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A simplified method for calculating spectral emission of nonequilibrium air plasmas in hypersonic shock-layers


A simplified method for calculating the spectral emission of nonequilibrium air plasmas is developed. In order to obtain the nonequilibrium energy level populations, the nonequilibrium coefficients are introduced into the Saha-Boltzmann equation. These nonequilibrium coefficients are calculated by using several significant radiative processes. An approach to the determination of nonequilibrium electronic energy level populations of diatomic molecules is also presented. Based on the method, spectral emission of atoms and molecules in a typical air plasma cell is investigated. The results reveal that there is a significant difference between the nonequilibrium and equilibrium emission. We apply the method to the nonequilibrium AVCO R-156 experiment. Good agreement with the NEQAIR code and the measured data is shown, indicating that the method is reasonable and has good accuracy.

1 Introduction

During the hypersonic reentry, a significant portion of the heating experienced by a vehicle may be due to the emission of air plasmas in the shock-layer. To investigate the radiative heating, it is highly important to calculate the spectral emission of air plasmas [1].

The air plasmas in the shock-layer may be equilibrium or nonequilibrium [2, 3]. For equilibrium air plasmas, their spectral emission can be easily calculated by means of equilibrium statistical and spectral theory. However, for nonequilibrium air plasmas, the prediction of their emission is a challenge.

For this reason, most of the physical models and computational codes in the literature have tried their best to address nonequilibrium radiation [4,5,6,7,8,9,10]. Full models, such as the collisional-radiative (CR) model, are well-known to be accurate [11, 12]. However, their computational cost is very high. To improve the computational efficiency, researchers tend to develop simplified models. These simplified models will be desirable and useful in engineering. The nonequilibrium air radiation (NEQAIR) code presented by Park is the most widely used. Although NEQAIR employed the so-called quasi-steady-state (QSS) assumption [13, 14], its computational cost is not low due to the consideration of many collisional and radiative processes in air plasmas [15].

In this paper, based on the main idea of the CR model, a simplified method for calculating nonequilibrium air plasma radiation is developed. The method uses a modified Saha-Boltzmann equation to represent the nonequilibrium energy level population. By solving the conservation equation containing significant radiative processes, the nonequilibrium population of bound electronic energy levels can be obtained. Based on a typical air plasma cell, the difference between the nonequilibrium and equilibrium emission of particles is analyzed. In addition, application of the method in nonequilibrium AVCO R-156 experiments is discussed.

2 Simplified method

In the following, we consider the 11-species air model that contains N, N+, O, O+, N2, \( {\mathrm{N}}_2^{+} \), O2, \( {\mathrm{O}}_2^{+} \), NO, NO+ and free electrons. The method for determining the bound electronic energy level populations is first explained, which is based on our previous work [16, 17]. Then, we apply the method to some typical plasmas and analyze the nonequilibrium radiation from atoms and molecules, respectively.

2.1 Bound electronic energy level populations

The main idea of the full CR model is that the population of each bound electronic energy level is affected by various collisional and radiative processes [1]. If the QSS assumption is used, the population of the i th bound electronic energy level ni is determined by

$$ {\left.\frac{dn_i}{dt}\right|}_{\mathrm{collisional}}+{\left.\frac{dn_i}{dt}\right|}_{\mathrm{radiative}}=0 $$

In Eq. (1), the collisional processes can be subdivided into cases where the impacting particles are electrons, ions, or neutral particles. The radiative processes include radiative transition, radiative recombination, and so on [1, 12]. The full CR model is accurate because it considers as many processes as possible. However, this leads to a large number of coupled rate-equations and a huge increase in the computational cost.

