A Two-Temperature Open-Source CFD Model for Hypersonic Reacting Flows, Part Two: Multi-Dimensional Analysis ^†

Abstract

hy2Foam is a newly-coded open-source two-temperature computational fluid dynamics (CFD) solver that has previously been validated for zero-dimensional test cases. It aims at (1) giving open-source access to a state-of-the-art hypersonic CFD solver to students and researchers; and (2) providing a foundation for a future hybrid CFD-DSMC (direct simulation Monte Carlo) code within the OpenFOAM framework. This paper focuses on the multi-dimensional verification of hy2Foam and firstly describes the different models implemented. In conjunction with employing the coupled vibration-dissociation-vibration (CVDV) chemistry–vibration model, novel use is made of the quantum-kinetic (QK) rates in a CFD solver. hy2Foam has been shown to produce results in good agreement with previously published data for a Mach 11 nitrogen flow over a blunted cone and with the dsmcFoam code for a Mach 20 cylinder flow for a binary reacting mixture. This latter case scenario provides a useful basis for other codes to compare against.

1. Introduction

The Knudsen number, defined as the ratio of the mean free path of the gas particles to the characteristic length of the problem, is commonly employed to gauge the degree of rarefaction of a gas. During the entry of a planetary atmosphere at hypervelocities, a craft will traverse the full range of the Knudsen number, from the free-molecular regime down to the continuum regime. Practically, this translates into the need for different numerical techniques to resolve the flow-field past such hypersonic bodies.

The direct simulation Monte Carlo (DSMC) method developed by Bird [1] is a particle-based methodology that is particularly well-suited for computing high Knudsen number flows, typically above 0.05, while the conventional computational fluid dynamics (CFD) approach that solves the Navier–Stokes–Fourier (NSF) equations is generally adopted for the lower range, below 0.005. In between lies an intermediate zone where DSMC is computationally prohibitive and where conventional CFD fails due to the presence of non-continuum regions within the flow-field. To extend the range of applicability of the NSF equations towards the continuum–transition regime, Park formulated the two-temperature CFD model [2], thus distinguishing the trans-rotational energy pool from the vibro-electronic energy pools and modelling the energy exchange processes via vibrational rate equations.

Open-source solvers dedicated to the study of the hypersonic regime are very scarce. Among such codes are hy2Foam and dsmcFoam. hy2Foam [3] is a new two-temperature CFD solver developed at the University of Strathclyde [4,5] within the framework of the open-source CFD platform OpenFOAM [6]. It aims at (1) giving open-source access to a state-of-the-art hypersonic CFD solver to students and researchers; and (2) providing a foundation for a future hybrid CFD-DSMC code [7].

For rarefied gas flow environments, OpenFOAM possesses a DSMC solver called dsmcFoam that has been developed and validated at the same University [8,9]. The two main features of dsmcFoam are that vibrational-translational post-collision energy redistribution is executed using a quantum version of the Larsen–Borgnakke procedure [8,10] and that chemical reactions are described by Quantum-Kinetic (QK) theory, as initially proposed by Bird [11].

The primary objective of this paper concerns the verification and validation of hy2Foam for multi-dimensional case scenarios. The secondary goal is the assessment, for a two-dimensional gas flow, of one of the conclusions formulated in [5], namely that the combination of a particular chemistry–vibration model with chemical rates derived from QK theory provides a satisfactory consistency between the CFD and DSMC codes.

2. Methodology

This section essentially describes the state-of-the-art numerical procedures that have been implemented in hy2Foam to solve hypervelocity flow-fields in near-thermal equilibrium.

2.1. Non-Equilibrium Navier–Stokes–Fourier Equations

The computation of transient hypersonic reacting flows in the continuum regime traditionally employs the non-equilibrium Navier–Stokes–Fourier (NSF) equations. These are shown below in flux–divergence form using a Cartesian coordinate system for a mixture composed of $N_s$ species, including $N_m$ molecules [12]:

$$\frac{\partial \mathcal{U}}{\partial t}+\frac{\partial \left({\mathcal{F}}_{i,inv}-{\mathcal{F}}_{i,vis}\right)}{\partial x_i}=\dot{\mathcal{W}}.$$

(1)

