INTRODUCTION

At the moment, there are many kinetic mechanisms to describe hydrogen combustion in oxygen. Some of them include more than a hundred reactions and several tens of chemical species; others may contain only several tens of reactions and 5–10 species. Two-stage mechanisms, where the induction period and the stage of chemical transformations are distinguished, have become very popular [1, 2]. However, it is not always possible to use these models in practice because of various problem such as determining the constants in a wide range of parameters and quantitative comparisons with experiments.

The choice of the chemical model to be used in numerical simulations of detonation wave (DW) propagation is an important and urgent issue. For example, Bedarev et al. [3, 4] studied various mechanisms for the description of hydrogen combustion in oxygen and noted that detailed mechanisms adequately reproduce the ignition delay and ensure good agreement with experimental data on the DW propagation velocity and (in mixtures strongly diluted by argon) detonation cell size. Nevertheless, the chemical model may significantly affect the computed results: the profiles of gas-dynamic quantities and the parameters of the cellular structure of the detonation wave computed by different models may be noticeably different. The reasons are the difference in the choice of chemical reactions included into the model and in the values of the coefficients determining the rates of these reactions.

In the present paper, we compare four detailed models of chemical kinetics, which describe hydrogen combustion in oxygen. The ignition delays for different initial temperatures and pressures are determined by solving the problem of an explosion in a constant volume. The one-dimensional DW structure [the Zel’dovich–von Neumann–Döring (ZND) solution] is obtained for different chemical models. The influence of nitrogen addition on the ignition delay and one-dimensional DW structure is estimated. Direct numerical simulations of DW propagation in a plane channel filled with hydrogen–oxygen and hydrogen–oxygen–nitrogen mixtures are performed.

1. GOVERNING EQUATIONS

Detonation in a gas mixture can be described by the Euler equations coupled with equations that describe the kinetics of chemical transformations:

$$\frac{\partial \rho Y_i }{\partial t} + \frac{\partial }{\partial x}\,({\rho uY_i } ) + \frac{\partial }{\partial y}\,( {\rho vY_i }) + \frac{\partial }{\partial z}\,( {\rho wY_i } ) {} {}=W_i \dot{\omega }_i ,\;\;$$
(1)
$$i=1,\, \ldots ,\, N-1.$$

Here \(t\), \(x\), \(y\), and \(z\) are the time and the spatial coordinates, respectively, \(\rho\) is the density of the mixture, \(u\) is its velocity, \(N\) is the number of chemical species of the mixture, \(Y_i\) is the mass fraction of the \(i\) th species, \(W_{i}\) is the molecular weight, and \(\dot {\omega }_i\) is the molar rate of production/destruction of the \(i\) th species in chemical reactions. The number of independent equations Eq. 1 is \(N - 1\) because the sum of the partial densities of different species is equal to the total density of the mixture.

In the constant-volume explosion problem, the following equations were integrated for determining the ignition delay:

$$\frac{dY_i }{dt} = \frac{W_i \dot {\omega }_i }{\rho },\;\; i = 1,\,\ldots ,\, N- 1,$$
(2)
$$\frac{dT}{dt} = - \frac{1}{\rho c_v }\,\sum \limits_{i = 1}^N e_iW_i \dot {\omega }_i$$
(3)

.

The integration was performed with the use of the RADAU5 code [5] with the time step chosen automatically. The ignition delay \(\tau _{\mathrm ign}\) was determined as the time period from the computation start to the instant when the highest temperature growth rate was achieved.

The ZND solution can be obtained for the DW with the velocity equal to or greater than the Chapman–Jouguet (CJ) velocity: \(D \ge D_\text{CJ}\); the conditions \(D = D_\text{CJ}\) and \(D > D_\text{CJ}\) correspond to the CJ and overdriven detonations, respectively. The CJ velocity was determined using the SDTools software package [6].

After finding the DW velocity, the following system of conservation laws and equations for concentrations of the mixture components written in a coordinate system where the DW is at rest is solved:

$$\frac{d}{dx}\,( {\rho u} ) = 0яя \to яя\rho u \equiv m = \mathrm{const};$$
(4)
$$\frac{d}{dx}\,( {\rho u^2 + p} ) = 0яя {}$$
$${}\to яя\rho u^2 + p = mu + p \equiv F_u = \mathrm{const};$$
(5)
$$\frac{d}{dx}\,( {\rho uH} ) = 0яя \to яя\rho uH \equiv F_e = \mathrm{const},$$
(6)
$$H = F_e / m = \mathrm{const};$$
$$\frac{d}{dx}\,( {\rho uY_i } ) = W_i {\omega _i }я \to \frac{dY_i }{dx} = \frac{W_i \dot {\omega }_i }{m},$$
(7)
$$i =1,\,\ldots ,\, N - 1,\;\; Y_N = 1 - \sum \limits_{i = 1}^{N - 1} Y_i ;$$
$$p = \rho \tilde {R}T = \frac{\tilde {R}Tm}{u}, \tilde {R} = \sum\limits_{i = 1}^N Y_i R_i ,\; R_i = \frac{R}{W_i}.$$
(8)

Here \(p\) is the pressure, \(T\) is the temperature, \(R\) is the universal gas constant, \(H \equiv h + {u^2}/{2}\) is the total enthalpy of the mixture, and \(h\) is the specific enthalpy of the mixture.

