VERIFICATION AND VALIDATION OF A PARALLEL 3D DIRECT SIMULATION MONTE CARLO SOLVER FOR ATMOSPHERIC ENTRY APPLICATIONS

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1 VERIFICATION AND VALIDATION OF A PARALLEL 3D DIRECT SIMULATION MONTE CARLO SOLVER FOR ATMOSPHERIC ENTRY APPLICATIONS Paul Nizenkov and Stefanos Fasoulas Institut für Raumfahrtsysteme (IRS), Universität Stuttgart, 569 Stuttgart, Germany nizenkov@irs.uni-stuttgart.de ABSTRACT The in-house Direct Simulation Monte Carlo (DSMC) solver is verified and validated, which enables parallel, three-dimensional simulations of rarefied gas flows on high-performance clusters. Theoretical aspects of the DSMC method and newly employed schemes enabling grid-independence are briefly discussed. Cases including theoretical calculations, experimental measurements and other numerical simulations are selected from literature and compared to the code. Simulation results of the hypersonic flow around a cylinder, rarefied gas flow around a blunted-cone as well as multiple points of the re-entry trajectory of the Orion capsule are presented in terms of drag and heat flux. An outlook on future code development and applications is given.. INTRODUCTION Rarefied, non-equilibrium gas flows as encountered during atmospheric aerocapture and entry manoeuvres pose a great challenge for the design of spacecraft and their thermal protection system. Even in the continuum flow regime, regions of transitional and free-molecular flow are present in the wake of the heat shield, where control elements and/or the payload might be located. Moreover, the complex rarefied flow environment with significant disturbances is outside of the capabilities of conventional continuum methods such as Navier-Stokes based computational fluid dynamics (CFD). A microscopic approach is required to enable an accurate description of such flows. It models fluid properties on a molecular level and is mathematically described by the Boltzmann equation. Based on these ideas, the Direct Simulation Monte Carlo (DSMC) method was introduced in the 96s by Graeme A. Bird and relies on statistical particle treatment and phenomenological modelling of collisions, energy relaxation and chemical reactions []. The DSMC method describes the fluid flow in a discrete manner with simulation particles, where every simulation particle represents a certain amount of real particles. The basic assumptions are that particle movement and collisions can be decoupled and only binary collisions have to be considered. While the DSMC method does not numerically solve the Boltzmann equation, it was shown that in the infinite particle number limit, the DSMC method approximates the Boltzmann equation []. Moreover, the method was further developed and validated in the last 5 years by different research groups. An excellent summary of current developments and capabilities of the method is given in [3]. The Institute of Space Systems (IRS) and the Institute of Aerodynamics and Gas Dynamics (IAG) at the University of Stuttgart cooperatively develop a coupled Particle in Cell (PIC) and DSMC solver called, which enables the simulation of reactive plasma flows [4]. Besides atmospheric entry simulation, the code is applied to the simulation of electric propulsion systems and highlyionized plasmas. The focus of the present paper lies solely on the DSMC implementation of the code, which is referred to as in the following.. SIMULATION The general simulation procedure during one time step is depicted in Fig.. A reservoir or free-stream flow is modelled by simulation particles which are inserted at the beginning of the simulation or every time step, respectively. The number of inserted particles is determined by the number density, weighting factor (ratio of real to simulation particles) and insertion volume. Particles are then moved through the computational domain and possible gas-surface interactions are considered. Subsequently, collision pairs are formed in each cell and a statistical decision process based on random numbers for chemical reactions and treatment of the internal degrees of freedom is conducted. During this process, particles are only associated with microscopic information such as individual internal energy and velocity. Finally, macroscopic values are sampled through spatial and temporal averaging of weighted particle properties before the next time step. The code utilizes several phenomenological models to describe a physical flow. Collision partners are found with the nearest neighbour scheme and an octree-sorting algorithm [5], avoiding the introduction of numerical diffusion if the fluid domain is not resolved with the mean free path. The sorting algorithm accelerates the search for the nearest collision partner and guarantees a physically correct determination of the collision probability, independent of the grid resolution. Collisions are treated with the

