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A review of the mathematical modeling of equilibrium and nonequilibrium hypersonic flows

Abstract

This paper systematically reviews the mathematical modeling based on the computational fluid dynamics (CFD) method of equilibrium and nonequilibrium hypersonic flows. First, some physicochemical phenomena in hypersonic flows (e.g., vibrational energy excitation and chemical reactions) and the flow characteristics at various altitudes (e.g., thermochemical equilibrium, chemical nonequilibrium, and thermochemical nonequilibrium) are reviewed. Second, the judgment rules of whether the CFD method can be applied to hypersonic flows are summarized for accurate numerical calculations. This study focuses on the related numerical models and calculation processes of the CFD method in a thermochemical equilibrium flow and two nonequilibrium flows. For the thermochemical equilibrium flow, the governing equations, chemical composition calculation methods, and related research on the thermodynamic and transport properties of air are reviewed. For the nonequilibrium flows, the governing equations that include one-, two-, and three-temperature models are reviewed. The one-temperature model is applied to a chemical nonequilibrium flow, whereas the two- and three-temperature models are applied to a thermochemical nonequilibrium flow. The associated calculations and numerical models of the thermodynamic and transport properties, chemical reaction sources, and energy transfers between different energy modes of the three models are presented in detail. Finally, the corresponding numerical models of two special wall boundary conditions commonly used in hypersonic flows (i.e., slip boundary conditions and catalytic walls) and related research, are reviewed.

1 Introduction

Hypersonic vehicles, such as missiles, re-entry bodies, launch vehicles, and deep space detectors, have become the primary tools for strategic strikes, missile defense, space shuttles, and space exploration [1, 2]. Their extremely high speeds provide not only quick reach and responsiveness, but also temperatures that are sufficiently high to excite the internal vibrational energy within molecules and induce a series of chemical reactions (dissociation, exchange and even ionization) in the gas [3, 4]. These physicochemical phenomena cause the properties of gas around hypersonic vehicles to disobey the “calorically perfect gas” [5]; for instance, the specific heats become functions of the temperature rather than constants. Moreover, considering the trajectories of hypersonic vehicles, the gas around vehicles always exhibits different characteristics at various altitudes [6]. The nonequilibrium phenomena occur when hypersonic vehicles fly at high altitudes.

These complex physicochemical phenomena and gas characteristics occurring at various altitudes significantly increase the difficulty of describing and solving hypersonic flow problems. Although particle methods [7], such as the Direct Simulation Monte Carlo (DSMC) method, can unify these phenomena and characteristics of hypersonic flows in a Boltzmann equation, the extreme computational effort prevents their application to large geometries and low altitudes [8].

To date, the computational fluid dynamics (CFD) method remains the primary numerical simulation method for modeling hypersonic flows at lower altitudes within continuum and near-continuum regimes owing to its high computational efficiency [9, 10]. To accurately describe these phenomena and characteristics by the CFD method, hypersonic flows can be classified into three types: thermochemical equilibrium, chemical nonequilibrium, and thermochemical nonequilibrium flows. For the thermochemical equilibrium flow, conventional Navier–Stokes (N-S) equations can be adopted, in which a single temperature is applied without coupling with the chemical reactions. For the nonequilibrium flow, three types of nonequilibrium N-S equations are applied to describe the corresponding nonequilibrium characteristics: the one-temperature model [11] for the chemical nonequilibrium flow, and the two- [12], and three-temperature models [13] for the thermochemical nonequilibrium flow.

Although the forms of these master equations are not complex, there are some challenges in their solution process, such as the accurate description of gas properties at high temperatures, chemical reaction processes, and energy transfers between various energy modes [14]. Since the 1950s, researchers have proposed many corresponding numerical models. However, understanding the derivation processes of these numerical models requires knowledge of statistical thermodynamics, kinetic theory, chemical kinetic methods, and even quantum mechanics, which can discourage readers from learning about the application of the CFD method in hypersonic flows.

The appropriate wall boundary conditions are vital to accurately predict the aerodynamic performance of hypersonic vehicles. Typically, two special wall boundary conditions are considered in hypersonic flows: slip boundary conditions and a catalytic wall. Recently, the effect of a catalytic wall on the prediction of the aerodynamic heat has become a research hotspot.

This paper introduces where and how CFD methods can be applied in hypersonic flows. It reviews two calculation processes and related numerical models of the CFD method for equilibrium and nonequilibrium flows, and two special wall boundary conditions. The purpose is to enable readers interested in hypersonic flows to systematically and quickly understand the numerical calculation process of the CFD method in this field. This paper is organized as follows. Section 2 introduces the physicochemical phenomena and characteristics of the hypersonic air flow at various altitudes. Section 3 summarizes the rules for determining the applicability of the CFD method in hypersonic flows. Section 4 summarizes the calculation process of the CFD method in the thermochemical equilibrium flow, numerical models of the chemical composition, and related studies on the thermodynamic and transport properties of hypersonic air flows. Section 5 summarizes the numerical calculation process of the CFD method in nonequilibrium flows, and reviews various numerical models of the chemical reactions, thermodynamic and transport properties, and energy transfers since the nineteenth century. Finally, Section 6 highlights the wall boundary conditions, including the slip boundary conditions and catalytic walls, and reviews related numerical models and applications.

2 Characteristics of hypersonic flows

Hypersonic flows are always accompanied by a series of physicochemical phenomena, including vibrational energy excitation and chemical reactions [15]. Figure 1 shows the temperature ranges of the vibrational excitation and some chemical reactions of air derived from [3] along with the chemical reactions of air reviewed by Park [16]. As seen in Fig. 1, the vibrational energy of the molecules is excited when the temperature exceeds 800 K. The O2 dissociation begins at 2500 K and ends at 4000 K, while the N2 dissociation starts at 4000 K and ends at 9000 K. The exchange reactions involving NO occur at 2500–9000 K. As the temperature further increases, ionization reactions begin to occur; for example, N and O are ionized into N+, O+, and e. There are some reactions that are not included in Fig. 1, such as associative ionization, exchange ionization, electron-impact ionization, and charge exchange reactions, which are listed in Table 2.

Fig. 1
figure 1

Temperature ranges of vibrational excitation and chemical reactions of air

Taking the Apollo reentry as an example, the gas around the vehicle exhibits different characteristics at various altitudes, as shown in Fig. 2 [17, 18]. During reentry into the atmosphere, the vehicle traverses different flow regimes, from the free-molecular regime down to the continuum regime. According to the altitude and flight speed, the flow around the vehicle can be classified into three types: thermochemical nonequilibrium flow, chemical nonequilibrium flow, and thermochemical equilibrium flow. Nonequilibrium flow occurs when the gas flow is at a low density and/or involves very small length scales [19]. At high altitudes, owing to the rarefied gas environment, the number of intermolecular collisions is insufficient to achieve a new equilibrium state [20]. Under this condition, the relaxation process of chemical reactions and/or energy transfers between different energy modes must be considered. When the flow is in both chemical and thermal nonequilibrium, it is referred to as a thermochemical nonequilibrium flow. When the flow is in thermal equilibrium but in chemical nonequilibrium, it is referred to as a chemical nonequilibrium flow. At a low altitude, the sufficient number of collisions because of the high density maintains the gas in the well-known equilibrium state. All internal energy modes equilibrate with each other, and the chemical reactions have been fully carried out. This flow around the vehicle is called a thermochemical equilibrium flow.

Fig. 2
figure 2

Characteristics of hypersonic flows [17, 18]

3 Applicability judgment rules of the CFD method

The Knudsen number Kn, which gauges the degree of rarefaction of a gas [21], is used to divide the flow regime. It is expressed as

$$Kn = \lambda /L,$$
(1)

where λ is the local mean free path and L is the characteristic flow length.

There are four commonly accepted flow regimes [8, 22], as shown in Fig. 3.

  • (1) Continuum flow, Kn < 0.001. The flow is considered to be continuous.

  • (2) Slip flow, 0.001 < Kn < 0.1. The flow velocity has a slightly tangential component at the boundaries of the surface of the body [23], and the transitional nonequilibrium is important near the surfaces.

  • (3) Transitional flow, 0.1 < Kn < 10. Insufficient intermolecular collisions cause the flow to depart from thermal equilibrium.

  • (4) Free-molecular flow, Kn > 10. The intermolecular collisions can be neglected, and the gas only interacts with the walls of the object.

Fig. 3
figure 3

Validity range of different flow equations

The Kn defined in Eq. (1) is applied to monatomic gases only considering the translational motion of gas molecules. However, for diatomic gases, there are also rotational and vibrational motions. Different relaxation times are required for these various motions to achieve a new equilibrium state, which easily results in the nonequilibrium phenomenon in the rarefied gas and high speeds. In rarefied gas dynamics, the temporal Knudsen number Kntemporal characterizes the rarefaction of various motions of the gas. It is expressed as

$$Kn_{{{\text{temporal}}}} \propto Kn\frac{{\tau_{i} }}{{\tau_{f} }},$$
(2)

where τi represents the relaxation times of the chemical reactions and various internal energies (e.g., translational, rotational, and vibrational energies), and τf represents the characteristic flow time.

The relaxation times of the chemical reactions (τc), vibrational excitation (τv), rotational excitation (τr), and translation (τt) increase in the following order: τt < τr < τv < τc [24, 25]. Thus, as shown in Fig. 2, during the Apollo atmospheric reentry, it gradually experienced thermochemical nonequilibrium flow, chemical nonequilibrium flow, and thermochemical equilibrium flow.

The common application range of the CFD method is Kn < 0.1 [26]. Conventional N-S equations are recommended when Kn < 0.001 [8], in which sufficient gas particles occupy an element. Under this condition, despite the presence of vibrational excitation and chemical reactions owing to the high temperature, the equilibrium state is quickly achieved. When 0.001 < Kn < 0.1, the chemical reactions and vibrational excitation require more time to achieve the equilibrium state, during which the flow moves downstream [3]; hence the flow field presents a nonequilibrium state. Thus, nonequilibrium N-S equations should be adopted when 0.001 < Kn < 0.1. When 0.001 < Kn < 0.1, the shear stress and heat flux predicted by the N-S equations with Newtonian and Fourier models are no longer accurate [27]. Thus, the slip boundary conditions should be considered [28].

As shown in Fig. 3, the Boltzmann equation can be applied to the whole Kn, which can accurately describe the microscopic molecular transport in various flow regimes. It is expressed as [29]

$$\frac{\partial f}{{\partial t}} + u_{i} \frac{\partial f}{{\partial x_{i} }} = Q(f,f),$$
(3)

where f = f(x, u, t) is the distribution function at position x and velocity u, and Q(f, f) is the collision operator.

There are two types of numerical methods for solving the Boltzmann equation: probabilistic methods such as the DSMC method [26], and deterministic methods such as the discrete velocity method (DVM) [30]. The DSMC method tracks the movements and collisions of individual molecules, and treats molecular collisions using stochastic rather than deterministic procedures to simulate gas flows at the molecular level. The DVM directly solves the Boltzmann equation by the regular numerical discretization of the particle velocity space. However, the conventional DVM performs a slow convergence rate in the near-continuum flow regime owing to the cell size and time step being constrained by the particle mean free path and mean collision time. To improve the efficiency of the DVM in the near-continuum flow regime, Xu [31] proposed the unified gas-kinetic scheme (UGKS). The UGKS couples the particle transport and collision, making the cell size and time step independent of the particle mean free path and collision time. Thus, it has become an efficient DVM-type multiscale method for flow simulation in the entire flow regime.

The predictions of the DSMC method and DVM are consistent with those obtained by the CFD method at a low Kn because the Boltzmann equation can be reduced to the N-S equations through the Chapman–Enskog theory [7]. However, the numerical solution of the Boltzmann equation, either by the DSMC method or DVM, has a high computational cost for large-scale complex geometric models at a low Kn. The N-S equations have a high computational efficiency but cannot accurately describe the characteristics of the rarefied gas in the slip and early transition regimes. The construction of appropriate macroscopic fluid equations to describe the rarefied gas at a lower computational cost has been discussed by many researchers. The moment method is the most effective approach, which reduces the Boltzmann equation to a set of moment equations by expanding the distribution function. Based on the moment method, well-known macroscopic fluid equations have been proposed, such as the Grad13 [32], R13 [33], and R26 [34] moment equations.

