 for transitional flow cases ( $ 0.1 < Kn \u221e < 1$) as estimated by Blanchard and Walberg.^{40}

The number of DSMC parcels varied from 10–500 × 10^{6} for the 140 and 80 km cases, respectively, required to satisfy the minimum number of particles per cell ( $ N i > 5$). The DSMC simulations were run on the *HSUper* cluster of *Helmut Schmidt University* on 256–2304 cores, depending on the number of cells and parcels.

### B. Continuum regime

*hy2Foam*solver developed by Casseau

*et al.*

^{38}which in its core relies on the

*rhoCentralFoam*solver from Ref. 47 and

*rhoReactingFoam*solver within the OpenFOAM framework.

*rhoCentralFoam*is a density-based shock capturing solver that discretizes convective fluxes by means of a total variation diminishing (TVD) scheme. The governing equations are the compressible Navier–Stokes equations that describe the conservation of mass, momentum and energy, expressed as

*ρ*denotes the density defined as $ \rho = p / ( R s T )$ according to the ideal-gas law,

**u**is the velocity vector,

*t*is the time,

*p*is the pressure, $ \tau \xaf \xaf$ is the viscous stress tensor,

*E*is the total energy,

*T*is the temperature, and

**J**is the heat flux vector. The vector of conserved quantities is given by

*hy2Foam*solver includes the addition of electro-vibrational energy modes implemented using Park's two-temperature model

^{48}and energy transfers between the vibrational and translational modes based on the Landau–Teller theory

^{49}and work of Ref. 6. This yields additional dimensions in the vector of conserved quantities considering multiple species with subscript

*s*as

^{38}

*m*, $F$ is the (inviscid) flux vector, and

*δ*is the Kronecker delta. From here, the non-equilibrium Navier–Stokes–Fourier (NSF) equation can be formulated for a fluid mixture consisting of

_{ij}*s*species and

*m*molecules through

*hy2Foam*can be found in the notes of Ref. 50 and the publications from Refs. 37 and 38.

Modeling reactions is achieved using finite-rate chemistry with Arrhenius rate coefficients based on Park's recommendation for Mars reentry two-temperature CFD problems.^{7,51} The complete list of reactions for the CFD cases can be found in Table IV. Species transport properties including viscosity, thermal conductivity, mixing rules, and diffusion were computed according to the works of Refs. 52–55, respectively.

The meshing approach for the CFD cases is outlined in Fig. 4. Similar to the DSMC approach, a baseline mesh was created including boundary layers with $ N BL = 20$, *k* = 1.15, and $ y + = 0.1$ based on the freestream properties. Then, due to the presence of discontinuities in the continuum regime, the mesh was iteratively refined at the bow shock. The maximum refinement level of the octree was set to 3, yielding a maximum of 20 × 10^{6} cells for the 30 km case in order to satisfy our requirements. Since a fully unstructured meshing approach was employed, choosing appropriate numerical methods was important to obtain robustness, sufficient accuracy, and low numerical diffusion. We used the *Crank–Nicolson* technique^{56} to advance the solution in time using a coefficient of 0.9 and the MUSCL scheme proposed by^{57} as the divergence scheme coupled with the *vanAlbada* limiter.^{58} The CFD simulations were performed with an adaptive time-stepping approach based on a fixed local CFL number of 0.1 to ensure numerical robustness. This resulted in a minimum time step of 5 × 10^{−9} s for the 30 km case and required us to run the *hy2Foam* simulations also on *HSUper* on up to 4608 cores.

The *k*–*ω* SST turbulence model proposed by Menter^{59} was employed to capture effects of residual viscosity and turbulent flow detachment that might occur around the heat shield and wake of the capsule at lower altitudes. Freestream flow properties are enforced at the farfield boundary (see Fig. 2) using a Dirichlet type *fixedValue* condition, while at the outlet Neumann type, *zeroGradient* conditions are used for all flow quantities. The *maxwellSlipU* wall boundary condition^{60} was utilized with an accomodation coefficient of $ \alpha MW = 1.0$, while the *smoluchowskiJumpT* boundary condition^{61} was employed for modeling temperature jumps occurring in regions of rarefaction close to the wall.

