Stability of Detonation Waves Propagating in Plane and Rectangular Channels

Abstract

Stability of detonation waves (DWs) propagating in a plane or rectangular channel with respect to two-dimensional and three-dimensional disturbances is considered. Accepting a simple hypothesis that the most unstable mode of the linear theory continues to dominate even in the nonlinear regime, one can derive a number of fairly definite predictions of the developed DW structure from the linear stability theory. In particular, the theory predicts the number of detonation cells formed in a channel of a specified size and the cell size, the minimum size of the channel in which the multifront DW structure can still exist, and the parameters at which the number of cells changes in a jump-like manner. All these predictions are qualitatively consistent with available experimental data and numerical results.

INTRODUCTION

When the multifront (cellular) structure of detonation waves (DWs) was discovered in the late 1950s, an assumption that it is induced by instability of a plane DW to transverse disturbances was put forward almost immediately [1]. The existence of this instability associated with a strong (exponential) dependence of the chemical reaction rate on temperature was first demonstrated for the piecewise-constant main flow (infinitely fast reaction) [2] and then for the main flow described by the one-dimensional Zeldovich–Neumann–Döring solution [3]. In the 1960s, linear stability of DWs was studied in much detail by Erpenbeck [4–7].

The interest to this problem was revived in the 1990–2000s. This fact can be partially explained by prospects of developing engines that can use detonation for fuel combustion [8]. Various engines, such as the pulse detonation engine [9] and the engine continuously rotating in an annular channel [10–12], and their experimental prototypes are now actively tested. Theoretically, they offer a number of essential advantages over conventional internal combustion engines. In particular, it is the use of detonation that can lead to creation of microscale engines [13] because usual internal combustion engines based on slow combustion have inacceptably high heat losses through the surface if they are reduced to millimeter (or even smaller) scales.

In 1990, an important paper [14] was published, where a simpler and more physically transparent method was used instead of Erpenbeck’s technically complicated approach based on the Laplace transform, which did not allow direct construction of neutral stability curves. Many investigations based on this method were later performed; a review of these studies can be found in [8].

Despite the long story of its development, the status of the DW stability theory differs significantly from that of the hydrodynamic stability theory of shear flows. In the latter case, the important role of linear instability in the transition to turbulence and in the formation of a developed turbulent flow is obvious. The conclusions of the hydrodynamic stability theory have been supported many times by direct experiments and numerical simulations; it forms the basis for engineering methods predicting the laminar–turbulent transition.

The situation with the DW stability theory is essentially different. Though the problem formulation and the result obtained raise no doubts, they have been confirmed neither by experiments nor by direct numerical simulations. In fact, the role of linear instability in the formation of the unsteady multifront DW structure remains unclear for several reasons. First, all investigations of DW stability were performed with the simplest model of chemical kinetics including one irreversible reaction that does not describe chemical conversions in any real reacting mixture and that is a purely model reaction. Naturally, direct comparisons of results with experimental data is impossible. Second, a plane wave propagating in an infinite space is usually considered in stability studies, whereas the effects caused by the presence of solid boundaries, which are always present in experiments, are ignored.

An important tool for finding the relationship between the linear stability theory and the unsteady multidimensional DW structure can be computational experiments: currently available supercomputers allow even three-dimensional computations of DW propagation with both one irreversible reaction and a detailed chemical model. Thus, there arises a problem that we try to solve in the present work: the idea is to study DW stability in sufficiently simple, but realistic geometry and to obtain possible predictions that could be tested by means of direct numerical simulations.

For this purpose, we consider stability of a DW propagating in a rectangular channel (in a plane channel as a particular case). We consider those features that follow from the presence of solid walls bounding the wave in the transverse directions. The characteristics of the multidimensional DW structure are predicted on the basis of the following simple hypothesis: the unstable mode with the greatest growth rate predicted by the linear theory continues to dominate even in the developed nonlinear regime. This assumption is based on the fact that disturbances grow exponentially in the linear regime, so that the mode whose growth rate is (even slightly) greater than the growth rate of other modes increases noticeably faster than other disturbances and is the first mode that enters the nonlinear stage of development. At least, the formulated hypothesis seems to be fairly reasonable for conditions where only one mode is unstable and the interval of unstable wave numbers is sufficiently narrow. As a whole, however, the validity of this assumption must be certainly verified through comparisons of predictions that follow from this hypothesis with experimental results and numerical simulations. Such comparisons can be the subject of further investigations.