Several flow regimes may simultaneously exist in the flow around hypersonic vehicles. Locally rarefied flow may exist in the shock layer, boundary layer, and wake of the body [35]. Because a constant characteristic length was employed, the Kn could not distinguish the flow regimes in different regions of hypersonic flows over vehicles. Boyd and Wang [36, 37] proposed the gradient-length local Knudsen number KnGLL to determine the regions where the CFD method can be applied. It is expressed as

$$Kn_{GLL,Q} = \frac{\lambda }{Q}\left| {\frac{dQ}{{dl}}} \right|,$$
(4)
$$Kn_{GLL} = \max (Kn_{GLL,D} ,Kn_{GLL,T} ,Kn_{GLL,V} ),$$
(5)

where Q denotes the flow properties (density D, temperature T, and velocity V), and l is the distance between two points in the flow field.

Boyd [36] investigated the one-dimensional normal shock waves and two-dimensional bow shock waves formed by the flow of argon and nitrogen over a sphere. Wang [37] conducted a numerical study of hypersonic nitrogen flows over an axisymmetric sharp cone tip and a hollow cylinder/flare configuration [37]. Both studies were conducted using the CFD and DSMC methods, and concluded that the continuum approach would break down when the value of KnGLL exceeded 0.05.

4 Thermochemical equilibrium flow

4.1 Conventional Navier–Stokes equations

As shown in Fig. 2, the thermochemical equilibrium flow mainly exists at a low altitude below 40 km and/or over hypersonic vehicles with a speed below 4 km/s [38], where sufficient collisions occur between particles to establish the equilibrium of various energy modes and make the chemical reactions independent of time. The various internal energies of the gas, including the translational, rotational, vibrational, and electronic energies, can be expressed by a single temperature [39]. The relevant chemical reactions have been fully carried out, which showed that the gas composition is dependent only on the temperature and pressure, which enables the uncoupling of the chemical reactions and flow equations [40]. Consequently, the assumption of thermochemical equilibrium allows the governing equations to be written in a form with a single temperature and without individual species concentrations [41]. In other words, conventional N-S equations can be applied.

The master equations of the thermochemical equilibrium flow are the same as those of the perfect gas, including the mass conservation, moment conservation, and energy conservation equations expressed as Eqs. (6)–(8), respectively.

$$\frac{\partial \rho }{{\partial t}} + \frac{{\partial \left( {\rho u_{j} } \right)}}{{\partial x_{j} }} = 0,$$
(6)
$$\frac{\partial }{\partial t}\left( {\rho u_{i} } \right) + \frac{\partial }{{\partial x_{j} }}\left( {\rho u_{i} u_{j} + p\delta_{ij} - \tau_{ij} } \right) = 0,$$
(7)
$$\frac{\partial E}{{\partial t}} + \frac{\partial }{{\partial x_{j} }}\left( {(E + p)u_{j} - \tau_{ij} u_{i} + q_{j} } \right) = 0,$$
(8)

where ρ is the density (kg/m3), ui and uj are the velocity vectors (m/s), δij is the Kronecker delta, τij is the shear stress tensor, p is the pressure (Pa), E is the total energy per unit volume (kg/(ms2)), and qj is the heat conduction vector.

Figure 4 shows the computation process of the CFD method for the thermochemical equilibrium flow. Accurate inputs of the gas properties, including the thermodynamic and transport properties, are the key for the predictions of aerodynamic parameters. The thermodynamic and transport properties can be obtained in three ways. The first is to directly obtain these from given tables, in which the properties are dependent only on the temperature and pressure, such as Hasen’s table for 7-species air and Peng–Pindroh’s table for 9-species air. The second method is to calculate these properties using curve-fitting formulas. The third method is to derive them from individual species by the corresponding mixing rules, which requires the chemical composition. Owing to the assumption of chemical equilibrium, the master equations of the chemical composition are independent of the flow equations. The three techniques for calculating the chemical composition are the equilibrium constant method [3], minimization of Gibbs free-energy method [42], and element potential method [43].

Fig. 4
figure 4

Diagram of the CFD method for the thermochemical equilibrium flow

4.2 Chemical compositions

For the equilibrium constant method, each occurring chemical reaction must be provided, and the related equilibrium constants are required. In contrast, the minimization of Gibbs free-energy and element potential methods only require knowledge of the elements of reactants and the species of products, which simplifies the calculation process and reduces the calculation time to a certain extent.

Some popular programs can directly provide the chemical composition, such as the CHEMKIN-II [44], NASA CEA [42], and STANJAN [45].

4.2.1 Equilibrium constant method

The chemical composition of a mixture includes NS species and NE elements. NR chemical reactions occur in the mixture. For the chemical reaction i, the related equilibrium constant Kp,i can be written as

$$\prod\limits_{i} {p_{j}^{{v_{j} }} } = K_{p,i} (T),$$
(9)

where vj is the stoichiometric mole number of species j which is negative for the reactants and positive for the products, pj is the partial pressure of species j, and Kp(T) is the equilibrium constant of the given chemical reaction which can be obtained from experiments or calculated by statistical thermodynamics.

According to Dalton’s law of partial pressure, the pressure of the mixture can be written as

$$p = \sum\limits_{j = 1}^{NS} {p_{j} }.$$
(10)

However, the number of unknowns is NS, and the number of equations is only NR + 1; thus, NSNR – 1 equations are required for a closed form. The number ratios of different elements are known quantities that can be expressed as functions of the partial pressure of the products as

$$\frac{{N_{m} }}{{N_{k} }} = \frac{{\sum\limits_{j = 1}^{NS} {n_{mj} } p_{j} }}{{\sum\limits_{j = 1}^{NS} {n_{kj} } p_{j} }},$$
(11)

where Nm and Nk are the numbers of elements m and k, respectively. nmj and nkj are the numbers of elements m and k in species j, respectively.

Finally, the partial pressure of species j, denoted as pj, can be solved, so the corresponding mass fraction can be easily calculated.

4.2.2 Minimization of Gibbs free-energy method

Gordon and McBride [42, 46] discussed the minimization of Gibbs free energy method in detail and applied it to the NASA Chemical Equilibrium and Application (CEA) computer program. Based on the mass conservation of elements and minimization of the Gibbs free energy theory, the final derived equations are

$$b_{i} = \sum\limits_{j = 1}^{NS} {n_{ij} } \eta_{j},$$
(12)
$$\mu_{j}^{0} (T) + {\mathcal{R}}T\ln (\eta_{i} ) + {\mathcal{R}}T\ln (p) + \sum\limits_{i = 1}^{NE} {\lambda_{i} } n_{ij} = 0,$$
(13)

where bi is the moles of element i per unit mass, nij is the stoichiometric coefficient, which represents the number of element i of species j, ηj is the moles of species j per unit mass, \(\mu_{j}^{0} (T)\) is the temperature-dependent part of the chemical potential per unit mole (J/mol), \({\mathcal{R}}\) is the universal gas constant (8314 J/(molK)), λi is the Lagrangian multiplier (J/mol).

As shown in Eqs. (12) and (13), the NS + NE equations must be solved to obtain ηj and λi. Finally, the chemical compositions can be obtained from ηj.

4.2.3 Element potential method

The element potential method is also based on the minimization of the Gibbs free energy, which attempts to do so by arbitrarily changing the quantity of each element. In this method, the element potential φi is proposed, which requires element conservation equations so that the number of equations to be solved is less than that of minimization of Gibbs free-energy method.

The mole fraction of species j, denoted as Xj, can be expressed as a function of the element potential and Gibbs free energy as [43]

$$X_{j} = \sum\limits_{j = 1}^{NS} {\exp } \left( {\frac{{ - g_{j}^{^\circ } }}{{{\mathcal{R}}T}} + \sum\limits_{i = 1}^{NE} {\varphi_{i} } n_{ij} } \right),$$
(14)

where \(g_{j}^{^\circ }\) is the molar Gibbs free energy of individual chemical species (J/mol), and φi is the element potential of element i.

According to the law of mass conservation, the moles of element i in a mixture ai can be written as

$$a_{i} = \sum\limits_{j = 1}^{NS} {n_{ij} } {\mathcal{N}}_{{{\text{tot}}}} \exp \left( {\frac{{ - g_{j}^{^\circ } }}{{{\mathcal{R}}T}} + \sum\limits_{i = 1}^{NE} {\varphi_{i} } n_{ij} } \right),$$
(15)

where \({\mathcal{N}}_{{{\text{tot}}}}\) is the total moles of the mixture.

The sum of the mole fractions of all species in the mixture is unity; therefore, an additional equation can be written as

$$\sum\limits_{j = 1}^{NS} {\exp } \left( {\frac{{ - g_{j}^{^\circ } }}{{{\mathcal{R}}T}} + \sum\limits_{i = 1}^{NE} {\varphi_{i} } n_{ij} } \right) = 1.$$
(16)

As shown in Eqs. (15) and (16), NE + 1 equations can be used to solve φi, and the corresponding mole fraction of each species can be calculated by Eq. (14).

4.3 Thermodynamic and transport properties of air

As shown in Fig. 4, there are three methods to obtain the transport and thermodynamic properties of the thermochemical equilibrium flow: obtained directly from the given tables, calculated by curve-fitting formulas, and directly calculated based on the statistical thermodynamics theory and collision integrals.

Based on statistical thermodynamics, the thermodynamic properties of individual species can be calculated using related partition functions (see Section 5.3). Based on the kinetic theory, the transport properties of individual species and a mixture can be directly calculated using collision integrals (see Section 5.4). The accuracy of thermodynamic and transport properties is strongly dependent on the appropriate partition functions and precise collision integrals. For a mixture, the accuracy of the gas composition is another important factor in calculating the thermodynamic and transport properties. Therefore, researchers have improved the accuracy of thermodynamic and transport property calculations by using more accurate collision integrals and considering more components existed in the gas. For air, the related researches on the thermodynamic and transport properties are summarized in Table 1.

Table 1 Thermodynamic and/or transport properties of the thermochemical equilibrium flow

For the convenience of engineering applications, some references present the thermodynamic and/or transport properties of air in a tabulated form, such as Hansen’s table [47] for 7-species air and Peng–Pindroh’s table [48] for 9-species air. However, the table-lookup process is often cumbersome when performing CFD calculations. Based on the tabulated values, simple closed-form equations can be obtained by the curve-fitting approach, which can be incorporated into existing numerical codes. Based on Peng–Pindroh’s [48] tabulated data, Srinivasan [49] developed improved curve-fitting formulas for the transport properties of 9-species air using Grabau-type transition functions. Based on the NASA RGAS data [57], Srinivasan [50] also curve-fitted the thermodynamic properties of 9-species air. Gupta [41, 51] calculated the thermodynamic and transport properties of 11-species air from 500 to 30,000 K at a pressure range of 10–4 to 102 atm, and curve-fitted these properties as functions of the temperature at a constant pressure.

When the air temperature exceeds 10,000 K, more ions are produced, which influence the thermodynamic and transport properties. Therefore, more compositions should be taken into account. Bacri [52, 53] calculated the thermodynamic and transport properties of 28-species air at 1 – 200 atm and 1000 – 30,000 K, and presented these properties in graphic forms. Murphy [54] calculated the transport properties of 45-species air at atmospheric pressure and 300 – 30,000 K, in which argon and carbon species were considered. Capitelli [55] calculated the transport properties of 19-species air at 50 – 100,000 K, and reported the corresponding transport properties at 50 – 30,000 K in a table. Capitelli also provided the curve-fitting expressions for these properties based on the calculated data. D’Angola [56] calculated the thermodynamic and transport properties of 19-species air at 0.01 – 100 atm and 50 – 60,000 K, and provided the corresponding curve-fitting expressions.

