Rarefied Reactive Gas Flows over Simple and Complex Geometries Using an Open-Source DSMC Solver

Abstract

During atmospheric reentry, a significant number of chemical reactions are produced inside the high-temperature shock wave formed upstream of the spacecraft. Chemical reactions can significantly alter the flowfield structure surrounding the vehicle and affect surface properties, including heat transfer, pressure, and skin friction coefficients. In this scenario, the primary goal of this investigation is to evaluate the Quantum-Kinetic chemistry model for computing rarefied reactive gas flow over simple and complex geometries. The results are compared with well-established reaction models available for the transitional flow regime. The study focuses on two configurations, a sphere and the Orion capsule, analyzed at different altitudes to assess the impact of chemical nonequilibrium across varying flow rarefaction levels. Including chemical reactions led to lower post-shock temperatures, broader shock structures, and significant species dissociation in both geometries. These effects strongly influenced the surface heat flux, pressure, and temperature distributions. Comparison with results from the literature confirmed the validity of the implemented QK model and highlighted the importance of including chemical kinetics when simulating hypersonic flows in the upper atmosphere.

1. Introduction

The development of new space transportation systems has created considerable interest in hypersonic flow simulations. The enthalpy can be large enough to promote dissociation and exchange reactions, ionization, and radiation in hypervelocity. The presence of one or more of these processes may significantly change the flow field properties surrounding the vehicle. In this scenario, developing computational codes that account for thermochemical nonequilibrium conditions is of fundamental importance for aerothermodynamics predictions [1,2,3,4].

Over the past decade, the lack of experimental facilities for investigating hypersonic flows has motivated a significant number of researchers to develop numerous numerical models of reactive gas flows. These models have been proposed and utilized to address various aspects of hypersonic problems in both continuum and rarefied flow regimes. Depending on the reentry conditions, a significant number of chemical reactions occur, which can affect the heat transfer and pressure loads acting on the vehicle’s surface and the flowfield structure around the spacecraft. Bird introduced the first chemical reactions model for rarefied flow applications in 1979 [5]. In the Total Collision Energy (TCE) chemistry model, reaction rates are calculated according to the Arrhenius law [6], which heavily depends on data derived from the equilibrium gas theory. In addition, the TCE model relies on experimental data collected under conditions that do not match those encountered in real flight, and it employs empirical fits extrapolated to flight conditions [7]. The TCE model has been successfully used at mature Direct Simulation Monte Carlo (DSMC) codes for aerothermodynamics computations; however, due to the particulate nature of the DSMC method, it may be inappropriate for rarefied hypersonic flows [8].

In 2008, a new collision-based chemical reactions model was introduced by Bird [9]. This model can predict equilibrium and nonequilibrium reaction rates using only kinetic theory and fundamental molecular properties—i.e., chemical reactions can be modeled without dependence on local macroscopic properties. The computed results using the collision-based model demonstrated good agreement with those obtained in experiments and analytical calculations [9,10]. The need for efficient reaction procedures in DSMC applications, which involve the computation of billions of collisions, led to the development of the Quantum-Kinetic (QK) chemistry model [11,12]. One of the main features of the QK model is the direct linking of the vibrational energy level of a molecule to the dissociation of that molecule [13].

The QK chemistry model has been utilized to investigate a wide range of aerothermodynamic problems, and a selection of these works is discussed herein.

Computational simulations were performed for a hypersonic rarefied flow around a 1.6 m diameter sphere using the DSMC method. The altitude considered in this work ranged from 200 to 90 km altitude and the velocity was kept constant at 7.5 km/s. According to Dogra et al. [14], a TCE-QK blended model, referred to as “new chemistry,” was implemented into the DS2V code and utilized for the computations. The results showed that little chemical reaction activity was present for altitudes higher than 120 km. However, most chemical reactions were activated at an altitude of 90 km, affecting the stagnation point heat transfer and surface properties, including heat transfer, pressure, and skin friction coefficients.

Gimelshein et al. [15] analyzed the accuracy of three DSMC chemistry models applied to 2D normal shock wave problems. The main objective of this investigation was to validate and distinguish differences between chemical reaction models. The DSMC code SMILE was employed in conjunction with the Variable Hard Sphere (VHS) collision model to perform the simulations. Three different chemical reaction models were used: (i) modified TCE to account for quantized vibrational energy; (ii) the Quantum-Kinetic (Q-K) model, and (iii) a DSMC implementation of a two-temperature kinetic model [9,10,16]. Two shock conditions were simulated using pure oxygen at 295 K at Mach numbers of 9.3 and 13.3 over a rectangular body. Additionally, the computational results were compared to those obtained in a shock tube for the ${\mathrm{O}}_2$ vibrational temperature. From this work, it was shown that most validation studies for dissociation models either adjust the overall dissociation rate coefficient or the degree of vibration–translation nonequilibrium is not known.

An extension of the Quantum-Kinetic model to compute ionized hypersonic flows was investigated by Liechty [17]. The new electronic energy level transition and chemistry model were evaluated by their ability to reproduce rates of chemical reactions. Simulations were run considering a test gas composed of 11 species in an adiabatic box 0.002 m on a side with 2 × ${10}^6$ particles. Furthermore, the gas temperature ranged from 10,000 K to 30,000 K at a number density of 1.0 × ${10}^{23} /{\mathrm{m}}^3$. According to Liechty [17], the analytic and sampled values compared favorably with the rates from the literature.