FORMULATION OF THE PROBLEM OF DETONATION WAVE STABILITY

Let us consider a DW described by the Euler equations supplemented with a simple model of chemical kinetics widely used in theoretical and numerical investigations of detonation:

$$\frac{\partial \rho \lambda }{\partial t} + \nabla \cdot \rho \lambda {\mathbf{u}} = \rho \omega,\qquad \omega = K( {1 - \lambda } )\exp\Big( - \frac{E_a}{T}\Big).$$

(1)

Here $t$ is the time, $\mathbf{u}$ is the gas velocity, $\rho$ and $T$ are the gas density and temperature, respectively, $\lambda$ is the variable that describes the degree of reaction completeness ($\lambda$ = 0 and 1 correspond to mixtures where the chemical reaction has not yet started and has completed, respectively), $E_{a}$ is the reaction activation energy, and $K$ is the pre-exponential factor.

The total energy of the unit volume of the mixture in this model has the form

$$ E = \frac{p}{\gamma - 1} + \frac{1}{2}\,\rho {\mathbf{u}}^2 - Q\rho \lambda,$$

(2)

where $p = \rho T$ is the pressure, $\gamma= 1.2$ is the ratio of specific heats assumed to be constant, and $Q$ is the specific heat release.

All equations in the paper are written in the dimensionless form. The coordinates are normalized to the reaction zone half-width$L_{1 / 2}^\ast$, i.e., to the distance from the leading front of the DW to the point where $\lambda = 1 / 2$; the density and velocity are normalized to the gas density $\rho _0^\ast$ and velocity of sound $c_0^\ast$ in the state ahead of the DW front; the time is normalized to $L_{1 / 2}^\ast / c_0^\ast$; the pressure and temperature are normalized to $\rho _0^\ast c_0^{\ast 2}$ and $c_0^{\ast 2} / R^\ast$, respectively; the activation energy and heat release are normalized to $c_0^{\ast 2}$. Here $R^\ast$ is the specific gas constant, the zero subscript refers to the state ahead of the DW front, and the asterisk superscript is used to indicate dimensional variables.

The one-dimensional solution of the Euler equations for a chemically reacting mixture, which describes the distribution of the parameters in a plane DW, is known as the Zeldovich–Neumann–Döring (ZND) solution. For the model considered here, it is written in quadratures (hereinafter, we use the reference system where the undisturbed DW front is at rest):

$$ x = \int\limits_{0}^\lambda {\exp} \bigg[ {\frac{E_a /u}{p_{0} / m + ( {u_{0} - u} )}} \bigg] \frac{ud\lambda}{K( {1 - \lambda } )},$$

(3)

$$ u( \lambda ) = \frac{1}{\gamma + 1}\bigg[\bigg(\frac{c_0^2 }{u_0 } + \gamma u_0 \bigg) -\sqrt {\bigg(u_0 - \frac{c_0^2 }{u_0 }\bigg)^2 - 2({\gamma ^2 - 1} )Q\lambda }\bigg],$$

(4)

$$ m = \rho _{0} u_{{0}}, \quad \rho = m / u, \quad p = p_{0} +m( {u_{0} - u} ).$$

(5)

The value of the pre-exponent $K$ is determined by the condition $\lambda = 1 / 2$ at the point $x = 1$ . Choosing the sign at the root in Eq. (4), we reject solutions that describe weak detonation, which cannot exist for thermodynamic reasons [15]. The limiting regime where the radicand vanishes corresponds to a self-sustained DW (Chapman–Jouguet detonation), and the DW Mach number in this case is

$$ \mbox{M}_{\rm{CJ}} = \sqrt {\frac{( {\gamma ^2 - 1})Q}{2c_{0}^2 }} + \sqrt {\frac{( {\gamma^2 - 1} )Q}{2c_{0}^2 } + 1}.$$

(6)

An overdriven DW pushed by a piston propagates with a Mach number greater than $\mbox{M}_{\rm CJ}$, and the overdrive factor is determined by the ratio $f = (\mbox{M} / \mbox{M}_{\rm{CJ} } )^2$. Thus, the ZND solution is completely characterized by three parameters: $E_a$, $Q$, and $f$.