Equations Eq. 5 and Eq. 6 yield

$$u + \tilde {R}\frac{T}{u} = \frac{F_u }{m},яяяяяu^2 = 2( {H - h} ),$$
(9)
$$\frac{u^2}{2} + h = \frac{u^2}{2} + \sum \limits_{i = 1}^N Y_ih_i ( T ) = H,$$

where \(h_{i}\) is the enthalpy of the corresponding chemical species. Expressing \(\tilde {R}T\) from Eq. 9 and eliminating the velocity, we obtain the following equation for the temperature:

$$f( T ) \equiv [ {2( {H - h} ) + \tilde {R}T}]^2 {}$$
$${}-2( {{F_u }/{m}} )^2( {H - h} ) = 0.$$
(10)

In calculating the one-dimensional DW structure, Eq. 7 are integrated numerically. Knowing the concentrations of the mixture components at a new point and, hence, the specific enthalpy of the mixture \(h\), one can solve the nonlinear equation Eq. 10 by this or that iterative method and find the temperature; after that, the velocity and all other gas-dynamic quantities are determined. In our computations, we used the fourth-order Runge–Kutta scheme for integrating Eq. 7, and the temperature was obtained using the Newton iterations.

2. DESCRIPTION OF THE CHEMICAL MODELS

In the present paper, we consider four chemical models, which describe hydrogen combustion in oxygen. Each model allows the presence of an inert gas, e.g., nitrogen or argon, as the third species, which does not directly participate in chemical reactions, but affects the reaction rates. The models include up to 20 forward and reverse chemical reactions. The reaction rates are determined from the Arrhenius law

$$k( T ) = AT^\beta \exp( { - E_a / RT} ).$$
(11)

Here the constants \(A\) and \(\beta\), as well as the activation energy \(E_{a}\) are defined by the chemical model. More detailed descriptions of the models under consideration are provided in Appendix.

The simplest model in the study is the ONERA model [7], which was specially developed to describe supersonic combustion. It includes 7 chemical species (H, O, H2, O2, OH, H2O, and N2) and 7 reactions.

In three other models, there are 9 chemical species: the species used in the ONERA model plus HO2 and H2O2. The Deiterding–Westbrook model described in [8] is an adapted version of the earlier Westbrook model [9], which was developed for detonation in carbon–hydrogen mixtures. The model includes 17 reactions.

Fig. 1
figure 1

Ignition delays in the 2H2 + O2mixture (solid curves) and 2H2 + O2 + 3N2 mixture (dashed curves) versus temperature calculated by the four models under study.

Full size image

The Wilson–MacCormack model [10] is a variation of the Jachimowski model [11] modified to describe supersonic combustion. The model includes 19 reactions, which is greater than that in the previous model by 2.

The last model considered is the Petersen–Hanson model [12], and it should be noted that its was specially developed to take into account the dependence of the chemical reaction rate on pressure. Although this dependence is present only in one reaction, it significantly affects the results. The model includes 20 chemical reactions.

In the first two models, the coefficients are explicitly specified for both forward and reverse reactions. In two other models, the coefficients for the reverse reactions are calculated using the chemical equilibrium constants. Generally speaking, this approach is more correct because it prevents possible inconsistency of the rates of forward and reverse reactions with thermodynamic data. In the present study, the data [13] were used in all cases for calculating the thermodynamic parameters of the mixture components.

3. RESULTS

3.1. Explosion in a Constant Volume

For each chemical model, we calculated the ignition delay \(\tau _{\mathrm ign}\) as a function of the initial temperature for stoichiometric mixtures 2H2 + O2 and 2H2 + O2 + 3N2 for the initial pressures \(p_{\mathrm ini}\) = 1, 10, and 50 atm. The results are shown in Fig. 1.

First of all, it should be noted that the dependence of \(\tau _{\mathrm ign}\) on temperature in the ONERA model is much simpler than those in three other models: in fact, it is an exponential dependence for all pressures. Nevertheless, at high temperatures the ONERA model agrees well with the Deiterding–Westbrook and Petersen–Hanson models; at pressures of 1 and 10 atm, the ONERA model is in good agreement with the Wilson–MacCormack model as well. With a further increase in pressure, the ignition delay \(\tau _{\mathrm ign}\) in the Wilson–MacCormack model in the high-temperature region decreases more slowly than that in other models. As a result, the value of \(\tau _{\mathrm ign}\) at \(p_{\mathrm ini}\) = 50 atm and \(T \approx 2000\) K calculated by the Wilson–MacCormack model is higher than the predictions of other models by 4–6 times.