The vector of conserved quantities, $\mathcal{U}$, is defined as

$$\mathcal{U}={\left(\rho ,{\rho }_s,\rho u,\rho v,\rho w,E_{ve,m},E\right)}^T,s\in N_s,m\in N_m,$$

(2)

where u, v, and w are the components of the velocity vector. ρ is the mass density of the fluid and ${\rho }_s$ is the partial density of species s. The flux vectors are split into inviscid and viscous contributions and are written as follows:

$${\mathcal{F}}_{i,inv}=\left(\begin{array}{c}\rho u_i\\ {\rho }_s{u}_i\phantom{\mathcal{J}}\\ \rho u_{i}u+{\delta }_{i1}p\\ \rho u_{i}v+{\delta }_{i2}p\\ \rho u_{i}w+{\delta }_{i3}p\\ E_{ve,m}{u}_i\phantom{\mathcal{J}}\\ (E+p)u_i\phantom{{\displaystyle \sum _{r\ne e}{h}_r{\mathcal{J}}_{r,i}}}\end{array}\right),$$

(3)

$${\mathcal{F}}_{i,vis}=\left(\begin{array}{c}0\\ -{\mathcal{J}}_{s,i}\\ {\tau }_{i1}\\ {\tau }_{i2}\\ {\tau }_{i3}\\ -q_{ve,i,m}-e_{ve,m}{\mathcal{J}}_{m,i}\\ {\displaystyle {\tau }_{ij}{u}^j-q_{tr,i}-q_{ve,i}-\sum _{r\ne e}{h}_r{\mathcal{J}}_{r,i}}\end{array}\right),$$

(4)

where the index i refers to one of the three dimensions of space and δ is the Kronecker delta. E and $E_{ve,m}$ represent, respectively, the total energy and the total vibro-electronic energy for molecule m. In the remainder of the article, the index $tr$ denotes the trans-rotational energy mode, $ve$ the vibro-electronic energy mode and e the electron energy mode. $h_s$ is the enthalpy per unit mass of species s, while the pressure, p, is recovered from the partial pressures using Dalton’s law

$${\displaystyle p=\sum _{s\ne e}{p}_s+p_e=\sum _{s\ne e}\left({\rho }_s{R}_s{T}_{tr}\right)+{\rho }_e{R}_e{T}_{ve,\mathrm{ref}},}$$

(5)

where $R_s$ is the specific gas constant. Finally, the electron temperature is set to the vibro-electronic temperature of a reference particle, $T_{ve,\mathrm{ref}}$.

In Equation (4), ${\tau }_{ij}$ represent the components of the viscous stress tensor, which can be written as follows:

$${\tau }_{ij}=\mu \left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right)+(\lambda +{\mu }_b)\frac{\partial u_k}{\partial x_k}{\delta }_{ij},$$

(6)

where μ is the shear viscosity, λ is the second viscosity coefficient, and ${\mu }_b$ is the bulk viscosity. This work assumes that Stokes’ hypothesis is valid, namely that ${\mu }_b=0$ and that the shear and second viscosity are not independent quantities but given by the relation $\lambda =-\frac{2}{3}\mu$. In hy2Foam, the species shear viscosities can be considered as temperature-independent, or either set to follow Blottner’s formula [13] or a power law. The spatial components of the heat conduction vector are assumed to follow Fourier’s law

$$q_{tr,i}=\sum _s{q}_{tr,i,s}=\sum _s-{\kappa }_{tr,s}\frac{\partial T_{tr}}{\partial x_i}\mathrm{for}s\in N_s\backslash \left\{e\right\}$$

(7)

and

$$q_{ve,i}=\sum _s{q}_{ve,i,s}=\sum _s-{\kappa }_{ve,s}\frac{\partial T_{ve,s}}{\partial x_i}\mathrm{for}s\in N_s.$$

(8)

The thermal diffusivities ${\kappa }_{tr,s}$ and ${\kappa }_{ve,s}$ can be considered as temperature-independent, or set to follow Eucken’s formula [12,14]. Mixture quantities are recovered from species quantities using a mixing rule. The Wilke [15], Gupta [16], and Armaly and Sutton [17] mixing rules have been implemented, and the latter one is being used in this work, as recommended in the review of Palmer and Wright [18].

