=Paper=
{{Paper
|id=Vol-2930/paper26
|storemode=property
|title=Modeling of Relaxation Processes in Air Flows behind Shock Waves
|pdfUrl=https://ceur-ws.org/Vol-2930/paper26.pdf
|volume=Vol-2930
|authors=Anton Karpenko,Semen Tolstoguzov,Konstantin Volkov
}}
==Modeling of Relaxation Processes in Air Flows behind Shock Waves==
https://ceur-ws.org/Vol-2930/paper26.pdf
Modeling of Relaxation Processes in Air Flows behind Shock
Waves
Anton G. Karpenkoa, Semen S. Tolstoguzova, band Konstantin N. Volkova, b
a
St Petersburg State University, St Petersburg, Universitetskii prospekt 28, 198504, Russia
b
Baltic State Technical University, St Petersburg, ul. 1ay Krasnoarmeiskay 1, 190005, Russia
Abstract
Non-equilibrium flows of a reacting five-component air mixture consisting of π2 , π2 , ππ,
π, π behind shock waves at different altitudes from the earthβs 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.
Keywords 1
Hypersonic, chemical kinetics in air, shockwave
1. 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 π2 , π2 , ππ, π, π in which the
various kinetic processes occur. These processes include chemical reactions of dissociation and
recombination and chemical exchange reactions. They are as follows:
β’ Dissociation and recombination
π2 + π β π + π + π, π2 + π β π + π + π, ππ + π β π + π + π; (1)
β’ Chemical exchange reactions
π2 + π β ππ + π, π2 + π β ππ + π. (2)
Here, π = π2 , π2 , ππ, π, π.
There are various approaches to the description of non-equilibrium flows including one-
temperature 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 one-
temperature 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.
The vibrational energy πππ of molecule π at level π is calculated based on the anharmonic Morse oscillator modelππ
VI International Conference Information Technologies and High Performance Computing (ITHPC-2021)
September 14-16, 2021, Khabarovsk, Russia
EMAIL: aspera.2003.ru@mail.ru (A. 1); semen.tolstoguzov96@mail.ru (A. 2); dsci@mail.ru (A. 3)
ORCID: 0000-0002-1250-9766 (A. 1); 0000-0001-9560-2693 (A. 2); 0000-0001-6055-2323 (A. 3)
Β©οΈ 2021 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
186
1 1
ππ0 + βπ(πππ β πππ π₯ππ β πππ π₯ππ π)π, ππ0 = βπ (2 πππ β 4 πππ π₯ππ ),
where h is Plank constant, c is speed of light, πππ and πππ π₯ππ are spectroscopic constants that
characterize the frequency and anharmonicity of molecular vibrations, π = π2 , π2 , ππ.
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
ππ ππ ππ
πππ (π) = exp (β π ), π£πππ
π = βπ exp (β π ).
πππ£πππ (π) ππ ππ
This study focuses on numerical simulation of non-equilibrium flows of a reacting five-
component 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.
2. 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 ππ2 , ππ2 , πππ , ππ , ππ , the gas temperature,
T, and the macroscopic velocity, v.
2.1. Governing equations
The system of governing equations for macro-parameters contains the equations of one-
temperature 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]
ππ ππ£ (3)
π£ π + ππ = π
2β2 + π
2β3;
ππ₯ ππ₯ π π
ππ£ ππ (4)
ππ£ ππ₯ + ππ₯ = 0;
ππΈ ππ£ (5)
π£ πγ° + (π + πΈ) ππ₯ = 0.
Here, π = βπ ππ ππ is density of the mixture, π = πππ is pressure, π = βπ ππ is numerical density
of the mixture, πΈ is total energy of a unit of volume. The total energy is represented as
πΈ = πΈ π‘π + πΈ πππ‘ + πΈ π£πππ + πΈ π ,
where πΈ γ±π , πΈ πππ‘ , πΈ π£πππ , πΈ π are translational, rotational, vibrational energy and the energy of
formation of particles of the mixture per unit volume.
2.2. 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
ππ£ = π(0) π£ (0); (6)
2 (7)
ππ£ 2 + π = π(0) (π£ (0) ) + π(0) ;
2
πΈ+π π£2 πΈ (0) +π(0) (π£ (0) ) (8)
π
+ 2
= π(0)
+ 2
.
The superscript 0 indicates the parameters before the shock front.
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2.3. Relaxation terms
To close the system (3)-(5) it is necessary to express the relaxation terms in terms of macro-
2β2 2β3 2β2
parameters, π
π and π
π . The terms π
π describe changes in the numerical densities of
molecules due to exchange reactions (2) and have the following form [7]:
π
π2β2
2
πβΆπ
= πππ ππ πππβπ 2
β ππ2 ππ πππβΆπ
2 βππ
, π
π2β2
2
= βππ ππ πππβπ
πβΆπ
2
β ππ2 ππ πππβΆπ
2 βππ
,
2β2 2β2 2β2 2β2 2β2 2β2 2β2 2β2 2β2
π
ππ = βπ
π2 β π
π2 , π
π = βπ
π2 + π
π2 , π
π = π
π2 β π
π2 ,
where πππβΆπ
2 βππ
and πππβΆπ
2 βππ
are temperature-dependent coefficients of the rate of direct exchange
πβΆπ πβΆπ
reactions, πππβπ2 and πππβπ2 are coefficients of the rate of reverse reactions (2).