In studying stability of a plane DW, the Euler equations are linearized around solution (3)–(5). In addition to perturbations of the gas-dynamic variables, the shape of the wave front is also distorted. If the wave front shape is defined as $x = \psi ({y,z,t} )$, it is convenient to use a curvilinear coordinate system and introduce a new independent variable $\xi = x - \psi( {y,z,t} )$ instead of $x$. All gas-dynamic variables and the wave front shape are written as

$$ q(\xi,y,z,t)=\bar {q}( \xi ) + \hat {q}(\xi ) \exp\,(\alpha t)\exp\,(i(k_y y + k_z z)),$$

(7)

$$ \psi (y,z,t) = 0 + \hat {\psi }\exp\,(\alpha t) \exp\,(i(k_y y+ k_z z)),$$

(8)

where $\bar {q}( \xi )$ is the undisturbed flow defined by the ZND solution, $\hat {q}( \xi )$ and $\hat {\psi }$ are the complex amplitude of disturbances, $k_y$ and $k_z$ are the wave numbers along the corresponding axes, and$\alpha = \alpha _r + i\alpha _i$ is a complex parameter whose real part is the disturbance growth rate coefficient. Substituting Eqs. (7) and (8) into the equations of motion and retaining only linear terms with respect to the disturbance amplitude, we obtain the following system of equations:

$$ \mathbf{A}\frac{d\mathbf{U}}{d\xi } + \mathbf{K}\mathbf{U} = \hat {\psi }\mathbf{G},$$

(9)

$${} \mathbf{A} = \left(\begin{array}{cccccc} \bar {u} & { - \bar {\beta }} & 0 & 0 & 0 & 0 \\[1mm] 0 & \bar {u} & 0 & 0 & \bar {\beta } & 0 \\[1mm] 0 & 0 & \bar {u} & 0 & 0 & 0 \\[1mm] 0 & 0 & 0 & \bar {u} & 0 & 0 \\[1mm] 0 & {\gamma \bar {p}} & 0 & 0 & \bar {u} & 0 \\[1mm] 0 & 0 & 0 & 0 & 0 & \bar {u} \end{array}\right),$$

$${} \mathbf{U}= \left(\begin{array}{c} \hat {\beta } \\[1mm] \hat {u} \\[1mm] \hat {v} \\[1mm] \hat {w} \\[1mm] \hat {p} \\[1mm] \hat {\lambda } \end{array}\right), \quad \mathbf{G} = \left(\begin{array}{c} {\alpha \bar {\beta }_\xi } \\[1mm] {\alpha \bar {u}_\xi } \\[1mm] {ik_y \bar {\beta }\bar {p}_\xi } \\[1mm] {ik_z \bar {\beta }\bar {p}_\xi } \\[1mm] {\alpha \bar {p}_\xi } \\[1mm] {\alpha \bar {\lambda }_\xi } \end{array} \right),$$

$${} \mathbf{K} = \left(\begin{array}{cccccc} \alpha - \bar {u}_\xi & \bar {\beta }_\xi &- ik_y \bar {\beta } & - ik_z \bar {\beta } & 0 & 0 \\[1mm] \bar {p}_\xi & \alpha + \bar {u}_\xi & 0 & 0 & 0 & 0 \\[1mm] 0 & 0 & \alpha & 0 & ik_y \bar {\beta } & 0 \\[1mm] 0 & 0 & 0 & \alpha & ik_z \bar {\beta } & 0 \\[1mm] - \bar {\sigma }_\beta & \bar {p}_\xi & ik_y \gamma \bar {p} & ik_z \gamma \bar {p} & \alpha + \gamma \bar {u}_\xi - \bar {\sigma }_p & - \bar {\sigma }_\lambda \\[1mm] - \bar {\omega }_\beta & \bar {\lambda }_\xi & 0 & 0 & - \bar {\omega }_p & \alpha - \bar {\omega }_\lambda \end{array}\right).$$

Here $\beta=1 / \rho$ is the specific volume and $\sigma=(\gamma - 1)\rho Q\omega$; the subscripts $\xi$, $\beta$, $p$, and $\lambda$ at the variables that refer to the main flow mean the derivatives with respect to the corresponding variables.

System (9) has to be supplemented with boundary conditions. The boundary conditions at the wave front at $\xi = 0$ are obtained by means of linearization of the Rankine–Hugoniot relations, which yields the following expressions (the subscript $s$ is used for the parameters immediately behind the shock wave):

$${} \hat {\beta }_s = \frac{4}{\gamma + 1}\frac{1}{\mbox{M}^2}\,\alpha \hat {\psi},\quad \hat {u}_s = \frac{2}{\gamma + 1}\frac{\mbox{M}^2 + 1}{\mbox{M}^2}\,\alpha\hat {\psi },$$

$$ \hat {v}_s = ik_y ( {\bar {u}_0 - \bar {u}_s })\hat {\psi },\quad \hat {w}_s = ik_z ( {\bar {u}_0 - \bar {u}_s } )\hat {\psi },$$