The diffusion fluxes, ${\mathcal{J}}_{s,i}$, are governed by Fick’s law with a correction term to ensure mass conservation [19]

$$\begin{array}{ccccc}{\mathcal{J}}_{s,i}& =& {\displaystyle {\mathcal{I}}_{s,i}-Y_s\sum _{r\ne e}{I}_{r,i},}\hfill & & \mathrm{for}s,r\in N_s\backslash \left\{e\right\},\hfill \\ & =& {\displaystyle {\mathcal{M}}_e\sum _{r\ne e}\frac{{\mathfrak{C}}_r\times {\mathcal{J}}_{r,i}}{{\mathcal{M}}_r}}\hfill & & \mathrm{otherwise},\hfill \end{array}$$

(9)

with

$${\mathcal{I}}_{s,i}=-\rho {\mathcal{D}}_s\frac{\partial Y_s}{\partial x_i},$$

(10)

where Y is the mass-fraction and with the effective diffusion coefficient of species s in the mixture defined as [20]

$${\displaystyle {\mathcal{D}}_s=\left(1-X_s\right){\left(\sum _{r\ne s}\frac{X_r}{{\mathcal{D}}_{s,r}}\right)}^{-1}\mathrm{for}s,r\in N_s\backslash \left\{e\right\},}$$

(11)

in which X is the molar-fraction, ${\mathfrak{C}}_r$ represents the charge of species r and $\mathcal{M}$ is the molecular weight. The binary diffusion coefficients, ${\mathcal{D}}_{s,r}$, are modelled using binary collision integrals ${\mathsf{\Delta }}_{s,r}$ as [21]

$${\displaystyle {\mathcal{D}}_{s,r}=\frac{k_B{T}_{tr}}{p{\mathsf{\Delta }}_{s,r}}.}$$

(12)

Finally, the source term vector, $\dot{\mathcal{W}}$, is written as

$$\dot{\mathcal{W}}={\left(0,{\dot{\omega }}_s,0,0,0,{\dot{\omega }}_{v,m},0\right)}^T,$$

(13)

where ${\dot{\omega }}_s$ is the net mass production of species s and ${\dot{\omega }}_{v,m}$, for $m\in N_m$, is defined as follows:

$$\begin{array}{lll}{\dot{\omega }}_{v,m}& =& Q_{m{,}_{V-T}}+Q_{m{,}_{V-V}}+Q_{m{,}_{C-V}}+Q_{m{,}_{e-V}},\mathrm{if}\mathrm{the}\mathrm{reference}\mathrm{molecule}\mathrm{for}e\mathrm{is}\mathrm{not}m,\hfill \\ & =& Q_{m{,}_{V-T}}+Q_{m{,}_{V-V}}+Q_{m{,}_{C-V}}+Q_{m{,}_{e-V}}+Q_{h-e}+Q_{e-i}+\mathit{U}·\nabla p_e\mathrm{otherwise},\hfill \end{array}$$

(14)

where $\mathit{U}$ represents the velocity vector. The meaning of the different vibrational source terms that appear in Equation (14) is given in the subsequent paragraphs.

2.2. Energy Transfers

The decomposition of the total energy, E, as shown in Equation (15), allows the isolation of different energy pools, each of them being described by a specific temperature and exchanging energy with the other pools:

$$\begin{array}{ccccccccccccccc}E& =& {\displaystyle \frac{1}{2}\rho \sum _i{u}_i^2}& +& {\displaystyle \sum _{s\ne e}{E}_{t,s}}& +& {\displaystyle \sum _{s\ne e}{E}_{r,s}}& +& {\displaystyle \sum _{s\ne e}{E}_{v,s}}& +& {\displaystyle \sum _{s\ne e}{E}_{el,s}}& +& E_e& +& {\displaystyle \sum _{s\ne e}{\rho }_s{h}_s^{°}}\end{array}.$$

(15)

In order of appearance in Equation (15) are the kinetic, translational (index t), rotational (r), vibrational (v), electronic ($el$), electron (e), and chemical energies. $u_i$ are the spatial components of the velocity vector and $h_s^{°}$ is the standard enthalpy of formation of species s. The relationship between the total energy of a specific mode and the energy per unit mass of species s is given in Equation (16) for the electronic energy mode

$${\displaystyle E_{el}=\sum _{s\ne e}{E}_{el,s}=\sum _{s\ne e}{\rho }_s{e}_{el,s}}$$