Scanlon et al. [8] presented an open-source implementation of Q-K chemistry modeling for the dsmcFoam code. An adiabatic box was used to investigate vibrational relaxation, dissociation, and exchange reaction rates in a five-species, 19-reaction air model. In addition, a hypersonic rarefied reacting flow over a 2D cylinder was used to test the dsmcFoam implementation of the chemistry model. The test cases demonstrated that the Q-K chemistry model was successfully implemented into the dsmcFoam code, yielding satisfactory concurrence with both analytical and computed results.

Chemically reacting hypersonic flow over 3D cavities was investigated by Palharini et al. [18,19], Palharini and Santos [20]. In this investigation, the reactive gas flow over cavities, with different length-to-depth ratios (L/D), was carried out using the dsmcFoam code. According to this investigation, the flowfield structure highly depends on the cavity L/D ratio. In addition, chemical reactions play a crucial role in determining the properties of the cavity’s surfaces.

Chemistry modeling for the Martian atmosphere was analyzed [21] using the Quantum-Kinetic chemistry model for eight species implemented into the dsmcFoam code. The chemical rates for dissociation and exchange reactions were validated by comparing the obtained results with the Arrhenius-based Total Collision Energy and available experimental data. According to the authors, a good agreement was found for ${\mathrm{CO}}_2$ reactions.

Chen et al. [22] investigated the variation in stagnation heat flux in a hypersonic cylinder flow using an in-house DSMC code derived from compilations based on the DSMC94. The Quantum-Kinetic (QK) chemistry model was employed to perform the high-temperature air reactions. This investigation found that the high-density effect in chemical reactions within the thermal boundary layer induced a significant increase in the stagnation heat flux with decreasing cylinder diameter for all cases. In addition, NO dissociation reaction was observed inside the thermal boundary layer when ${\mathrm{Ma}}_{\infty }$ > 30.

Trivedi et al. [23] performed a numerical investigation of combustion processes for laminar one-dimensional hydrogen-air flames using the DSMC method. The combustion reactions were performed employing the Q-K chemistry model. To achieve the correct flame speed, it was necessary to adjust the activation energy for key reactions. The computed results showed a reasonable agreement with those obtained from Direct Numerical Simulation (DNS) using a conventional continuum approach.

The Q-K model was developed considering the harmonic oscillator (HO) model for treating vibrational excitation. According to the literature [24], this is a reasonable hypothesis, as only a few vibrational levels are excited, which poses limitations for high-enthalpy flows. To overcome this problem, Civrais et al. [25] proposed the use of an anharmonic oscillator (aHO) model. The aHO was implemented into the dsmcFoam code, and the computed reaction rates were compared with analytical expressions, demonstrating excellent agreement between the results and verifying the successful implementation of the model into the computational code. According to the authors, the dissociation reactions are more likely to occur than with the original Q-K model. In addition, a reasonable agreement was found between the aHO model and experimental measurements, high-fidelity calculations, and established chemistry models for both thermal equilibrium and non-equilibrium conditions.

At high altitudes, the collisional time and length scales are large in comparison to those associated with the flow geometry and macroscopic variable gradients. Individual particles experience few collisions over the macroscopic time scales, and the molecular velocity distribution may diverge significantly from the equilibrium distribution. In doing so, continuum governing equations tend to break down, and computational fluid dynamics (CFD) techniques and Arrhenius-based chemistry models become inappropriate in rarefied conditions [26,27]. A particle-based approach is necessary, and the direct simulation Monte Carlo (DSMC) technique was used in the present investigation to compute hypersonic reactive flows using the Quantum-Kinetic (Q-K) chemistry model. The computational code used in the present investigation is called dsmcFoam code. The Q-K chemistry model employed during the computations has been implemented, tested, and updated in collaboration with different universities and research institutes [8,25,28,29]. In this scenario, the main focus of the present study is twofold. First, to extend the computational work performed by Scanlon et al. [8] by applying the Q-K chemistry model to study the impact of reactive gas flows over simple and complex geometries. Second is the continued effort to provide an aerothermodynamic characterization of basic shapes in the transitional flow regime for entry flow environments under thermochemical non-equilibrium conditions.

2. Numerical Procedure

The direct simulation Monte Carlo method (DSMC) was primarily developed by Bird [30] between 1960 and 1980 and has become one of the most important numerical techniques for solving rarefied gas flows in the transition regime. The DSMC method is based on physical concepts of rarefied gases and on the physical assumptions that form the basis for deriving the Boltzmann equation [31]. However, the DSMC method is not derived directly from the Boltzmann equation. As both the DSMC method and the Boltzmann equation are based on classical kinetic theory, the DSMC method is subject to the same restrictions as the Boltzmann equation, i.e., the assumption of molecular chaos and restrictions related to dilute gases.

The DSMC method models the flow as a collection of particles or molecules. Each particle has a position, velocity, and internal energy. The state of the particle is stored and modified with time as the particles move, collide, and interact with the surface in the simulated physical domain. The assumption of a dilute gas—i.e., the mean molecular diameter is much smaller than the mean molecular distance in the gas—allows the molecular motion to be decoupled from molecular collisions. The particle movement is modeled deterministically, while collisions are treated statistically. Since it is impossible to simulate the real number of particles in the computational domain, a small number of representative particles is used, each representing a large number of real particles. Simulations can vary from thousands to millions of DSMC simulator particles in rarefied flow problems.

