Modeling of a Reduced Hybrid H 2 –Air Kinetic Scheme Integrating the Effect of Hypersonic Reactive Air with Supersonic Combustion

: A hybrid H 2 –air kinetic scheme of 11 species and 15 reactions is developed, which is capable of simulating the high-temperature air reaction ﬂows and H 2 –O 2 combustion ﬂows respectively or simultaneously. Based on the Gupta scheme, the mole fraction varying with a Mach number at speciﬁc conditions is analyzed, and the weakly-ionized 7-species 7-reaction scheme is selected. The effect of nitrogenous species on the H 2 –O 2 combustion is analyzed by a zero-dimensional simulation of steady-state and unsteady-state combustion under speciﬁed conditions, and the selected dominant nitrogenous reaction N + OH = NO + H is distinguished by the production rate of the nitrogenous species. The thermodynamic properties are veriﬁed by comparison using the NIST–JANAF database. The reaction rate coefﬁcients of the dominant reaction of the hybrid kinetic scheme distinguished by a sensitivity analysis are corrected. The proposed kinetic scheme is validated by a zero-dimensional calculation of the ignition delay time and two-dimensional computational ﬂuid dynamics (CFD) simulation with ﬁnite-rate chemistry on the shock-induced sub-detonative and super-detonative combustion. The ignition delay time of the hybrid kinetic scheme is almost in the middle between the Shang scheme and Jachimowski scheme, and all the calculated ignition delay times are acceptably greater than the experiments due to the errors of the experiments and numerical models. The clearly captured bow shock wave and combustion front using the hybrid kinetic scheme and Shang scheme are almost the same, which is strongly consistent with the schlieren image. In addition, a good agreement of the ﬂow characteristics and mass fraction of the species along the stagnation line is also obtained, which indicates the accuracy and reasonableness of the hybrid kinetic scheme to simulate hybrid H 2 –air reactive ﬂows.


Introduction
Attention has widely been paid to the performance of air-breathing hypersonic vehicles and their propulsion systems in recent years [1][2][3][4]. Since increasing flight speed is one of the primary objectives, the accurate prediction of the crucial aeroheating environment where the dissociation and ionization of air species due to the effect of high temperature is necessary to be considered. In the combustor of the propulsion system, generally powered by a scramjet engine, the oxidant for the injected fuel is practically the mixture of the products or intermediates of high-temperature air chemical reactions rather than the hypothetical non-reactive N 2 -O 2 mixture in previous investigations. In this case, a hybrid kinetic scheme considering both high-temperature air reactions and combustion reactions is necessary to be modeled.
The flight test is significant for obtaining the pressure, temperature, and heat flux under real working conditions, while the flow characteristics and the composition of out in recent studies. Bussing [35] studied the chemical reaction effect on the leading edge and the combustion of the scramjet engine for an air-breathing hypersonic vehicle employing H 2 as the fuel, but the interaction between the two chemical reaction systems was not considered. Zhao [36] proposed a hybrid C 2 H 4 -air kinetic scheme where the thermodynamic database of air reactions and combustion reactions are coupled. The intereffects between the air reactions and combustion reactions were discussed based on the simulation of the hypersonic flow field with a sonic fuel jet. However, the accuracy of the mixed reaction model needs to be improved due to the fact that the reaction results and thermodynamic properties are not well-matched with references. A critical difference between air chemical reactions and combustion reactions is the temperature range of the thermodynamic properties, where the combustion species are always given ranging 300 K to 6000 K, while the maximum temperature is up to 15,000 K or even 30,000 K for air species, which makes the coupling much more difficult rather than simply combining the species and reaction equations.
Fuels with fast ignition properties are required within high-speed combustions, where H 2 and small hydrocarbons, such as C 2 H 4 , are favorable candidates. Larger hydrocarbons have an advantage from energy content and ease of storage; however, most investigations of supersonic combustions are simulated using H 2 , which is also selected in this work. In this paper, a hybrid H 2 -air kinetic scheme is modeled based on the widely accepted high-temperature air kinetic scheme of Gupta and the combustion kinetic scheme of Shang and Jachimowski. The mole fractions of the species at different Mach numbers are analyzed for air reactions. The steady-state and unsteady-state combustion under specific conditions is simulated to filter the trace species. The effect of the nitrogenous reactions on the H 2 -O 2 combustion is studied. The thermodynamic properties are verified according to the NIST-JANAF database, and then a sensitivity analysis is performed to correct the reaction rate coefficients of the dominant reaction for integrating the reaction systems. The hybrid kinetic scheme is validated by a zero-dimensional simulation of the combustion delay time compared with the kinetic scheme of Shang and Jachimowski as well as the experiment results. An analysis and validation are also carried out for the two-dimensional CFD simulation of the shock-induced sub-detonative and super-detonative combustion comparing the structure of flow field and distribution of species with the Shang scheme and experiment data.

