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A. I. Lopato
chemical energy of the fuel into useful work. Thus, one can note the relevance of
studies on detonation processes in gases.
Conducting natural experiments with the study of DWs is often associated with
certain problems. The problems include the measurement of flow parameters and
the stability of installations and sensors to high pressures and temperatures in areas
with shock and detonation waves. In addition, the range of investigated properties
of DWs is limited by a set of acceptable installations and constructions that can be
used for conducting natural experiments. Numerical calculations are devoid of such
problems and make it possible to obtain flow patterns with DWs in a sufficiently wide
set of research areas with a degree of accuracy determined by a number of factors
including the adequacy of the mathematical model and the numerical method, the
approximation order of the numerical scheme, and the stability of the numerical
method.
The chapter is organized as follows. Related work is highlighted in Sect. 8.2.
Section 8.3 provides the mathematical model of the studied problem. The computational algorithm of the second approximation order is described in Sect. 8.4.
Section 8.5 presents the results of verification and numerical experiments. Section 8.6
concludes the chapter.
8.2 Related Work
Since DW, in general, is a multi-dimensional object, mathematical modeling requires
taking into account multidimensional effects with a complex kinetics model of chemical reactions. On the other hand, the mathematical model corresponding to the onedimensional structure of DW is simpler but provides a relatively rich spectrum of
dynamic features that deserve the detailed study and have relevance to multidimensional effects of DW, for example, cellular structures. As is known from numerical
and experimental studies, the propagation of DW is associated with the formation of
a complex nonlinear oscillating process including pulsations of parameters behind
the front of DW, which are investigated in a number of numerical studies. In [2, 3],
different modes of parameters pulsations of the one-dimensional DW were obtained
depending on the activation energy of the considered model mixture. The mathematical model in [4] included the one-step irreversible reaction and Arrhenius kinetics
with parameters values that were proposed apparently for the first time. A spectral
Fourier analysis of the peak pressure pulsations on time was carried out with the
allocation of dominant frequencies. The comparison with the results of theoretical
and numerical studies on some quantitative characteristics such as limit cycle size in
the case of the weakly unstable detonation was performed.
In [5, 6], detonation in the model hydrogen–air mixture [7] was considered. Highfrequency (HF) and low-frequency modes of detonation propagation were obtained
in [5]. It is shown that with an increase in the approximation order of the numerical
method, the front of the reaction zone is less smeared and the scale of the reaction
zone is better resolved, which leads to that the instabilities are captured correctly.
A. I. Lopato
chemical energy of the fuel into useful work. Thus, one can note the relevance of
studies on detonation processes in gases.
Conducting natural experiments with the study of DWs is often associated with
certain problems. The problems include the measurement of flow parameters and
the stability of installations and sensors to high pressures and temperatures in areas
with shock and detonation waves. In addition, the range of investigated properties
of DWs is limited by a set of acceptable installations and constructions that can be
used for conducting natural experiments. Numerical calculations are devoid of such
problems and make it possible to obtain flow patterns with DWs in a sufficiently wide
set of research areas with a degree of accuracy determined by a number of factors
including the adequacy of the mathematical model and the numerical method, the
approximation order of the numerical scheme, and the stability of the numerical
method.
The chapter is organized as follows. Related work is highlighted in Sect. 8.2.
Section 8.3 provides the mathematical model of the studied problem. The computational algorithm of the second approximation order is described in Sect. 8.4.
Section 8.5 presents the results of verification and numerical experiments. Section 8.6
concludes the chapter.
8.2 Related Work
Since DW, in general, is a multi-dimensional object, mathematical modeling requires
taking into account multidimensional effects with a complex kinetics model of chemical reactions. On the other hand, the mathematical model corresponding to the onedimensional structure of DW is simpler but provides a relatively rich spectrum of
dynamic features that deserve the detailed study and have relevance to multidimensional effects of DW, for example, cellular structures. As is known from numerical
and experimental studies, the propagation of DW is associated with the formation of
a complex nonlinear oscillating process including pulsations of parameters behind
the front of DW, which are investigated in a number of numerical studies. In [2, 3],
different modes of parameters pulsations of the one-dimensional DW were obtained
depending on the activation energy of the considered model mixture. The mathematical model in [4] included the one-step irreversible reaction and Arrhenius kinetics
with parameters values that were proposed apparently for the first time. A spectral
Fourier analysis of the peak pressure pulsations on time was carried out with the
allocation of dominant frequencies. The comparison with the results of theoretical
and numerical studies on some quantitative characteristics such as limit cycle size in
the case of the weakly unstable detonation was performed.
In [5, 6], detonation in the model hydrogen–air mixture [7] was considered. Highfrequency (HF) and low-frequency modes of detonation propagation were obtained
in [5]. It is shown that with an increase in the approximation order of the numerical
method, the front of the reaction zone is less smeared and the scale of the reaction
zone is better resolved, which leads to that the instabilities are captured correctly.