A computational grid, representing the physical space to be investigated, is required for executing the method. Each cell provides a convenient reference for sampling the macroscopic gas properties and for selecting potential collision pairs.

The linear dimensions of the cells should be small in comparison with the length of the macroscopic flow gradients normal to the streamwise directions, which means that the cell dimensions should be the order of or smaller than the local mean free path Bird [30], Alexander et al. [32]. Another requirement of the DSMC method is the minimum number of simulated particles in each cell. As mentioned earlier, the DSMC method employs a cell-based system for sampling macroscopic properties and selecting collision partners. As the collision rate is a function of the number of particles in the cells, each cell should have the largest possible number of particles. However, the possible number of collision partners is a function of the number of particles in each cell. In this scenario, the greater the number of particles, the greater the number of possible collision pairs. As a result, it is necessary to determine the optimal number of particles in each cell, which is sufficient to promote statistical accuracy while maintaining a realistic computational expenditure.

To solve this conflict, Bird [33] introduced the option of subdividing the cells into an arbitrary number of sub-cells to select collision pairs. This procedure enhances the method’s accuracy by ensuring that collisions occur only between particles that are near neighbors. Thus, each cell should have a minimum number of around 20 to 30 particles [30].

Another requirement of the DSMC method is setting an appropriate time step, $\Delta t$. The trajectories of the particles in physical space are calculated under the assumption of the decoupling between the particle motion and the intermolecular collisions. The time step should be sufficiently small compared to the local mean collision time [34,35].

DSMC Algorithm

The DSMC code employed in this paper is the open-source code dsmcFoam [36,37], which has been developed within the framework of the open-source CFD toolbox OpenFOAM [38]. Since its release as part of the general OpenFOAM software suite in 2010, dsmcFoam has been subject to continuous development by researchers at the universities of Glasgow and Strathclyde in the UK, culminating in a new set of chemistry modules for rarefied, reacting flows [8] using state-of-the-art Quantum-Kinetic (Q-K) procedures [11] to describe dissociation and exchange reactions for a five-species air mixture containing ${\mathrm{N}}_2$, ${\mathrm{O}}_2$, NO, N, and O. The other principal characteristics that the code enjoys include the vibrational internal energy mode and VHS collision procedures with ongoing developments in ionization and CFD-DSMC hybridization.

In 2011, Graeme Bird, the originator of the DSMC methodology, published a paper dealing with a novel approach to DSMC chemistry modeling [11]. In this new model, reaction rates were based on microscopic collision considerations, as opposed to the conventional Total Collision Energy (TCE) approach [5], which relied on macroscopic temperature data, often extrapolated from low-temperature experiments under equilibrium conditions. The new approach, called the Quantum-Kinetic (Q-K) method, is the method adopted in this paper for a five-species air mixture consisting of ${\mathrm{N}}_2$, ${\mathrm{O}}_2$, NO, N, and O. The Q-K method has the benefits of only limited reliance on macroscopic temperature. There is no requirement for the gas to be in thermodynamic equilibrium to determine reaction rates.

In the Q-K approach, knowledge of the vibrational state of each molecule is required; thus, the first development in the Q-K representation of chemistry in the dsmcFoam code was to incorporate the vibrational mode. For dissociation reactions involving particles A and B, if the energy in a collision event is greater than the bond energy of the molecule, then the molecule will split. We define the collision energy, ${\mathrm{E}}_c$, as the sum of the relative translational energy of the colliding pair and the vibrational energy of the particle to be split—i.e.,

$$E_c=E_{tr,AB}+E_{v,A}$$

(1)

The maximum possible quantum level, i, that can be achieved during a collision is

$$i_{max}=\left[{\displaystyle \frac{E_c}{\kappa {\theta }_v}}\right],$$

(2)

where $\kappa$ is the Boltzmann constant and ${\theta }_v$ is the characteristic vibrational temperature for species A. The brackets indicate a floor integer value for the quantum level. If ${\mathrm{i}}_{max}$ is beyond the dissociation limit, i.e.,

$$i_{max}>{\displaystyle \frac{{\theta }_d}{{\theta }_v}}$$

(3)

then the molecule A must split before any vibrational or rotational relaxation can occur. ${\theta }_d$ in this instance is the characteristic dissociation temperature of the molecule.

One novel feature of the Q-K approach is that the gas does not need to be in equilibrium during the reaction processing. However, if it is assumed so, analytical rates can be derived, against which the DSMC coding can be tested. This has been carried out in detail for various heat bath scenarios by [8] in their assessment of the Q-K chemistry modules in dsmcFoam.

Exchange reactions are processed in the Q-K approach as being probable if the collision energy ${\mathrm{E}}_c$ is greater than the activation energy for the reaction ${\mathrm{E}}_a$, with a probability of

$$P={\left(1-{\displaystyle \frac{E_a}{E_c}}\right)}^{3/2-\omega }{\displaystyle /}\sum _{i=0}^{i_{max}}{\left(1-{\displaystyle \frac{i\kappa {\theta }_v}{E_c}}\right)}^{3/2-\omega },$$

(4)

where $\omega$ is the temperature exponent of viscosity—thus resulting, for a five-species air mixture, in the possibility of 15 dissociation reactions and 4 exchange reactions as shown in

Table 1. Chemical reactions and their respective heat of formation (${\mathrm{H}}_f$) and activation energy (${\mathrm{E}}_a$).