Finite-Rate Chemistry Model
The chemical reactions described in this paper are based on the Arrhenius formula [12]. Considering a kinetic scheme of NS species and NR reactions, the general system of chemical reactions can be written as: where X i is the mole fraction of species or catalyst i, X i and β i,r are the stoichiometric number of species i in the rth reaction, k f ,r and k b,r are the forward and reverse reaction rates of the rth reaction. The net mass generation rate per unit volume of specie i is given as: .
where γ j is the mole fraction. Considering the three-body catalytic effect, the mole fraction is written as: where Z (j−NS),i is the three-body catalytic efficiency determined by corresponding chemical reactions. Forward chemical reaction rates are usually calculated by the Arrhenius empirical formula [37]: where A f ,r , n f ,r , and Ea f ,r are the pre-exponential factor, temperature index, and activation energy of the rth forward reaction, respectively. These three coefficients are generally obtained by theoretical calculation or experiment data. For the reverse chemical reaction rate, if the reaction rate coefficient cannot be obtained directly, it can be calculated by the chemical equilibrium constant from the forward reaction rate: where K eq,r is the chemical equilibrium constant of the rth reaction: where p 0 is the reference pressure, S is the molar entropy, H is the molar enthalpy, and the superscript 0 indicates the standard state. ∆ denotes the change after the end of reactions. The change of Gibbs free energy is given in the exponential form:

Sensitivity Analysis
Sensitivity analysis is applied in this paper to determine the dominant combustion reaction to be corrected. The generation rate g(y, A) of an arbitrary parameter y, for instance, the temperature or mole fraction of species, are defined as: g(y, A) = ∂y ∂t (8) where the t denotes time, and A is a vector of the pre-exponential factors. The matrix of sensitivity coefficient vector E is given as: The time derivative of the matrix of sensitivity coefficient is given as: where the J = ∂g/∂y is the Jacobian matrix of equation. The effect of the pre-exponential factor on the reaction rate is greater when the value of the sensitivity coefficient is larger.

Computation Fluid Dynamics Method
The governing equations of the axisymmetric reactive flow in this study are the multi-species Euler equations where the viscosity is neglected, which is given as: where the conservation term W, inviscid flux F c , axisymmetric inviscid source A c , and the source term Q are given as: where the ρ is the density, p is the pressure, E and H are the total energy and total enthalpy of unit mass, u and v are the components of velocity on the axial and radial direction, and . ω indicates the production rate of species. The ideal gas equation of state of the mixture is employed, which is given as: where the p i is partial pressure of species i, R i is the corresponding gas constant, and T is the temperature. The specific heat capacity at constant pressure, specific enthalpy, and specific entropy of species i are described by piecewise polynomials given as: where a 1i~a7i are the fitting constants. The multi-species lattice Boltzmann flux solver (LBFS) developed by Yang [38][39][40] is employed for spatial discretization, which shows great ability in shock wave capturing and numerical stability. An explicit five-step Runge-Kutta iteration is utilized for the temporal discretization. The CFD simulation with finite-rate chemistry performed in-house is validated in previous studies for hypersonic reactive and non-reactive flows [38][39][40][41][42], which is not detailly mentioned in this paper.

Hybrid H 2 -Air Kinetic Scheme Modelling
The main difficulties to couple the chemical reactions of high-temperature air and H 2 -O 2 combustion are classified as follows: (c) The effect of nitrogenous species and reactions on combustion reactions is necessary to be analyzed and included. Meanwhile, the reaction rate coefficients of sensitive reactions need to be corrected for the hybrid kinetic scheme.
The process of modeling for the hybrid H 2 -air kinetic scheme is illustrated in Figure 1. Firstly, the species and reactions are analyzed and selected. Then, the thermodynamic properties are verified, and the reaction rate coefficients of sensitive reactions are corrected. Finally, the hybrid kinetic scheme is validated by numerical simulations.
Energies 2021, 14, x FOR PEER REVIEW 6 of 28 reactions need to be corrected for the hybrid kinetic scheme.
The process of modeling for the hybrid H2-air kinetic scheme is illustrated in Figure  1. Firstly, the species and reactions are analyzed and selected. Then, the thermodynamic properties are verified, and the reaction rate coefficients of sensitive reactions are corrected. Finally, the hybrid kinetic scheme is validated by numerical simulations.

