Introduction

An important problem in physical and chemical gas dynamics is the study of non-equilibrium kinetics in air flows behind shock waves. The impact of non-equilibrium kinetic processes on distributions of flow quantities is required, for example to correctly predict stand-off distance of shock wave in super-and hypersonic flows when spacecrafts enter the Earth's atmosphere.

Air is considered as a five-component mixture consisting of 𝑁 2 , 𝑂 2 , 𝑁𝑂, 𝑁, 𝑂 in which the various kinetic processes occur. These processes include chemical reactions of dissociation and recombination and chemical exchange reactions. They are as follows:

• Dissociation and recombination 𝑁 2 + 𝑀 ⇄ 𝑁 + 𝑁 + 𝑀, 𝑂 2 + 𝑀 ⇄ 𝑂 + 𝑂 + 𝑀, 𝑁𝑂 + 𝑀 ⇄ 𝑁 + 𝑂 + 𝑀;

(1) • Chemical exchange reactions 𝑁 2 + 𝑂 ⇄ 𝑁𝑂 + 𝑁, 𝑂 2 + 𝑁 ⇄ 𝑁𝑂 + 𝑂.

(2) Here, 𝑀 = 𝑁 2 , 𝑂 2 , 𝑁𝑂, 𝑁, 𝑂.

There are various approaches to the description of non-equilibrium flows including onetemperature approach [1], multi-temperature approach [2,3], and multi-level approach [4]. The most detailed description of non-equilibrium kinetics is given by the multi-level approach. However, the computational costs of this approach are extremely high due to the need to solve a large number of equations [5]. Therefore, the study uses a more simple from the computational point of view onetemperature approximation, which requires less time for calculations and is often used to solve applied problems of hypersonic flows [6,7].

The sudy does not take into account the processes of ionization and electronic excitation, since the temperature ranges at which the influence of these processes on the flow parameters is weaker than the influence of vibrational excitation and chemical reactions is considered. For the populations of the vibrational levels of air components, assuming that the distribution of molecules over the vibrational levels is quasi-stationary, and corresponds to the Boltzmann distribution, the flowing relationships are written This study focuses on numerical simulation of non-equilibrium flows of a reacting fivecomponent air mixture behind a shock wave at different heights from the earth's surface and at different velocities of the incoming flow. To assess the applicability of equilibrium thermodynamic models for calculating macro-parameters behind a shock wave, the lengths of relaxation zones are considered for various initial conditions.

The vibrational energy 𝜀

Mathematical model

Within the framework of the one-temperature approach, the determining macro-parameters of the flow are the numerical densities of molecules and atoms 𝑛 𝑁 2 , 𝑛 𝑂 2 , 𝑛 𝑁𝑂 , 𝑛 𝑁 , 𝑛 𝑂 , the gas temperature, T, and the macroscopic velocity, v.

Governing equations

The system of governing equations for macro-parameters contains the equations of onetemperature non-equilibrium chemical kinetics, the equations of conservation of momentum and total energy. In the case of a stationary one-dimensional flow of an inviscid and non-heat-conducting mixture, these equations have the form [7] 𝑣

𝑑𝑛 𝑀 𝑑𝑥 + 𝑛 𝑀 𝑑𝑣 𝑑𝑥 = 𝑅 𝑀 2↔2 + 𝑅 𝑀 2↔3 ;(3)𝜌𝑣 𝑑𝑣 𝑑𝑥 + 𝑑𝑝 𝑑𝑥 = 0;(4)𝑣 𝑑𝐸 𝑑-+ (𝑝 + 𝐸) 𝑑𝑣 𝑑𝑥 = 0.(5)

Here, 𝜌 = ∑ 𝑛 𝑀 𝑚 𝑀 𝑀 is density of the mixture, 𝑝 = 𝑛𝑘𝑇 is pressure, 𝑛 = ∑ 𝑛 𝑀 𝑀 is numerical density of the mixture, 𝐸 is total energy of a unit of volume. The total energy is represented as 𝐸 = 𝐸 𝑡𝑟 + 𝐸 𝑟𝑜𝑡 + 𝐸 𝑣𝑖𝑏𝑟 + 𝐸 𝑓 , where 𝐸 〱𝑟 , 𝐸 𝑟𝑜𝑡 , 𝐸 𝑣𝑖𝑏𝑟 , 𝐸 𝑓 are translational, rotational, vibrational energy and the energy of formation of particles of the mixture per unit volume.

