Non-equilibrium flows of a reacting five-component air mixture consisting of 2, 2, , , behind shock waves at different altitudes from the earth's surface at different speeds of the incoming flow are numerically investigated. One-temperature mathematical model of non-equilibrium air flows is applied. The distributions of flow quantities behind the shock wave fronts are obtained and analyzed. The relaxation lengths of flow quantities are compared for various Mach numbers and altitudes.
where h is Plank constant, c is speed of light, and characterize the frequency and anharmonicity of molecular vibrations, = 2, 2, are spectroscopic constants that
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 ( ) = ( ) exp (− ), = ∑ exp (− ).
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.
Within the framework of the one-temperature approach, the determining macro-parameters of the flow are the numerical densities of molecules and atoms 2, 2 , , , , the gas temperature, 2.1.
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]
(
= 0. = + +
+ , = (0) (0); 2 + = (0)( (0))2 + (0) ; + + 2 2 = (0)+ (0) (0) + ( (0))2 2 .
Here, = ∑ is density of the mixture, is pressure, = ∑ is numerical density of the mixture, is total energy of a unit of volume. The total energy is represented as where 〱 , ,
, formation of particles of the mixture per unit volume. 2.2.
are translational, rotational, vibrational energy and the energy of
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 The superscript 0 indicates the parameters before the shock front. 2.3. Relaxation terms reactions, and 2→
⟶ ⟶ → 2
and
⟶
molecules due to exchange reactions (
To close the system (
2↔2 − 22↔2,
are temperature-dependent coefficients of the rate of direct exchange
→ 2 are coefficients of the rate of reverse reactions (
2, of molecules 2, 2, , , are gas temperature-dependent coefficients of the rate of dissociation
in collision with a particle M,
, 2, , 2, , are coefficients of
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 , = exp (− ), case of decay reactions, recommended in [8] are used. where ( ) is activation energy in the case of exchange reactions and dissociation energy in the and are constant factors that are determined in a number of studies based on experimental data or detailed numerical calculations. The dissociation rate coefficients
The coefficients of the rate of recombination and reverse exchange reactions are calculated using the relations following from the principle of detailed balance [7]. These relations are as follows , 2 =
, 2 = 2, 2, ,
= , →
→ 2 = 2→→ → → 2 = 2→ → ( 2
2 2 ( 2 2 2 ℎ2 ℎ2 ( 2 ℎ2
3/2 ) 2
3/2 ) 2
3/2 ) ( ) 2 ( )exp ( 2), ( ) 2 ( )exp ( 2), ( )
( )exp ( ), ( 2 ) 2 ( ) 2 (( )) exp ( 2 ( ) ( 2 ) 2 ( ) 2 ( ) ( ) ( ) exp ( 2 − ), − ), where ( ) is statistical sum of rotational degrees of freedom, ( ) is statistical sum of the vibrational degrees of freedom.
The results are obtained using numerical methods. At the beginning, a system of equations (
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 2and 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 2 and 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.
Figures 5 and 6 demonstrate that the numerical densities of and components increase with distance from the shock front. An increase in the Mach number leads to an increase in the concentration of and components in the air mixture behind the shock wave, the sharpness of the concentration change increases. As the altitude increases, the concentration of nitrogen atoms N decreases, and the concentration of oxygen atoms decreases at 10 km, but already at 24 km there is an increase in the concentration. Figure 7 shows the change in the numerical density of NO component depending on the distance from the shock wave. The numerical density of the molecules increases as they move away from the front. The numerical density also increases as the Mach number increases. As the altitude increases, the concentration of nitric oxide molecules decreases. It is possible to note a non-monotonic change in the numerical density of nitrogen oxide molecules with parameters before the shock wave equal to the parameters of the atmosphere at an altitude of 60 km.
Distributions in the flow parameters of the air flow behind shock waves at different altitudes from the Earth's surface at various Mach numbers of the incoming flow are numerically computed and investigated.
The study has shown that when moving away from the shock wave front, the concentrations of the components of the air mixture 2 and 2 decrease, and the concentrations of , and increase. As the velocity of the incoming flow increases, the temperature and velocity behind the shock wave increase, the concentrations of 2 and 2 molecules decrease, and the concentrations of , and increase. At low pressures, there is a non-monotonic change in the numerical density of nitrogen oxide 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.
The study was financially supported by the Russian Science Foundation (project No. 19-7110019).
The studies were carried out using the resources of the Center for Shared Use of Scientific Equipment "Center for Processing and Storage of Scientific Data of the Far Eastern Branch of the Russian Academy of Sciences", funded by the Russian Federation represented by the Ministry of Science and Higher Education of the Russian Federation under project No. 075-15-2021-663.