2 Particle movement Boundary treatment Sampling Δt Nearest neighbour Octree-sorting Particle pairing Collision process Chemical reactions Relaxation Figure. Schematic of a DSMC time step in. variable hard sphere (VHS) model, where the collisional cross-section depends on the relative translational energy and probabilities are calculated by the Natural Sample Size method [6]. Energy transfer between kinetic and internal modes is treated with the Larsen-Borgnakke model []. While rotational energy modes are assumed to be continuous, vibrational modes are quantized with the simple harmonic oscillator. The probabilities for rotational and vibrational relaxation are fixed values of P rot =. and P vib =.4 based on empirical considerations. Chemical reactions are enabled through the use of the extended Arrhenius equation. Surface interactions are modelled by the Maxwell model with surface accommodation coefficients, currently without any catalytic effects. The probabilistic nature of the method introduces statistical fluctuations for steady-state problems. An increase of the sample size (longer sampling duration and/or more simulation particles) can reduce these fluctuations. The quality of simulation results is established by several measures. To avoid a non-physical time step, maximal collision probabilities are monitored (P max < ) and the time step adjusted accordingly. The weighting factor is chosen to achieve a meaningful statistical ensemble. Specifically, a sufficient number of simulation particles per cell in low density regions (with respect to the free-stream) has to be ensured. Although a generally applicable value cannot be given, mean values of N part,cell > 5 have proven to produce acceptable results. To achieve an accurate transient simulation, an increased number of particles has to be present at each time step in lieu of temporal averaging. For steady-state problems, convergence is determined based on the transient development of the variable of interest. 3. VERIFICATION AND VALIDATION This section presents cases collected from literature to verify and validate different capabilities of the code. In Section 3., the chemistry model is verified by the means of comparison to semi-empirical and numerical solutions of reservoir simulations, while the following sections present more complex geometries such as a cylinder, a spherically blunted cone and the Orion capsule. In addition, simulated flow conditions increase in difficulty starting with a monatomic gas advancing to a diatomic gas with internal energy relaxation and finally high-temperature air including chemical reactions. 3.. Chemistry Model The chemistry module of the DSMC code is verified based on simple reservoir simulations to exclude the influence of a complex geometry. In the simulation of entry manoeuvres, dissociation and recombination of molecules is of great importance. The first case focuses on such a process for a single species A A + A. The degree of dissociation can be determined by α = n A n A + n A, () where n A and n A are the number densities of atoms and molecules, respectively. The equilibrium composition of a dissociating gas can be described as a function of density and temperature [] α α = ρ ( d ρ exp T ) d, () T where the subscript d denotes species characteristic values, for example T d = 3 K and ρ d =.3 5 kg m 3 for nitrogen. Although the characteristic density ρ d is dependent on the prevalent temperature, differences are found to be negligible and a constant value is chosen []. For the nitrogen reservoir calculation with, a cube with a side length of 5 m and specular walls is simulated. The viscosity coefficient ω =. and the characteristic temperature T ref = K of the simulated species have to be equal due to the VHS model, resulting in reference diameters of d ref = 3 m and d ref = 4. m for atomic and diatomic nitrogen, respectively. The characteristic vibrational temperature for diatomic nitrogen is set to Θ vib = 3395 K. The density ratio ρ d /ρ is chosen as, resulting in a number density of n =.9 3 m 3. The employed reaction rates are summarized in Tab.. A range of temperatures was simulated including two dissociative and recombination reactions each. After thermal and chemical equilibrium was observed, the degree of dissociation was calculated with Eq. and compared to the equilibrium composition resulting from Eq.. Generally, good agreement between the theoretical and numerical solution, as shown in Fig., is observed. While a slightly earlier onset of dissociation can be observed for the DSMC Table. Arrhenius coefficients for nitrogen dissociation and recombination. A [ m 3 s, m 6 s ] b [ ] E a [K] N + N N + N N + N + N N + N + N. 39.6

3 Degree of dissociation α [ ] Theory Temperature T [K] Figure. Degree of dissociation of diatomic nitrogen at equilibrium, ρ d /ρ =. simulation, differences are within reason considering uncertainties in the input parameters such as the characteristic values for Eq. and the Arrhenius coefficients for the DSMC simulation. The next test case considers a transient reservoir simulation of five-species air (N,O, N, O, NO) with 34 potential chemical reaction paths and an initial composition of 9% N and % O. The case is representative for chemical processes occurring during high-enthalpy atmospheric entry. Three different initial conditions (T = K, K, and 3 K at p =.35 kpa) were simulated with and compared to a numerical solution which can be found by means of simple first-order Euler integration [9], if thermal equilibrium is assumed (equilibrium between different energy modes). Simulation parameters such as number density n, cube side length λ and total initial number of particles are given in Tab.. While the same species parameters are utilized for N and N as in the previous test case, the reference diameter for O, NO and O at T ref = K and ω =. is d ref = 3.96 m, 4. m, and 3. m, respectively. Characteristic vibrational temperatures are Θ vib,o = 39 K and Θ vib,no = K. Arrhenius coefficients are set according to []. To augment thermal equilibrium, the rotational and vibrational relaxation probabilities were set to unity. Since this does not guarantee thermal equilibrium, total temperature of