(10)

$${} \hat {p}_s = - \frac{4\bar {\rho }_0 \bar {u}_0 }{\gamma + 1}\,\alpha \hat{\psi },\quad \hat {\lambda }_s = 0.$$

The boundary condition for $\xi \to \infty$ is the absence of disturbances (acoustic waves) propagating from infinity toward the shock wave (because the problem of stability of the DW itself, without any external disturbances, is solved). The corresponding condition can be obtained by considering system (9) at large values of $\xi$, where all coefficients in the equations become constant. Then, the asymptotic solutions have the form $\exp\,(\kappa \xi)$, where $\kappa$ are the eigenvalues of the matrix $\mathbf{A}^{ - 1}\mathbf{K}$. One of these eigenvalues yields a solution growing at infinity; in fact, it corresponds to an acoustic wave arriving from infinity and decaying when approaching the wave front. To avoid such disturbances, the solution has to be subjected to the condition of its orthogonality to the eigenvector corresponding to this eigenvalue.

Using a simplified assumption that the chemical reaction is completed, i.e., $\lambda=1$, we obtain the equation

$$ F( \alpha ) = \alpha \hat {u} - ik_y \bar {u}_\infty \hat {v} -ik_z \bar {u}_\infty \hat {w} {} + \frac{\bar {\beta }_\infty }{\bar {c}_\infty}\, \sqrt {\alpha ^2 + ( {k_y^2 + k_z^2 } )( {\bar {c}_\infty^2 - \bar {u}_\infty ^2 } )} \hat {p} = 0$$

(11)

(the subscript $\infty$ refers to the ZND solution as $\xi \to \infty$). System (9) with the boundary conditions (10) and (11) forms an eigenvalue problem with respect to the parameter $\alpha$. Its solution determines the plane DW stability characteristics: the DW is unstable at $\mathrm{Re}\,(\alpha)>0$.

STABILITY OF THE DETONATION WAVE IN AN INFINITE SPACE

For any prescribed values of $k_y$ and $k_z$, system (9), which describes the DW stability to three-dimensional disturbances in an infinite space, can be reduced to a simpler system for two-dimensional disturbances. For this purpose, it is sufficient to turn the coordinate system in the plane $y$,$z$ and introduce new coordinates $\tilde {y}$ and $\tilde {z}$ so that the direction of the axis $\tilde{y}$ should coincide with the direction of the vector $(k_y,k_z)$. Then, the transverse wave number of the two-dimensional disturbance is equal to the length of this vector: $k = \sqrt {k_y^2 + k_z^2}$.

In the numerical solution of the stability problem, we choose the initial approximation for $\alpha$, which allows us to use conditions (10) for determining the initial values of disturbances at $\xi = 0$(the DW front disturbance amplitude $\hat {\psi }$ in the linear problem is arbitrary and it taken to be equal to zero in what follows for simplicity). Then system (9) is integrated from $\xi = 0$ to some sufficiently large value $\xi = L_\xi$(chosen so that all parameters of the main flow should definitely reach constant values). After that, it is necessary to satisfy condition (11) at $\xi = L_\xi$. As this condition is obviously not satisfied for an arbitrary value of $\alpha$, the equation $F(\alpha ) = 0$is solved by iterations (the Newton iterative method is used), and all above-described actions are repeated at each iteration until the solution converges.

The initial approximation for iterations by the Newton method can be found in the following way. A sufficiently large domain of the complex plane $( {\alpha _r,\alpha _i } )$is covered by a fine mesh consisting of points where the values of the function $F( \alpha ) = 0$ are calculated. Those places of the complex plane where both the real and imaginary parts of the function $F$ vanish more or less simultaneously yield the initial approximations for finding $\alpha$.

Fig. 1

Growth rate coefficients of disturbances for $Q = 50$, $E_a = 50$, and $f=1.2$.

A typical result of such a study of stability is illustrated in Fig. 1, which shows the growth rate coefficient of disturbances as a function of the wave number $k$ for $Q = 50$, $E_a = 50$, and $f =1.2$. System (9) was integrated by the fourth-order Runge–Kutta method; the mesh on the interval $[0,L_\xi]$, $L_\xi = 6$, contained 1600 points.

It is seen from Fig. 1 that nine modes of disturbances are unstable for these parameters. There is an upper boundary of existence of unstable disturbances $k_\mathrm{stab}$(in the present case, $k_\mathrm{stab} =12.675$); thus, all disturbances with $k \ge k_\mathrm{stab}$ are steady. It is also seen that the greatest growth rate coefficient is observed for mode 2; the wave number of the corresponding disturbance is $k_{\max}=1.79$. The first five modes remain unstable at $k = 0$. This means that the flow is unstable to one-dimensional disturbances at which the wave front remains plane, but oscillates back and forth with respect to the equilibrium position.