## IV. RESULTS AND DISCUSSION

### A. Rarefied and transitional regime

Due to signal strength (i.e., forces on the capsule) and measurement noise, only drag coefficient flight data are available for the rarefied and transitional flow regime and also subject to noise above 120 km. A comparison of predictions of *c _{d}* from DSMC simulations with flight data is outlined in Fig. 5. The magnitude of mean relative error throughout all altitudes remains small at an average of $ \delta \xaf = 1.8 %$. With the exception of the 110 km case, the drag coefficient is generally slightly underpredicted.

In Fig. 6, contours of different flow quantities are illustrated in the *xy-*plane cross section for the 80, 100, and 140 km cases. The first column indicates contours at 140 km altitude. It is clearly visible that no bow shock is present at 140 km altitude which can be attributed to very-low-density, rarefied flow around the capsule and partially specular surface–gas interactions. In the area in front of the capsule, multiple regions of very hot gas are formed ( $ T t r \u226b 10 \u2009 000 \u2009 K$), in which some of the specularly reflected gas particles collide with the incoming freestream. In this flow regime, however, most of the incoming particles in the freestream impact the capsule's heat shield almost undisturbed due to their long mean free path. As a result, the drag coefficient reaches its maximum value at the 140 km case. The wake of the capsule is characterized by an area of extremely low gas density, indicated through a very long mean free path of the particles.

As the capsule loses altitude, the flow becomes transitional and a (thickened) bow shock starts to appear which can be observed at the Mach number contours of the 100 km case. Consequently, the hot temperature region forms directly behind the bow shock and in front of the capsule. Effects of dissociation and increased density reduce the maximum translational temperature to $ max ( T t r ) = 15 \u2009 000 \u2009 K$. Looking at the 80 km case, it is evident that the fundamental flow physics do not change as drastical compared to the 100 km case since the flow is still transitional. The most significant changes include the thinning of the bow shock, as the surrounding flow becomes increasingly continuous. In addition, a clearly recognizable recompression shock occurs behind the capsule, which is noticeable by a hot temperature zone in the wake and a shorter and more homogenously distributed mean free path in this region.

### B. Continuum regime

When the capsule traversed 70 km altitude, the flow became strictly continuous since $ Kn \u221e < 0.01$. This section is concerned with predictions of the *hy2Foam* solver for altitudes between 70 and 30 km. For this range, flight data with small measurement errors is available as well as numerical data from a previous publication by Edquist^{20} where the LAURA code was used (48–31 km). Note that after a threshold of $ a x > 0.05 \xb7 g E$, the RCS thrusters, which held the capsule at a constant $ \alpha = 11.1 \xb0$, were deactivated to allow the vehicle to freely adjust the angle of attack based on its natural trim angle. The altitude roughly coincided with the beginning of the continuum regime at 70 km when the capsule underwent pitching motion around its center of gravity.

The angle of attack data is shown in Fig. 7(a), and we took the raw data for each altitude since we compare our results to raw acceleration data from which the aerodynamic coefficients were derived. In Figs. 7(b) and 7(c), results of the drag and lift coefficients are compared with flight data and previous numerical results, respectively. In general, predictions of the *hy2Foam* setup yield slightly lower values of the drag coefficient with a mean relative error of $ \delta \xaf ( c d ) = 1.4 %$, while the predicted lift coefficient does not indicate a clear tendency and exhibits a similar mean relative error of $ \delta \xaf ( c l ) = 2.1 %$. It is visible that the *hy2Foam* simulations show an improvement in both accuracy and trend compared to the LAURA results from Ref. 40.