5 Nonequilibrium flows

As shown in Fig. 2, nonequilibrium flows (thermochemical nonequilibrium flow and chemical nonequilibrium flow) exist at a high attitude owing to the rarefied gas environment. How to accurately describe the nonequilibrium characteristics, such as the nonequilibrium of different energies and inadequate chemical reactions, with numerical models is critical to the prediction of aerodynamic parameters.

To date, various CFD solvers have been developed by some universities and scientific research institutions for nonequilibrium flows in the near-continuum regime. These include some proprietary solvers, such as NASA Langley code LAURA [58], NASA Ames code DPLR [59], Le-MANS [60] developed by the University of Michigan, US3D [61] developed by the University of Minnesota, and the DLR-TAU [62] developed by the German Aerospace Center. Open-source solvers have also been developed, such as the Eilmer [63] developed by the University of Queensland, COOLFluiD [64] created by the Von Karman Institute, and Hy2Foam [65, 66] developed by the University of Strathclyde. Meanwhile, software packages have been developed to provide the gas properties of nonequilibrium flows and species production rates, such as the CEA [46] developed by the Lewis Research Center, Cantera [67] sponsored by NumFOCUS, KAPPA [68] created by the Saint Petersburg State University, and Mutation++ [14] developed by the von Karman Institute for Fluid Dynamics.

5.1 Nonequilibrium Navier–stokes equations

In general, there are three types of nonequilibrium equations to describe nonequilibrium flows: one-temperature model [11], two-temperature model [12], and three-temperature model [13]. These three models are based on the assumption of chemical nonequilibrium, which requires the continuity equations of individual species. Based on the molecular dynamics theory, diatomic molecules have various energy modes, including translational, rotational, vibrational, and electronic modes [69], which should be expressed in terms of their respective temperatures. If these energies achieve equilibrium within the flow characteristic time, i.e., the thermal equilibrium state, then the one-temperature model can be applied; otherwise, the multitemperature models (two- and three-temperature models) should be used. In other words, the one-temperature model should be applied to a chemical nonequilibrium flow, while a multitemperature model should be applied to a thermochemical nonequilibrium flow.

In the one-temperature model, only one temperature T is required to describe all the energies that exist in the gas because of the assumption of thermal equilibrium. In the three-temperature model [13, 70, 71], the first temperature is the translational-rotational temperature Ttr, which characterizes the translational energy of the heavy species and the rotational energy of the molecules, in which the translational and rotational temperatures are assumed to be in equilibrium at all times. The second temperature is the vibrational temperature Tv, which characterizes the vibrational energy of molecules. The third temperature is the electron-electronic temperature Tele, which characterizes the free electronic translational energy and electronic excitation energy of the heavy species. The two-temperature model [12, 72] is derived from the three-temperature model by assuming that the vibrational temperature is equal to the electron-electronic temperature. One temperature is the translational-rotational temperature Ttr; the other is the vibrational-electron-electronic temperature Tve that describes the vibrational, electron translational, and electronic excitation energies.

Regardless of the model used to solve the nonequilibrium flows, the conservation of the mass of individual species (Eq. (17)), momentum (Eq. (18)), and total energy (Eq. (19)) must be ensured. The vibrational-electronic energy conservation equation (Eq. (20)) must be added to the two-temperature model, while the vibrational (Eq. (21)) and electron-electronic energy ((Eq. (22)) conservation equations should be added to the three-temperature model. Therefore, the master equations for the three models are as follows:

  • (1) One-temperature model → Eqs. (17)–(19).

  • (2) Two-temperature model → Eqs. (17)–(20).

  • (3) Three-temperature model → Eqs. (17)–(19), (21), and (22).

Figure 5 shows the calculation process of the CFD method for nonequilibrium flows. The master equations are nonequilibrium N-S equations, in which the numerical model (one-, two-, or three-temperature model) can be determined by the corresponding nonequilibrium characteristics. The gas properties, including the thermodynamic and transport properties, can be treated as the input parameters of the master equations and calculated by various techniques. The chemical reaction source can be obtained by different chemical kinetic models. The sources of the vibrational-electronic, vibrational, and electron-electronic energy equations can be obtained by the corresponding energy transfer model. These related numerical models of gas properties and sources presented in Fig. 5 will be reviewed in detail in the following sections.

$$\frac{{\partial \rho_{s} }}{\partial t} + \frac{\partial }{{\partial x_{j} }}\left( {\rho_{s} u_{j} + J_{s,j} } \right) = \dot{w}_{s}$$
(17)
$$\frac{\partial }{\partial t}\left( {\rho u_{i} } \right) + \frac{\partial }{{\partial x_{j} }}\left( {\rho u_{i} u_{j} + p\delta_{ij} - \tau_{ij} } \right) = 0$$
(18)
$$\frac{\partial E}{{\partial t}} + \frac{\partial }{{\partial x_{j} }}\left( {(E + p)u_{j} - \tau_{ij} u_{i} + q_{j} + \sum\limits_{s = 1}^{ns} {J_{s,j} } h_{s} } \right) = 0$$
(19)
$$\frac{{\partial E_{ve} }}{\partial t} + \frac{\partial }{{\partial x_{j} }}\left( {E_{ve} u_{j} + q_{ve,j} + \sum\limits_{s = 1}^{ns} {J_{s,j} } h_{ve,s} } \right) + p_{e} \frac{{\partial u_{j} }}{{\partial x_{j} }} = Q_{ve}$$
(20)
$$\frac{{\partial E_{v} }}{\partial t} + \frac{\partial }{{\partial x_{j} }}\left( {E_{v} u_{j} + q_{v,j} + \sum\limits_{s = 1}^{ns} {J_{s,j} } h_{v,s} } \right) = Q_{v}$$
(21)
$$\frac{{\partial E_{e} }}{\partial t} + \frac{\partial }{{\partial x_{j} }}\left[ {\left( {E_{e} + p_{e} } \right)u_{j} + q_{e,j} + \sum\limits_{s} {J_{s,j} h_{e,s} } } \right] - u_{j} \frac{{\partial p_{e} }}{{\partial x_{j} }} = Q_{e}$$
(22)
Fig. 5
figure 5

Diagram of the calculation process of the CFD method for nonequilibrium flows

5.2 Chemical reactions

The net source of the chemical species s due to chemical reactions denoted by \(\dot{w}_{s}\) (kg/(m3s)) can be expressed in terms of the reaction rates as [73]

$$\dot{w}_{s} = M_{s} \sum\limits_{r = 1}^{NR} {\left( {\nu_{s,r}^{\prime \prime } - \nu_{s,r}^{\prime } } \right)} \left[ {k_{f,r} \prod\limits_{s = 1}^{NS} {\left( {\frac{{\rho_{s} }}{{M_{s} }}} \right)^{{\nu_{s,r}^{\prime } }} } - k_{b,r} \prod\limits_{s = 1}^{NS} {\left( {\frac{{\rho_{s} }}{{M_{s} }}} \right)^{{\nu_{s,r}^{\prime \prime } }} } } \right],$$
(23)

where \(\nu_{s,r}^{\prime \prime }\) and \(\nu_{s,r}^{\prime }\) are the forward and backward stoichiometric coefficients of species s in the reaction r, kf and kb represent the forward and backward reaction rate coefficients (m3s−1mol−1), respectively, \(M_{s}\) is the molecular weight of species s (kg/mol).

Since the 1970s, various chemical kinetic models have been proposed to evaluate the kf and kb of air. Blottner [74] developed a chemical kinetic model for 7-species air (N2, O2, NO, N, O, NO+, and e), with seven elementary reactions. Dunn and Kang [75] proposed a chemical model for 11-species air (N2, O2, NO, N, O, NO+, N2+, O2+, N+, O+, and e), which includes 26 elementary reactions. In the Dunn–Kang model, both the kf and kb were evaluated by the Arrhenius law. Based on the Blottner and Dunn–Kang models, Gupta [51] proposed a chemical kinetic model for 11-species air containing 20 elementary reactions. The forward reaction rate coefficients of the first seven reactions were obtained from the Blottner model, while those of the remaining reactions were obtained from the Dunn–Kang model. Gupta considered that the backward reaction rate was dependent on the forward reaction rate, and proposed an equilibrium constant method to calculate the backward reaction rate coefficients. Park proposed the Park1985 [76], Park1987 [77], Park1989 [78], Park1991 [79], and Park1993 [16] models for 11-species air, and the Park2001 [80] model that considers the ablating heat shield. Similar to the Gupta model, the backward reaction rate coefficients in the models were calculated by the equilibrium constant method.

All of the above chemical kinetic models have clear parameter tables in formula form, which can be directly used to calculate the reaction rate coefficients. However, some reactions were excluded from these models, especially ionization reactions. Owing to the uncertainty of the measurements, it is impractical to obtain the reaction rate coefficients experimentally. A feasible method is to obtain the reaction rates at several temperatures by the DSMC method, and then curve-fitting these results as a function of the temperature, such as Ozawa’s modified model [81] and the Q-K model [82].

The forward rate coefficient kf in the above models is assumed to follow the Arrhenius law expressed as

$$k_{f} = A_{f} \times T_{c,f}^{{B_{f} }} \exp ( - \frac{{T_{a} }}{{T_{c,f} }}),$$
(24)

where Af is a pre-exponential factor, Bf is the temperature exponent, Ta is the temperature of activation derived from the activation energy, and Tc,f is the controlling temperature of the forward reaction.

The backward rate coefficient kb can be calculated by two methods. One method assumes that the kb follows the Arrhenius law and is independent of the kf, such as the Dunn–Kang model. It is written as [75]

$$k_{b} = A_{b} \times T_{c,b}^{{B_{b} }} \exp ( - \frac{{T_{b} }}{{T_{c,b} }}).$$
(25)

However, this form is inappropriate for high velocities [51]. The other method assumes that the kb is a function of the kf, such as Gupta model and Park models, and is written as [51, 76]

$$k_{b} = \frac{{k_{f} }}{{K_{eq} }},$$
(26)

where Keq is the equilibrium constant calculated as a function of the temperature. Two expressions proposed by Gupta (Eq. (27)) and Park (Eq. (28)) are commonly used in relevant research, and are written as [51, 76]

$$K_{{{\text{eq}}}} = \exp (A_{1} Z^{5} + A_{2} Z^{4} + A_{3} Z^{3} + A_{4} Z^{2} + A_{5} Z + A_{6} ),$$
(27)
$$K_{eq} = \exp \left( {A_{1} /Z + A_{2} + A_{3} \ln Z + A_{4} Z + A_{5} Z^{2} } \right),$$
(28)

where Z = 104/T, and the equilibrium constant coefficient A can be obtained from [51] and [76].

Table 2 summarizes the parameters of the forward rate coefficients evaluated by the Dunn–Kang (1973) [75], Gupta (1990) [51], Park1985 [76], Park1987 [77], Park1989 [78], Park1991 [79], and Park1993 [16] models. As seen in Table 2, the chemical reactions in air can be classified as Dissociation (No. 1–32), NO Exchange (No. 33–34), Associative ionization (No. 35–37), Exchange ionization (No. 38), NO Ionization (No. 39–40), Charge exchange (No. 41–54), and Electron-impact ionization (No. 55–56). Compared with the Dunn–Kang and Gupta models, Park’s models considered ions as a third body for dissociation reactions. Although many models have been proposed by Park, the Park1989 and Park1991 models are basically the same as the Park1993 model except for a few reactions.

Table 2 Forward reaction rate coefficients for air (cm3mole-1sec-1)

Figure 6 shows the comparison of the forward rate coefficients of the 5-species air (N2, O2, NO, N, and O) of the Dunn–Kang, Gupta, and Park1993 models. No obvious difference is observed between the forward rate coefficients of the three models for the chemical reactions in the 5-species air. The O2 dissociation most likely occurs in the air. The forward rate coefficient obtained by the Gupta model is smaller than those calculated by the Dunn–Kang and Park1993 models.