${\mathbf{N}}^{\circ }$	Reaction	${\mathbf{H}}_{\mathit{f}}$ ($\times {10}^{-19}$ J)	${\mathbf{E}}_{\mathit{a}}$ ($\times {10}^{-19}$ J)
1	${\mathrm{O}}_2$ + N → O + O + N	8.197	8.197
2	${\mathrm{O}}_2$ + NO → O + O + NO	8.197	8.197
3	${\mathrm{O}}_2$ + ${\mathrm{N}}_2$ → O + O + ${\mathrm{N}}_2$	8.197	8.197
4	${\mathrm{O}}_2$ + ${\mathrm{O}}_2$ → O + O + ${\mathrm{O}}_2$	8.197	8.197
5	${\mathrm{O}}_2$ + O → O + O + O	8.197	8.197
6	${\mathrm{N}}_2$ + O → N + N + O	15.67	15.67
7	${\mathrm{N}}_2$ + ${\mathrm{O}}_2$ → N + N + ${\mathrm{O}}_2$	15.67	15.67
8	${\mathrm{N}}_2$ + NO → N + N + NO	15.67	15.67
9	${\mathrm{N}}_2$ + ${\mathrm{N}}_2$ → N + N + ${\mathrm{N}}_2$	15.67	15.67
10	${\mathrm{N}}_2$ + N → N + N + N	15.67	15.67
11	NO + ${\mathrm{N}}_2$ → N + O + ${\mathrm{N}}_2$	10.43	10.43
12	NO + ${\mathrm{O}}_2$ → N + O + ${\mathrm{O}}_2$	10.43	10.43
13	NO + NO → N + O + NO	10.43	10.43
14	NO + O → N + O + O	10.43	10.43
15	NO + N → N + O + N	10.43	10.43
16	NO + O → ${\mathrm{O}}_2$ + N	2.719	2.719
17	${\mathrm{N}}_2$ + O → NO + N	5.175	5.175
18	${\mathrm{O}}_2$ + N → NO + O	−2.719	0
19	NO + N → ${\mathrm{N}}_2$ + O	−5.175	0

[8].

3. Results and Discussions

To validate the set of chemical reactions implemented into the dsmcFoam code, it is suitable to test this new code feature and compare the results with those available in the open literature. In pursuit of this goal, simulations of rarefied hypersonic reacting and non-reacting gas flows are performed over simple and complex geometries. The first case considered is the reactive hypersonic flow over a 1.6-meter-diameter sphere at 90 and 130 km altitudes, respectively. Next, the influence of chemical reactions on the flow structure and aerodynamic forces acting on the Orion capsule are presented for the altitudes of 95 and 105 km. In the present work, all the simulations were performed using the Q-K chemistry model accounting for the five-species model (${\mathrm{N}}_2$, ${\mathrm{O}}_2$, NO, O, N) and 19 chemical reactions (Table 1).

3.1. Reactive Rarefied Hypersonic Flow over 3D Sphere

Determining the gas flow over a sphere is one of the simplest and most fundamental problems in the field of rarefied gas dynamics. It has been extensively used in numerical simulations and experiments as a validation test case [14,39,40]. In this section, the results obtained by the dsmcFoam code for a reactive gas flow over a sphere at 90 and 130 km altitude are compared with those available in the work developed by Dogra et al. [14]. The same gas species are used to simulate reactive flows; however, Dogra’s work employs 32 chemical reactions.

In the present investigation, the 1.6-meter-diameter sphere is immersed in a flow composed of molecular nitrogen, oxygen, and atomic oxygen. The species concentration and the freestream conditions employed in this work are shown in

Table 2. Freestream conditions for 1.6 m sphere simulations [14].

Parameters	130 km	90 km
Number density (${\mathrm{n}}_{\infty }$, ${\mathrm{m}}^{-3}$)	1.946 × ${10}^{17}$	7.177 × ${10}^{19}$
Density (${\rho }_{\infty }$, $\mathrm{kg}/{\mathrm{m}}^3$)	8.221 × ${10}^{-9}$	1.380 × ${10}^{-6}$
Pressure (${\mathrm{p}}_{\infty }$, Pa)	1.343 × ${10}^{-3}$	1.862 × ${10}^{-1}$
Mean free path (${\lambda }_{\infty }$, m)	7.724	1.600 × ${10}^{-2}$
Velocity (${\mathrm{V}}_{\infty }$, m/s)	7500	7500
Temperature (${\mathrm{T}}_{\infty }$, K)	500	188
Wall temperature (${\mathrm{T}}_w$, K)	760	951
Overall Knudsen (${\mathrm{Kn}}_d$)	4.812	0.010
Atmospheric composition
Molecular nitrogen (${\mathrm{N}}_2$)	0.691	0.788
Molecular oxygen (${\mathrm{O}}_2$)	0.071	0.209
Atomic oxygen (O)	0.238	0.004

[14]. Molecular collision is modeled by using the variable hard sphere (VHS) molecular model [30].

Table 3. Gas properties [30].