(16)

and the different energies per unit mass of species are detailed below for each mode:

$$e_t=\frac{3}{2}{R}_s{T}_{tr},$$

(17)

$$e_r=R_s{T}_{tr},$$

(18)

$$e_{v,s}=R_s\frac{{\theta }_{v,s}}{exp\left(\frac{{\theta }_{v,s}}{T_{ve,s}}\right)-1},$$

(19)

$$e_e=R_e{T}_{ve,\mathrm{ref}},$$

(20)

$$e_{el,s}=R_s\frac{{\displaystyle \sum _{i=1}^{\infty }{g}_{i,s}{\theta }_{el,i,s}exp\left(-{\theta }_{el,i,s}/T_{ve,s}\right)}}{{\displaystyle \sum _{i=0}^{\infty }{g}_{i,s}exp\left(-{\theta }_{el,i,s}/T_{ve,s}\right)}}.$$

(21)

The simple harmonic oscillator model [11] is utilized for the vibrational mode, with ${\theta }_{v,s}$ being the characteristic vibrational temperature for species s. ${\theta }_{el,i,s}$ and $g_{i,s}$ are the characteristic electronic temperature and the degeneracy degree of a given electronic state i for species s, respectively. The two-temperature model formulation adopted in hy2Foam then consists in rewriting Equation (15) as follows:

$$\begin{array}{ccccccccccc}E& =& {\displaystyle \frac{1}{2}\rho \sum _i{u}_i^2}& +& E_{tr}& +& {\displaystyle \sum _{s\ne e}{E}_{ve,s}}& +& E_e& +& {\displaystyle \sum _{s\ne e}{\rho }_s{h}_s^{°}}\end{array}$$

(22)

with

$$E_{tr}=E_t+E_r$$

(23)

and

$$E_{ve,s}=E_{v,s}+E_{el,s}.$$

(24)

The vibrational source terms introduced into the NSF equations in the preceding paragraph are now described in sequence. The energy exchange between the trans-rotational and the vibro–electronic energy modes (V–T), designated by $Q_{m{,}_{V-T}}$, is dictated by the Landau–Teller equation [22] and may be written as

$$Q_{m{,}_{V-T}}={\rho }_m\frac{\partial e_{ve,m}\left(T_{ve,m}\right)}{\partial t}={\rho }_m\frac{e_{ve,m}\left(T_{tr}\right)-e_{ve,m}\left(T_{ve,m}\right)}{{\tau }_{m{,}_{V-T}}}m\in N_m,$$

(25)

where ${\tau }_{m{,}_{V-T}}$ is the molar-averaged V–T relaxation time. This former quantity is evaluated as the summation of the Millikan and White semi-empirical correlation [23] and Park’s correction factor [2], and later denoted as MWP. It becomes

$${\tau }_{m{,}_{V-T}}=\frac{{\displaystyle \sum _{s\ne e}{X}_s}}{{\displaystyle \sum _{s\ne e}{X}_s/{\tau }_{m-s{,}_{V-T}}}}m\in N_m,$$

(26)

where X is the species molar-fraction and ${\tau }_{m-s{,}_{V-T}}$ is the interspecies relaxation time expressed as

$${\tau }_{m-s{,}_{V-T}}=\frac{1}{p}exp\left[A_{m,s}\left(T_{tr}^{-1/3}-B_{m,s}\right)-18.42\right]+\frac{1}{{\overline{c}}_m{\sigma }_{v,m}{n}_{m,s}}\mathrm{with}p\mathrm{in}\mathrm{atm}.$$

(27)

For a given colliding pair ($m,s$), the tabulated values of $A_{m,s}$ and $B_{m,s}$ used in this paper can be found in [4]. The collision-limited relaxation time introduced by Park [2] is a function of the average molecular speed, ${\overline{c}}_m$, the limited collision cross-section, ${\sigma }_{v,m}$, and the number density of the colliding pair, $n_{m,s}$.

The second V–T model tested in this work is the SSH theory named after Schwartz, Slawsky, and Herzfeld [24]. The coefficients of this latter model are taken from the work of Thivet [25] and a blended model is created with the MWP formulation for molecule-atom collisions.