Fig. 6
figure 6

Forward rate coefficients of the 5-species air of the Dunn–Kang, Park1993, and Gupta models

The influence of the chemical kinetic model on aerodynamic properties has been widely investigated. Wang et al. [83] assessed the performance of four chemical reaction models (Dunn–Kang, Gupta, Park1987, and Park1991 models) of the heat transfer acting on three typical hypersonic vehicles (ELECTRE vehicle, Apollo command module, and Space Shuttle Orbiter), as shown in Fig. 7. Their results showed that the predictions of the heat flux with a complex geometry were more sensitive to the choice of chemical kinetic models. Hao et al. [84] predicted the electron and ion distributions on the RAM-C II vehicle and FIRE II capsule using the Park1989 and Gupta models. Their results showed that the different chemical reaction models significantly affected the ion and electron distributions, as shown in Fig. 8. Niu et al. [85] investigated the influence of three models (Gupta, Park1993, and Ozawa’s modified models) and two controlling temperatures (Ttr0.5Tve0.5 and Ttr0.7Tve0.3) on the species concentration and distribution in the shock layer over the BSUV II and RAM-C II vehicles. Their results showed that the NO concentration increased, and the electron density decreased as the Ttr weight factor increased. The Ozawa model with the Ttr0.7Tve0.3 controlling temperature was more effective in predicting the electron formation compared with the other models, as shown in Fig. 9.

Fig. 7
figure 7

Heat flux distributions of ELECTRE, Apollo, and Space Shuttle Orbiter with different chemical reaction models [83]

Fig. 8
figure 8

Electron number density for FIRE II [84]

Fig. 9
figure 9

Electron number density of BSUV-II [85]. (a) Peak electron number density along the surface and (b) electron number density at the location of the electrostatic probe rake

5.3 Thermodynamic properties

The internal energy of molecules can be regarded as the expression of energy storage and release in various internal modes of molecular motion. Namely, the translation, rotation, vibration of the molecular, as well as the kinetic and potential energies of the electrons around the nucleus [40]. Figure 10 shows these energy modes of a diatomic molecule. In a thermal nonequilibrium flow, the different energy modes should be expressed in terms of their associated temperatures to distinguish their relevant contributions to the basic thermodynamic properties [86]. The thermodynamic properties of individual species, such as the enthalpy, entropy, and specific heats, can be calculated using the formulas from statistical thermodynamics [87], in which the partition-function approach is employed.

Fig. 10
figure 10

Energy modes for a diatomic molecule [40]

According to quantum physics, the energy of a single atom or molecule is a discrete value, which can be calculated by its energy state i. For a system consisting of N particles, the sensible energy is the sum of the energy of the particles in each energy state. This is expressed as

$$E = \sum\limits_{i = 0}^{\infty } {\varepsilon_{i} } N_{i},$$
(29)

where Ni is the number of particles in energy state i, and εi is the energy of level i of a particle.

If the distribution of Ni is known, then the sensible internal energy of the system can be calculated. Although many possible distributions of Ni may exist in the system, the most probable macroscopic state exists among all the possible ones, i.e., the equilibrium macroscopic state. This state is given by the Boltzmann distribution as [40]

$$N_{i}^{*} = \frac{{Ng_{i} \exp \left( { - \varepsilon_{i} /kT} \right)}}{Q}$$
(30)

with

$$Q = \sum\limits_{i = 0}^{\infty } {g_{i} } \exp \left( { - \varepsilon_{i} /kT} \right),$$
(31)

where Q is the partition function, and gi is the degeneracy representing the statistical weight in energy state i of the molecule.

The energy per unit mass of an individual species can be directly derived from the internal partition function given by

$$e = RT^{2} \left( {\frac{\partial \ln Q}{{\partial T}}} \right).$$
(32)

According to the related partition function, the rotational (er,s), vibrational (ev,s), and electronic (eel,s) energies per unit mass of the heavy species s are expressed as [19]

$$e_{r,s} = R_{s} T_{r,s},$$
(33)
$$e_{v,s} = R_{s} \frac{{\theta_{v,s} }}{{\exp (\theta_{v,s} /T_{v,s} ) - 1}},$$
(34)
$$e_{el,s} = R_{s} \frac{{\theta_{el,1,s} (g_{1,s} /g_{0,s} )\exp \left( { - \theta_{el,1,s} /T_{el,s} } \right)}}{{1 + (g_{1,s} /g_{0,s} )\exp \left( { - \theta_{el,1,s} /T_{el,s} } \right)}},$$
(35)

where \(\theta_{v} = h\nu /k\) and \(\theta_{el} { = }\varepsilon_{i}^{el} /k\) are the characteristic vibrational and electronic temperatures, respectively; and Rs is the specific gas constant of species s (J/(kgK)). Obviously, the values of er,s and ev,s are zero for atoms and atom ions. Equation (33) was derived from a rigid rotor, and is only valid for homonuclear and heteronuclear molecules with a smaller degeneracy. In Eq. (35), only the first two terms of the partition function were considered, and the electronic energy of the ground state was set to zero.

The expression of ev,s in Eq. (34) was derived from the harmonic oscillator model, where the value Δε = εi-εi-1 remains constant for all i. However, the value of Δε varied at high temperatures (highly excited states). The anharmonic oscillator model provides a more precise approximation of the real molecular spectra because it considers their refinement for high vibrational levels. In the anharmonic oscillator model, the vibrational energy of a diatomic molecule at level i is written as [24]

$$\frac{{\varepsilon_{i}^{V} }}{hc} = \omega_{e} (i + \frac{1}{2}) - \omega_{e} x_{e} (i + \frac{1}{2})^{2} + \omega_{e} y_{e} (i + \frac{1}{2})^{3} + ...,$$
(36)

where ωe, ωexe, and ωeye are the spectroscopic constants. If only the first term on the right side of Eq. (36) is considered, it becomes the harmonic oscillator model.

Based on the kinetic theory, et,s is the translational energy per unit mass of species s written as

$${e}_{t,s}=\frac{1}{2}\left({c\mathrm{^{\prime}}}_{1}^{2}+{c\mathrm{^{\prime}}}_{2}^{2}+{c\mathrm{^{\prime}}}_{3}^{2}\right)=\frac{3}{2}{R}_{s}{T}_{t,s}.$$
(37)

According to Eqs. (33)–(35) and (37), the heat capacities at a constant volume (J/(kgK)) of the heavy species s in the translational (Cvt,s), rotational (Cvr,s), vibrational (Cvv,s), and electronic (Cvel,s) energy modes can be derived as follows:

$$Cv_{t,s} = \frac{3}{2}R_{s},$$
(38)
$$Cv_{r,s} = R_{s},$$
(39)
$$Cv_{v,s} = R_{s} \frac{{\theta_{v,s} /2T_{v,s} }}{{\sinh (\theta_{v,s} /2T_{v,s} )}},$$
(40)
$$Cv_{el,s} = R_{s} \frac{{\left( { - \theta_{el,1,s} /T_{el,s} } \right)^{2} (g_{1,s} /g_{0,s} )\exp \left( { - \theta_{el,1,s} /T_{el,s} } \right)}}{{\left[ {1 + (g_{1,s} /g_{0,s} )\exp \left( { - \theta_{el,1,s} /T_{el,s} } \right)} \right]^{2} }}.$$
(41)

According to Eqs. (38)–(41), the total contribution of the translational and rotational motions to the specific heat is 2.5R for molecules and 1.5R for atoms, respectively.

The heat capacities at constant pressure (J/(kgK)) of species s in different energy modes are defined as

$$Cp_{t,s} = Cv_{t,s} + R_{s},$$
(42)

\(Cp_{r,s} = Cv_{r,s}\), \(Cp_{v,s} = Cv_{v,s}\), \(Cp_{el,s} = Cv_{el,s}.\) (43)

The enthalpy at various energy modes for species s can be written as

$$h_{t,s} = Cp_{t,s} T_{t},$$
(44)

\(h_{r,s} = e_{r,s}\), \(h_{v,s} = e_{v,s}\), \(h_{el,s} = e_{el,s}.\) (45)

The electron heat capacities at constant volume and pressure are expressed as [71]

$$Cv_{e} = \frac{3}{2}R_{e},$$
(46)
$$Cp_{e} = \frac{5}{2}R_{e}.$$
(47)

5.4 Transport properties

The transport properties (e.g., viscosity, thermal conductivity, and diffusion coefficient) determine the reliability of the viscous stress, heat conduction, and species diffusion results, particularly in the boundary layers and shock waves. These transport properties can be directly calculated by collision integrals or certain empirical formulas. The following subsections will review the corresponding calculation methods for the transport properties of individual and multicomponent species.

5.4.1 Viscous stress and heat conduction

As shown in the nonequilibrium N-S equations, τij represents the components of the viscous stress tensor written as

$$\tau_{ij} = \mu \left( {\frac{{\partial u_{i} }}{{\partial x_{j} }} + \frac{{\partial u_{j} }}{{\partial x_{i} }}} \right) - \frac{2}{3}\mu \frac{{\partial u_{k} }}{{\partial x_{k} }}\delta_{ij},$$
(48)

where μ is the viscosity (kg/(ms)).

The total heat conduction vector qj is the sum of the heat conduction in different energy modes, each of which is assumed to follow Fourier’s law:

$$q_{j} = - \kappa_{t} \frac{{\partial T_{t} }}{{\partial x_{j} }} - \kappa_{r} \frac{{\partial T_{r} }}{{\partial x_{j} }} - \kappa_{v} \frac{{\partial T_{v} }}{{\partial x_{j} }} - \kappa_{el} \frac{{\partial T_{el} }}{{\partial x_{j} }} - \kappa_{e} \frac{{\partial T_{e} }}{{\partial x_{j} }},$$
(49)

where κt, κr, κv, κel, and κe are the translational, rotational, vibrational, electronic, and electron thermal conductivities (J/(msK)), respectively.

For the one-temperature model, Eq. (49) can be written as

$$q_{j} = - (\kappa_{t} + \kappa_{r} + \kappa_{v} + \kappa_{el} + \kappa_{e} )\frac{\partial T}{{\partial x_{j} }} = - \kappa \frac{\partial T}{{\partial x_{j} }}.$$
(50)

For the two-temperature model, Eq. (49) can be written as

$$q_{j} = q_{tr,j} + q_{ve,j} { = } - (\kappa_{t} + \kappa_{r} )\frac{{\partial T_{tr} }}{{\partial x_{j} }} - (\kappa_{v} + \kappa_{el} + \kappa_{e} )\frac{{\partial T_{ve} }}{{\partial x_{j} }} = - \kappa_{tr} \frac{{\partial T_{tr} }}{{\partial x_{j} }} - \kappa_{ve} \frac{{\partial T_{ve} }}{{\partial x_{j} }}.$$
(51)

For the three-temperature model, Eq. (49) can be written as

$$q_{j} = q_{tr,j} + q_{v,j} + q_{e,j} { = } - (\kappa_{t} + \kappa_{r} )\frac{{\partial T_{tr} }}{{\partial x_{j} }} - \kappa_{v} \frac{{\partial T_{v} }}{{\partial x_{j} }} - (\kappa_{el} + \kappa_{e} )\frac{{\partial T_{e} }}{{\partial x_{j} }} = - \kappa_{tr} \frac{{\partial T_{tr} }}{{\partial x_{j} }} - \kappa_{v} \frac{{\partial T_{v} }}{{\partial x_{j} }} - \kappa_{ele} \frac{{\partial T_{ele} }}{{\partial x_{j} }},$$
(52)

where κ, κtr, κve, κv, and κele are the total, translational-rotational, vibrational-electron-electronic, vibrational, and electron-electronic thermal conductivities, respectively.

Because air is a mixture of gases, the viscosity and thermal conductivities applied in air are mixed quantities. The viscosity and thermal conductivities of a mixture can be obtained by two methods: one is derived from individual species quantities using mixing rules, and the other is directly calculated by collision integrals.

5.4.2 Viscosity of individual species

Based on the kinetic theory, the viscosity of the heavy species s can be calculated by solving the Boltzmann equation using the Chapman–Enskog method, which is expressed as [38]

$$\mu_{s} = \frac{5}{16}\frac{{\sqrt {\pi m_{s} kT} }}{{\overline{Q}_{s,s}^{(2,2)} }},$$
(53)

where ms is the mass of species s (kg/particle), k is the Boltzmann’s constant, \(\overline{Q}_{s,s}^{(2,2)}\) is collision integral for ss colliding pairs (average collision cross-section) (m2), which is the average over a Maxwellian distribution of the collision cross-section for the ss colliding pairs (m2) [54].