Gas Specie	m (kg)	d (m)	$\mathit{\omega }$	${\mathit{\theta }}_{\mathit{v}}$ (K)	${\mathit{\theta }}_{\mathit{d}}$ (K)
${\mathrm{O}}_2$	53.12 × ${10}^{27}$	4.07 × ${10}^{-10}$	0.77	2256	59,500
${\mathrm{N}}_2$	46.50 × ${10}^{27}$	4.17 × ${10}^{-10}$	0.74	3371	113,500
NO	49.88 × ${10}^{27}$	4.20 × ${10}^{-10}$	0.79	2719	75,500
0	26.56 × ${10}^{27}$	3.00 × ${10}^{-10}$	0.80	-	-
N	23.25 × ${10}^{27}$	3.00 × ${10}^{-10}$	0.80	-	-

shows the gas properties, molecular mass (m) and diameter (d), viscosity index ($\omega$), and vibrational (${\theta }_v$) and dissociation (${\theta }_v$) temperatures for each of the gas species considered in the present investigation [30]. Internal energy exchange is modeled using the Larsen–Borgnakke distribution function [41] and the no-time-counter (NTC) collision sampling technique [42]. The gas–surface interaction was modeled by assuming diffuse reflection with complete thermal accommodation at the specific surface temperature. Following DSMC good practice [8], the cell size was set as one-third of the mean free path. The time step was calculated as one-fifth of molecular residence time and set as 2.3 × ${10}^{-6}$ and 6.7 × ${10}^{-4}$ for a 90 and 130 km altitude, respectively. The DSMC method is transient, and approximately 1 million time steps are required for both simulations to decrease the statistical error and obtain smooth curves for the macroscopic properties.

Figure 1 shows the computational mesh and boundary conditions employed in the 95 km altitude case. The computational domains were generated using the snappyHexMesh tool for a quarter section of a sphere. The dimensions were carefully calculated to minimize computational costs and prevent interaction between the shock wave and domain boundaries. Following the snappy process, computational domains with 2.3 million and 446,000 cells were generated for 95 km and 130 km altitude cases, respectively. The cell size was chosen to be one-third of the mean free path, with eight subcells per cell, to facilitate the near-neighbor collision procedure. At the spline, a velocity of 7500 m/s was specified, whereas a vacuum condition is imposed at the downstream outflow boundary [30]. The sides of the quarter section of the computational domain were defined as symmetry planes. A diffuse wall with full thermal and moment accommodation was chosen as a wall boundary condition. In addition, the temperature of the sphere wall was maintained at 188 K and 350 K for the cases at 95 km and 130 km altitude, respectively. Figure 1 shows a slice of the computational domain with boundary conditions employed in the computations of inert and reactive gas flows over the sphere at 95 km altitude.

The surface quantities obtained using the dsmcFoam are compared to those available in Reference [14] in Figure 2. In this set of plots, heat transfer (${\mathrm{C}}_h$), pressure (${\mathrm{C}}_p$), and skin friction (${\mathrm{C}}_f$) coefficients are measured along the sphere’s surface for both altitudes considered, i.e., 90 and 130 km. A significant influence of rarefaction effects on ${\mathrm{C}}_h$ and ${\mathrm{C}}_f$ is clearly noticed; however, similar impacts are not observed for ${\mathrm{C}}_p$.

Still referring to Figure 2, it is noticed that ${\mathrm{C}}_h$ is maximum at the stagnation point ($\theta =0^{\circ }$) and decreases to a minimum value at the wake region at $\theta ={90}^{\circ }$. At the stagnation point, a difference of 8.5% and 5.0% is found for ${\mathrm{C}}_h$ at 90 and 130 km, respectively. Due to the lack of collisions at sufficiently high altitudes, most chemical reactions are unlikely to occur, thereby reducing the differences between the computed results at 130 km altitude. According to Figure 2, ${\mathrm{C}}_p$ follows a similar trend to that described for the heat transfer coefficient; however, the maximum difference between the computational results is 3% at $\theta =0^{\circ }$. In contrast to ${\mathrm{C}}_h$ and ${\mathrm{C}}_p$, the skin friction coefficient at $\theta =0^{\circ }$, due to the flow expansion around the sphere, ${\mathrm{C}}_f$ reaches its maximum value at $\approx \theta ={45}^{\circ }$ and decreases to a minimum value at the wake. Comparing the results obtained with the dsmcFoam code with those found in Reference [14], the maximum difference in ${\mathrm{C}}_f$ is 1.5% at 90 km altitude. In addition, a good agreement is observed between the dsmcFoam and the results and those obtained by Dogra et al. [14].

Figure 3 shows the nondimensional macroscopic properties’ distribution along the stagnation streamline at 90 and 130 km altitude. The macroscopic property ratios considered in this section are the mole fraction (${\rho }_{specie}$/${\rho }_{total}$), temperature ratio ($\mathrm{T}/{\mathrm{T}}_{\infty }$), and density ratio ($\rho$/${\rho }_{\infty }$). In addition, full symbols represent the computational data obtained using the dsmcFoam code, and empty symbols give the computational results obtained by Dogra et al. [14] for a reactive gas flow over a 1.6 m sphere.

According to Figure 3, despite the differences in the chemical reaction models and the number of reactions used during the simulations for both codes, a good agreement is observed between the computed data obtained by Dogra et al. [14] and the dsmcFoam. At an altitude of 90 km, where most reactions are active due to intense intermolecular collisions, the mole fraction and density distribution along the stagnation streamline are very similar. We also noticed a significant degree of thermal non-equilibrium in the temperature distribution along the stagnation streamline. The translational, rotational, and vibrational temperature ratios follow the same trend; however, differences of 10% and 40% are observed in the translational and vibrational peak temperatures within the shock wave. Considering the 130 km altitude cases, due to the lack of intermolecular collisions in such a rarefied flow, it was observed that chemical reactions have no significant influence on the mole fraction, temperature, and density ratio distribution along the stagnation streamline.