As the temperature decreases, the results predicted by the ONERA model become more and more different from those of other models because of the simplified character of the dependence of \(\tau _{\mathrm ign}\) on temperature in this model. In the middle range of temperature (\(T \approx 1000\) K), the differences reach three orders of magnitude. The remaining three models demonstrate good qualitative and, to a certain degree, quantitative agreement with each other in this temperature range. With a further decrease in temperature, the Petersen–Hanson model starts to display some differences because it includes a reaction with a pressure-dependent rate. Apparently, if this dependence is taken into account, the results calculated by this model for different values of \(p_{\mathrm ini}\) in the low-temperature range are closer to each other as compared to the data obtained with the use of other chemical mechanisms.

Table 1. Velocity of detonation wave propagation in the Chapman–Jouguet regime
Full size table

In experimental studies of DW propagation, a certain amount of an inert gas is often added to the mixture to obtain a more regular cellular pattern and ensure more precise measurements of the detonation cell size, which increases due to dilution of the hydrogen–oxygen mixture. To study the influence of dilution of the mixture with an inert gas, we computed the ignition delays in a stoichiometric mixture of hydrogen and oxygen with addition of an identical number of nitrogen moles (2H2 + O2 + 3N2).

The dilution of the mixture produces a common effect in all models: addition of nitrogen usually increases the ignition delay (see Fig. 1). However, in the Wilson–MacCormack and Petersen–Hanson models, there are temperature ranges (near the inflection corresponding to the drastic change in the slope of the curve of \(\tau _{\mathrm ign}\) as a function of temperature) where the ignition delay in the mixture without nitrogen is longer.

Thus, the Deiterding–Westbrook, Wilson–MacCormack, and Petersen–Hanson models are in good overall agreement with each other in terms of ignition delay predictions. The ONERA model, which includes a smaller number of chemical species and a significantly reduced set of chemical reactions, yields an essentially different dependence of the ignition delay on temperature, and its predictions at moderate and low temperatures are significantly different from those of other models. However, being specially developed for supersonic flow simulations, this model ensures reasonable results at high temperatures. Simultaneously, in the range of high pressures and high temperatures, i.e., at parameters most typical for detonation combustion, the ONERA model is inconsistent with the Wilson–MacCormack model: the ignition delays predicted by the ONERA model are several times higher.

3.2. Zel’dovich–von Neumann–Döring Solution

Fig. 2
figure 2

Pressure and temperature versus the distance in the ZND solution for all models under study. Stoichiometric mixture 2H2 + O2.

Full size image
Fig. 3
figure 3

Pressure and temperature versus the distance in the ZND solution for all models under study. Stoichiometric hydrogen–oxygen mixture diluted by 50% of nitrogen: 2H2 + O2 + 3N2.

Full size image

The problem of the one-dimensional structure of a plane detonation wave was used for comparisons of the detailed chemical models under study. The first parameter used for comparisons was the detonation propaga- tion velocity in the CJ regime \(D_\text{CJ}\). As mentioned before, it was determined using the SDTools free software [6]. The results are summarized in Table 1.

As is seen from Table 1, it is only the ONERA model that predicts a slightly different CJ velocity; the values obtained by other models agree with high accuracy. It is an expected result because the equilibrium composition of the mixture at the given pressure and temperature corresponds to the minimum value of the thermodynamic Gibbs potential. Thus, for models with an identical set of chemical components, the equilibrium composition at the CJ point and, hence, the velocity are identical if the same dependences of the thermodynamic functions of individual species on temperature are used.

After that, we computed the one-dimensional DW structure (ZND solution) in the case of the CJ detonation for all chemical models and two types of stoichiometric mixtures: hydrogen–oxygen mixture (2H2 + O2) and hydrogen–oxygen–nitrogen mixture (2H2 + O2 + 3N2) with the initial pressure of 1 atm and initial temperature of 300 K. The pressure and temperature are plotted in Fig. 2 as functions of the distance from the shock wave front.

It is seen that the smallest distance between the shock wave front and the point where the combustion process begins is predicted by the ONERA model, whereas the Deiterding–Westbrook and Petersen–Hanson models yield a slightly greater distance. At the beginning, the curves constructed by these two models almost coincide; however, further downstream, the pressure and temperature predicted by the Petersen–Hanson model change more slowly and reach their equilibrium values at a greater distance from the shock wave.

The behaviors of the pressure and temperature profiles in the Wilson–MacCormack model are significantly different. Although the chemical reactions start approximately at the same distance from the shock wave, the combustion region is several times longer and the point with the maximum heat release is located noticeably farther from the shock wave front. This behavior is consistent with the calculated ignition delay in the previous Section. Indeed, the temperature behind the shock wave front is 1769 K, and the pressure is approximately 33 atm. These values correspond to the high-pressure and high-temperature region in Fig. 1, where the ignition delay predicted by this model is several times higher than the values predicted by the other chemical mechanisms.

Addition of 50% of nitrogen to the stoichiometric mixture of hydrogen and oxygen causes significant changes in the DW structure Fig. 3. The values of the main parameters behind the shock wave become smaller: the pressure decreases from 33 to 29 atm, and the temperature decreases from 1760 to 1590 K. The distance between the wave front and the ignition point significantly increases, and the heat release region proper becomes more extended. The value far downstream (near the CJ point) also change: the pressure decreases from 19 to 16.7 atm, and the temperature decreases from 3680 to 3100 K. We can say that addition of an inert gas makes the detonation less violent, and it occurs at greater scales in space and time.