As expressed in Eq. (53), the accuracy of the viscosity is dependent on the \(\overline{Q}_{s,s}^{(2,2)}\), whose values are usually presented in a tabulated form. For improved computational efficiency, various curve-fitting formulas are proposed such as the Gupta [51], Palmer [88], and Capitelli models [89]. The Gupta and Palmer curve-fitting models were derived from the same collision integral data. The Gupta and Capitelli models are expressed in Eq. (54) [51] and Eq. (55) [89], respectively.

$$\overline{Q}_{s,r}^{(2,2)} = \exp (D)T^{{(A(\ln T)^{2} + B\ln T + C)}}.$$
(54)
$$\begin{gathered} \, \overline{Q}_{s,r}^{(2,2)} = \frac{{a_{1} + a_{2} T^{{a_{3} }} }}{{a_{4} + a_{5} T^{{a_{6} }} }}{\text{ for heavy particles - heavy particles}}, \hfill \\ \hfill \\ \overline{Q}_{s,r}^{(2,2)} = \frac{{a_{3} (\ln T)^{{a_{6} }} \exp [(\ln T - a_{1} )/a_{2} ]}}{{\exp [(\ln T - a_{1} )/a_{2} ] + \exp [ - (\ln T - a_{1} )/a_{2} ]}}+ a_{7} \exp \left\{ { - [(\ln T - a_{8} )/a_{9} ]^{2} } \right\} + a_{4} \left[ {1 + (\ln T)^{{a_{5} }} } \right]{\text{ for electron - heavy particles}}. \hfill \\ \end{gathered}$$
(55)

Another widely used method for calculating the viscosity of individual species is the curve-fitting method, in which the viscosity is a function of the temperature. These include the Blottner model [90] and Gupta model [51] presented in Eqs. (56) and (57), respectively.

$$\mu_{s} = 0.1 \times {\text{exp}}\left( {\left( {A_{B} \ln \left( T \right) + B_{B} } \right)\ln \left( T \right) + C_{B} } \right).$$
(56)
$$\mu_{s} = 0.1 \times \left[ {\exp \left( {C_{G} } \right)} \right]T^{{\left[ {A_{G} \ln T + B_{G} } \right]}}.$$
(57)

The above numerical models for calculating the viscosity considers only one temperature. It is indisputable that T is the temperature of the flow field in the one-temperature model. However, for two- or three-temperature models, T is generally defined as the translational temperature [18, 51].

5.4.3 Thermal conductivities of individual species

Based on the kinetic theory, the translational thermal conductivity of the heavy species s is expressed as [51]

$$\kappa_{t,s} = \frac{75}{{64}}\frac{{k\sqrt {\pi kT/m_{s} } }}{{\overline{Q}_{s,s}^{(2,2)} }} = \frac{15}{4}\mu_{s} R_{s} { = }1.5\mu_{s} Cp_{t,s} { = }2.5\mu_{s} Cv_{t,s}.$$
(58)

The internal thermal conductivity resulting from the diffusion of the internal excitation energy of the species s is written as [51, 88, 91]

$$\kappa_{{{{int}},s}} = \frac{3}{8}\frac{{Cp_{{{{int}} ,s}} }}{{R_{s} }}\frac{{k\sqrt {\pi kT/m_{s} } }}{{\overline{Q}_{s,s}^{(1,1)} }} = \mu_{s} Cp_{{{{int}} ,s}} \left( {\frac{{\rho_{s} D_{s,s} }}{{\mu_{s} }}} \right) = \mu_{s} Cp_{{{{int}} ,s}} \frac{1}{{Sc_{s,s} }},$$
(59)

where Ds,s is the self-diffusion coefficient, and Sc is the Schmidt number.

According to Eq. (59), the rotational, vibrational, and electronic thermal conductivities of species s can be written as

$$\kappa_{r,s} = \mu_{s} Cv_{r,s} \frac{1}{{Sc_{s,s} }} ,\quad \kappa_{v,s} = \mu_{s} Cv_{v,s} \frac{1}{{Sc_{s,s} }}, \quad\kappa_{el,s} = \mu_{s} Cv_{el,s} \frac{1}{{Sc_{s,s} }}.$$
(60)

For the one-temperature model, the total thermal conductivity consists of the translational thermal conductivity (κt) and internal thermal conductivity (κint) written as

$$\kappa_{s} = \kappa_{t,s} + \kappa_{{{int} ,s}}.$$
(61)

For the two-temperature model, the translational-rotational (κtr,s) and vibrational-electron-electronic (κve,s) thermal conductivities of the heavy species s are written as

$$\kappa_{tr,s} = \kappa_{t,s} + \kappa_{r,s},$$
(62)
$$\kappa_{ve,s} = \kappa_{v,s} + \kappa_{el,s}.$$
(63)

For the three-temperature model, the translational-rotational thermal conductivity κtr,s is given by Eq. (62). The vibrational (κv,s) and electronic thermal (κel,s) conductivities are expressed in Eq. (60).

5.4.4 Mixing rules

For a mixture, the viscosity and thermal conductivity in each energy mode can be calculated by individual species quantities with appropriate mixing rules. In the 1950s, Wilke’s mixing rule [92] was developed by simplifying the full first-order Chapman–Enskog relation. This mixing rule sets the collision integral ratio as 5/3, and assumes that all binary interactions have the same cross-section [93]. The mixture viscosity and thermal conductivity in each energy mode with Wilke’s mixing rule are expressed as follows [94]:

$$Q = \sum\limits_{s} {\frac{{Q_{s} X_{s} }}{{\phi_{s} }}},$$
(64)

where Q represents the viscosity (μ) and thermal conductivity (κ, κtr, κve, κv, κel) quantities, X is the molar fraction, and ϕs is a scaling factor, defined as

$$\phi_{s} = X_{s} + \sum\limits_{r \ne s} {X_{r} } \left[ {1 + \sqrt {\frac{{\mu_{s} }}{{\mu_{r} }}} \left( {\frac{{M_{r} }}{{M_{s} }}} \right)^{1/4} } \right]^{2} \left[ {\sqrt {8\left( {1 + \frac{{M_{s} }}{{M_{r} }}} \right)} } \right]^{ - 1}.$$
(65)

However, Wilke’s mixing rule is only applicable to neutral gases, and the environmental temperature is limited to 10,000 K. Armaly and Sutton [95] proposed a mixing rule for partially ionized gas mixtures, in which different types of particle interactions are considered to obtain a more accurate approximation of the multicomponent transport properties. The ϕs is expressed as [95]

$$\phi_{s} = X_{s} + \sum\limits_{r \ne s} {X_{r} } \left[ {\frac{5}{{3{\mathcal{A}}_{s,r}^{*} }} + \frac{{M_{r} }}{{M_{s} }}} \right]\left[ {1 + \frac{{M_{r} }}{{M_{s} }}} \right]^{ - 1} \left[ {{\mathcal{F}}_{s,r} + {\mathcal{B}}_{s,r} \sqrt {\frac{{\mu_{s} }}{{\mu_{r} }}} \left( {\frac{{M_{r} }}{{M_{s} }}} \right)^{1/4} } \right]^{2} \left[ {\sqrt {8\left( {1 + \frac{{M_{s} }}{{M_{r} }}} \right)} } \right]^{ - 1},$$
(66)

where the coefficients \({\mathcal{A}}^{*}\), \({\mathcal{B}}\), and \({\mathcal{F}}\) depend on the type of particle interaction. The recommend value of \({\mathcal{A}}^{*}\) is 1.1 for the interactions between atom and its own ion, and 1.25 for other interactions [93]. \({\mathcal{B}}\) is recommended the value of 0.78 for the neutral–neutral interactions, 0.15 for the neutral-ion interactions, and 1.0 for the ion-ion, ion–electron, and electron–electron interactions [93]. \({\mathcal{F}}\) is usually assumed equal to 1.0 for all interactions [93].

5.4.5 Viscosity and thermal conductivities of a mixture based on collision integrals

Gupta [51] and Gnoffo [71] used collision integrals to directly calculate the mixture viscosity and thermal conductivity of weakly ionized flows. Because the interactions of each collision pair in a multicomponent mixture are considered, this model is more physically accurate than mixing rule models that use an approximation of the Chapman–Enskog formula [88].

The calculation of the collision terms among the heavy species is based on the translational temperature Tt, whereas that between electrons and other species is based on the electron temperature Te [71]. The mixture viscosity calculated by collision integrals is expressed as [51, 71]

$$\mu = \sum\limits_{s \ne e} {\left[ {\frac{{m_{s} X_{s} }}{{\sum\limits_{r \ne e} {X_{r} } \Delta_{s,r}^{(2)} \left( {T_{t}} \right) + X_{e} \Delta_{s,e}^{(2)} \left( {T_{e}} \right)}}} \right]} + \frac{{m_{e} X_{e} }}{{\sum\limits_{r} {X_{r} } \Delta_{e,r}^{(2)} \left( {T_{e}} \right)}}.$$
(67)

The translational (κt), rotational (κr), vibrational(κv), electronic excitation (κel), and electron translation (κe) thermal conductivities of the mixture are defined in Eqs. (68)–(72) [51].

$$\kappa_{t} = \frac{15}{4}k\sum\limits_{s \ne e} {\frac{{X_{s} }}{{\sum\limits_{r \ne e} {\alpha_{s,r} } X_{r} \Delta_{s,r}^{(2)} \left( {T_{t}} \right) + 3.54X_{e} \Delta_{s,e}^{(2)} \left( {T_{e}} \right)}}},$$
(68)
$$\kappa_{r} = k\sum\limits_{{s = mol}} {\frac{{X_{s} }}{{\sum\limits_{r \ne e} {X_{r} } \Delta_{s,r}^{(1)} \left( {T_{t}} \right) + X_{e} \Delta_{s,e}^{(1)} \left( {T_{e}} \right)}}},$$
(69)
$$\kappa_{v} = k\sum\limits_{{s = mol}} {\frac{{(Cp_{v,s} /R_{s} )X_{s} }}{{\sum\limits_{r \ne e} {X_{r} } \Delta_{s,r}^{(1)} \left( {T_{t} } \right) + X_{e} \Delta_{s,e}^{(1)} \left( {T_{e}} \right)}}},$$
(70)
$$\kappa_{el} = k\sum\limits_{{s = mol}} {\frac{{(Cp_{el,s} /R_{s} )X_{s} }}{{\sum\limits_{r \ne e} {X_{r} } \Delta_{s,r}^{(1)} \left( {T_{t}} \right) + X_{e} \Delta_{s,e}^{(1)} \left( {T_{e}} \right)}}},$$
(71)
$$\kappa_{e} = \frac{15}{4}k\frac{{X_{e} }}{{\sum\limits_{r \ne e} {1.45} X_{r} \Delta_{e,r}^{(2)} \left( {T_{e}} \right) + X_{e} \Delta_{e,e}^{(2)} \left( {T_{e}} \right)}},$$
(72)

where the αs,r is defined as [71, 88]

$$\alpha_{s,r} = 1 + \frac{{(1 - M_{s} /M_{r} )(0.45 - 2.54M_{s} /M_{r} )}}{{(1 + M_{s} /M_{r} )^{2} }},$$
(73)

where \(\Delta_{s,r}^{(1)}\) and \(\Delta_{s,r}^{(2)}\) are the collision terms (m·s) of the heavy species s and r, respectively. These can be calculated by the related collision integrals defined as [71]

$$\Delta_{s,r}^{(1)} (T) = \frac{8}{3}\left[ {\frac{{2M_{s} M_{r} }}{{\pi {\mathcal{R}}T\left( {M_{s} + M_{r} } \right)}}} \right]^{1/2} \overline{Q}_{s,s}^{(1,1)},$$
(74)
$$\Delta_{s,r}^{(2)} (T) = \frac{16}{5}\left[ {\frac{{2M_{s} M_{r} }}{{\pi {\mathcal{R}}T\left( {M_{s} + M_{r} } \right)}}} \right]^{1/2} \overline{Q}_{s,s}^{(2,2)}.$$
(75)

According to Eq. (67) and Eqs. (68)–(72), the mixture viscosity and thermal conductivities of different master equations can be easily obtained by replacing the corresponding control temperature in the collision terms. For the one-temperature model, both the heavy-species translation temperature Tt and electron translational temperature Te can be replaced by the temperature T. For the two-temperature model, Tt and Te should be replaced by the translational-rotational temperature Ttr and vibrational-electron-electronic temperature Tve, respectively. For the three-temperature model, Tt and Te should be replaced by Ttr and the electron-electronic temperature Tel, respectively.