Contours of essential flow quantities for the 70, 50, and 30 km cases are illustrated in Fig. 8. At 70 km altitude, it is apparent that the bow shock has become much thinner and forms a discontinuity around the capsule. The translational temperature increases to almost 9500 K as a step function when the freestream fluid traverses the bow shock and subsequently undergoes thermochemical non-equilibrium. A thin region of hot compressed gas appears behind the bow shock and initiates dissociation of CO_{2} molecules into CO and O, indicated by the plot in the last row. When the capsule flew below 50 km, the process of maximum deceleration occurred and a large amount of CO_{2} dissociates due to the combination of high translational-vibrational energy and pressure in front of the capsule. The recompression shock becomes more apparent in the wake and its distance to the capsule reduces. At 30 km altitude, the bow shock moves slightly farther away from the heat shield and becomes wider due to the lower Mach number. The maximum temperature of $ T t r \u2248 T v = 3000 \u2009 K$ results in a drastic reduction of CO_{2} dissociation, hence the mass fraction of CO decreases to a maximum of 2%.

*c*is given as

_{p}*c*as well as noticable difference between the translational and vibrational temperature in the stagnation region (see 50 km case in Fig. 8). The relative error of predicted

_{p}*c*decreases at higher altitudes, suggesting the presence of strong (local) turbulent effects occurring at lower altitudes in the stagnation region between the bow shock and the front surface.

_{p}Detailed views of the pressure coefficient distribution on the front surface of the capsule are presented in Fig. 10. The assumption that turbulent effects play an important role in the stagnation zone at lower altitudes is supported by an asymmetry of the pressure distribution visible at 30 km. Furthermore, the maximum obtained mass fraction CO at this altitude is 1.2%, while the vibrational temperature distribution is very similar to the translational temperature ( $ T t r max \u2243 T v max = 2900 \u2009 K$), implying that the thermo-chemical flow characteristics are close to equilibrium and cannot attribute to strong local pressure or temperature fluctuations in the stagnation region.

## V. CONCLUSION

The *hyStrath* framework was employed to simulate both the hypersonic rarefied and continuum flow regime around the Viking 1 capsule in order to evaluate the framework for predicting the lift and drag coefficients during hypersonic reentry. Using a fully unstructured octree adaptive meshing approach, multi-temperature model, and specialized wall boundary conditions, very good agreement with flight data was achieved. In the high-altitude rarefied and transitional regime, where the *dsmcFoam+* solver was employed, the formation of highly non-equilibrium flow and eventual formation of a (thickened) bow shock ( $ h \u2272 100 \u2009 km$) that drastically influence the flow physics and, therefore, the aerodynamic coefficients around the capsule, could be captured. Only the drag coefficient could be evaluated with flight data in the rarefied and transitional regime and showed excellent agreement with flight data analyzed by Blanchard and Walberg.^{40} The *hy2Foam* solver used for simulating the hypersonic continuum regime during Viking 1 reentry enjoyed improved validation due to the better availability of data for the continuum hypersonic coefficients. Here, the predictions for both the lift and drag coefficients could be, on the one hand, improved compared to a previous publication by Edquist^{20} and, on the other hand, the maximum altitude for employing a continuum solver increased from 48 to 70 km. Additionally, stagnation pressure coefficients obtained from a combination of flight data pressure gauges and expansion tube ground tests could be reproduced, indicating that the non-equilibrium flow in the stagnation region could be accurately modeled using the *hy2Foam* solver. Note that the SST model cannot fully capture the highly transient and turbulent effects in the stagnation zone, which become increasingly significant at lower altitudes and freestream Reynolds numbers. This is a finding that provides a basis for future studies of turbulent effects in the stagnation zone, especially using large eddy simulation (LES).

The present study serves as a basis for further research on reentry vehicles not only bound to Mars reentry. Next steps include evaluation of the framework and meshing approach for predicting aerothermal loads on Mars, Earth, and Titan reentry vehicles as well as simulating currently developed vehicles and future designs.

## ACKNOWLEDGMENTS

Computational resources (HPC-cluster HSUper) have been provided by the project hpc.bw, funded by dtec.bw—Digitalization and Technology Research Center of the Bundeswehr. dtec.bw is funded by the European Union—NextGenerationEU.