3.2. Reactive Rarefied Hypersonic Flow over Orion Capsule

Rarefied hypersonic reacting flow is used to study the Orion capsule during reentry at altitudes of 95 and 105 km, respectively. In this investigation, the results obtained using the dsmcFoam-QK (19 species reaction model) and dsmcFoam-NR (no reactions) are compared with numerical solutions provided by Wilmoth et al. [43] and Moss et al. [44].

DSMC simulations were performed by Wilmoth et al. [43] using the DS2V code [30] with both traditional rate-based chemistry models (TCE) and a mixed TCE-QK model labeled “new chemistry” [43]. In this mixed chemistry model, exchange and recombination reactions were not available during the simulations and were treated using traditional rate-based methods. However, dissociation reactions were performed using the new Q-K collision-based methodology. Additionally, the DAC code was used to calculate surface heat flux and compare it to the value obtained by the DS2V code.

Moss et al. [44] conducted a series of numerical simulations to characterize Orion’s aerodynamics across various conditions, from free molecular to continuum hypersonic. For the rarefied portion of the Earth’s atmosphere, two DSMC mature codes, DS3V and DAC, were employed, both utilizing a five-species reacting air gas model.

In the present work, advantage was taken of Orion’s symmetry to reduce computational costs. In this way, the computational mesh was prepared using a quarter-section model with symmetry boundary conditions applied at the perpendicular planes, as depicted in Figure 4. The Orion dimensions used in the present investigation can be found at reference [44]. The OpenFOAM mesh utility called snappyHexMesh was used to "snap” the mesh onto the Orion CAD geometry, creating hexahedral cells on the surface. After this process, a total of 15.06 × ${10}^6$ and 1.293 × ${10}^6$ cells were employed by the dsmcFoam calculation for the 95 and 105 km altitudes, respectively. The gas–surface interaction was modeled by assuming diffuse reflection with complete thermal accommodation at the specific surface temperature.

The computational mesh was populated with 94 million and 17.5 million DSMC particles at altitudes of 95 and 105 km, respectively. Freestream boundary conditions were applied at the inlet, top, and sides of the computational domain. The flow at the downstream outflow boundary was supersonic, and vacuum conditions were specified. For all computations, the outflow boundary was located 2.0 body diameters (10 m) downstream of the forebody stagnation point. Additionally, for the two cases considered, the wall temperature was maintained at 951 K and 760 K for altitudes of 95 km and 105 km, respectively. The capsule surface was assumed to be noncatalytic with diffusive reflection and full thermal and momentum accommodation.

The freestream conditions are similar to those previously analyzed by Moss et al. [44] for the Orion capsule. At both altitudes considered in this investigation, a freestream velocity of 7600 m/s was set. The reentry freestream conditions used in the present work, as well as the gas properties, are shown in

Table 4. Reentry freestream conditions for the Orion capsule.

Parameters	95 km	105 km
Number density (${\mathrm{n}}_{\infty }$, ${\mathrm{m}}^{-3}$)	2.9047 × ${10}^{19}$	4.9759 × ${10}^{18}$
Density (${\rho }_{\infty }$, $\mathrm{kg}/{\mathrm{m}}^3$)	1.3800 × ${10}^{-6}$	2.3004 × ${10}^{-7}$
Pressure (${\mathrm{p}}_{\infty }$, Pa)	7.5793 × ${10}^{-2}$	1.4495 × ${10}^{-2}$
Mean free path (${\lambda }_{\infty }$, m)	5.4315 × ${10}^{-2}$	3.1633 × ${10}^{-1}$
Velocity (${\mathrm{V}}_{\infty }$, m/s)	7600	7600
Temperature (${\mathrm{T}}_{\infty }$, K)	189	211
Overall Knudsen (${\mathrm{Kn}}_d$)	0.0108	0.0629
Atmospheric composition
Molecular nitrogen (${\mathrm{N}}_2$)	0.78685	0.78187
Molecular oxygen (${\mathrm{O}}_2$)	0.19719	0.15280
Atomic oxygen (O)	0.01595	0.06533

.

Figure 5 compares the molecular and atomic mole fractions extracted along the stagnation streamline. We clearly noticed an excellent agreement between the dsmcFoam-QK and the DS2V (new chemistry) for the two altitudes considered. Nevertheless, the production of the molecular species NO through exchange reactions in the dsmcFoam calculations seems to be slightly underpredicted when compared with DS2V computations. Considering that Wilmoth’s simulations were performed using a mixed TCE-QK chemistry model, a good concurrence was achieved with the dsmcFoam-QK code.