Vibrational–vibrational (V–V) energy transfer is denoted by $Q_{m{,}_{V-V}}$ and modelled according to the formulation given by Knab et al. [26,27], as described in [4]. In Equation (14), $Q_{h-e}$, $Q_{m{,}_{e-V}}$, and $Q_{e-i}$ stand for the energy exchange between free-electrons and heavy-particles, free-electrons and the vibrational mode, and the vibrational energy removal due to electron impact ionisation, respectively. They are assumed to be zero in this work. The term in $\mathit{U}·\nabla p_e$ represents an approximation to the work done on the electrons by the electric field induced by the electron pressure gradient [28]. Finally, the vibro-electronic energy added or removed by chemical reactions to species m, and represented by $Q_{m,{}_{C-V}}$, is modelled either with the Park TTv model [2] or the coupled vibration-dissociation-vibration (CVDV) model in hy2Foam [4].

2.3. Departure from the Continuum Regime

The local gradient-length Knudsen number introduced by Boyd [29] is often used as a breakdown parameter in the literature (e.g., in [30,31,32,33]). For a local macroscopic flow quantity ϕ, it is written as

$$K{n}_{GLL-\varphi }=\frac{\lambda }{\varphi }|\nabla \varphi |,$$

(28)

where λ is the local mean free path of the gas molecules and ϕ can either be the gas density, the temperature, or the magnitude of the velocity vector $\left|\mathit{U}\right|$ (or $max\left(\left|\mathit{U}\right|,a\right)$ in the denominator, where a is the speed of sound for low-speed regions [34]). The degree of local continuum breakdown is then evaluated as the maximum for all of the aforementioned flow quantities

$$K{n}_{GLL}=max\left(K{n}_{GLL-\rho },K{n}_{GLL-T_{tr}},K{n}_{GLL-T_{ve}},K{n}_{GLL-\left|\mathit{U}\right|}\right),$$

(29)

and the regions where the continuum assumption does not hold are identified when $K{n}_{GLL}$ exceeds a given breakdown value, $K{n}_{Br}$, typically being taken equal to 0.05. This parameter is key in hybrid CFD-DSMC codes.

3. Results and Discussion

3.1. Mach 11.3 Blunted Cone

In this section, a non-reacting nitrogen flow past a blunted cone is examined at Mach 11.3. The case is composed of a 6.35 mm-radius nose followed by a flat plate forming a 25° angle with the free-stream flow direction and whose streamwise extension is 5 cm. The 2D axisymmetric mesh that has been employed is shown in Figure 1. The structured grid is aligned with the bow shock and consists of 600 by 200 cells. The first spacing at the wall surface is set to $2\mathsf{\mu }$m by default and to $10\mathsf{\mu }$m for comparison purposes.

The initial conditions of this case scenario are given in

Table 1. Initial conditions for the Mach 11.3 blunted cone.

Quantity	Value	Unit
Free-stream velocity, $U_{\infty }$	2764.5	m/s
Free-stream pressure, $p_{\infty }$	21.9139	Pa
Free-stream density, ${\rho }_{\infty }$	$5.113\times {10}^{-4}$	kg/m${}^3$
Free-stream temperature, $T_{\infty }$	144.4	K
Free-stream mean-free-path, ${\lambda }_{\infty }$	$1.01\times {10}^{-4}$	m
Overall Knudsen number, $K{n}_{ov}$	0.002	-
Wall temperature, $T_w$	297.2	K

. The free-stream velocity corresponding to a Mach number of 11.3 is 2764.5 m/s. The free-stream temperature and pressure are 144.4 K and 21.9 Pa, respectively. The wall is assumed to be isothermally heated to a temperature of 297.2 K. This simulation uses the MWP formulation for V–T energy exchange and the Blottner and Eucken formulas to compute transport properties. The variable hard sphere model is chosen for the calculation of the mean-free-path and the Knudsen number is computed using the streamwise extent of the cone as the characteristic length. The non-equilibrium boundary conditions employed at the cone surface are the first-order Smoluchowski temperature jump [35] and Maxwell velocity slip [36], with the accommodation coefficients being taken as equal to unity. The set-up for this case can be found in the Supplementary Materials.