It should be expected that larger and more regular detonation cells would appear in numerical simulations of propagation of the multifront DW, as it is observed in real physical experiments.

3.3. Direct Numerical Simulation of Detonation Wave Propagation in a Plane Channel

It is well known that the DW front is unstable to transverse perturbations. Two-dimensional direct numerical simulations allow one to study the influence of the chemical model on the evolution of this instability and on the formation of the cellular structure of the detonation wave. Such simulations were performed by the numerical code developed by the authors for solving the Euler equations for a chemically reacting gas [14]. The code involves a third-order TVD scheme for approximation of convective terms and a semi-implicit Runge–Kutta scheme ASIRK-2C for integrating the equations with respect to time. Owing to the use of the semi-implicit scheme, it is possible to attenuate the constraints on the time step, which follow from the stability conditions for stiff chemical source terms. The code is written in the C++ language and is parallelized with the MPI, OpenMP, and CUDA technologies. It allows computations to be performed both on usual multiprocessor and multicore computers and also on computers with a hybrid architecture including graphics processing units (GPUs) or Xeon Phi co-processors.

All computations were performed for the initial temperature of the mixture equal to 300 K. Figure 4 illustrates the onset of instability of the propagating plane wave to transverse perturbations. The figure shows the time evolution of the DW front coordinate obtained in computing the wave propagation in a channel 1 cm wide filled with a stoichiometric hydrogen–oxygen mixture at the initial pressure of 1 atm. The computations were performed with the ONERA and Wilson–MacCormack models for a DW with the overdrive ratio \(f\) = 1.4 and theoretical velocity \(D( f ) =D_\text{CJ} \sqrt f\).

The case with an overdriven DW was chosen to avoid the emergence of one-dimensional fluctuations, which complicate the pattern of instability development. It is well known that overdriven DWs are more stable (see, e.g., [15, 16]). In particular, as can be seen from the one-dimensional computations [17], an increase in the overdrive ratio up to \(f\) = 1.4 guarantees suppression of instability to streamwise perturbations. Moreover, in the overdriven regime, the oscillations of the front velocity have a more “ordered" character, so that it is easier to trace the changes in the front position with time in one figure.

The ZND solution for the overdriven wave was used as the initial flow field. The computation was performed in a coordinate system moving with the velocity \(D(f)\), i.e., in theory, the DW position is unchanged during the computation. The computational domain length was 1.5 cm, which was much greater than the reaction region width. Uniform free-stream conditions were imposed at the inflow boundary, and the parameters corresponding to the equilibrium values far downstream from the DW front were set at the outflow boundary. The computation was performed on a grid with 600 \(\times\) 400 cells.

Fig. 4
figure 4

Time evolution of the DW front averaged over the channel width (curve 1) and near one of the walls (curve 2).

Full size image

As is seen from Fig. 4, after a certain small shift at the very beginning of the computation, the DW front position does remain unchanged for a long time. However, transverse perturbations appear on the wave front afterwards.

Figure 4 shows both the front position near one of the channel walls and the front position averaged over the channel width. Their difference clearly indicates that the leading front of the DW is far from being plane. It should be noted that the DW velocity after the emergence of transverse waves becomes slightly higher than \(D(f)\) predicted by the one-dimensional theory; therefore, in the coordinate system used here, it slowly moves in the upstream direction. For the ONERA model, this excess of velocity is approximately 3.25 m/s, which is smaller than 0.1% of the DW propagation velocity.

Fig. 5
figure 5

Linear and nonlinear stages of formation of the cellular DW structure. Petersen–Hanson model (\(p_{\mathrm ini}\) = 0.1 atm and \(f\) = 1).

Full size image

In the computation with the Wilson–MacCormack model, the transverse perturbations at the wave front arise noticeably earlier than in the case with the ONERA model. After that, the DW also starts to move upstream, and its velocity is higher than the theoretical velocity by 12.3 m/s. However, the velocity decreases at a certain instant, which is manifested in the lower part of Fig. 4 as an inflection on the corresponding curve at\(t\) \(\approx\)47.5 \(\mu\)s. Since that instant and up to the end of the computation, the velocity is higher than the value predicted by the one-dimensional theory approximately by 6.15 m/s.

It should be noted that the fact that the multifront DW velocity can be greater than the value calculated in the absence of transverse fluctuations was mentioned back in the classical monographs [18, 19]. Detonation propagation with a velocity slightly higher than the CJ velocity was also observed in physical experiments.

The emergence and growth of transverse waves rapidly lead to the formation of an unsteady two-dimensional cellular structure. The detonation cells are rather small at first, but they merge with time, and their size increases. For the ONERA model, the time instant \(t\) \(\approx\)50 \(\mu\)s, when large cells start to prevail, corresponds to certain changes in the character of oscillations of the front position (see Fig. 4). For the Wilson–MacCormack model, it coincides with the instant of the DW velocity reduction mentioned above.