2β3
The terms π
π describe the processes of dissociation and recombination (1) and take the form
π π
2β3
π
π2 = βπ ππ (ππ 2 π
ππππ,π2 β ππ π
2 π2 ,πππ π
), π
π2β3
2
= βπ ππ (ππ2 ππππ,π
π
2
β πππ2 πππ2 ,πππ π ),
2β3 π π π 2β3
π
ππ = βπ ππ (ππ ππ ππππ,ππ β πππ πππ,πππ π ), π
π2β3 = β2π
π
2β3
2
β π
ππ ,
π
π2β3 = β2π
π2β32
2β3
β π
ππ ,
π π π
where ππ2 ,πππ π , ππ2 ,πππ π , πππ,πππ π are gas temperature-dependent coefficients of the rate of dissociation
π π π
of molecules π2 , π2 , ππ in collision with a particle M, ππππ,π 2
, ππππ,π2
, ππππ,ππ are coefficients of
the rate of recombination of atoms to form molecules π2 , π2 , ππ.
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
π πΈ
ππ,ππ = π΄π π π exp (β πππΌ ),
where πΈπΌ (π·π ) is activation energy in the case of exchange reactions and dissociation energy in the
case of decay reactions, π΄π and π 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.
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
3/2
ππ2 β2 π·π2
π
ππππ,π2
= πππ2 ,πππ π ( 2 2πππ ) πππππ‘
2
(π)πππ£πππ
2
(π) exp ( ),
ππ ππ
3/2
ππ β2 π·π
π
ππππ,π2
= πππ2 ,πππ π ( π22 2πππ ) πππππ‘
2
(π)πππ£πππ
2
(π) exp ( 2 ),
ππ
π
3/2
π π π β2 πππ‘ (π)π π£πππ (π) π·
ππππ,ππ = πππ,πππ π (π ππ
π 2πππ
) πππ ππ
ππ
exp ( ππ ),
π π
πππ‘ (π)ππ£πππ (π)
πβπ ππ ππ ππ π π·π2 βπ·ππ
πππβπ2
= πππβπ ( 2 π ) πππ‘
2 βππ π
2 2
(π)ππ£πππ (π)
exp ( ππ
),
ππ π πππ ππ
πππ‘ (π)ππ£πππ (π)
πβπ ππ ππ ππ π π·π2 βπ·ππ
πππβπ2
= πππβπ ( 2 π ) πππ‘
2 βππ π
2 2
(π)ππ£πππ (π)
exp ( ππ
),
ππ π πππ ππ
πππ (π»)
where ππ is statistical sum of rotational degrees of freedom, πππππ
π (π») is statistical sum of the
vibrational degrees of freedom.
3. Results and discussion
The results are obtained using numerical methods. At the beginning, a system of equations (6)β(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)β(5) using the implicit
Geer method [9].
188
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 45β 32β²33β²β², 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 1β7 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.
Figure 1: The dependence of the temperature on the distance behind the shock wave front for
altitudes h=0 km (a), h=10 km (b), h=24 km (c), h=60 km (d)
Figure 2: The dependence of the velocity on the distance behind the shock wave front for altitudes
h=0 km (a), h=10 km (b), h=24 km (c), h=60 km (d)
189
When moving away from the wave front, the numerical densities of the molecules π2 and π2
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 π2 and π2 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.
Figure 3: The dependence of the π2 concentration on the distance behind the shock wave front for
altitudes h=0 km (a), h=10 km (b), h=24 km (c), h=60 km (d)
Figure 4: The dependence of the π2 concentration on the distance behind the shock wave front for
altitudes h=0 km (a), h=10 km (b), h=24 km (c), h=60 km (d)
Figures 5 and 6 demonstrate that the numerical densities of π and π components increase with
distance from the shock front. An increase in the Mach number leads to an increase in the
190
concentration of π and π 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 π decreases at 10 km, but already at 24 km there is
an increase in the concentration.
Figure 5: The dependence of the π concentration on the distance behind the shock wave front for
altitudes h=0 km (a), h=10 km (b), h=24 km (c), h=60 km (d)
Figure 6: The dependence of the π concentration on the distance behind the shock wave front for
altitudes h=0 km (a), h=10 km (b), h=24 km (c), h=60 km (d)
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
191
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.
Figure 7: The dependence of the ππ concentration on the distance behind the shock wave front for
altitudes h=0 km (a), h=10 km (b), h=24 km (c), h=60 km (d)
4. Conclusion
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 π2 and π2 decrease, and the concentrations of π, π and ππ increase.
As the velocity of the incoming flow increases, the temperature and velocity behind the shock wave
increase, the concentrations of π2 and π2 molecules decrease, and the concentrations of π,
π and ππ increase. At low pressures, there is a non-monotonic change in the numerical density of
nitrogen oxide ππ 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.
5. Acknowledgements
The study was financially supported by the Russian Science Foundation (project No. 19-71-
10019).
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.
192
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