Kinetic Scheme of High-Temperature Air
The high-temperature air is assumed to be the mixture of N2 and O2, while other species in the air are neglected in this paper. The kinetic schemes of Park [9], Dunn & Kang [8], and Gupta [12] are listed in Table 1, where the same reactions are merged regardless of different reaction rate coefficients, catalysts, and catalyst coefficients.

Kinetic Scheme of High-Temperature Air
The high-temperature air is assumed to be the mixture of N 2 and O 2 , while other species in the air are neglected in this paper. The kinetic schemes of Park [9], Dunn & Kang [8], and Gupta [12] are listed in Table 1 Zero-dimensional simulations are performed for the N 2 -O 2 mixture with initial mole fractions of 0.79 and 0.21 under the constant pressure of 1197 pa and constant temperature of 226.5 K, which is consistent with the conditions at an altitude of 30 km [27]. The mole fractions of the species behind a normal shock wave varying with a Mach number from 1 to 20 are shown in Figure 2. The dominant species are shown in Figure 2a with a linear axis. The dissociation reaction of N 2 and O 2 initiates at approximately Mach 7, where the NO and O are produced, while the production of species N starts at about Mach 9. The peak mole fractions of species NO, O, and N occur at about Mach 9, 10, and 15, respectively. As the Mach number further increases, the decrement of the mole fraction is due to the dilution by rapidly produced N and O for the species N, and consumption to ionization reactions for the species O and N. Non-ionized species O 2 , N 2 , O, N, and NO mainly exist when the Mach number is lower than 16, while ionized species O + and N + are considerably produced as the Mach number increases. The mole fraction is also given with the logarithmic axis in Figure 2b. When the Mach number is greater than 12, the ionized species N + , O + , N 2 + , and O 2 + are significantly produced, while the mole fraction of N 2 + and O 2 + is still small (10 −4 order). It is important to mention that the temperature is practically lower in the real flight environment than the simulation under the same Mach number where the mole fraction of ions is smaller, which is caused by the non-negligible heat absorption effect of chemical reactions.

No. Reaction Equation Park Dunn & Kang Gupta
Dissociation reactions Zero-dimensional simulations are performed for the N2-O2 mixture with initial mole fractions of 0.79 and 0.21 under the constant pressure of 1197 pa and constant temperature of 226.5 K, which is consistent with the conditions at an altitude of 30 km [27]. The mole fractions of the species behind a normal shock wave varying with a Mach number from 1 to 20 are shown in Figure 2. The dominant species are shown in Figure 2a with a linear axis. The dissociation reaction of N2 and O2 initiates at approximately Mach 7, where the NO and O are produced, while the production of species N starts at about Mach 9. The peak mole fractions of species NO, O, and N occur at about Mach 9, 10, and 15, respectively. As the Mach number further increases, the decrement of the mole fraction is due to the dilution by rapidly produced N and O for the species N, and consumption to ionization reactions for the species O and N. Non-ionized species O2, N2, O, N, and NO mainly exist when the Mach number is lower than 16, while ionized species O + and N + are considerably produced as the Mach number increases. The mole fraction is also given with the logarithmic axis in Figure 2b. When the Mach number is greater than 12, the ionized species N + , O + , N2 + , and O2 + are significantly produced, while the mole fraction of N2 + and O2 + is still small (10 −4 order). It is important to mention that the temperature is practically lower in the real flight environment than the simulation under the same Mach number where the mole fraction of ions is smaller, which is caused by the non-negligible heat absorption effect of chemical reactions.
(a) Since the single-temperature model is adopted where the highest temperature does not exceed approximately 10,000 K, in addition, the practical flight altitude is higher than the assumed 30 km where the effect of aeroheating is weaker than the simulation, and, furthermore, the high-temperature region is very small compared with the size of the airvehicle, and the hypersonic reactive flow field is considered as weak-ionized air in this study, where the species O2, N2, O, N, NO, NO + , and e − are included in the hybrid kinetic scheme according to the reduced kinetic scheme of Gupta consisting of 7 species and 7 reactions as listed in Table 2.