Numerical simulations of all stages of the development of the multifront structure were performed for the CJ detonation (\(f\) = 1) with the use of all four models of chemical kinetics under study for the initial pressure \(p_{\mathrm ini}\) = 0.1 atm. As was mentioned above, the computations were performed for a stoichiometric mixture of hydrogen and oxygen (2H2 + O2) and for a mixture of hydrogen, oxygen, and nitrogen (2H2 + O2 + 3N2). The channel height was 2 cm, and its length was varied in the interval from 6 to 30 cm depending on the model and composition of the mixture. The resolution in each case was chosen to provide 24 computational cells between the shock wave and the point with the maximum heat release. Thus, the final resolution was up to 400 cells in 1 cm and up to 2 000 000 cells in the entire computational domain. Figure 5 shows the pressure fields at different time instants for the two mixtures obtained with the use of the Petersen–Hanson model.

Table 2. Number of detonation cells depending on the model and initial composition of the mixture
Full size table
Fig. 6
figure 6

Number of detonation cells versus time in the computation with the Petersen–Hanson model.

Full size image

The upper row of Fig. 5 shows typical patterns of multifront detonation in the 2H2 + O2 mixture at two stages of the cellular structure formation. The early, linear, stage at which the growth of instability leads to the formation of numerous small cells is presented on the left. The right part of the figure illustrates the second, essentially nonlinear, stage at which small cells merge until their mean size exceeds the initial size by several times. The lower row shows the same stages for the 2H2 + O2 + 3N2 mixture. It is clearly seen that the size of the cells being formed depends on the composition of the mixture. If the mixture is diluted by nitrogen, the cell size increases approximately by a factor of 3: 5 cells emerge at the linear stage instead of 14, which transform to one half-cell at the end of the nonlinear stage, while the non-diluted mixture yields 1.5 cells. As in the case with the ZND solution, the maximum pressure and other parameters are noticeably different as well.

Thus, addition of an inert gas leads to enhancement of the spatial and temperature scales of detonation. This effect has been well known long ago from experiments. This is quantitatively illustrated in Fig. 6, which shows the change in the number of detonation cells with time. The number of cells was calculated by applying the Fourier transform to the DW front shape.

It is clearly seen from Fig. 6 that the cellular structure in the mixture without nitrogen starts to form much earlier (\(\approx\)15 \(\mu\)s as compared to \(\approx\)60 \(\mu\)s). The transition to the second stage with a drastic increase in the cell size also occurs earlier.

Dilution with nitrogen exerts an identical qualitative effect on the cellular structure of the DW for all chemical models. The quantitative results are summarized in Table 2. It should be noted that the number and, hence, the size of cells formed at the initial stage of the cellular structure formation depend to a large extent on the model used. It was demonstrated earlier [20–22] that the initial cell size for the simplest chemical model with one irreversible reaction is equal to the wave length of the transverse perturbation, which has the maximum growth rate in accordance with the linear theory of stability. It may be assumed that this statement is also valid for situations with detailed chemical mechanisms. Therefore, the differences observed at the initial stage of the cellular structure formation can be attributed to the differences in the characteristics of stability of the ZND solution for these mechanisms. Unfortunately, it is impossible to verify this assumption because there are no publications that describe computations of linear stability of the DW with detailed chemical models.

The growth and merging of detonation cells at later stages are obviously associated with nonlinear interactions activated when the perturbations reach noticeable amplitudes. The exact mechanism that controls these processes is currently unknown. In computations performed with the simplest model, the size of fully developed detonation cells is approximately twice the cell size at the linear stage [21].

It follows from Table 2 that this feature is not observed in computations with detailed chemical mechanisms. Indeed, despite significant differences in the number of cells at the initial stage, all models yield an approximately identical number of cells at the end of the computation. It should be mentioned, however, that it is desirable to repeat the computations for a wider channel in order to be absolutely sure that this conclusion is valid.

The results of Table 2 can be used to compare the predicted transverse size of fully developed detonation cells with experimental data. The following results were obtained for the non-diluted stoichiometric mixture of hydrogen and oxygen: 1–13 cm (ONERA model), 1.3–1.4 cm (Deiterding–Westbrook model), 1–2 cm (Wilson–MacCormack model), and 1.3 cm (Petersen–Hanson model). The experimental data [23] for the same parameters yield the cell size of 1.5–1.6 cm.

For the stoichiometric hydrogen–oxygen mixture diluted with nitrogen, all models predict the same detonation cell size: 4 cm. We could not find experimental data for such parameters. However, with extrapolation of data for the closest conditions available [24], the detonation cell size at the pressure of 0.1 atm can be expected to be 5–8 cm, which is also fairly close to the computed values.

CONCLUSIONS

Four models of hydrogen combustion in oxygen were compared. The computation of the ignition delay showed that the dependence of the ignition delay on temperature in the ONERA model has a much simpler, exponential, character as compared to other models. Nevertheless, these differences are manifested to the major extent at moderate and low temperatures, whereas the ignition delays predicted by this model agree well with the data obtained by the Deiterding–Westbrook and Petersen–Hanson models. At the same time, the Wilson–MacCormack model predicts several-fold higher ignition delays in the range of high temperatures and high pressures (most typical for detonation combustion), which differ from the results of other models.