Support in the improvement of the *dsmcFoam+* code and general assistance in the implementation of thermodynamic properties and reactions by Dr. Vincent Casseau is gratefully acknowledged.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Maximilian Maigler:** Conceptualization (lead); Data curation (equal); Formal analysis (lead); Investigation (equal); Methodology (equal); Project administration (lead); Software (lead); Validation (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (equal). **Valentina Pessina:** Conceptualization (supporting); Data curation (lead); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (supporting); Writing – review & editing (equal). **Jochen Schein:** Funding acquisition (lead); Project administration (equal); Supervision (lead).

## DATA AVAILABILITY

Raw data were generated at the *HSUper* large scale facility. Derived data supporting the findings of this study are available from the corresponding author upon reasonable request.

### APPENDIX A: META INFORMATION

Atmospheric freestream properties and telemetry of investigated flow cases (Table I).

h, km
. | Flow regimea . | $ Kn \u221e$ . | α, °
. | $ u \u221e$, ms^{-1}
. | $ p \u221e$, Pa . | $ T \u221e$, K . | CO_{2}, %
. | N_{2}, %
. | Ar, % . |
---|---|---|---|---|---|---|---|---|---|

30 | C | 3 × 10^{−5} | 12.1 | 2700 | 3.8 × 10 | 171 | 96 | 2 | 2 |

40 | C | 9 × 10^{−5} | 10.8 | 4013 | 1.1 × 10 | 152 | 96 | 2 | 2 |

50 | C | 3 × 10^{−4} | 10.5 | 4450 | 2.6 × 10 | 143 | 96 | 3 | 1 |

60 | C | 2 × 10^{−3} | 11.0 | 4550 | 6.6 × 10^{−1} | 166 | 96 | 3 | 1 |

70 | C | 5 × 10^{−3} | 12.2 | 4561 | 1.6 × 10^{−1} | 145 | 96 | 3 | 1 |

80 | T | 3 × 10^{−2} | 11.1 | 4560 | 3.4 × 10^{−2} | 152 | 96 | 3 | 1 |

90 | T | 5 × 10^{−2} | 11.1 | 4554 | 2.2 × 10^{−2} | 168 | 96 | 3 | 1 |

100 | T | 1 × 10^{−1} | 11.1 | 4547 | 8.1 × 10^{−3} | 161 | 96 | 3 | 1 |

110 | T | 6 × 10^{−1} | 11.1 | 4540 | 2.1 × 10^{−3} | 189 | 96 | 3 | 1 |

120 | T | 1 × 10^{0} | 11.1 | 4532 | 1.3 × 10^{−3} | 198 | 96 | 3 | 1 |

130 | R | 8 × 10^{0} | 11.1 | 4523 | 9.1 × 10^{−5} | 122 | 96 | 3 | 1 |

140 | R | 4 × 10^{1} | 11.1 | 4514 | 2.3 × 10^{−5} | 167 | 94 | 5 | 1 |

h, km
. | Flow regimea . | $ Kn \u221e$ . | α, °
. | $ u \u221e$, ms^{-1}
. | $ p \u221e$, Pa . | $ T \u221e$, K . | CO_{2}, %
. | N_{2}, %
. | Ar, % . |
---|---|---|---|---|---|---|---|---|---|