The hy2Foam solver uses the numerics present in the official OpenFOAM application rhoCentralFoam [37], namely the KNP central-upwind schemes of Kurganov, Noelle and Petrova [38]. Hence, hy2Foam is first order accurate in time and second order accurate in space. Improving the numerics of the solver to better capture the near-wall region notably is an ongoing area of research. Convergence was achieved after 2.8 h of computations on the ARCHIE-WeSt (Academic and Research Computer Hosting Industry Enterprise in the West of Scotland) High Performance Computer [39]. The run used 24 Intel Xeon X5650 2.66 GHz cores (Santa Clara, CA, USA) with 48 GB RAM and 4xQDR Infiniband Interconnect computer-networking communications.

This particular configuration has already been studied by Wang and Boyd [40], using the MONACO DSMC code and a Navier–Stokes CFD solver from the University of Michigan. The results from these simulations are reported in the subsequent graphs with the denomination DSMC: MONACO and CFD: Michigan. Moreover, experimental data is also shown in Figure 2d,f and correspond to the run 31 of the CUBRC (CUBRC Inc., Buffalo, NY, USA) experiments [41].

The emphasis is placed on stagnation line data and then on surface properties such as the pressure coefficient, ${\mathcal{C}}_p$, friction coefficient, ${\mathcal{C}}_f$, and Stanton number, $St$, respectively, given by

$${\mathcal{C}}_p=\frac{p-p_{\infty }}{0.5{\rho }_{\infty }{U}_{\infty }^2},$$

(30)

$${\mathcal{C}}_f=\frac{\tau }{0.5{\rho }_{\infty }{U}_{\infty }^2},$$

(31)

$$St=\frac{q}{0.5{\rho }_{\infty }{U}_{\infty }^3},$$

(32)

those essential aerothermodynamic quantities being shown in Figure 2d–f.

In Figure 2a–c, the pressure, temperature, and velocity stagnation line solutions given by the CFD code of Wang and Boyd are very similar to the ones produced by hy2Foam. It can be seen in Figure 2a that the vibrational mode is barely excited for this case scenario. The single-temperature model version of hy2Foam gave the same results. The small discrepancies concerning the shock stand-off distance can easily be explained by the difference in grid point density along the symmetry axis (indeed, the spatial extension of the domain normal to the body adopted here is about 40% larger than the one in [40]) and by the use of a different viscosity model (the power law was used in [40]). As it is the case in most simulations of hypersonic flow-fields, the bow shock thickness given by the NSF equations is clearly under-predicted, as shown by the different DSMC profiles.

Once again in Figure 2d–f, an excellent agreement is found between hy2Foam and the Michigan NSF solver as the profiles of the surface quantities are shown to be superimposed with a first spacing of $10\mathsf{\mu }$m. If the mesh is further refined to $2\mathsf{\mu }$m, a small decrease in the Stanton number is observed. In conclusion, it is thought that the mesh used in [40] had a first spacing close to $10\mathsf{\mu }$m.

The variable hard sphere mean-free-path computed by hy2Foam in the wall vicinity is of the order of $2\mathsf{\mu }$m. Good DSMC practice dictates that the mean-free-path to cell-size ratio should exceed one so one could argue that the DSMC mesh employed might not be fine enough near the wall to accurately capture the surface aerothermodynamic coefficients along the body, e.g., the peak amplitude of the skin-friction.

Finally, this case is a good illustration that a hypersonic simulation can now be carried out using open-source packages all the way from pre- to post-processing using Gmsh (version 2.11.0) [42] as a mesher, hy2Foam as a solver, Paraview [43] (version 4.1.0, Sandia Corporation, Albuquerque, NM, USA and Kitware Inc., Clifton Park, NY, USA) as a visualization utility, and Gnuplot [44] (version 5.0) as a grapher.

3.2. Mach 20 Cylinder

This section focuses on the hypersonic Mach 20 flow of nitrogen over a two-dimensional circular cylinder of radius R = 1 m. A symmetry plane exists about the $y=0$ plane allowing the modelling solely of the upper half of the domain to be considered. The streamwise extent of the computational domain spans from −1.8 m to 5 m and the initial conditions are listed in

Table 2. Initial conditions for the Mach 20 cylinder.