The differences are also observed in computations of the one-dimensional DW structure. For the initial pressure of 1 atm, the distance from the wave front to the point of ignition and the combustion region length computed with the Wilson–MacCormack model are much greater than the corresponding values predicted by other models.

Dilution of the mixture with inert nitrogen produced a qualitatively identical effect for all models. Addition of nitrogen resulted in an increase in the ignition delay and made the detonation less violent by reducing the maximum pressure and temperature behind the DW front and extending the profile of the ZND solution in space. Despite the presence of a pressure-dependent reaction in the Petersen–Hanson model, the results predicted by this model do not differ much from those of other models.

All the models under consideration were verified by their authors with the use of more detailed models, which, in turn, were designed so that they ensured a consistent description of available experimental data. However, the computations show that the models predict different values of parameters in some ranges of temperature and pressure, even of such a basic parameter as the ignition delay. It also seems that the Wilson–MacCormack model is not able to provide an accurate description of detonation combustion at initial pressures close to the atmospheric value. Nevertheless, this model has been often used recently for modeling one-dimensional pulsed instability [25–27].

Direct numerical simulations of DW propagation in a channel performed with different models revealed significant differences in the sizes of detonation cells formed at the initial, linear, stage of the cellular structure formation. It can be assumed that the reason is the differences in the characteristics of linear stability of the ZND solution in situations with different models, namely, different wave length of transverse perturbations with the maximum growth rate. Unfortunately, the available computations of stability characteristics are confined to the simplest model with one irreversible reaction. There are no computations with detailed models, apparently, because the corresponding linearized equations are too cumbersome.

The sizes of fully developed detonation cells formed at the second, nonlinear, stage of the cellular structure formation are in good agreement with each other (in computations with different models) and with available experimental data. However, it should be noted that more precise comparisons require computations in significantly wider channels accommodating a large number of detonation cells because the cell size in narrow channels may significantly change and adapt to the channel width so that the channel could accommodate an integer number of half-cells.

The earliest stages of the formation of a cellular structure from an initially plane DW are demonstrated in more detail by an example of propagation of an overdriven DW. It is shown that the multifront structure starts to form as a result of gradual growth of small transverse perturbations, which most probably appear in computations due to accumulation of rounding errors. It was found that the mean velocity of the DW after the cellular structure formation is slightly higher than the velocity predicted by the one-dimensional theory. This observation is consistent with experimental data.

This work was supported by the Russian Foundation for Basic Research (Grant Nos. 16-57-48007, 18-08-01442, and 18-33-00740).

APPENDIX

ONERA Model

ELEMENTS H O N SPECIES H O H2 O2 OH H2O N2

No.

Reaction

\(A\), mol, cm3, s

\(\beta\)

\(E_{a}/R\), K

1

H2 + O2 \(\to\)2OH

1.700\(\,\cdot \,\)1013

0

24 043.8

H2 + O2 \(\leftarrow\)2OH

4.032\(\,\cdot \,\)1010

0.3168

14 553.8

2

H + O2 \(\to\)OH + O

1.987\(\,\cdot \,\)1014

0

8455.6

H + O2 \(\leftarrow\)OH + O

8.930\(\,\cdot \,\)1011

0.3383

\(-\)117

3

H2 + OH \(\to\)H2O + H

1.024\(\,\cdot \,\)1008

1.6

1659.8

H2 + OH \(\leftarrow\)H2O + H

7.964\(\,\cdot \,\)1008

1.528

9300

4

H2 + O \(\to\)OH + H

5.119\(\,\cdot \,\)1004

2.67

3163.3

H2 + O \(\leftarrow\)OH + H

2.701\(\,\cdot \,\)1004

2.649

2239.6

5

2OH \(\to\)H2O + O

1.506\(\,\cdot \,\)1009

1.14

49.8

2OH \(\leftarrow\)H2O + O

2.22\(\,\cdot \,\)1010

1.089

8613.2

6

H + OH + M \(\to\)H2O + M

2.212\(\,\cdot \,\)1022

\(-\)2.0

0

H + OH + M \(\leftarrow\)H2O + M

8.936\(\,\cdot \,\)1022

\(-\)1.835

59 742.6

H2O/6.5/O2/0.4/N2/1.0/

7

2H + M \(\to\)H2 + M

9.791\(\,\cdot \,\)1016

\(-\)0.6

0

2H + M \(\leftarrow\)H2 + M

5.086\(\,\cdot \,\)1016

\(-\)0.3624

52 104.9

H2/2.5/H2O/12.0/N2/1.0/

Deiterding–Westbrook Model

ELEMENTS H O N SPECIES H O H2 O2 OH H2O HO2 H2O2N2

No.