Kinetic Scheme of Combustion
In this paper, H2 is employed as the fuel of the hybrid kinetic scheme. The H2-O2 reactions are analyzed first. The kinetic schemes of Shang [22] with 7 species and 7 reactions, Kee [24] with 11 species and 23 reactions, and Jachimowski with 13 species and 33 reactions were previously validated to perform well under different conditions. Evans [42] found that, although a little bit better performance on the trend of the ignition delay time is obtained considering more species, the results are almost the same employing the kinetic scheme only including basic species with a more detailed kinetic scheme. Balancing the accuracy and computational cost, the Shang scheme is employed, which shows great agreement with the experiments [8] and is validated within the hybrid kinetic scheme in the next chapter.
Nitrogenous species are produced in high-temperature air reactions, which impact the combustion reactions and further influence the performance of scramjet engines. Slack [34] found that adding 5% NO can shorten the ignition delay time of H2-O2 mixtures with a chemical equivalent ratio. Therefore, the effects of the nitrogenous species and reactions Since the single-temperature model is adopted where the highest temperature does not exceed approximately 10,000 K, in addition, the practical flight altitude is higher than the assumed 30 km where the effect of aeroheating is weaker than the simulation, and, furthermore, the high-temperature region is very small compared with the size of the air-vehicle, and the hypersonic reactive flow field is considered as weak-ionized air in this study, where the species O 2 , N 2 , O, N, NO, NO + , and e − are included in the hybrid kinetic scheme according to the reduced kinetic scheme of Gupta consisting of 7 species and 7 reactions as listed in Table 2.

Kinetic Scheme of Combustion
In this paper, H 2 is employed as the fuel of the hybrid kinetic scheme. The H 2 -O 2 reactions are analyzed first. The kinetic schemes of Shang [22] with 7 species and 7 reactions, Kee [24] with 11 species and 23 reactions, and Jachimowski with 13 species and 33 reactions were previously validated to perform well under different conditions. Evans [42] found that, although a little bit better performance on the trend of the ignition delay time is obtained considering more species, the results are almost the same employing the kinetic scheme only including basic species with a more detailed kinetic scheme. Balancing the accuracy and computational cost, the Shang scheme is employed, which shows great agreement with the experiments [8] and is validated within the hybrid kinetic scheme in the next chapter.
Nitrogenous species are produced in high-temperature air reactions, which impact the combustion reactions and further influence the performance of scramjet engines. Slack [34] found that adding 5% NO can shorten the ignition delay time of H 2 -O 2 mixtures with a chemical equivalent ratio. Therefore, the effects of the nitrogenous species and reac-  Table 3 are significantly considered based on the kinetic scheme of Jachimowski [28]. The temperature of the combustion flow in the combustor under working conditions is 1000 K~2500 K, while the temperature of the ambient of fuel jet is locally up to 4000 K~6000 K. Therefore, the temperature range of 1000 K~6000 K is mainly focused for the combustion species. The steady-state combustion with a chemical equivalent ratio of H 2 -air mixture under the constant pressure of 1 atm and 2 atm varying with the temperature is investigated, and the mole fractions of the products are shown in Figure 3. The mole fractions of H 2 O 2 , HO 2 , HNO, and NO 2 are less than 10 −5 , which is neglected in the hybrid kinetic scheme. The mole fractions of NO and N are relatively higher, especially when the temperature is higher than 3000 K, which indicates that the effect of the nitrogenous products performs significantly on the combustion and coupled with the influence of nitrogenous species produced in high-temperature air reactions.
To ensure the reasonableness of the neglection of the trace species in the steady-state combustion, which might be the active species to influence the combustion, unsteady-state combustion under the specific temperature of 1000 K and 2000 K is also performed, where the evolution of the mole fraction of species is shown in Figure 4. The mole fraction of H 2 O 2 and HO 2 is less than 10 −4 , and the mole fraction of HNO and NO 2 is less than 10 −8 , which is consistent with the result of the steady-state combustion, where the effect on the ignition and combustion is negligible. In addition, the increment of trace species is after the ignition illustrated by the peak mole fraction of OH, which further indicates the reasonableness of the neglection.
Energies 2021, 14, x FOR PEER REVIEW 9 of 28 listed in Table 3 are significantly considered based on the kinetic scheme of Jachimowski [28]. The temperature of the combustion flow in the combustor under working conditions is 1000 K~2500 K, while the temperature of the ambient of fuel jet is locally up to 4000 K~6000 K. Therefore, the temperature range of 1000 K~6000 K is mainly focused for the combustion species. The steady-state combustion with a chemical equivalent ratio of H2air mixture under the constant pressure of 1 atm and 2 atm varying with the temperature is investigated, and the mole fractions of the products are shown in Figure 3. The mole fractions of H2O2, HO2, HNO, and NO2 are less than 10 −5 , which is neglected in the hybrid kinetic scheme. The mole fractions of NO and N are relatively higher, especially when the temperature is higher than 3000 K, which indicates that the effect of the nitrogenous products performs significantly on the combustion and coupled with the influence of nitrogenous species produced in high-temperature air reactions.
(a) To ensure the reasonableness of the neglection of the trace species in the steady-state combustion, which might be the active species to influence the combustion, unsteadystate combustion under the specific temperature of 1000 K and 2000 K is also performed, where the evolution of the mole fraction of species is shown in Figure 4. The mole fraction of H2O2 and HO2 is less than 10 −4 , and the mole fraction of HNO and NO2 is less than 10 −8 , which is consistent with the result of the steady-state combustion, where the effect on the ignition and combustion is negligible. In addition, the increment of trace species is after the ignition illustrated by the peak mole fraction of OH, which further indicates the reasonableness of the neglection.   The analysis of the chemical reaction rates of the steady-state combustion of the H 2 -air mixture is analyzed to distinguish the dominant reactions for the main products N 2 , N, and NO, where the pressure is 1 atm and temperature is 2000 K. The results show that the dominant nitrogenous reactions are N + O 2 = NO + O, N + NO = N 2 + O, and N + OH = NO + H, namely thermal NO reactions, which are consistent with the reference [37]. The productivities of these species are shown in Figure 5. The N is not mainly produced by the decomposition reaction N + N + M = N 2 + M but the exchange reaction N + OH = NO + H, which is included in the hybrid kinetic scheme.  Table 4.   Table 4.