Quantity	Value	Unit
Free-stream velocity, $U_{\infty }$	6047	m/s
Free-stream pressure, $p_{\infty }$	0.89	Pa
Free-stream density, ${\rho }_{\infty }$	$1.363\times {10}^{-5}$	kg/m${}^3$
Free-stream temperature, $T_{\infty }$	220	K
Free-stream mean-free-path, ${\lambda }_{\infty }$	$4.45\times {10}^{-3}$	m
Overall Knudsen number, $K{n}_{ov}$	0.0022	-
Wall temperature, $T_w$	1000	K

.

A free-stream velocity of 6047 m·s${}^{-1}$ is applied with a free-stream pressure of 0.89 Pa and temperature $T_{\infty }$ = 220 K. Such a temperature is high enough to result in a vibrationally-excited and chemically active flow-field. The cylinder wall is held at a uniform temperature of 1000 K. The overall Knudsen number of 0.0022 lies in the lower range of the continuum-transition regime; however, the gas locally may lie in the transition regime.

The mesh used in this investigation is shown in Figure 3 and was constructed using Ansys ICEM (Integrated Computer Engineering and Manufacturing) CFD (Canonsburg, PA, USA) [45]. This mesh consists of 155,000 cells with the first cell spacing at the cylinder wall set to 2 microns. Both Maxwellian velocity slip and Smoluchowski temperature jump boundary conditions were applied at the walls with the accommodation coefficient equal to 1.

For the shear viscosity and the thermal conductivity, the Blottner and Eucken formulae were applied, respectively, while the mixing rule employed was that of Armaly and Sutton. Both MWP and SSH formulations are successively used for V–T energy transfer for comparison purposes. The different set-ups are summarised in

Table 3. Computational fluid dynamics (CFD) simulations performed.

Run Number	V–T Transfer	Electronic Mode	CV Model	Rates
1	MWP	no	CVDV	QK
2	SSH	no	CVDV	QK
3	MWP	no	Park TTv	Park

together with a run identification number.

Two chemical reactions were also considered, these being the irreversible molecule–molecule and molecule–atom dissociation of nitrogen

$$\begin{array}{lll}{\mathrm{N}}_2+{\mathrm{N}}_2& \to & 2\mathrm{N}+{\mathrm{N}}_2,\\ {\mathrm{N}}_2+\mathrm{N}& \to & 2\mathrm{N}+\mathrm{N}.\end{array}$$

The Arrhenius rate constants are shown in

Table 4. Parameters for the evaluation of the forward rate constant.

Reaction Rate	Reaction	Arrhenius Law Constants
	Colliding Partner	A	β	${\mathit{T}}_{\mathit{a}}$
Quantum-Kinetics (QK)	N2	$2.47\times {10}^{18}$	−0.62	113,500
	N	$6.02\times {10}^{18}$	−0.68	113,500
Park	N2	$7.0\times {10}^{21}$	−1.6	113,200
	N	$3.0\times {10}^{22}$	−1.6	113,200

in which the units of A and $T_a$ are given in m${}^3$·mol${}^{-1}$·s${}^{-1}$ and Kelvin, respectively. They are derived from the QK theory [8] and Park’s rates for use in a two-temperature CFD solver [2].

Finally, two configurations associating a chemistry–vibration model with a set of chemical rates have been studied. The first one is called CVDV-QK and has been tested for zero-dimensional heat bath scenarios only [4]. The second configuration, subsequently named Park, is the one conventionally used in hypersonics and combines the Park TTv model with Park’s rates. The set-up for this case can be found in the Supplementary Materials.

The DSMC configuration is provided in [5], and good DSMC practice was satisfied for both the mesh and time-step. In the DSMC mesh, a total of 5.5 million cells were employed, which were filled with over 80 million equivalent DSMC particles at steady-state. Non-reacting case scenarios were also executed in [5], and the analysis will not be repeated here, although some of the results will be reported in the following graphs and tables.

The departure from local thermodynamic equilibrium is shown in Figure 4a for the CFD run number 2 using the local gradient-length Knudsen number. Dark grey and black colours indicate $K{n}_{GLL}$ values beyond 0.05 and cover, as expected, the bow shock and the near-wake areas. Similar results were found for the other runs.