each species was calculated with T tot = 3T tra + T rot + ξ vib T vib 5 + ξ vib, (3) where ξ vib is the corresponding vibrational degree of freedom and subscripts tra, rot and vib denote the transla- Table. Parameters for the reservoir simulation of fivespecies air at p =.35 kpa. T [K] n [ m 3 ] λ [m] N part [ ] tional, rotational, and vibrational energy mode. Total reservoir temperature was determined by the particle-weighted mean of species temperatures. Fig. 3 shows the transient development of the species composition at an initial temperature of 3 K. The number density of the respective species is normalized by the initial number density. Excellent agreement is found between the numerical solution and particle simulation. An early onset of O and N dissociation followed by formation of NO can be observed as expected for high-temperature air. Moreover, the deviation from thermal equilibrium does not seem to have a great effect on the species composition when compared to the numerical solution. Statistical noise can be seen in the O and NO concentration towards chemical equilibrium due to very low particle numbers of the respective species and consequently an insufficient ensemble size. However, the simulation result appears to be oscillating around the numerical solution. ns/n [ ] N 3 4 O O NO N Numerical 9 6 Time [s] Figure 3. Species composition of a five-species air reservoir during chemical relaxation (T = 3 K). The transient species composition of a reservoir with an initial temperature of K is shown in Fig. 4. Dissociation of both diatomic species is considerably slower compared to the previous test case. Consequently, results for the atomic nitrogen are accompanied by statistical noise due to low particle numbers. Nonetheless, the particle simulation agrees very well with the numerical solution. The temperature development is of great interest in order to verify the implementation of the chemistry as well as relaxation module. Fig. 5 illustrates the transient reservoir temperature for the three different initial conditions. The equilibrium temperature predicted by the numerical solution is compared to the total reservoir temperature determined by Eq. 3. The agreement between both computations is excellent for all three initial temperatures. In conclusion, the chemistry module is able to reproduce Arrhenius type reaction rates and the corresponding energy variation in a reservoir simulation. Transient species composition and temperature are in excellent agreement with theoretical results.

4 N ns/n [ ] 3 4 O O N NO Numerical 9 6 Time [s] Figure 4. Species composition of a five-species air reservoir during chemical relaxation (T = K). Temperature Teq,Ttot [K] K K K Numerical 9 6 Time [s] Figure 5. Temperature comparison of a five-species air reservoir during chemical relaxation. 3.. Cylinder Simulation results of a simple, two-dimensional cylinder exposed to a Mach argon flow (u = 634. m s, T = K) are compared to other simulations []. The usage of the monatomic argon eliminates the possibility of internal energy relaxation and simplifies the code-to-code verification process. Results of a Navier- Stokes solver DPLR, developed at NASA s Ames Research Center, and the DSMC code MONACO, developed by Iain D. Boyd and his research group at the University of Michigan, are utilized for the verification. Two different flow regimes, characterized by the Knudsen number, Kn =. and Kn =.5, are investigated. While the first test case is in the continuum regime with a number density of n = 4.4 m 3, the second test case represents a rarefied gas flow at a number density of n = m 3. Species parameters are chosen according to []. The cylinder diameter is.34 m (= in) with complete diffuse reflection as well as thermal accommodation and a constant wall temperature of 5 K. The nomenclature and geometry are given in Fig. 6. M Φ Figure 6. Definition of cylinder test case. At first, the drag of the cylinder given per unit length as a two-dimensional problem is considered. Excellent agreement between all three simulations can be seen in Tab. 3 at the lower Knudsen number Kn =.. Although both DSMC simulations compute a lower total drag per length with respect to the CFD simulation, the difference is negligible. For the rarefied gas flow at Kn =.5, both DSMC simulations compare very well but a strong deviation of the CFD results can be observed due to continuum breakdown. Table 3. Comparison of the total drag/length [N m ]. Kn MONACO DPLR < % 4. < %.5.. < %.64 % In the following, the heat flux along the cylinder angle Φ is compared. As depicted in Fig., a good agreement between all codes as expected due to flow conditions close to the continuum regime is achieved. The peak heat flux is lower by < % and 4% than the other DSMC and CFD code, respectively. The results of the test case in the transitional flow regime at Kn =.5 can be seen in Fig.. Again, good agreement is achieved between the DSMC codes with a deviation of just % at the stagnation point, while the CFD code overpredicts the heat flux due to the breakdown of the continuum assumption. The peak heat flux at the stagnation point is overpredicted by 35%, with an increase of the relative deviation further downstream in accordance with an increasingly rarefied gas flow. In conclusion, the first results of an incident flow suggest a correct implementation of the elastic collision module as well as wall treatment routines in and allow the advancement to more complex problems. 4 3 MONACO DPLR Φ [ ] Figure. Heat flux along cylinder surface, Kn =..