One of the most critical parameters in developing a reliable thermal protection system (TPS) is accurately predicting the heat flux to the spacecraft surface, which has a direct impact on TPS design and material selection. To investigate the influence of reacting and non-reacting flows over the Orion capsule surface at 95 and 105 km of altitude, comparisons of surface heat flux measurements using the dsmcFoam, DS2V, and DAC codes are presented in Figure 6. According to the left-hand-side plot, at 95 km altitude, the heat transfer coefficients (${\mathrm{C}}_h$) calculated by both codes are in close agreement. Although the trends are similar, it is evident that the heat flux predicted using the dsmcFoam-QK is in excess when compared with the DS2V calculations. Comparing the dsmcFoam computations at 95 km altitude for reacting and non-reacting flows, a reduction of 36.8% in the ${\mathrm{C}}_h$ is observed. According to the right-hand plot of Figure 6, it is evident that the heat transfer coefficient predicted by dsmcFoam-QK is in very close agreement with DS2V (old chemistry). Furthermore, despite following a similar pattern to the Q-K approach, the DS2V (new chemistry) appears to underpredict ${\mathrm{C}}_h$.

The sensitivity of the aerodynamic forces and moments to chemical reactions is presented in

Table 5. dsmcFoam aerodynamic calculations of rarefied hypersonic non-reacting (NR) and reacting (Q-K) flow over Orion at 95 km altitude and $0^{\circ }$ angle of attack.

Parameter	dsmcFoam (NR)	dsmcFoam (Q-K)	Difference (%)
Drag (${\mathrm{C}}_D$)	1.566	1.587	1.28
Lift (${\mathrm{C}}_L$)	0.000	0.000	0.00
Axial force (${\mathrm{C}}_D$)	1.566	1.587	1.28
Normal force (${\mathrm{C}}_N$)	0.000	0.000	0.00
Pitching moment (${\mathrm{C}}_{m,cg}$)	−0.07518	−0.07616	1.30
Pitching moment (${\mathrm{C}}_{m,0}$)	0.00000	0.00000	0.00

for 95 km altitude. According to the simulated data, the DSMC solutions show that the Orion aerodynamics are insensitive to the inclusion of chemical reactions. Despite the chemical insensitivity, reactive flows play a vital role in reducing the shock wave temperature and heat flux at the vehicle surface, as previously discussed.

Table 6. dsmcFoam and DAC aerodynamic calculations of rarefied hypersonic over Orion at 105 km altitude and $0^{\circ }$ angle of attack.

Parameter	dsmcFoam (NR)	dsmcFoam (Q-K)	DAC
Drag (${\mathrm{C}}_D$)	1.7007	1.7092	1.709
Lift (${\mathrm{C}}_L$)	0.0000	0.000	0.000
Axial force (${\mathrm{C}}_D$)	1.7007	1.7092	1.709
Normal force (${\mathrm{C}}_N$)	0.000	0.000	0.000
Pitching moment (${\mathrm{C}}_{m,cg}$)	−0.08163	−0.08204	−0.0820
Pitching moment (${\mathrm{C}}_{m,0}$)	0.0000	0.0000	0.0000

and

Table 7. dsmcFoam, DAC, DS3V, and LAURA aerodynamic calculations of rarefied hypersonic over Orion at 105 km altitude and ${-26}^{\circ }$ angle of attack.

Parameter	dsmcFoam (NR)	dsmcFoam (Q-K)	DAC	DS3V	LAURA
Drag (${\mathrm{C}}_D$)	1.479	1.480	1.449	1.460	1.753
Lift (${\mathrm{C}}_L$)	0.390	0.393	0.347	0.349	0.342
Axial force (${\mathrm{C}}_D$)	1.500	1.503	1.455	1.465	1.726
Normal force (${\mathrm{C}}_N$)	−0.298	−0.296	−0.323	−0.327	−0.461
Pitching moment (${\mathrm{C}}_{m,cg}$)	−0.04830	−0.04841	−0.0412	−0.0431	−0.0356
Pitching moment (${\mathrm{C}}_{m,0}$)	0.102	0.114	0.114	0.069
Centre of pressure (${\mathrm{x}}_{xp,m}$)	1.733	1.737	1.779	1.752	1.849

show the dsmcFoam aerodynamic calculations for the Orion capsule at 105 km altitude at two angles of attack, $0^{\circ }$ and ${-26}^{\circ }$, respectively. Included in these tables are the results from DAC and DS3V codes, which both use the DSMC technique, and LAURA, which provides solutions to the Navier–Stokes equations [44]. For both angles of attack investigated, good agreement is shown between the dsmcFoam, DAC, and DS3V. However, due to the rarefaction effects, the results obtained by LAURA show significant differences compared with the DSMC solutions. In general, the results demonstrate a high level of agreement and consistency among the various computational tools.

Reacting vs. Non-Reacting

The effect of chemical reactions on the computed results is of particular interest. The flowfield structure for the Orion capsule is shown in Figure 7, where the overall temperature ratio (${\mathrm{T}}_{ov}/{\mathrm{T}}_{\infty }$) is presented for non-reacting ($dsmcFoam$-NR) and reacting ($dsmcFoam$-QK) conditions. Due to the endothermic nature of the chemical reactions, a significant decrease in the shock wave temperature is visible. A temperature reduction of 25.3% and 18.5% at altitudes of 95 and 105 km, respectively, demonstrates that the presence of chemical reactions has a significant impact on the shock wave stand-off distance, shock thickness, and temperature distribution at the wake region.

Notably, the temperature plots exhibit a significant degree of thermal non-equilibrium, with the overall kinetic temperature, as well as the translational, rotational, and vibrational temperatures, being presented. Furthermore, as the altitude changes, it is possible to observe an alteration in the curve shape, i.e., from a steep gradient at 95 km altitude to shallower gradients at 105 km altitude. This behavior demonstrates the diffuse nature of the shock wave when the degree of gas rarefaction increases.