Reaction

\(A\), mol, cm3, s

\(\beta\)

\(E_{a}\), cal/mol

1

H + O2 \(\to\)O + OH

1.86\(\,\cdot \,\)1014

0

16 790

H + O2 \(\leftarrow\)O + OH

1.48\(\,\cdot \,\)1013

0

680

2

H2 + O \(\to\)H + OH

1.82\(\,\cdot \,\)1010

1.0

8900

H2 + O \(\leftarrow\)H + OH

8.32\(\,\cdot \,\)1009

1.0

6950

3

H2O + O \(\to\)2OH

3.39\(\,\cdot \,\)1013

0

18 350

H2O + O \(\leftarrow\)2OH

3.16\(\,\cdot \,\)1012

0

1100

4

H2O + H \(\to\)H2 + OH

9.55\(\,\cdot \,\)1013

0

20 300

H2O + H \(\leftarrow\)H2 + OH

2.19\(\,\cdot \,\)1013

0

5150

5

H2O2 + OH\(\to\)H2O + HO2

1.00\(\,\cdot \,\)1013

0

1800

H2O2 + OH\(\leftarrow\)H2O + HO2

2.82\(\,\cdot \,\)1013

0

32 790

6

HO2 + O \(\to\)OH + O2

5.01\(\,\cdot \,\)1013

0

1000

HO2 + O \(\leftarrow\)OH + O2

6.46\(\,\cdot \,\)1013

0

56 160

7

HO2 + H \(\to\)2OH

2.51\(\,\cdot \,\)1014

0

1900

HO2 + H \(\leftarrow\)2OH

1.20\(\,\cdot \,\)1013

0

40 100

8

HO2 + H \(\to\)H2 + O2

2.51\(\,\cdot \,\)1013

0

700

HO2 + H \(\leftarrow\)H2 + O2

5.50\(\,\cdot \,\)1013

0

57 800

9

HO2 + OH \(\to\)H2O + O2

5.01\(\,\cdot \,\)1013

0

1000

HO2 + OH \(\leftarrow\)H2O + O2

6.31\(\,\cdot \,\)1014

0

73 860

10

H2O2 + O2 \(\to\)2HO2

3.98\(\,\cdot \,\)1013

0

42 640

H2O2 + O2 \(\leftarrow\)2HO2

1.00\(\,\cdot \,\)1013

0

1000

11

H2O2 + H\(\to\)HO2 + H2

1.70\(\,\cdot \,\)1012

0

3750

H2O2 + H\(\leftarrow\)HO2 + H2

7.24\(\,\cdot \,\)1011

0

18 700

12

H2O + M \(\to\)H + OH + M

2.19\(\,\cdot \,\)1016

0

105 000

H2O + M \(\leftarrow\)H + OH + M

1.41\(\,\cdot \,\)1023

\(-\)2.0

0

H2O/6.5/O2/0.4/N2/1.0/

13

H + O2 + M \(\to\)HO2 + M

1.66\(\,\cdot \,\)1015

0

\(-\)1000

H + O2 + M \(\leftarrow\)HO2 + M

2.29\(\,\cdot \,\)1015

0

45 900

H2O/6.5/O2/0.4/N2/1.0/

14

H2O2 + M\(\to\)2OH + M

1.20\(\,\cdot \,\)1017

0

45 500

H2O2 + M\(\leftarrow\)2OH + M

9.12\(\,\cdot \,\)1014

0

\(-\)5070

H2O/6.5/O2/0.4/N2/1.0/

15

O + H + M \(\to\)OH + M

1.00\(\,\cdot \,\)1016

0

0

O + H + M \(\leftarrow\)OH + M

7.94\(\,\cdot \,\)1019

\(-\)1.0

103 720

H2O/6.5/O2/0.4/N2/1.0/

16

O2 + M \(\to\)2O + M

5.13\(\,\cdot \,\)1015

0

115 000

O2 + M \(\leftarrow\)2O + M

4.68\(\,\cdot \,\)1015

\(-\)0.28

0

H2O/6.5/O2/0.4/N2/1.0/

17

H2 + M \(\to\)2H + M

2.19\(\,\cdot \,\)1014

0

96 000

H2 + M \(\leftarrow\)2H + M

3.02\(\,\cdot \,\)1015

0

0

H2O/6.5/O2/0.4/N2/1.0/

Wilson–MacCormack Model

ELEMENTS H O N SPECIES H O H2 O2 OH H2O HO2 H2O2N2

No.

Reaction

\(A\), mol, cm3, s

\(\beta\)