5 6 5 4 MONACO DPLR Φ [ ] Figure. Heat flux along cylinder surface, Kn = Blunted Cone The next validation case is a spherically blunted cone geometry in a Mach molecular nitrogen flow. As opposed to the previous case, relaxation of internal energy modes such as rotation and vibration is now possible. al results [] are available for three different inflow conditions and different angles of attack (α =,,, and 3 ). The experimental test conditions are given in [] and the derived free-stream conditions for the simulation are shown in Tab. 4. The VHS model parameters of nitrogen are d ref = 4. m as the reference diameter at a temperature of T ref = 3 K and a viscosity coefficient of ω =.4. The characteristic vibrational temperature is set to Θ v = 3395 K. Table 4. Free-stream conditions of the nitrogen flow. Case v [ m s ] T [K] n [m 3 ] Set Set Set The model geometry and positions of the thermocouples can be seen in Fig. 9, while details of the experimental setup can be found in []. al measurements were taken on the forebody (points -4), the base (points 5-6) and the sting (-9). A global uncertainty of ±% is given in [] for all experimental measurements of the heat flux except when stated otherwise. The temperature of the thin-walled model used in the experiments fluctuated between 9 K and 3 K, thus a constant wall temperature of 3 K was chosen for the simulation. Full thermal and complete diffuse reflection was assumed. For the simulation without an angle of attack, only / (= 3 slice) of the axisymmetric fluid domain was simulated while for the other cases half of the fluid domain was modelled, taking advantage of the symmetry in the xy-plane. Beginning with the first set of test conditions (n = 3.5 m 3 ), the heat flux along the surface of the model at α = is shown in Fig.. Labels of the experimental data points correspond to the positions of the thermocouples and the path variable S is normalized with the nose radius R n =.5 m. Good qualitative agreement can be seen on the forebody of the blunted cone M [mm] R b 5. R c.5 R j. R n.5 R s 6.5 y S R b 3 R c R n z x Figure 9. blunted cone geometry and positions of thermocouples. R j R s # S/R n as well as on the sting in the wake. While the error at the first three measurement points is within the experimental uncertainty of ±%, the heat flux is overpredicted by 9% at the fourth thermocouple. al results of thermocouples 5 and 6 were given as a maximal value (<. kw m ) since experimental accuracy cannot be guaranteed due to very low heating rates. On the sting, the error increases from 6% to 4% for the last three thermocouples Figure. Comparison of heat flux along surface of blunted cone (Set, α = ). Fig. shows the comparison for the second set of test conditions (n =.5 m 3 ) at α =. Again, good qualitative agreement with the experiment can be observed on the forebody, base and sting of the model. The error compared to the first four thermocouples is below %. Although the measurements on the base are still very low and expected to be equal or less than.4 kw m, the error is below %. Heating rates at thermocouples and deviate by less than 3%. The last measurement cannot be matched by the simulation with almost an error of 5%. Since the correct tendency is predicted for the other flow conditions, especially at the sting, it is not clear whether the deviation is due to inaccu-

6 rate simulation results or uncertainties in the experimental measurements. Except for the last two measurements points, the particle simulation was able to stay within the experimental uncertainty for the second set of flow conditions. The heat flux at the most dense flow conditions (n =. m 3 ) at α = is shown in Fig.. Although, the heat flux on the forebody is overpredicted by % to 4%, good quantitative agreement can be seen on the base with errors below 6%. Errors below % and thereby within the experimental uncertainty are determined on the sting of the model. As noted in [], high wall-temperature gradients along the forebody may have possibly led to local conduction effects resulting in underestimated heating rates with the strongest deviation up to thermocouple 4. This assertion corresponds to the simulation results of and an other DSMC code [3] Figure. Comparison of heat flux along surface of blunted cone (Set, α = ) Figure. Comparison of heat flux along surface of blunted cone (Set 3, α = ). Finally, Fig. 3 illustrates results of full, three-dimensional simulations of Set at angles of attack α = and α = 3. As logarithmic plots can be deceiving and the values for thermocouples 5 and 6 can be excluded from the comparison of Set, regular axes are utilized. Good qualitative agreement can be observed for both angles of attack. Excellent error values below 5% are achieved on the forebody at α = and acceptable agreement with errors below 3% can be seen on the sting. A very different heat flux behaviour is observed at α = 3, with the maximum heat