Thermodynamic Properties
The thermodynamic properties proposed by Gupta [12], where the temperature ranges from 300 K to 15,000 K, are employed for the high-temperature air species (O 2 , N 2 , O, N, NO, NO + , e − ) of the hybrid kinetic scheme, and the thermodynamic properties published by Esch [43] are utilized for the combustion species (H 2 , H, OH, H 2 O). The verification is carried out by comparing with the data of the NIST-JANAF database [44], where the specific heat at a constant pressure and the specific enthalpy consist well with the reference as shown in Figures 6 and 7

Thermodynamic Properties
The thermodynamic properties proposed by Gupta [12], where the temperature ranges from 300 K to 15,000 K, are employed for the high-temperature air species (O2, N2, O, N, NO, NO + , e − ) of the hybrid kinetic scheme, and the thermodynamic properties published by Esch [43] are utilized for the combustion species (H2, H, OH, H2O). The verification is carried out by comparing with the data of the NIST-JANAF database [44], where the specific heat at a constant pressure and the specific enthalpy consist well with the reference as shown in Figure 6 and Figure 7, respectively.

Sensitivity Analysis and Reaction Rate Correction
The reaction rate of the high-temperature air reaction and combustion reaction with the same reaction equation is not the same, which makes the combination of the two kinetic schemes puzzling. In this study, based on the high-temperature air reactions, a sensitivity analysis is carried out for the combustion reactions, and the reaction rate coefficients of the most sensitive reaction are corrected to match the physical results for further validation.
The sensitivity coefficient is calculated for specific temperatures (1000 K, 1500 K, 2000 K), pressures (1 atm, 2 atm, 4 atm), and equivalence ratios (0.5, 1, 2). The reactions with the first eight greatest sensitivity coefficients are shown in Figure 8 [28,37]. The production of free radicals increases as the initial temperature decreases or the initial pressure and the equivalence ratio increase.

Sensitivity Analysis and Reaction Rate Correction
The reaction rate of the high-temperature air reaction and combustion reaction the same reaction equation is not the same, which makes the combination of the tw netic schemes puzzling. In this study, based on the high-temperature air reactions, a sitivity analysis is carried out for the combustion reactions, and the reaction rate c cients of the most sensitive reaction are corrected to match the physical results for fu validation.
The sensitivity coefficient is calculated for specific temperatures (1000 K, 1500 K, K), pressures (1 atm, 2 atm, 4 atm), and equivalence ratios (0.5, 1, 2). The reactions the first eight greatest sensitivity coefficients are shown in Figure 8 successively, w the reactions with the positive sensitivity coefficient indicate the enhancement of the bustion. The same reaction equations with a consistent sequence are obtained. The c termination reaction H + OH + M = H2O + M and chain-amplification reactions H + O + OH and O + H2 = OH + H are illustrated to be the dominant reactions, which is consi with the regulation of the chain reactions [28,37]. The production of free radicals incr as the initial temperature decreases or the initial pressure and the equivalence rat crease. The reaction rate coefficients of the most sensitive reaction H + O2 = O + OH fr Shang scheme are selected to be corrected, where the coefficients of the kinetic sch Miller [44] from the NIST-JANAF database take place, which are listed in Table  validated in the followed-up cases.