For general hypersonic air plasmas, the departure from thermal equilibrium is usually not very large. By introducing a nonequilibrium coefficient into the Saha-Boltzmann equation [16, 17], there will be

$$ {n}_i={b}_i{n}_e{n}^{+}\frac{g_i}{2{Q}^{+}}{\left(\frac{h^2}{2\pi {m}_e{kT}_e}\right)}^{3/2}\exp \left(\frac{I_i}{kT_e}\right) $$

where bi is the nonequilibrium coefficient for the population of the i th level, ne and n+ are the number density of free electrons and charged particles, Q+ is the partition sum of charged particles, gi and Ii are the degeneracy and ionization energy of the i th level, Te is electronic temperature, me is electron mass, h is Planck constant, and k is Boltzmann constant.

To calculate the bi in Eq. (2), three most significant radiative processes, that is, radiative recombination, spontaneous transitions and photo excitation from the ground state, are considered. After some simplifications are made [16], this leads to

$$ {n}_i\sum \limits_{j=2}^{i-1}{A}_{ij}\approx \sum \limits_{j=i+1}^{\infty }{n}_j{A}_{ji}+{n}_e{n}_0^{+}{R}_i $$

where nj is the population of the j th level, Aij and Aji are the transition probabilities, \( {n}_0^{+} \) is the number density of ground-state charged particles, and Ri is the radiative recombination coefficient of the i th level. Ri has the form [1]:

$$ {R}_i=5.20\times {10}^{-14}p\sqrt{\frac{I_i}{I_H}}{\left(\frac{I_i}{kT_e}\right)}^{3/2}{e}^{\frac{I_i}{kT_e}}{E}_1\left(\frac{I_i}{kT_e}\right){G}_p $$

where p is the primary quantum number of the i th level, IH is the ionization energy of atomic hydrogen, and Gp the Gaunt coefficient (Gp = 1.0 in the following calculations).

Based on Eqs. (2) and (3), bi can be calculated, and then the bound electronic energy level populations are obtained.

Specifically, the ionization in nonequilibrium air plasmas is usually highly insufficient [18]. Thus, for several lowest levels, the Boltzmann distribution characterized by Te can be directly assumed [19], as NEQAIR code did. For several highest levels, the Saha distribution characterized by Te can be used [17]. For the other levels, the population should be calculated by Eqs. (2) and (3).

2.2 Atomic emission

When the speed of a vehicle is high in earth reentry, the diatomic species behind the shock may be highly dissociated [20]. In this situation, spectral emission of N and O may be dominant [21, 22]. Therefore, calculation of atomic radiation is highly important.

We employ Park’s electronic level system [2]. The nonequilibrium air plasmas is typically optically thick for radiation below 200 nm. It means that the atomic lines corresponding to the transitions from high levels to levels 1–3 can be totally absorbed. Thus, in Eq. (3), the ground state(s) is not level 1 but are levels 1–3. Eq. (3) transforms to

$$ {n}_i\sum \limits_{j=4}^{i-1}{A}_{ij}=\sum \limits_{j=i+1}^{j_{\mathrm{max}}}{n}_j{A}_{ji}+{n}_e{n}_1^{+}{R}_i $$

where jmax equals to 22 (for N) or 19 (for O).

To improve the efficiency, it is not necessary to calculate the nonequilibrium population of all levels. Table 1 lists the atomic lines based on NEQAIR [7]. As mentioned above, atomic lines below 200 nm are almost totally absorbed. For the prediction of radiative heating, the atomic lines above 200 nm are of great importance. Most of the strong atomic lines above 200 nm are related to levels 6–12. For these levels, we need surely to calculate their nonequilibrium populations according to Eqs. (2) and (3). For other lower or higher levels, the use of the equilibrium population would lead to small errors in the spectral emission calculation.

Table 1 Atomic lines analysis

To investigate the difference between the nonequilibrium and equilibrium radiation of atoms, we chose an arbitrary air plasma cell. Table 2 lists the condition of the air plasma cell [9]. In the spectral calculations, we only consider bound-bound transitions of N and O. For the shape of each emission line, the Voigt profile is assumed [3].