30 | C | 3 × 10^{−5} | 12.1 | 2700 | 3.8 × 10 | 171 | 96 | 2 | 2 |

40 | C | 9 × 10^{−5} | 10.8 | 4013 | 1.1 × 10 | 152 | 96 | 2 | 2 |

50 | C | 3 × 10^{−4} | 10.5 | 4450 | 2.6 × 10 | 143 | 96 | 3 | 1 |

60 | C | 2 × 10^{−3} | 11.0 | 4550 | 6.6 × 10^{−1} | 166 | 96 | 3 | 1 |

70 | C | 5 × 10^{−3} | 12.2 | 4561 | 1.6 × 10^{−1} | 145 | 96 | 3 | 1 |

80 | T | 3 × 10^{−2} | 11.1 | 4560 | 3.4 × 10^{−2} | 152 | 96 | 3 | 1 |

90 | T | 5 × 10^{−2} | 11.1 | 4554 | 2.2 × 10^{−2} | 168 | 96 | 3 | 1 |

100 | T | 1 × 10^{−1} | 11.1 | 4547 | 8.1 × 10^{−3} | 161 | 96 | 3 | 1 |

110 | T | 6 × 10^{−1} | 11.1 | 4540 | 2.1 × 10^{−3} | 189 | 96 | 3 | 1 |

120 | T | 1 × 10^{0} | 11.1 | 4532 | 1.3 × 10^{−3} | 198 | 96 | 3 | 1 |

130 | R | 8 × 10^{0} | 11.1 | 4523 | 9.1 × 10^{−5} | 122 | 96 | 3 | 1 |

140 | R | 4 × 10^{1} | 11.1 | 4514 | 2.3 × 10^{−5} | 167 | 94 | 5 | 1 |

^{a}

**C**: Continuum (*hy2Foam*), **T**: Transitional (*dsmcFoam+*), **R**: Rarefied (*dsmcFoam+*).

### APPENDIX B: dsmcFoam+ SETUP

Q-K reactions used in *dsmcFoam+* simulations (Table II). Collision-averaged VSS model parameters based on Ref. 46 with $ T ref = 273 \u2009 K$ (Table III).