5.4.6 Species diffusion

Accurate knowledge of the species diffusion in multicomponent gas mixtures is important for predicting surface properties, especially the surface heat transfer. Four models can be used to calculate the mass diffusion flux Js,j: the Fick model, Modified Fick model, Self-consistent effective binary diffusion (SCEBD) model, and Stefan–Maxwell model.

The mass diffusion flux is proportional to the gradient of the mass fraction of the heavy species s based on the Fick model written as [96]

$$J_{s \ne e,j} = - \rho D_{s} \frac{{\partial Y_{s} }}{{\partial x_{j} }},$$
(76)

where Ys is the mass fraction of species s, and Ds is the effective diffusion coefficient of species s.

The electron mass diffusion flux is calculated by assuming ambipolar diffusion to ensure the charge neutrality of the flow. This is given by [96]

$$J_{e,j} = - \frac{1}{{q_{e} }}\sum\limits_{s \ne e} {q_{s} } J_{s,j},$$
(77)

where qs is the charge per unit mass of species s.

To ensure that the sum of the mass diffusion fluxes is zero, Modified Fick model was proposed. The mass diffusion flux of the heavy species s is written as [94]

$$J_{s \ne e,j} = - \rho D_{s} \frac{{\partial Y_{s} }}{{\partial x_{j} }} - Y_{s} \sum\limits_{r \ne e} {( - \rho D_{r} \frac{{\partial Y_{r} }}{{\partial x_{j} }}} ).$$
(78)

The SCEBD model was developed for multicomponent plasma flows. It uses small electron mass approximation to simplify the mass diffusion flux equations. The mass diffusion flux of the heavy species s is expressed as [97]

$$J_{s \ne e,j} = - \frac{{pM_{s} D_{s} }}{{RT_{tr} }}\frac{{\partial \left( {p_{s} /p} \right)}}{{\partial x_{j} }} + Y_{s} \sum\limits_{r \ne e} {\frac{{pM_{r} D_{r} }}{{RT_{tr} }}} \frac{{\partial \left( {p_{r} /p} \right)}}{{\partial x_{j} }} + \frac{1}{{RT_{tr} }}\left[ {M_{s} q_{s} \rho_{s} D_{s} - Y_{s} \sum\limits_{r \ne e} {M_{r} } q_{r} \rho_{r} D_{r} } \right]E,$$
(79)

where Ms and Mr are the molar weights of the species s and r, respectively. The electric field E is defined as

$$E = \frac{p}{{q_{e} \rho_{e} }}\frac{{\partial \left( {p_{e} /p} \right)}}{{\partial x_{j} }}.$$
(80)

The Stefan–Maxwell model was derived by solving the mole fraction gradient. The mass diffusion flux in terms of the mass fraction gradient is expressed as [98]

$$J_{s,j} = - \rho D_{s} \frac{{\partial Y_{s} }}{{\partial x_{j} }} + \frac{{Y_{s} }}{{\left( {1 - X_{s} } \right)}}D_{s} \sum\limits_{r \ne s} {\left( {\rho \frac{M}{{M_{r} }}\frac{{\partial Y_{r} }}{{\partial x_{j} }} + \frac{M}{{M_{r} }}\frac{{J_{r} }}{{D_{s,r} }}} \right)}.$$
(81)

Equation (81) can be solved by an iterative method. First, the mass flux of each heavy species s is calculated at iteration N by [94]

$$J_{s \ne e,j}^{N} = - \rho D_{s} \frac{{\partial Y_{s} }}{{\partial x_{j} }} + \frac{{Y_{s} }}{{\left( {1 - X_{s} } \right)}}D_{s} \sum\limits_{r \ne s,e} {\left( {\rho \frac{M}{{M_{r} }}\frac{{\partial Y_{r} }}{{\partial x_{j} }} + \frac{M}{{M_{r} }}\frac{{J_{r}^{N - 1} }}{{D_{s,r} }}} \right)},$$
(82)

then the entire set is corrected to iteration N + 1 using the closure equation [94]

$$J_{s \ne e,j}^{N + 1} = J_{s}^{N} - Y_{s} \sum\limits_{r \ne e} {J_{r}^{N} }.$$
(83)

In the four diffusion models mentioned above, the effective diffusion coefficient of species s in a mixture Ds can be derived from two numerical models. The first is the constant Lewis number model (CLN), in which a single diffusion coefficient is applied to all species. It is widely used in multicomponent flows consisting of species with similar diffusion properties [18]. In the one-temperature model, the effective diffusion coefficient is written as [99]

$$D_{s} = D = \frac{\kappa Le}{{\rho Cp}},$$
(84)

and in the two- and three-temperature models, it is expressed as [94]

$$D_{s} = D = \frac{{\kappa_{tr} Le}}{{\rho Cp_{tr} }},$$
(85)

where Le is the Lewis number and usually set to 1.4.

The second model is the binary diffusion (BD) model in which Ds is defined as a function of the binary diffusion coefficient as follows [98]:

$$D_{s} = \left( {1 - X_{s} } \right)\left( {\sum\limits_{r \ne s} {\frac{{X_{r} }}{{D_{s,r} }}} } \right)^{ - 1},$$
(86)

where Ds,r is the binary diffusion coefficient of the interactions of the heavy species s and r, which can be directly calculated by the collision terms written as [94]

$$D_{s,r} = \frac{{kT_{t} }}{{p\Delta_{s,r}^{(1)} \left( {T_{t} } \right)}},\; s,r \ne e.$$
(87)

For the interactions between electrons and other particles, the binary diffusion coefficient De,r is expressed as

$$D_{e,r} = \frac{{kT_{e} }}{{p\Delta_{e,r}^{(1)} \left( {T_{e} } \right)}},$$
(88)

where Tt is the translational temperature, which should be replaced by T in the one-temperature model, and by Ttr in the two- and three-temperature models. Te is the electron temperature, which should be replaced by T, Tve, and Tele in the one-, two-, and three-temperature models, respectively.

Gosse [99] compared the surface heat transfers over a sphere with a diameter of 0.23 m calculated by the constant Lewis number Fick diffusion model with those calculated by the binary diffusion SCEBD model and binary diffusion Stefan–Maxwell model. As shown in Fig. 11, the surface heat transfers predicted by the binary diffusion SCEBD and binary diffusion Stefan–Maxwell models are lower than those calculated by the constant Lewis number Fick model. Alkandry [94] investigated the performance of the Fick, modified Fick, SCEBD, and Stefan–Maxwell models with a BD model on the total, convective, and diffusion heat transfers over the Stardust Sample Return Capsule. The finite catalytic wall condition was applied. The results showed that the heat transfer predicted by the Fick model is higher than that predicted by the other models. The heat transfer values calculated by the modified Fick, SCEBD, and Stefan–Maxwell models with the BD model are in good agreement, as shown in Fig. 12.

Fig. 11
figure 11

Heat transfer at the stagnation point of a sphere with different diffusion models [99]

Fig. 12
figure 12

Total, convective, and diffusion heat transfers at the stagnation point of the Stardust Sample Return Capsule with different diffusion models [94]

5.5 Energy transfers

In the thermal nonequilibrium flow, the total energy is the sum of various energies that can exchange with each other. As shown in the conservation equations of the vibrational-electronic energy, vibrational energy, and electron energy (Eqs. (20)–(22)), the source terms Qv, Qe, and Qve are expressed as [70]

$$Q_{v} = \sum\limits_{s = mol} {Q_{s,V - T} } + \sum\limits_{s = mol} {Q_{s,e - V} } + \sum\limits_{s = mol} {Q_{s,C - V} },$$
(89)
$$Q_{e} = \sum\limits_{s \ne e} {Q_{s,T - e} } + Q_{e,C - e} - \sum\limits_{s = mol} {Q_{s,e - V} } + \sum\limits_{s = ion} {Q_{s,e - i} },$$
(90)
$$Q_{ve} = \sum\limits_{s = mol} {Q_{s,V - T} } + \sum\limits_{s = mol} {Q_{s,C - V} } + \sum\limits_{s \ne e} {Q_{s,T - e} } + Q_{e,C - e} + \sum\limits_{s = ion} {Q_{s,e - i} },$$
(91)

where Qs,V-T is the energy transfer between vibrational and translational modes, Qs,e-V is the energy transfer due to inelastic collisions between electron and molecules, Qs,C-V is the energy lost/gained due to the molecular depletion/production, Qs,T-e is the energy transfer due to the elastic collision between electrons and heavy particles, Qe,C-e is the electronic energy lost/gained due to the electronic depletion/production, and Qs,e-i is the energy transfer due to the electron impact ionization.

5.5.1 Vibrational-translational energy transfer

Two formulas can be used to calculate the energy exchange between the translational and vibrational energy modes (Qs,V-T): the classical Landau–Teller formula and modified Landau–Teller formula [100] given by Eqs. (92) and (93), respectively. Compared with the classical Landau–Teller formula, the modified Landau–Teller formula can be used in a strong nonequilibrium environment, such as the arbitrary deviation from the thermal equilibrium [101].

$$Q_{s,V - T} = \rho_{s} \frac{{e_{v,s} \left( {T_{t} } \right) - e_{v,s} \left( {T_{v} } \right)}}{{\tau_{s,V - T} }} = \rho_{s} \frac{{Cv_{v,s} (T_{t} - T_{v} )}}{{\tau_{s,V - T} }},$$
(92)
$$Q_{s,V - T} = \frac{{T_{t} }}{{T_{v} }}(T_{t} - T_{v} )\rho_{s} \frac{{Cv_{v,s} }}{{\tau_{s,V - T} }},$$
(93)

where τs,V-T is the average relaxation time between the translational and vibrational energies of molecule s. For a mixture, τs,V-T is given by

$$\tau_{s,V - T} = \frac{{\sum\limits_{r = mol} {X_{r} } }}{{\sum\limits_{r = mol} {X_{r} } /\tau_{s - r,V - T} }}.$$
(94)

Millikan and White proposed a semi-empirical model, the M-W formula, to evaluate the V-T relaxation time between species s and r (τs-r,V-T). However, this formula is linearly dependent on the Ttr−1/3, which results in the underestimation of the V-T relaxation times at high temperatures [102]. Thus, a series of correction methods based on the Millikan–White model have been proposed. The most widely used method is Park’s correction, which takes into account the inaccurate estimation of the collision cross-sections at high temperatures. The model of the Millikan–White formula with Park’s correction is called the M-W–P model written as [16]

$$\tau_{s - r,V - T} = \tau_{s - r,V - T}^{MW} + \tau_{s - r,V - T}^{P},$$
(95)

with

$$\tau_{s - r,V - T}^{MW} = \frac{1}{p}\exp \left[ {A_{s,r} \left( {T_{tr}^{ - 1/3} - B_{s,r} } \right) - 18.42} \right]$$
(96)

and

$$\tau_{s - r,V - T}^{P} = \frac{1}{{\overline{c}_{s} \sigma_{v,s} n_{s,r} }},$$
(97)

where the unit of p is atm. The coefficients As,r and Bs,r can be calculated by the expressions [16] in Eqs. (98) and (99); \(\overline{c}_{s}\) is the average molecular speed (m/s) defined as \(\sqrt {8RT/\pi M_{s} }\); ns,r is the number density of the colliding pair of species s and r (m−3); and σv,s is the limited collision cross-section (m2) calculated by Eq. (100).

$$A_{s,r} = 1.16 \times 10^{ - 9/2} \sqrt {\frac{{M_{s} M_{r} }}{{M_{s} + M_{r} }}} \theta_{v,s}^{4/3},$$
(98)
$$B_{s,r} = 0.015 \times 10^{ - 3/4} \left( {\frac{{M_{s} M_{r} }}{{M_{s} + M_{r} }}} \right)^{1/4},$$
(99)
$$\sigma_{v,s} { = }\sigma_{v,s}^{^{\prime}} (50000/T)^{2},$$
(100)

where the values of \(\sigma_{v,s}^{^{\prime}}\) for N2, O2, and NO are usually set as 3 × 10–21 m2 [16].