The density profile for the initial species along the stagnation streamline is also shown in Figure 8. At 95 km altitude, the number of density profiles is identical for reacting and non-reacting flows at location X < −0.65 m. However, for X ≈ −0.60 m, significant changes are evident for the number density ratio as the flow moves towards the Orion’s surface. Due to its low dissociation threshold, molecular oxygen is the first species to dissociate, resulting in a decrease in ${\mathrm{O}}_2$ concentration and an increase in atomic oxygen across the shock layer. In contrast, a slight reduction in the molecular nitrogen number density is observed when the results of inert gas flow are compared. Since the molecular nitrogen dissociation temperature is approximately double that of ${\mathrm{O}}_2$, just a more severe reentry condition with higher enthalpy is able to fully dissociate the ${\mathrm{N}}_2$. At an altitude of 105 km, the dissociation process is significantly less intense than at an altitude of 95 km. Consequently, a slight decrease in ${\mathrm{O}}_2$ and an increase in atomic oxygen concentration are close to the Orion surface, and no appreciable changes in the temperature, pressure, and velocity are noticed for reaching flow at this altitude.

Mach number, number density, and translational temperature ratio contours are depicted in Figure 9 displays the hypersonic reacting flow over the Orion capsule at 95 and 105 km altitude, respectively. In this set of contour plots, total number density and translational temperature are normalized by the freestream conditions. According to the Mach number contours, the diffuse shock effect is evident throughout the entire computational domain, particularly in the wake region. While the sonic line at the forebody is located at almost the same position for both altitudes studied, the same phenomenon is not observed in the wake region. At the 95 km altitude, low Mach number values are observed at the wake region, and the position of the sonic line extends far downstream of the Orion capsule. However, for the 105 km altitude case, the sonic line is located closer to the vehicle’s afterbody.

Upon examining the number density and translational temperature ratio, the extremely diffuse nature of the shock wave is evident at an altitude of 105 km, and the shock structure extends well upstream of the Orion body. On the other hand, at an altitude of 95 km, the shock is confined to a region much closer to the vehicle’s heat shield. The number density in the near wake is relatively low for both altitudes, with magnitudes as low as 10% of the freestream value. Along the forebody, as the flow is compressed against the thermal protection system, a significant temperature increase is observed. In the wake region, the temperature decreases during the expansion process, and the highest value, which is 35 times the freestream temperature, can be found for the 105 km altitude case.

At the initial phase of the simulations, the atmospheric gas surrounding the Orion capsule is primarily composed of molecular nitrogen, oxygen, and atomic oxygen. If the spacecraft reentry occurs at a sufficiently high speed, dissociation, and exchange reactions may take place, introducing new molecular and atomic species into the simulations. To demonstrate this, normalized number density contours for each individual species are shown in Figure 10. According to this group of contour plots, the highest concentrations of nitric oxide (NO) and atomic oxygen occur in the forebody region of the Orion Command Module, a consequence of the high temperatures inside the shock wave, which promote chemical reactions. Reactant species are also found in the afterbody region; however, their concentration is at least one order of magnitude lower than that of the forebody.

The study conducted above demonstrates that an accurate definition of an aero heating environment and the presence of chemical reactions play critical roles in capturing the correct physics during spacecraft reentry simulations.

4. Conclusions

This study examined rarefied hypersonic flows over two reentry geometries, a sphere and the Orion Command Module, at transitional altitudes using the Direct Simulation Monte Carlo (DSMC) method implemented via dsmcFoam. Simulations were performed at 90 and 130 km for the sphere and at 95 and 105 km for Orion, with and without chemical reactions, to evaluate the impact of chemical non-equilibrium effects on shock structure and surface quantities under varying degrees of rarefaction.

The results demonstrate that chemical activity has a significant impact on the flowfield, particularly at lower altitudes. For the spherical case at 90 km, reactions led to broader shocks, reduced translational temperatures by approximately 30% within the shock layer, and increased stand-off distances, in contrast to the sharper and more compressed shock observed in the non-reacting flow. At 130 km, these effects were less intense due to the lower atmospheric density, but remained evident in both thermal and compositional fields. Flow profiles along the stagnation line and over the surface confirmed the strong influence of chemical reactions on number density, pressure, and temperature distributions.

The Orion capsule simulations further highlighted the relevance of detailed chemical modeling in hypersonic reentry flows. The inclusion of the Q-Kinetic (QK) model at 95 km resulted in enhanced dissociation of $O_2$ and $N_2$, as well as the formation of NO and atomic species, which modified shock behavior and reduced peak temperatures near the forebody. Surface quantities such as pressure and heat flux were also strongly affected, particularly in the stagnation region and along the shoulder, revealing the sensitivity of flow–surface interactions to both altitude and geometry.

Comparison with previous studies in the literature confirmed the validity of the numerical approach, with good agreement observed for key flow characteristics and surface properties. These results validate the implementation of the chemical kinetics model in dsmcFoam and support the use of DSMC for simulating transitional reentry regimes, where conventional continuum-based methods fail to capture essential non-equilibrium effects.

The study underscores the importance of accurately modeling chemical non-equilibrium in high-altitude reentry flows. Neglecting these effects can lead to significant overestimation of thermal and aerodynamic loads, as the energy consumed by dissociation is not accounted for in simplified models. This is particularly critical for small satellites and deployable reentry systems with limited design margins.