Table 2 The condition of the air plasma cell

Figure 1 shows spectral emission of the two atomic species. In Figs. 1(a) and (b), the electronic energy level population of N and O is assumed to be Boltzmannian with Te. However, in Figs. 1(c) and (d), the population of N and O is calculated by means of the above-mentioned method. One can immediately find that the characteristics of the spectrum are significantly different. For the emission of N, we draw a comparison between Figs. 1(a) and (c). It is shown that the Boltzmannian assumption will cause the emission to be overestimated. For O, a similar phenomenon is seen by a comparison between Figs. 1(b) and (d). The results imply that the use of Boltzmannian distribution could lead to a considerable error in the prediction of radiative heating.

Fig. 1
figure 1

Spectral emission of N (a) and O (b) assuming a Boltzmannian distribution with Te. Nonequilibrium emission of N (c) and O (d) using the present method

2.3 Diatomic molecular emission

For diatomic molecules in air plasmas, the relaxation of electronic energy levels is quicker than vibrational levels and dissociation processes [23]. In order to make use of Eq. (3), we need to determine the radiative recombination coefficient.

We first attempt to found a relationship between the primary quantum number and the ionization energy of a certain electronic level. If the Hydrogen-like approximation is used, this relationship will be

$$ {\omega}_i={\left({I}_H/{I}_i\right)}^{1/2} $$

where ωi is the primary quantum number of the i th level of molecules. Considering the important electronic energy levels of diatomic molecules for air plasma radiation, we can obtain Table 3 for the three diatomic molecules [24, 25].

Table 3 The values of ωi for diatomic molecules

Assuming that the electronic, vibrational and rotational energy modes of diatomic molecules are independent, the energy of the i th level can be expressed as [13].

$$ E\left(e,v,J\right)=E(e)+G(v)+F(J) $$

where e, v, J are the quantum number of electronic, vibrational and rotational energy levels. In this case, the degeneracy of the i th level is

$$ {g}_i={g}_e\sum \limits_{v,J}{g}_J\exp \left(-\frac{G(v)}{kT_v}-\frac{F(J)}{kT_r}\right) $$

To investigate the difference between the nonequilibrium and equilibrium radiation of diatomic molecules, we reconsider the air plasma cell listed in Table 2. We only calculate N2 as an example. The first positive system (\( B{{}^3\Pi}_g-A{{}^3\Sigma}_u^{+} \)) and the second positive system (C3Πu − B3Πg) of N2 are studied.

We first calculate the nonequilibrium electronic level population of N2. According to Table 3 and Eq. (4), radiative recombination coefficients can be obtained. Based on the transition probabilities in the literature [12], we substitute Eq. (2) into Eq. (3) and calculate the nonequilibrium coefficients, which are listed in Table 4. These nonequilibrium coefficients suggest that, compared with the population determined by the Saha-Boltzmann equation, the population of B3Πg is 5.89 times larger, whereas the population of C3Πu is 0.25 times smaller.

Table 4 The nonequilibrium coefficients

Figure 2 shows the spectral emission of N2. Here, we only consider the bound-bound transitions, and the Voigt profile is used as the lineshape. The results in Fig. 2(a) are based on the multi-temperature (multi-T) model, where the population of electronic, vibrational, and rotational energy levels is assumed to be Boltzmannian distribution characterized by Te, Tv, and Tr, respectively. The results in Fig. 2(b) are based on our method. Specifically, the population of electronic levels is calculated by the above-mentioned method, whereas the population of vibrational and rotational levels is assumed to be Boltzmannian distribution characterized by Tv and Tr, respectively.

Fig. 2
figure 2

Emission of N2 based on the multi-T model (a) and the present method (b)

Compared with Fig. 2(b), the intensity of the first positive system in Fig. 2(a) is significantly larger, while the intensity of the second positive system is slightly smaller. The results indicate that the multi-T model overestimate the emission of N2. The overestimation is attributed to the error in the population of electronic levels.

3 Application in hypersonic flow

To verify the application of the present method in hypersonic nonequilibrium air plasmas, we chose to study the cell conditions listed in Ref. [13]. We did not discuss the equilibrium states, but focused the attention on the nonequilibrium AVCO R-156 experiments.