Schwartz et al. proposed a more quantitative calculation model of the V-T relaxation time called the S-S-H model [103], which considers the probability of energy transfer between the vibrational energy and translational energy. The V-T relaxation time τs-r,V-T is related to the probability P10 of 1 → 0 transitions by the Landau–Teller relation [104].

$$\tau_{s - r,V - T} = \frac{1}{{ZP_{10} (1 - e^{ - h\omega /kT} )}},$$
(101)

where Z is the number of collisions a molecule experiences per second, h is the Planck's constant, and ω is the frequency of the transition 1 → 0.

The probability P10 can be derived from the rate coefficient k10 (cm3/s) for the deactivation of the lowest excited vibrational level. The related expression is written as

$$P_{10} = k_{10} \left( {2d_{s,r}^{2} \sqrt {\frac{2\pi kT}{{m_{s,r} }}} } \right)^{ - 1},$$
(102)

where ds,r is the collision diameter, and ms,r is the reduced mass of the colliding pairs s and r.

The rate coefficient of the specific colliding pair is expressed as an approximation based on the experimental data.

$$k_{10} = AT^{n} \exp \left\{ { - \frac{B}{{T^{1/3} }} + \frac{C}{{T^{m} }}} \right\} \times \left[ {1 - D \times \exp \left\{ { - \frac{{E_{10} }}{T}} \right\}} \right]^{ - 1},$$
(103)

where the recommended values of the parameters n, m, A, B, C, and D are referred to reference [104].

However, both the M-W-P and S-S-H models of the V-T relaxation time are simplified models that do not consider the reactive interactions. Recently, a kinetic model based on the rigorous kinetic theory of the V-T relaxation time was discussed as integrals of the relative velocity and scattering angles of the vibrational quantum obtained/lost during the elementary V-T transition [100]. Kustova [102, 105] derived a detailed calculation process of the V-T relaxation time with this model. Oblapenko [106] calculated the V-T relaxation times of N2, O2, and NO in collisions with air species with this model and compared the results with those of the M-W model, experimental data, and quasiclassical trajectory calculations (QCT). The comparison showed that the results of the rigorous-kinetic model of the V-T relaxation time is consistent with the QCT and experimental data, both quantitatively and qualitatively, over a wide temperature range.

The rigorous-kinetic model of the V-T relaxation time of the collisions between molecules s and particles r is given by

$$\tau_{{_{s - r,V - T} }} = \left[ {\frac{4kn}{{m_{s} Cv_{s} }}\left\langle {\left( {\Delta {\mathcal{E}}_{s}^{v} } \right)^{2} } \right\rangle_{{_{s - r,V - T} }} } \right]^{ - 1},$$
(104)

where n is the number density of the mixture (m−3), ms is the mass of species s (kg/particle), Cvs is the specific vibrational heat of species s, \(\Delta {\mathcal{E}}_{s}^{v}\) is the reduced vibrational energy of species s and is defined as \(\Delta {\mathcal{E}}_{s}^{v} = (\varepsilon_{s}^{i^{\prime}} - \varepsilon_{s}^{i} )/kT\), in which i and i denote the vibrational levels of species s before and after the V-T transition, respectively.

5.5.2 Chemical-vibrational energy transfer

The internal energies contribute to overcoming the activation threshold of chemical reactions; therefore, molecules in higher vibrational states are more likely to dissociate [107]. Figure 13 shows the relationship between the vibrational energy at various states and the energy required for the dissociation of diatomic molecules. The total energy required for molecular dissociation is denoted by εd, while the vibrational energy at the level i state is denoted by εvj. The energy (Δε) required for dissociation decreases as the vibrational energy level increases. This indicates that the vibrational energies and chemical reaction processes are coupled with each other. Coupled chemical-vibrational models have been proposed for calculating the chemical-vibrational energy exchange rate Qs,C-V. These include the Park model [108], Macheret–Fridman model [109], and coupled vibration-dissociation-vibration (CVDV) model [110].

Fig. 13
figure 13

Schematic of the coupled vibration-dissociation of a molecule [111, 112]

In the Park model, Qs,C-V is the vibrational energy gained or removed by the dissociation reactions of the diatomic molecule s, which is expressed as [108]

$$Q_{s,C - V} = \dot{w}_{s} \hat{D}_{s},$$
(105)

where \(\hat{D}_{s}\) denotes the vibrational energy per unit mass of the diatomic molecule s (J/kg), which can be calculated by the preferential model for molecules at higher vibrational energy levels and the non-preferential model for other molecules, written as Eqs. (106) and (107), respectively [18].

$$\hat{D}_{s} = \alpha_{s} \varepsilon_{d,s},$$
(106)
$$\hat{D}_{s} = e_{v,s},$$
(107)

where εd,s is the dissociation energy/potential of molecule s; αs is a constant usually set as 0.3 [111]; and ev,s is the average vibrational energy of molecule s.

Macheret and Fridman developed a semi-empirical model for homonuclear molecules based on the assumption of impulsive collisions. Taking into account two dissociation regimes stemming from the upper and lower vibrational states, the Qs,C-V is expressed as [113]

$$Q_{s,C - V} = \left( {\dot{w}_{s,b} E_{b,s} - \dot{w}_{s,f} E_{f,s} } \right),$$
(108)

where Ef and Eb are the forward and backward weighted average vibrational energy, respectively, expressed as [109]

$$E_{f,s} = \left( {Z_{l,s} \alpha k\theta_{d,s} \left( {T_{v} /T_{a} } \right)^{2} + Z_{h,s} k\theta_{d,s} } \right)/Z_{{{\text{M}} - {\text{F,s}}}},$$
(109)
$$E_{b,s} = (1 - L + \alpha L)k\theta_{d,s},$$
(110)

with

$$Z_{{{\text{M}} - {\text{F,s}}}} = Z_{l,s} + Z_{h,s},$$
(111)
$$Z_{l,s} = L \times \exp \left[ { - \theta_{d,s} \left( {\frac{1}{{T_{a} }} - \frac{1}{{T_{t} }}} \right)} \right],$$
(112)
$$Z_{h,s} = \frac{{1 - \exp \left( { - \theta_{v,s} /T_{v} } \right)}}{{1 - \exp \left( { - \theta_{v,s} /T_{t} } \right)}}(1 - L) \times \exp \left[ { - \theta_{d,s} \left( {\frac{1}{{T_{v} }} - \frac{1}{{T_{t} }}} \right)} \right],$$
(113)
$${\kern 1pt} \alpha = \left( {\frac{{m_{s} }}{{m_{s} + m_{r} }}} \right)^{2},$$
(114)

where ZM-F, Zl, and Zh denote the nonequilibrium factors, θv is the characteristic vibrational temperature, θd is the characteristic dissociation temperature, L depends on the type of collision pairs, and Ta is the average temperature defined as

$$T_{a} = \alpha T_{v} + (1 - \alpha )T_{t}.$$
(115)

The CVDV model is another popular model used to describe the removal of the preferential energy from the upper vibrational energy states via dissociation. It assumes that the dissociation probability is exponentially related to the vibrational energy level. The Qs,C-V is expressed as [110]

$$Q_{s,C - V} = \dot{w}_{b,s} \overline{E}\left( { - U} \right) - \dot{w}_{f,s} \overline{E}\left( {T_{F} } \right),$$
(116)

where \(\overline{E}\left( { - U} \right)\) and \(\overline{E}\left( {T_{F} } \right)\) are the weighted average vibrational energies gained by recombination and removed by dissociation, respectively. Both can be simplified by the assumption of truncated harmonic oscillators [113] written as

$$\overline{E}(T) = \frac{{k\theta_{v,s} }}{{\exp \left( {\theta_{v,s} /T} \right) - 1}} - \frac{{k\theta_{d,s} }}{{\exp \left( {\theta_{d,s} /T} \right) - 1}}.$$
(117)

The relation of TF and U is defined by

$$\frac{1}{{T_{F} }} = \frac{1}{{T_{v} }} - \frac{1}{{T_{t} }} - \frac{1}{U}.$$
(118)

If the CVDV model is adopted, then the forward rate coefficient kf in Eq. (24) should be replaced by [18]

$$k_{f} = \frac{{Z(T_{t} )Z(T_{F} )}}{{Z(T_{v} )Z( - U)}}A_{f} \times T_{c,f}^{{B_{f} }} \exp ( - \frac{{T_{a} }}{{T_{c,f} }}).$$
(119)

5.5.3 Other energy transfers

The energy exchange between the vibrational and electronic energy modes Qs,e-V can also be expressed by the classical Landau–Teller formula as

$$Q_{s,e - V} = \rho_{s} \frac{{e_{v,s} \left( {T_{e} } \right) - e_{v,s} \left( {T_{v} } \right)}}{{\tau_{s,e - V} }} = \rho_{s} \frac{{Cv_{v,s} (T_{e} - T_{v} )}}{{\tau_{s,e - V} }},$$
(120)

where τs,e-V is the relaxation time of the vibrational-electronic energy exchange, which is usually given in a tabular form [13].

The energy transfer due to elastic collisions between electrons and heavy particles Qs,T-e is expressed as [71]

$$Q_{s,T - e} = 3\rho_{e} {\mathcal{R}}(T_{t} - T_{e} )\frac{{\nu_{e,s} }}{{M_{s} }},$$
(121)

where νe,s is the effective collision frequency between an electron and the heavy species s written as

$$\nu_{e,s} = \frac{8}{3}\left( {\frac{\pi }{{m_{e} }}} \right)^{1/2} n_{s} e^{4} \frac{1}{{\left( {2kT_{e} } \right)^{3/2} }}\ln \left( {\frac{{k^{3} T_{e}^{3} }}{{\pi n_{e} e^{6} }}} \right),$$
(122)

where n represents the number density, and e is the electronic charge.

The energy loss due to electron-impact ionization Qs,e-i, in which free electrons strike neutral particles and produce ions and electrons, is expressed as [71]

$$Q_{s,e - i} = \dot{w}_{s}^{e} \hat{I}_{s},$$
(123)

where \(\dot{w}_{s}^{e}\) represents the rate of species s produced by the electron impact ionization reactions (kg/(m3s)), and \(\hat{I}_{s}\) is the first ionization energy of the species s (J/kg).

The electronic energy increase or loss due to chemical reactions Qs,C-e is written as

$$Q_{e,C - e} = \dot{w}_{e} e_{e}.$$
(124)

6 Boundary conditions

6.1 Slip boundary conditions

The no-slip boundary conditions commonly used in the CFD method assume that the velocity and temperature of the gas near the wall are equal to those on the wall, and that transport processes near the wall are not affected by wall collisions. However, from a microscopic perspective, the transport of the momentum, heat, and mass of the gas near the wall is not the same as that in the interior of the gas [114]. Hence, the results based on the no-slip assumption are inaccurate for rarefied gas, where the mean free path cannot be ignored [115]. When the mean free path becomes comparable to the characteristic length of the flow, the flow should be treated at the level of a velocity distribution function, such that the kinetic Boltzmann equation or DSMC method [116] is adopted. However, both methods require significant computational effort at a low Kn, which is difficult for practical applications. Therefore, it is important to find ways to expand the application scope of the CFD method. The slip boundary conditions extend the CFD method to moderately rarefied gas. As shown in Fig. 3, the slip boundary conditions extend the application range of the CFD method from Kn < 0.001 to Kn < 0.1.