Ignition and Combustion Characteristics
A zero-dimensional simulation is performed to calculate the ignition delay time is defined as the length of time from the initial condition to the maximum temperatur ent, where the initial temperature of 1500 K and pressure of 0.5 atm and 1 atm are sim The proposed hybrid H2-air kinetic scheme is validated by comparing with the Shang and Jachimowski scheme, as well as the experiment [28], as shown in Figure 9. The ignition time is presented as the temperature and pressure increase. The results of the kinetic scheme are between the Shang scheme and Jachimowski scheme, and the igni lay time of the numerical results is greater than the experimental data to a certain which is due to the errors caused by the reduction process of these reduced kinetic sc and, meanwhile, the experimental errors cannot be neglected. Even so, the error of the i delay time for the steady-state combustion is acceptable, and a good agreement is pre The hybrid kinetic scheme is also employed for the steady-state combustion c tion at an initial temperature of 1500 K and pressure of 1 atm to validate the reliab the combustion characteristics. The mole fraction of the species and steady-state t ature varying with the equivalent ratio are shown in Figure 10. The results show agreement between these three kinetic schemes. Despite the reaction rate coeffic the nitrogenous reactions of the Gupta scheme and the Jachimowski scheme listed ble 6 being unequal, similar results are obtained due to the mole fraction of nitro The reaction rate coefficients of the most sensitive reaction H + O 2 = O + OH from the Shang scheme are selected to be corrected, where the coefficients of the kinetic scheme of Miller [44] from the NIST-JANAF database take place, which are listed in Table 5 and validated in the followed-up cases.

Ignition and Combustion Characteristics
A zero-dimensional simulation is performed to calculate the ignition delay time, which is defined as the length of time from the initial condition to the maximum temperature gradient, where the initial temperature of 1500 K and pressure of 0.5 atm and 1 atm are simulated. The proposed hybrid H 2 -air kinetic scheme is validated by comparing with the Shang scheme and Jachimowski scheme, as well as the experiment [28], as shown in Figure 9. The shorter ignition time is presented as the temperature and pressure increase. The results of the hybrid kinetic scheme are between the Shang scheme and Jachimowski scheme, and the ignition delay time of the numerical results is greater than the experimental data to a certain degree, which is due to the errors caused by the reduction process of these reduced kinetic schemes, and, meanwhile, the experimental errors cannot be neglected. Even so, the error of the ignition delay time for the steady-state combustion is acceptable, and a good agreement is presented.  The hybrid kinetic scheme is also employed for the steady-state combustion calculation at an initial temperature of 1500 K and pressure of 1 atm to validate the reliability of the combustion characteristics. The mole fraction of the species and steady-state temperature varying with the equivalent ratio are shown in Figure 10. The results show great agreement between these three kinetic schemes. Despite the reaction rate coefficients of the nitrogenous reactions of the Gupta scheme and the Jachimowski scheme listed in Table 6 being unequal, similar results are obtained due to the mole fraction of nitrogenous species being relatively lower, which indicates the reasonableness and reliability of the hybrid H 2 -air kinetic scheme.

Shock-Induced Sub-Detonative Combustion
The simulation of the shock-induced sub-detonative combustion is performed for a semi-sphere with a cylinder body, where the diameter is 7.5 mm. The number of nodes along the wall and in the direction perpendicular to the wall of the structured mesh is 100 and 70, and the cells are clustered at the bow shock wave as shown in Figure 11. The axisymmetric multi-species Euler equations are solved. The adiabatic, non-catalytic, no-slip condition is applied to the wall. The non-reflection condition is applied to the far-field. The detailed parameters of the mainstream are listed in Table 7, where the Mach number, pressure, temperature, and mole fractions of the species are given. along the wall and in the direction perpendicular to the wall of the structured mesh is 100 and 70, and the cells are clustered at the bow shock wave as shown in Figure 11. The axisymmetric multi-species Euler equations are solved. The adiabatic, non-catalytic, noslip condition is applied to the wall. The non-reflection condition is applied to the farfield. The detailed parameters of the mainstream are listed in Table 7, where the Mach number, pressure, temperature, and mole fractions of the species are given. Figure 11. Structured mesh for shock-induced sub-detonative combustion calculation. The temperature distribution of the present scheme is given compared with the schlieren image of the experiment [45] as well as the result of the Shang scheme as shown in Figure 12. The non-coincident bow shock wave and combustion front are clearly captured, which, due to the velocity of the H2-O2 detonation (2550 m/s), are greater than the velocity of the mainstream (1892 m/s). Great agreement of the bow shock wave and combustion front is presented by the two numerical results as well as the schlieren image, except for the combustion front of the experiment being slightly expanded close to the outlet of the computational domain.
The distributions of the pressure, temperature, density, and mass fraction of the species along the stagnation line are also compared with the result of the Shang scheme and published reference [46] as shown in Figures 13 and 14. The location of the bow shock wave and combustion front obtained by the present scheme is slightly closer to the wall, which is because the relatively slower reaction rates result in a longer induction time of the combustion. Despite the small differences between the results, the trend and maximum values of these parameters consist well, which indicates the accuracy and reasonableness of the proposed hybrid kinetic scheme.  The temperature distribution of the present scheme is given compared with the schlieren image of the experiment [45] as well as the result of the Shang scheme as shown in Figure 12. The non-coincident bow shock wave and combustion front are clearly captured, which, due to the velocity of the H 2 -O 2 detonation (2550 m/s), are greater than the velocity of the mainstream (1892 m/s). Great agreement of the bow shock wave and combustion front is presented by the two numerical results as well as the schlieren image, except for the combustion front of the experiment being slightly expanded close to the outlet of the computational domain.  The distributions of the pressure, temperature, density, and mass fraction of the species along the stagnation line are also compared with the result of the Shang scheme and published reference [46] as shown in Figures 13 and 14. The location of the bow shock wave and combustion front obtained by the present scheme is slightly closer to the wall, which is because the relatively slower reaction rates result in a longer induction time of the combustion. Despite the small differences between the results, the trend and maximum values of these parameters consist well, which indicates the accuracy and reasonableness of the proposed hybrid kinetic scheme.