Based on the nonequilibrium cell condition in Ref. [13], we calculated the fine spectra with a high resolution (0.001 nm). The results are shown in Fig. 3(a). Here, the bound-bound, bond-free, and free-free transitions of particles are considered. It is seen that atomic lines are dominant due to the high temperature. Regarding to the molecular bands, the emission of NO and O2 makes the spectral intensity below 200 nm relatively strong. Meanwhile, the peak intensity at ~ 400 nm is mainly due to the emission of \( {\mathrm{N}}_2^{+} \). However, the emission of N2 is relatively weak.

Fig. 3
figure 3

Emission of the AVCO R-156 nonequilibrium air plasma. (a) The calculated fine spectra. (b) A comparison with the NEQAIR II and the measured data

Figure 3(b) shows a comparison with the computational results of NEQAIR code and the nonequilibrium AVCO R-156 experimental spectra. The resolution of our calculated spectrum is converted to 20 nm, which is the same as the resolution of the NEQAIR results and the measured data. It is clearly seen that the present results are in good agreement with the NEQAIR and the measured data. This indicates that the present method is reasonable, and has a good accuracy. In the spectral region of 650–700 nm, there is a deviation between our results and the experimental data. It can be attributed to the following two reasons. One is that the transition probabilities related to the atomic lines within 663–665 nm are inaccurate to a certain extent, as Table 5 shows [26]. The other is that, owing to the lack of transition probabilities, the spontaneous transitions from the upper levels to the 12th level are not included in the present calculation.

Table 5 Transition probabilities related to atomic lines within 663–665 nm

Since it is not necessary to solve the rate equations of energy level populations, the present method is simpler than most of QSS models including the NEQAIR code. It is applicable for the 11-species air plasmas that contains N, N+, O, O+, N2, \( {\mathrm{N}}_2^{+} \), O2, \( {\mathrm{O}}_2^{+} \), NO, NO+ and free electrons. In fact, if the corresponding gas components are added, this method is also suitable for a variety of hypersonic applications.

4 Conclusions

We developed a simplified method for calculating spectral emission of nonequilibrium air plasmas. The method introduces nonequilibrium coefficients into the population of each energy level governed by the Saha-Boltzmann equation. By considering the three significant radiative processes, the nonequilibrium coefficients can be calculated. Thus, the nonequilibrium populations can be obtained. For diatomic molecules, an approach to the determination of the radiative recombination coefficient and the energy level degeneracy is presented. Based on the approach, we can also calculate the nonequilibrium coefficients. A typical air plasma cell is chosen to investigate the difference between the nonequilibrium and equilibrium emission of atoms and molecules. It is suggested that the multi-T model overestimate the spectral emission in comparison with our results. Application of the present method to the nonequilibrium AVCO R-156 flowfield was shown. The results indicate that the present method agrees well with the NEQAIR code and the experimental data. This work offers a simple method to the calculation of nonequilibrium air plasma radiation, which will be useful in hypersonic engineering.

Availability of data and materials

Data supporting the results of this article can be found, including data generated during the study. According to reasonable requests, the minimal datasets necessary to interpret and replicate the results of this article can be obtained from the corresponding author.


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The authors would like to thank Prof. Junbo Yang and Prof. Juncai Yang for their helpful work. The authors are thankful to the reviewers for their valuable comments to improve the quality of the manuscript.


This work is supported by the National Numerical Windtunnel project (NNW2019ZT3-B07) and the National Natural Science Foundation of China (11674395).

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XH proposed the method for calculating the bound electronic energy level population of atoms and molecules with the help of TJ. CG collected the quantum data of atoms and molecules. XH performed the calculation and data analysis with the help of CG and TJ. XH and CG wrote the manuscript. TJ reviewed and processed the manuscript. All authors read and approved the final manuscript.

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Correspondence to Tao Jiang.

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He, X., Gao, C. & Jiang, T. A simplified method for calculating spectral emission of nonequilibrium air plasmas in hypersonic shock-layers. Adv. Aerodyn. 3, 15 (2021).

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