Reaction type . | Nr. . | Reaction . |
---|---|---|

Dissociation | 1 | CO_{2} + CO_{2} → CO + O + CO_{2} |

2 | CO_{2} + CO → CO + O + CO | |

3 | CO_{2} + N_{2} → CO + O + N_{2} | |

4 | CO_{2} + Ar → CO + O + Ar | |

5 | N_{2} + N_{2} → N + N + N_{2} | |

6 | N_{2} + CO_{2} → N + N + CO_{2} | |

7 | N_{2} + O_{2} → N + N + O_{2} | |

8 | N_{2} + Ar → N + N + Ar | |

9 | O_{2} + O_{2} → O + O + O_{2} | |

10 | O_{2} + CO_{2} → O + O + CO_{2} | |

11 | O_{2} + N_{2} → O + O + N_{2} | |

12 | O_{2} + Ar → O + O + Ar | |

Exchange | 13 | NO + O $\u21cc$ N + O_{2} |

14 | N_{2} + O $\u21cc$ NO + N | |

15 | CO + O $\u21cc$ C + O_{2} | |

16 | CO_{2} + O $\u21cc$ CO + O_{2} |

Reaction type . | Nr. . | Reaction . |
---|---|---|

Dissociation | 1 | CO_{2} + CO_{2} → CO + O + CO_{2} |

2 | CO_{2} + CO → CO + O + CO | |

3 | CO_{2} + N_{2} → CO + O + N_{2} | |

4 | CO_{2} + Ar → CO + O + Ar | |

5 | N_{2} + N_{2} → N + N + N_{2} | |

6 | N_{2} + CO_{2} → N + N + CO_{2} | |

7 | N_{2} + O_{2} → N + N + O_{2} | |

8 | N_{2} + Ar → N + N + Ar | |

9 | O_{2} + O_{2} → O + O + O_{2} | |

10 | O_{2} + CO_{2} → O + O + CO_{2} | |

11 | O_{2} + N_{2} → O + O + N_{2} | |

12 | O_{2} + Ar → O + O + Ar | |

Exchange | 13 | NO + O $\u21cc$ N + O_{2} |

14 | N_{2} + O $\u21cc$ NO + N | |

15 | CO + O $\u21cc$ C + O_{2} | |

16 | CO_{2} + O $\u21cc$ CO + O_{2} |

Species . | $ d ref$ (Å) . | ω (K)
. | $ \alpha VSS$ . |
---|---|---|---|

CO_{2} | 4.647 | 0.693 | 1.373 |

CO | 4.101 | 0.726 | 1.341 |

NO | 3.983 | 0.716 | 1.425 |

N_{2} | 3.911 | 0.693 | 1.351 |

O_{2} | 3.773 | 0.702 | 1.391 |

O | 3.340 | 0.772 | 1.471 |

N | 3.402 | 0.753 | 1.477 |

C | 4.042 | 0.811 | 1.490 |

Ar | 3.832 | 0.700 | 1.384 |

Species . | $ d ref$ (Å) . | ω (K)
. | $ \alpha VSS$ . |
---|---|---|---|

CO_{2} | 4.647 | 0.693 | 1.373 |

CO | 4.101 | 0.726 | 1.341 |

NO | 3.983 | 0.716 | 1.425 |

N_{2} | 3.911 | 0.693 | 1.351 |

O_{2} | 3.773 | 0.702 | 1.391 |

O | 3.340 | 0.772 | 1.471 |

N | 3.402 | 0.753 | 1.477 |

C | 4.042 | 0.811 | 1.490 |

Ar | 3.832 | 0.700 | 1.384 |

### APPENDIX C: hy2Foam SETUP

Reaction type . | Nr. . | Reaction . | A, m^{3} mol^{−1} s^{−1}
. | β
. | T, K
. _{a} |
---|---|---|---|---|---|

Dissociation | 1 | CO_{2} + M → CO + O + M | 6.9 × 10^{17} | −1.50 | 63 275 |

2 | CO + M → C + O + M | 2.3 × 10^{16} | −1.00 | 129 000 | |

3 | O_{2} + M → O + O + M | 2.0 × 10^{18} | −1.50 | 59 750 | |

4 | N_{2} + M → N + N + M | 7.0 × 10^{18} | −1.60 | 113 200 | |

5 | NO + M → N + O + M | 8.4 × 10^{9} | 0.00 | 19 450 | |

Exchange | 6 | CO + O $\u21cc$ C + O_{2} | 3.9 × 10^{10} | −0.18 | 69 200 |

7 | CO_{2} + O $\u21cc$ CO + O_{2} | 2.1 × 10^{10} | 0.00 | 27 800 | |

8 | N_{2} + O $\u21cc$ NO + N | 6.4 × 10^{14} | −1.00 | 38 370 | |

9 | NO + O $\u21cc$ N + O_{2} | 8.4 × 10^{19} | 0.00 | 19 450 |

Reaction type . | Nr. . | Reaction . | A, m^{3} mol^{−1} s^{−1}
. | β
. | T, K
. _{a} |
---|---|---|---|---|---|

Dissociation | 1 | CO_{2} + M → CO + O + M | 6.9 × 10^{17} | −1.50 | 63 275 |

2 | CO + M → C + O + M | 2.3 × 10^{16} | −1.00 | 129 000 | |

3 | O_{2} + M → O + O + M | 2.0 × 10^{18} | −1.50 | 59 750 | |

4 | N_{2} + M → N + N + M | 7.0 × 10^{18} | −1.60 | 113 200 | |

5 | NO + M → N + O + M | 8.4 × 10^{9} | 0.00 | 19 450 | |

Exchange | 6 | CO + O $\u21cc$ C + O_{2} | 3.9 × 10^{10} | −0.18 | 69 200 |

7 | CO_{2} + O $\u21cc$ CO + O_{2} | 2.1 × 10^{10} | 0.00 | 27 800 | |

8 | N_{2} + O $\u21cc$ NO + N | 6.4 × 10^{14} | −1.00 | 38 370 | |

9 | NO + O $\u21cc$ N + O_{2} | 8.4 × 10^{19} | 0.00 | 19 450 |

## REFERENCES

*Molecular Gas Dynamics and the Direct Simulation of Gas Flows*

*International Symposium on Rarefied Gas Dynamics, ARC-E-DAA-TN59250*(

*ECCOMAS CFD 2006: Proceedings of the European Conference on Computational Fluid Dynamics, Egmond aan Zee, The Netherlands, September 5-8, 2006*

*44th AIAA Thermophysics Conference*

*7th European Conference for Aeronautics and Space Sciences*

*Non-Equilibrium Dynamics: From Physical Models to Hypersonic Flights*(The von Karman Institute for Fluid Dynamics, Belgium, 2009).

*Upwind and High-Resolution Schemes*

_{2}flows, and calibration results in Langley 6-inch expansion tube