flux shifted to the shoulder of the forebody. While the error stays below 3% for the first four thermocouples, better agreement is achieved at the sting with the last two thermocouples within the experimental uncertainty. Concluding, this test case demonstrated the ability of to model flows of diatomic molecules with internal degrees of freedom around complex, threedimensional geometry (a) α = (b) α = 3 Figure 3. Comparison of heat flux along surface of blunted cone at different angles of attack (Set ) Orion Capsule The last comparison is based on the Orion Crew Module (CM) and the comparison with other DSMC and CFD simulations [4]. The free-stream conditions given in Tab. 5 feature a hypersonic flow with a wide range of flow conditions from the continuum to the rarefied gas regime. Fig. 4 shows the geometric definition of the capsule. The simulation included high-temperature chemistry with five-species air (parameters identical to the reservoir test case in Section 3.) and 34 chemical reactions (Arrhenius coefficients []) while ionization was neglected as it was found to have little influence on aerodynamic coefficients. The initial atmospheric composition for the corresponding altitude can be found in [4]. Full diffuse reflection and

7 Table 5. Free-stream conditions and wall temperature for Orion CM (u = 6 m s, α = 6 ) [4]. Alt. [km] n [m 3 ] T [K] Kn [-] T w [K] [m] R b.546 R n 6.35 R s.55 R as.5 L 3.3 β 3.5 α R b S R s R n β R as Drag coefficient [ ] LAURA DS3V DAC 4 6 Altitude [km] Figure 5. Drag coefficient over altitude of Orion CM (u = 6 m s, α = 6 ). To summarize, the presented DSMC code was applied to a complex geometry such as the Orion re-entry capsule. Results of the drag coefficient showed very good agreement with well-established CFD and DSMC codes, free-stream conditions ranging from the continuum to rarefied gas flow regime. It should be noted that although the DSMC method is applicable in the continuum regime, it comes with high computational effort due to high particle numbers and a low required time step. For example, the total simulation duration of the most dense flow at 5 km altitude with simulation particles and a time step of t = 4 9 s was over hours on 4 Intel Xeon CPU E5-6 v3 processors (96 cores). Figure 4. Geometric definition of the Orion CM. complete thermal accommodation was assumed with an isothermal wall temperature T w presented in Tab. 5. The drag coefficient was calculated based on the base radius R b, free-stream velocity u and density ρ. The study [4] included simulations with two DSMC codes, DS3V and DAC, and the CFD code LAURA. DS3V, which is freely available, is developed by Graeme A. Bird and DAC by G. J. LeBeau from NASA s Johnson Space Center. The CFD code LAURA is developed by NASA s Langley Research Center. Details regarding these codes and respective references are available in [4]. While calculations with the continuum approach were conducted from 5 km up to 5 km altitude, particle-based simulations ranged from 5 km to km altitude. The results of the simulations in terms of drag coefficient over altitude compared to are presented in Fig. 5. Excellent agreement of the drag coefficient is observed between the DSMC codes at all altitudes. The deviation from the DAC and DS3V results is below % and 3%, respectively. Furthermore, excellent agreement with the CFD code is achieved at lower altitudes of 5 km and 95 km, with a deviation of less than %. At higher Knudsen number, the breakdown of continuum is evident and the difference increases to roughly %. 4. CONCLUSION The capabilities of were presented by means of several verification and validation cases. Through reservoir simulations, the chemistry module was compared to semi-empirical and numerical relations, showing excellent agreement. Following that, more complex geometries and flow conditions were investigated. First, simulations of hypersonic flow of argon in the continuum and transitional regime around a cylinder were compared to results of other DSMC as well as CFD codes. Very good agreement with other DSMC simulations is achieved whereas good agreement with CFD results is only observed in the continuum regime. Second, a comparison with experimental measurements of the heat flux on a spherically blunted cone and its sting in the wake was made. The results of the threedimensional simulation of the rarefied, hypersonic nitrogen flow compared well on the forebody and sting of the model. Finally, simulation of multiple points of Orion s reentry trajectory comprised a full, three-dimensional case with high-temperature air chemistry and was compared to other CFD and DSMC simulations. To summarize the capabilities of the presented DSMC code: Full, three-dimensional simulation, Parallel computation on high-performance clusters, Chemistry modelling of diatomic molecules, and Grid-independence enabled through nearest neighbour scheme with octree-sorting algorithm.