Shock-Induced Super-Detonative Combustion
The shock-induced super-detonative combustion is also simulated for validation. The structured computational mesh with 150 nodes along the wall and 150 nodes in the normal direction of the wall and cells are clustered at the bow shock wave and combustion front as shown in Figure 15. The same numerical method is employed as the previous case. Detailed parameters of the boundaries are given in Table 8. A stoichiometric hydrogen-air mixture with X(N 2 ):X(O 2 ):X(H 2 ) = 3.76:1:2 is applied [27].

Shock-Induced Super-Detonative Combustion
The shock-induced super-detonative combustion is also simulated for validation. The structured computational mesh with 150 nodes along the wall and 150 nodes in the normal direction of the wall and cells are clustered at the bow shock wave and combustion front as shown in Figure 15. The same numerical method is employed as the previous case. Detailed parameters of the boundaries are given in Table 8. A stoichiometric hydrogen-air mixture with X(N2):X(O2):X(H2) = 3.76:1:2 is applied [27]. The temperature distribution of the present scheme and Shang scheme as well as the schlieren image is shown in Figure 16. A typical flow field structure is obtained, and the bow shock wave and combustion front are greatly captured. The bow shock wave and combustion front are coincident in front of the stagnation point of the configuration and then progressively separated as they flow downstream. Since the velocity of the mainstream (2605 m/s) is greater than the detonation velocity (2055 m/s), the strength of the shock wave is sufficiently strong for the ignition, which makes the location of the bow shock wave consistent with the combustion front at the stagnation region. The separation of the bow shock wave and the combustion front is due to the decay of the shock wave, which is insufficient for the ignition, where the induction time of the combustion reactions is lengthened. The results show that the temperature distribution of the numerical study is almost the same, which fits perfectly with the schlieren image of the experiment [45].

Shock-Induced Super-Detonative Combustion
The shock-induced super-detonative combustion is also simulated for validation. The structured computational mesh with 150 nodes along the wall and 150 nodes in the normal direction of the wall and cells are clustered at the bow shock wave and combustion front as shown in Figure 15. The same numerical method is employed as the previous case. Detailed parameters of the boundaries are given in Table 8. A stoichiometric hydrogen-air mixture with X(N2):X(O2):X(H2) = 3.76:1:2 is applied [27].    The temperature distribution of the present scheme and Shang scheme as well as the schlieren image is shown in Figure 16. A typical flow field structure is obtained, and the bow shock wave and combustion front are greatly captured. The bow shock wave and combustion front are coincident in front of the stagnation point of the configuration and then progressively separated as they flow downstream. Since the velocity of the mainstream (2605 m/s) is greater than the detonation velocity (2055 m/s), the strength of the shock wave is sufficiently strong for the ignition, which makes the location of the bow shock wave consistent with the combustion front at the stagnation region. The separation of the bow shock wave and the combustion front is due to the decay of the shock wave, which is insufficient for the ignition, where the induction time of the combustion reactions is lengthened. The results show that the temperature distribution of the numerical study is almost the same, which fits perfectly with the schlieren image of the experiment [45].
The distributions of the pressure, temperature, density, and mass fraction of the species along the stagnation line are also compared with the result of the Shang scheme and published reference [46] as shown in Figure 17 and Figure 18, respectively. A pressure peak and a density peak, namely the Von Neumann peaks, are presented behind the combustion front due to the massive heat-releasing caused by the violent combustion with fast combustion reactions, which makes the remaining energy insufficient to maintain the strength of the combustion near the wall, where the pressure and density decrease. Perfect agreement is obtained between the results, which also indicates the accuracy and reasonableness of the proposed hybrid kinetic scheme.
(a) The distributions of the pressure, temperature, density, and mass fraction of the species along the stagnation line are also compared with the result of the Shang scheme and published reference [46] as shown in Figures 17 and 18, respectively. A pressure peak and a density peak, namely the Von Neumann peaks, are presented behind the combustion front due to the massive heat-releasing caused by the violent combustion with fast combustion reactions, which makes the remaining energy insufficient to maintain the strength of the combustion near the wall, where the pressure and density decrease. Perfect agreement is obtained between the results, which also indicates the accuracy and reasonableness of the proposed hybrid kinetic scheme.