Based on the kinetic theory, Maxwell [117] proposed a model for the slip velocity us at the solid surface expressed as [118]

$$u_{s} - u_{w} = \sigma_{P} \left. {\frac{{v_{m} \mu }}{p}\frac{{\partial u_{x} }}{\partial n}} \right|_{0} + \sigma_{T} \left. {\frac{\mu }{\rho T}\frac{\partial T}{{\partial x}}} \right|_{0},$$
(125)

where μ is the shear viscosity; vm is the most probable speed of the gas, defined as \(\sqrt {2kT/m}\); x and n are the Cartesian coordinates in the parallel and normal directions to the solid surface, respectively; and σP and σT are the viscous and thermal slip coefficients, respectively. The second term on the right-hand side is related to thermal creep; hence, it can be neglected for an isothermal wall. Under the diffuse-specular boundary condition, Maxwell derived the σP and σT as [118]

$$\sigma_{P} = \frac{{2 - \alpha_{M} }}{{\alpha_{M} }}, \quad \sigma_{T} = \frac{3}{4},$$
(126)

where αM is the probability of diffuse scattering, and the range of αM is 0–1.

Based on the analysis of the Maxwell model for momentum slip, Smoluchowski proposed the temperature slip model, which assumes that the energy brought to the boundary by approaching molecules is responsible for the heat conducted through the boundary [119]. This is expressed as

$$T_{s} - T_{w} = \left. {\frac{{2 - \alpha_{T} }}{{\alpha_{T} }}\frac{2\gamma }{{(\gamma + 1){Pr} }}\lambda \frac{\partial T}{{\partial n}}} \right|_{0},$$
(127)

where αT is the fraction of molecules reflected by the wall temperature, γ is the ratio of specific heats, and Pr is the Prandtl number. λ is the molecular mean free path defined as [119]

$$\lambda = \frac{\mu }{\rho }\sqrt {\frac{\pi }{2RT}},$$
(128)

where R is the specific gas constant (J/(kgK)).

In the N-S equations, the shear stress was assumed to vary linearly with the velocity gradient on the wall, where the rarefied flow near the surface was not considered. However, a nonlinear velocity profile and finite slip velocity developed near the surface owing to infrequent gas–gas interactions [120]. This boundary layer with a nonlinear velocity distribution is called the Knudsen layer, with a thickness of several mean free paths of gas molecules. To accurately predict the velocity profile in the Knudsen layer, Lockerby [121] proposed a correction by modeling the local gas viscosity in the Knudsen layer as

$$\mu = \mu /\Psi (n/\lambda ),$$
(129)

where the “wall-function” \(\Psi \left( {n/\lambda } \right)\) is expressed as

$$\Psi \left( {n/\lambda } \right) \approx 1 + 0.7\left( {1 + n/\lambda } \right)^{ - 3}.$$
(130)

The Knudsen layer function is another method to calculate the velocity profile in the Knudsen layer, which describes the relationship between the defect velocity ud and the normal distance to the surface n. Recently, Su et al. [120] and Wang et al. [118] solved the linearized Boltzmann equation with a diffuse-specular boundary condition to obtain a series of velocities in the Knudsen layer, which were fitted to the Knudsen layer function. The defect velocity ud, which describes the deviation of the linearly extrapolated velocity in the bulk region from the true velocity [120] inside the Knudsen layer, is defined as [118]

$$\frac{{u_{d} (n)}}{{u_{d} (0)}} = \sum\limits_{i = 0}^{2} {\sum\limits_{j = 0}^{2} {c_{i,j} } } n^{i} (n\ln n)^{j},$$
(131)
$$u_{d} (0) = c_{1} \left[ {1 - \exp \left( { - c_{2} \alpha_{M} } \right)} \right],$$
(132)

where ci,j, c1, and c2 are the fitting coefficients from [118].

6.2 Catalytic wall

In a hypersonic air flow, a large number of atoms, ions, and electrons are produced around the shock wave owing to the high temperature. These atoms, ions, and electrons recombine into molecules at the surface owing to the low surface temperature. These recombination reactions are exothermic reactions that release heat to the surface, which significantly increases the surface heat flux [122].

In the 1990s, Scott [123] proposed a phenomenological model (global one-step model [124]) for catalytic atom recombination. Scott assumed that the atoms diffused to the surface and were specularly reflected from the surface or completely adsorbed by the surface. The recombination coefficient γs was proposed to define the ratio of the recombining atoms and the incident atoms impinging on the surface. This is expressed as

$$\gamma_{s} = \frac{{\left| {N_{s,re} } \right|}}{{\left| {N_{s,tot} } \right|}}.$$
(133)

The recombination coefficient is limited to the range of 0–1. γs = 1 and γs = 0 denote a fully catalytic and noncatalytic wall, respectively. The value of γs is usually obtained experimentally; for related experiment investigations, please refer to [125,126,127].

Owing to the recombination reactions, the normal component of the net mass flux of atom s consumed on the surface is expressed as

$$\dot{m}_{s} = - k_{s} (\rho_{w} Y_{w,s} )^{P},$$
(134)

where P is the chemical order of the reaction (usually set as 1), Yw,s is the mass fraction of atom s on the surface, and ks is the recombination rate derived from the Hertz–Knudsen relation written as

$$k_{s} = \gamma_{s} \sqrt {\frac{{kT_{w} }}{{2\pi m_{s} }}}.$$
(135)

The net mass flux generated by the diffusion of atom s to the surface is given by

$$J_{s} = \rho_{w} D_{w,s} \left( {\frac{{\partial Y_{w,s} }}{\partial n}} \right).$$
(136)

According to the mass balance on the surface, the net mass flux of species s formed at the surface is zero. The mass flux due to the recombination reactions must be balanced by that of the diffusion to the surface [128], which is expressed as

$$\dot{m}_{s} + J_{s} = 0.$$
(137)

Substituting Eq. (134) and Eq. (136) into Eq. (137) with P = 1, the final form is written as

$$- k_{s} \rho_{w} Y_{w,s} + \rho_{w} D_{w,s} \left( {\frac{{\partial Y_{w,s} }}{\partial n}} \right) = 0.$$
(138)

Many related investigations [129,130,131,132,133] have confirmed that catalytic recombination reactions on the wall can significantly increase the aerothermal load on the surface of vehicles. The diffusive heat flux caused by recombination reactions is the main reason for the increasing aerodynamic heat [132], as shown in Fig. 14. The catalytic recombination coefficient plays a significant role in the prediction of the aerodynamic heat. The heat flux increases with the improvement of the catalytic recombination coefficient, as shown in Figs. 15 and 16. However, the heat flux does not continuously increase as the catalytic recombination coefficients increases [133], whereas a strong sensitivity of the catalytic heating augmentation for weakly and moderately catalytic walls exists [134]. Different catalytic recombination coefficients of different atoms have been considered by Yang and Park [124]. The surface plot of the ratio of the diffusive heat flux and convective heat flux is shown in Fig. 17.

Fig. 14
figure 14

Heat flux on a calorimeter probe for the mixtures: (a) O2-Ar and (b) N2-Ar [132]

Fig. 15
figure 15

Heat flux for RAM-C II [132]

Fig. 16
figure 16

Heat flux on a calorimeter probe [133]

Fig. 17
figure 17

Surface plot of oxygen and nitrogen catalytic efficiency on the diffusion heat transfer [124]

The phenomenological model suffers from oversimplification; it only considers the net effect of the surface and ignores the detailed chemical reaction mechanisms [135]. Moreover, a frozen boundary layer is assumed in this model, in which the timescale of the atomic diffusion to the surface is considered to be much shorter than that of the recombination reactions [127]. Figure 18 shows the three interaction mechanisms between the gas and surface [136, 137], where s denotes the surface and A(s) denotes an atom adsorbed on the surface.

Fig. 18
figure 18

Mechanisms of catalytic recombination reaction on the surface

(1) Adsorption/desorption: \(A + (s) \leftrightarrow A(s)\).

(2) Eley–Rideal (E–R) recombination/associated dissociative adsorption: \(A + B{\text{(s)}} \leftrightarrow AB + (s)\).

(3) Langmuir–Hinshelwood (L–H) recombination/associated dissociative adsorption: \(A(s) + B{\text{(s)}} \leftrightarrow AB + 2(s)\).

It can be seen that atoms diffuse near the solid wall, and some of them are directly absorbed by the surface. In the E–R recombination, the atoms in the gas phase directly recombine with the atoms absorbed on the surface to form molecules, which are then desorbed from the surface. In the L–H recombination, the atoms absorbed on the surface recombine with each other, and are then desorbed from the surface in the form of molecules.

Based on the interaction mechanisms presented in Fig. 18, a gas/surface finite-rate model has been proposed and widely discussed [136, 138, 138], which enables the definition of an arbitrary number of physical interactions and chemical reactions between the gas and surface. The mass flux of species s on the surface is defined as

$$\dot{m}_{s} = M_{s} \sum\limits_{i = 1}^{nr} {\left\{ {\left( {v_{si}^{\prime \prime } - v_{si}^{\prime } } \right)\left( {k_{fi} \prod\limits_{s = 1}^{ns} {\left[ {X_{s} } \right]^{{v_{si}^{\prime } }} } - k_{bi} \prod\limits_{s = 1}^{ns} {\left[ {X_{s} } \right]^{{v_{si}^{\prime \prime } }} } } \right)} \right\}}.$$
(139)

This formula is similar to the finite-rate formulation of the gas phase chemistry in Eq. (23) [138]. However, this formula describes the species production/loss on the surface; thus, the unit of the forward and backward rate coefficients is m2/(smole).

7 Conclusion

This paper is a primer for readers interested in hypersonic flows. It introduces the physical phenomena inherent in hypersonic flows and the flow characteristics of three types of hypersonic flows (thermochemical equilibrium flow, chemical nonequilibrium flow, and thermochemical nonequilibrium flow). Then, the Kn ranges for applying the CFD method, conventional N-S equations, and nonequilibrium N-S equations are summarized. State-of-the-art mathematical modeling based on the CFD method and related calculation processes for the three hypersonic flows are reviewed in detail.

For the thermochemical equilibrium flow, conventional N-S equations can be used, in which the individual species concentrations are not explicitly required. The calculation accuracy of the aerodynamic parameters is dependent on the inputs of the transport and thermodynamic properties. Related studies on these properties are summarized in a Table. Three methods for calculating the chemical compositions of the thermochemical equilibrium flow are also reviewed. For the two nonequilibrium flows, the flow equations should be coupled with chemical reactions. Three types of nonequilibrium N-S equations are reviewed: one-, two-, and three-temperature models. The one-temperature model is applied to the chemical nonequilibrium flow, while two- and three-temperature models are applied to the thermochemical nonequilibrium flow. Several chemical kinetic models, derivations of the thermodynamic and transport properties of an individual species or mixture, and numerical models of energy transfers between different energy modes are presented in detail.

Two special wall boundary conditions that frequently appear in hypersonic flows (i.e., slip boundary condition and catalytic wall) are introduced. The corresponding numerical models and application research are reviewed.

Availability of data and materials

Not applicable.

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Acknowledgements

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Funding

This research was supported by the Key Laboratory of Hypersonic Aerodynamic Force and Heat Technology of the AVIC Aerodynamics Research Institute, National Natural Science Foundation of China (Grant Nos. 31371873, 31000665, 51176027, and 31300408), Special Program for Applied Research on Super Computation of the NSFC-Guangdong Joint Fund (the second phase) of China and CAST-BISEE (Beijing Institute of Spacecraft Environment Engineering) innovation fund.

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WZ designed the framework of this paper, and was a major contributor in writing the manuscript. ZZ participated in the design of this paper, and was a major contributor in revising the manuscript. XW revised Section 5.3 (Thermodynamic properties) in the manuscript. TS polished the language of the manuscript. All authors read and approved the final manuscript.

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Correspondence to Zhijun Zhang.

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Zhang, W., Zhang, Z., Wang, X. et al. A review of the mathematical modeling of equilibrium and nonequilibrium hypersonic flows. Adv. Aerodyn. 4, 38 (2022). https://doi.org/10.1186/s42774-022-00125-x

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Keywords

  • Mathematical modeling
  • Hypersonic flows
  • Chemical nonequilibrium flow
  • Thermochemical nonequilibrium flow
  • Thermochemical equilibrium flow