8 The code is subject to on-going research, some of which is applicable to atmospheric entry. This includes the extension to polyatomic molecules required for the simulation of Mars and Titan entry. Automatic convergence detection and variable relaxation probabilities are implemented as part of student research projects. Moreover, coupling with the Low Diffusion method, applicable in the continuum regime, will enable the efficient simulation of flows with large density gradients, such as the flow around blunted entry vehicles. In addition, the application of the coupled PIC-DSMC solver to electromagnetic shielding concepts is envisioned. ACKNOWLEDGEMENTS The authors would like to thank Airbus Defence and Space, namely Dr. Martin Konopka and Peter Noeding, for supporting the research in the framework of a research training group. The authors are also grateful for the financial support by the Landesgraduiertenförderung Baden-Württemberg. Computational resources have been provided by the High Performance Computing Center Stuttgart (HLRS) of the University of Stuttgart. REFERENCES [9] Haas, B.L. & McDonald, J.D. (993). Validation of chemistry models employed in a particle simulation method. Journal of Thermophysics and Heat Transfer (), 4 4. [] Haas, B.L. (99). Fundamentals of chemistry modeling applicable to a vectorized particle simulation, in 5th Joint Thermophysics and Heat Transfer Conference, AIAA, Reston, Virigina. [] Lofthouse, A.J., Boyd, I.D. & Wright, M.J. (). Effects of continuum breakdown on hypersonic aerothermodynamics. Physics of Fluids 9(). [] Allègre, J., Bisch, D. & Lengrand, J.C. (99). al Rarefied Heat Transfer at Hypersonic Conditions over -Degree Blunted Cone. Journal of Spacecraft and Rockets 34(6), 4. [3] Moss, J.N., Dogra, V.K., Price, J.M. & Hash, D.B. (995). Comparison of DSMC and experimental results for hypersonic external flows, in 3th AIAA Thermophysics Conference, AIAA, Reston, Virigina. [4] Moss, J.N., Boyles, K. & Greene, F.A. (6). Orion Aerodynamics for Hypersonic Free Molecular to Continuum Conditions, in 4th AIAA/AHI Space Planes and Hypersonic Systems and Technologies Conference, AIAA, Reston, Virigina, pp [] Bird, G.A. (994). Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Oxford University Press, Oxford. [] Wagner, W. (99). A convergence proof for Bird s direct simulation Monte Carlo method for the Boltzmann equation. Journal of Statistical Physics 66(3-4), 44. [3] Boyd, I.D. (4). Computation of Hypersonic Flows Using the Direct Simulation Monte Carlo Method. Journal of Spacecraft and Rockets (Articles in Advance), 6. [4] Munz, C.D., Auweter-Kurtz, M., Fasoulas, S. et al. (4). Coupled Particle-In-Cell and Direct Simulation Monte Carlo method for simulating reactive plasma flows. Comptes Rendus Mécanique 34(-), [5] Pfeiffer, M., Mirza, A. & Fasoulas, S. (3). A gridindependent particle pairing strategy for DSMC. Journal of Computational Physics 46, 36. [6] Baganoff, D. & Mcdonald, J.D. (99). A collisionselection rule for a particle simulation method suited to vector computers. Physics of Fluids A: Fluid Dynamics (), [] Borgnakke, C. & Larsen, P.S. (95). Statistical collision model for Monte Carlo simulation of polyatomic gas mixture. Journal of Computational Physics (4), [] Lighthill, M.J. (95). Dynamics of a dissociating gas Part I Equilibrium flow. Journal of Fluid Mechanics (), 3.