\(E_{a}\), cal/mol

1

H2 + O2 \(\leftrightarrow\)HO2 + H

1.0\(\,\cdot \,\)1014

0

56 000

2

H + O2 \(\leftrightarrow\)OH + O

2.6\(\,\cdot \,\)1014

0

16 800

3

O + H2 \(\leftrightarrow\)OH + H

1.8\(\,\cdot \,\)1010

1.0

8900

4

OH + H2 \(\leftrightarrow\)H + H2O

2.2\(\,\cdot \,\)1013

0

5150

5

2OH \(\leftrightarrow\)O + H2O

6.3\(\,\cdot \,\)1012

\(-\)2.0

1090

6

H + OH + M \(\leftrightarrow\)H2O + M

2.2\(\,\cdot \,\)1022

\(-\)1.0

0

H2O/6.0/N2/1.0/

7

2H + M \(\leftrightarrow\)H2 + M

6.4\(\,\cdot \,\)1017

\(-\)0.6

0

H2/2.0/H2O/6.0/N2/1.0/

8

H + O + M \(\leftrightarrow\)OH + M

6.0\(\,\cdot \,\)1016

0

0

H2O/5.0/N2/1.0/

9

H + O2 + M \(\leftrightarrow\)HO2 + M

2.1\(\,\cdot \,\)1015

0

\(-\)1000

H2/2.0/H2O/16.0/N2/1.0/

10

2O + M \(\leftrightarrow\)O2 + M

6.0\(\,\cdot \,\)1013

0

\(-\)1800

11

HO2 + H \(\leftrightarrow\)2OH

1.4\(\,\cdot \,\)1014

0

1080

12

HO2 + H \(\leftrightarrow\)H2O + O

1.0\(\,\cdot \,\)1013

0

1080

13

HO2 + O \(\leftrightarrow\)O2 + OH

1.5\(\,\cdot \,\)1013

0

950

14

HO2 + OH \(\leftrightarrow\)H2O + O2

8.0\(\,\cdot \,\)1012

0

0

15

2HO2 \(\leftrightarrow\)H2O2 + O2

2.0\(\,\cdot \,\)1012

0

0

16

H + H2O2 \(\leftrightarrow\)H2 + HO2

1.4\(\,\cdot \,\)1012

0

3600

17

O + H2O2 \(\leftrightarrow\)OH + HO2

1.4\(\,\cdot \,\)1013

0

6400

18

OH + H2O2 \(\leftrightarrow\)H2O + HO2

6.1\(\,\cdot \,\)1012

0

1430

19

H2O2 + M\(\leftrightarrow\)2OH + M

1.2\(\,\cdot \,\)1017

0

45 500

H2O/15.0/N2/1.0/

Petersen–Hanson Model

ELEMENTS H O N SPECIES H O H2 O2 OH H2O HO2 H2O2N2

No.

Reaction

\(A\), mol, cm3, s

\(\beta\)

\(E_{a}\), cal/mol

1

O + H\(_{2} \leftrightarrow\)H + OH

5.00\(\,\cdot \,\)1004

2.7

6290

2

H + O2 + M \(\leftrightarrow\)HO2 + M

2.80\(\,\cdot \,\)1018

\(-\)0.9

0

O2/0/H2O/0/N2/0/

3

H + 2O2 \(\leftrightarrow\)HO2 + O2

3.00\(\,\cdot \,\)1020

\(-\)1.7

0

4

H + O2 + H2O\(\leftrightarrow\)HO2 + H2O

9.38\(\,\cdot \,\)1018

\(-\)0.8

0

5

H + O2 \(\leftrightarrow\)O + OH

8.30\(\,\cdot \,\)1013

0

14 413

6

H + HO2 \(\leftrightarrow\)O2 + H2

2.80\(\,\cdot \,\)1013

0

1068

7

H + HO2 \(\leftrightarrow\)2OH

1.34\(\,\cdot \,\)1014

0

635

8

H + H2O2 \(\leftrightarrow\)HO2 + H2

1.21\(\,\cdot \,\)1007

2.0

5200

9

OH + H2 \(\leftrightarrow\)H2O + H

2.16\(\,\cdot \,\)1008

1.5

3430

10

2OH( + M) \(\leftrightarrow\)H2O2( + M)

7.40\(\,\cdot \,\)1013

\(-\)0.4

0

LOW / 2.30\(\,\cdot \,\)1018 \(-\)0.9\(-\)1 700 /

TROE / 0.7346, 94.0, 1 756.0, 5 182.0 /

H2/2.0/H2O/6.0/N2/1.0/

11

OH + HO2 \(\leftrightarrow\)O2 + H2O

2.90\(\,\cdot \,\)1013

0

\(-\)500

12

OH + H2O2 \(\leftrightarrow\)HO2 + H2O

1.75\(\,\cdot \,\)1012

0

320

13

OH + H2O2 \(\leftrightarrow\)HO2 + H2O

5.80\(\,\cdot \,\)1014

0

9560

14

2HO2 \(\leftrightarrow\)O2 + H2O2

1.30\(\,\cdot \,\)1011

0

\(-\)1630

15

2HO2 \(\leftrightarrow\)O2 + H2O2

4.20\(\,\cdot \,\)1014

0

12 000

16

2O + M \(\leftrightarrow\)O2 + M

1.20\(\,\cdot \,\)1017

\(-\)1.0

0

H2/2.4/H2O/15.4/N2/1.0/

17

O + H + M \(\leftrightarrow\)OH + M

5.00\(\,\cdot \,\)1017

\(-\)1.0

0

H2/2.0/H2O/6.0/N2/1.0/

18

H + OH + M \(\leftrightarrow\)H2O + M

2.20\(\,\cdot \,\)1022

\(-\)2.0

0

H2/0.73/H2O/3.65/N2/1.0/

19

2H + M \(\leftrightarrow\)H2 + M

1.00\(\,\cdot \,\)1018

\(-\)1.0

0

H2/1.7/H2O/7.0/N2/1.0/

20

H + O2 + N2 \(\leftrightarrow\)HO2 + N2

2.60\(\,\cdot \,\)1019

\(-\)1.2

0