The Mach and trans-rotational temperature fields given by hy2Foam and dsmcFoam in Figure 4b,c compare well in the compression region, for $x<0$. In the wake of the cylinder, however, $T_{tr}$ is generally smaller using DSMC. Unlike the non-reacting simulation that demonstrated a good agreement for the vibrational temperature field in the compression area [5], the discrepancies in Figure 4d are much larger this time in the whole domain when compared with DSMC. This can be explained by the application of the QK theory to capture the chemistry–vibration coupling and the use of the quantum Larsen–Borgnakke method in dsmcFoam, which promotes a quicker energy harmonisation in expansion regions (where $T_v>T_{tr}$) as reported in [4] using a zero-dimensional analysis. The Park combination used in run 1 is the instance that provides less accurate results with local vibrational temperatures above 10,000 K.

Figure 5a–c compares the stagnation line profiles of Mach number, temperature, and number density given by hy2Foam and dsmcFoam. It is normally evident that the bow shock is more diffuse when using the DSMC method compared with CFD. However, it is evident that the shock stand-off distances are almost identical using both solvers and are approximately equal to 0.25 m, which is about 5 cm closer to the body than for the non-reacting case [5]. The peak in trans-rotational temperature is correctly determined using the CVDV-QK combination for both runs 1 and 2 and is slightly over-predicted using the Park combination. As shown previously in Figure 4d, the trend in vibrational temperature shows a steeper increase across the shock wave. In Figure 5c, the evolution of the species number densities for all runs is in satisfactory agreement with dsmcFoam outside of the $K{n}_{GLL}>0.01$ band. The early production of atomic nitrogen within the shock is not captured in the CFD solver due to the slight difference in shock thickness prediction.

Surface properties of pressure coefficient, skin friction and heat transfer are shown in Figure 5d–f for both non-reacting (NR) and reacting simulations. There is a reasonable agreement between the CFD and DSMC solvers for ${\mathcal{C}}_p$ and ${\mathcal{C}}_f$.

The drag coefficient for each simulation is provided in

Table 5. Aerothermodynamic coefficients.

CFD Run Number	${\mathit{C}}_{\mathit{D}}$		${\mathit{C}}_{\mathit{H}}(\mathbf{kW})$
	CFD	DSMC	CFD	DSMC
NR	1.3	1.286	106	115
1	1.302	1.284	81.0	63.3
2	1.302		80.5
3	1.304		88.1

$C_D$ is the drag coefficient and $C_H$ represents the integrated heat flux.

, and this coefficient estimated by hy2Foam represents less than 2% error when compared with dsmcFoam, and the reacting environment does not significantly affect its magnitude. In addition, the values of hy2Foam and dsmcFoam are reasonably close to the one predicted by the Newtonian theory that is $4/3$. The integrated heat flux, $C_H$, is however showing larger discrepancies between the two codes. There are several factors that could explain this observation: (1) a $K{n}_{GLL}$ number greater than 0.1 all around the cylinder; (2) a different treatment of the vibration-translational energy transfer between CFD and DSMC codes; and (3) the use of the KNP numerical schemes in hy2Foam that are known for being too dissipative in the near-wall region. It is also shown that the CVDV-QK association is producing a 27% larger integrated heat flux as compared with dsmcFoam, while the Park combination overpredicts $C_H$ by 39%.

4. Conclusions

The newly-coded open-source two-temperature CFD solver hy2Foam has been extended to simulate hypersonic multi-dimensional case scenarios. hy2Foam has shown to perform as well as a Navier–Stokes CFD solver from the University of Michigan for a Mach 11 blunted cone, and to be in satisfactory agreement with the dsmcFoam code for a Mach 20 flow of a binary reacting mixture around a circular cylinder. For this latter case, the aerothermodynamic loads were better estimated using the novel CVDV-QK model combination rather than the conventional Park combination, when compared to the dsmcFoam code. This result reaffirms the predictions found using zero-dimensional cases in [4]. Finally, it is considered that the cylinder case scenario presented in this paper provides a useful basis for other codes to compare against.

Future work for the extension of hy2Foam will include the development of an 11-species plasma model for application in weakly-ionized flow environments. The discrepancies observed between the CFD and DSMC codes for $K{n}_{GLL}$ numbers outwith the continuum regime will motivate the development of an open-source hypersonic hybrid hydrodynamic-molecular gas flow solver using both the dsmcFoam and hy2Foam codes [7].