Conclusions
A hybrid H2-air kinetic scheme of 11 species and 15 reactions was developed, which is capable of simulating the high-temperature air reaction flows and H2-O2 combustion flows respectively or simultaneously. The species and reactions of high-temperature air were selected by analyzing the mole fractions of the species varying with the Mach

Conclusions
A hybrid H2-air kinetic scheme of 11 species and 15 reactions was developed, which is capable of simulating the high-temperature air reaction flows and H2-O2 combustion flows respectively or simultaneously. The species and reactions of high-temperature air were selected by analyzing the mole fractions of the species varying with the Mach

Conclusions
A hybrid H 2 -air kinetic scheme of 11 species and 15 reactions was developed, which is capable of simulating the high-temperature air reaction flows and H 2 -O 2 combustion flows respectively or simultaneously. The species and reactions of high-temperature air were selected by analyzing the mole fractions of the species varying with the Mach number at specific conditions. The nitrogenous species of the combustion reactions were screened by the simulations of the steady-state and unsteady-state combustion and the dominant nitrogenous reaction with non-negligible reaction rates being selected. The thermodynamic properties were verified by comparison with the NIST-JANAF database. The reaction rate coefficients of the most sensitive reaction distinguished by the sensitivity analysis were corrected. A zero-dimensional simulation was performed to validate the ignition delay time and combustion characteristics, and two-dimensional CFD simulations were carried out to validate the performance of the shock-induced sub-detonative combustion and super-detonative combustion. The conclusions are as follows: (1) The high-temperature air reactions are initiated as the Mach number is greater than 7.
The peak mole fraction of NO, O, and N occurs when the Mach number is about 9, 10, and 15, respectively, and then decreases as the Mach number increases. When the Mach number is greater than 16, the species N + and O + are produced considerably. Considering the working condition and numerical cost, the weakly-ionized 7-species 7-reactions kinetic scheme of Gupta is selected to be the high-temperature air reactions of the hybrid kinetic scheme. (2) The species H 2 O 2 , HO 2 , HNO, and NO 2 of the Jachimowski scheme are trace species in the steady-state and unsteady-state combustion, which are neglected in the hybrid kinetic scheme, while the N and NO are mainly considered. The reaction N + OH = NO + H with the greatest production rate of the nitrogenous species is selected to evaluate the impact on H 2 -O 2 combustion. (3) The specific heat at constant pressure and the specific enthalpy employed in the hybrid kinetic scheme is verified. A strong agreement is obtained compared with the NIST-JANAF database. (4) The ignition delay time decreases as the initial temperature and pressure increase, where the results calculated by the hybrid kinetic scheme are between the Shang scheme and the Jachimowski scheme. Despite the ignition delay time of the experiment being slightly shorter than the numerical simulations, the acceptable error indicates the reasonableness of the hybrid kinetic scheme. (5) Almost the same results are obtained by the hybrid kinetic scheme and Shang scheme from the simulation of the shock wave induced sub-detonative combustion and superdetonative combustion. The separated bow shock wave and combustion front of the sub-detonation and the progressively separating bow shock wave and combustion front of the super-detonation are well captured, which consists well with the schlieren image. The distribution of the flow characteristics and mass fraction of species also match well with the reference, which indicates the accuracy and reasonableness of the hybrid kinetic scheme. Funding: This research and APC was funded by National Numerical Wind-tunnel project of China, grant number NNW2019ZT7-B30 and NNW2018-ZT3B08.

Data Availability Statement:
The data that support the findings of this study are available from the corresponding author upon reasonable request.