Initial conditions

The one-temperature approach assumes that only the chemical composition is preserved at the shock wave front, so the relation on the shock wave are valid. The equations take the form 𝜌𝑣 = 𝜌 (0) 𝑣 (0) ; ( 6)

𝜌𝑣 2 + 𝑝 = 𝜌 (0) (𝑣 (0) ) 2 + 𝑝 (0) ; (7) 𝐸+𝑝 𝜌 + 𝑣 2 2 = 𝐸 (0) +𝑝 (0) 𝜌 (0) + (𝑣 (0) ) 2 2 . (8)

The superscript 0 indicates the parameters before the shock front.

Relaxation terms

To close the system (3)-( 5) it is necessary to express the relaxation terms in terms of macroparameters, 𝑅 𝑀 2↔2 and 𝑅 𝑀 2↔3 . The terms 𝑅 𝑀 2↔2 describe changes in the numerical densities of molecules due to exchange reactions (2) and have the following form [7]: To calculate the one-temperature coefficients of the rate of direct exchange and dissociation reactions, the Arrhenius law was used, which is valid in a thermally equilibrium gas 𝑘 𝑐,𝑒𝑞 𝑀 = 𝐴 𝑀 𝑇 𝑛 exp (− 𝐸 𝛼 𝑘𝑇 ), where 𝐸 𝛼 (𝐷 𝑐 ) is activation energy in the case of exchange reactions and dissociation energy in the case of decay reactions, 𝐴 𝑀 and 𝑛 are constant factors that are determined in a number of studies based on experimental data or detailed numerical calculations. The dissociation rate coefficients recommended in [8] are used.

𝑅 𝑁 2 2↔2 =

The coefficients of the rate of recombination and reverse exchange reactions are calculated using the relations following from the principle of detailed balance [7]

Results and discussion

The results are obtained using numerical methods. At the beginning, a system of equations ( 6)−( 8), is solved to find the gas-dynamic parameters of the mixture behind the shock wave front. Further, the obtained values are used as initial data for solving the system of equations ( 3)−( 5) using the implicit Geer method [9].

To determine the parameters at different altitudes of flight, data from standard atmosphere [10] are applied. They set the average numerical values of the main atmospheric parameters for altitudes from -2 000 m to 1 200 000 m or latitude 45 ∘ 32′33′′, corresponding to the average level of solar activity.

The considered flight speeds vary from M=10 to M=16. The upper limit is due to the fact that at Mach numbers above M=16 the temperature and pressure immediately behind the shock layer have values at which the ionization processes begin to significantly affect the flow [11].

Figures 1−7 show the dependences of temperature and velocity on the distance behind the shock wave front, respectively, at different flight altitudes. Fragments a correspond to altitude of h=0 km, fragments b correspond to altitude of h=10 km, fragments c correspond to altitude of h=24 km, and fragments d correspond to altitude of h=60 km.

It can be seen from Figures 1 and 2 that with an increase in the Mach number of the incoming flow, the changes in temperature and velocity become much more significant and pass much faster. As the altitude increases, the temperature and velocity values behind the shock wave decrease. When moving away from the wave front, the numerical densities of the molecules 𝑁 2 and 𝑂 2 decrease, as can be seen in Figures 3 and 4. With an increase in the Mach number, the concentration of components in the air mixture 𝑁 2 and 𝑂 2 falls behind the shock wave, the sharpness of the concentration change increases. At an altitude of 10 km (Figure 3b and 4b), the concentration of components increases, but already at an altitude of 24 km (Figures 3c and 4c), the concentration of components decreases and continues to decrease with increasing altitude. The study has shown that when moving away from the shock wave front, the concentrations of the components of the air mixture 𝑁 2 and 𝑂 2 decrease, and the concentrations of 𝑁, 𝑂 and 𝑁𝑂 increase. As the velocity of the incoming flow increases, the temperature and velocity behind the shock wave increase, the concentrations of 𝑁 2 and 𝑂 2 molecules decrease, and the concentrations of 𝑁, 𝑂 and 𝑁𝑂 increase. At low pressures, there is a non-monotonic change in the numerical density of nitrogen oxide 𝑁𝑂 molecules.

A comparison is made of the lengths of the zones of variation of the flow parameters behind the shock waves. Studies have shown that the length of the zone of change in the concentration of the mixture components can exceed the length of the zone of change in the flow macro-parameters behind the shock wave. In cases where there is no need to obtain the results of changes in the concentration of mixture components, the zone of numerical calculations of the flow can be reduced to the length of the zone of change in gas dynamic variables.