The number of unstable modes and their growth rate coefficients vary depending on the activation energy, heat release, and overdrive factor. Thus, there are 20 stable modes for $Q = 50$, $E_a = 50$, and $f=1$(Chapman–Jouguet detonation). The DW stability characteristics for various values of the parameters $Q$, $E_a$, and $f$ were studied in much detail by Lee and Stewart [16]; our results agree very well with their data.

STABILITY OF THE DETONATION WAVE IN A PLANE CHANNEL

Let us consider the stability of the DW propagating in a plane channel of width $L$. The presence of the solid walls means that the velocity component normal to the walls has to vanish on the channel walls. Thus, the wave number in the corresponding direction can take only a series of discrete values: $k = \pi n / L$($n = {0,1,2,}\,\ldots$). We are interested only in those disturbances whose wave numbers fall within the domain of existence of unstable modes: from $k$ = 0 to $k=k_\mathrm{stab}$. Obviously, for a sufficiently small size of the channel, even the disturbance wave number $k_1 = \pi / L$ with $n=1$ exceeds $k_\mathrm{stab}$, so that it is stable. In this case, all unstable disturbances reduce to one-dimensional oscillations with $k=0$ , which means that the cellular structure cannot exist in channels with $L < \pi / k_\mathrm{stab}$ . This conclusion is in complete agreement with the experimental observation that the cellular DW structure does not exist in narrow channels.

If now the channel size is increased, more and more harmonics of the fundamental wave number $k_1$ fall into the instability region. One of these harmonics having the number $n = N$ has the greatest growth rate coefficient. As the channel width is increased, the value of $N$ also increases; in a sufficiently wide channel, the wave number of the corresponding harmonic is close to $k_{\max}$, and the number $N$ itself is approximately equal to $k_{\max} L / \pi$.

Fig. 2

Number of transverse disturbances (doubled number of detonation cells) (a) and transverse size of detonation cells (b) in a plane channel versus the channel width ($Q = 50$, $E_a = 50$, and $f=1.2$).

Let us now take into account the assumption formulated in Introduction that the multifront DW properties are determined by the disturbance with the maximum growth rate coefficient even in the nonlinear regime. Then we can immediately predict the transverse size of the detonation cell $a$ and the number of cells formed in the channel of a prescribed size. As is shown in Fig. 1, it is sufficient for this purpose to design a discrete set of admissible wave numbers and to determine which of them corresponds to the maximum growth rate coefficient (see the points of intersection of the vertical lines in Fig. 1 with the bold envelope line).

Figure 2 shows the thus-obtained values of $N$ and the detonation cell sizes $a = 2L / N$ as functions of the channel width. It should be noted that the number of full detonation cells is equal to one half of the value of $N$: DWs with a half-integer number of cells can exist in the channel (which is again consistent with experimental observations). It follow from Fig. 2 that the cellular structure “adapts" to changes in the channel size by means of changing the transverse cell size and retaining the number of cells. However, this is only possible within certain limits: at a certain time, the growth rate coefficients of two harmonics become identical; at this time, the cell size changes in a jump-like manner, and the number of cells also changes (by “one half" of the cell). This behavior is also in agreement with experimental observations.

Concerning the detonation cell size, it is seen from Fig. 2b that it can change within fairly wide limits. As the channel size is increased, however, the cell size approaches a certain constant value. This value $a_\infty = 2\pi / k_{\max}$ is completely determined by $k_{\max}$, which is the value of the wave number corresponding to the maximum growth rate coefficient of disturbances.

STABILITY OF THE DETONATION WAVE IN A RECTANGULAR CHANNEL

In a rectangular channel of width $L_y$ and height $L_z$, the boundary conditions impose “rules" for choosing wave numbers in both transverse directions:

$${} k_y = \pi n / L_y\quad (n = 0,\,1,\,2,\,\ldots) ;$$

$${} k_z = \pi m / L_z\quad (m = 0,\,1,\,2,\,\ldots) .$$

The corresponding values are shown in Fig. 3 as a rectangular grid with light points. However, as three-dimensional equations of DW stability can be reduced to two-dimensional equations with the wave number $k = \sqrt {k_y^2 + k_z^2 }$, the growth rate coefficient (shown by gray scale hues in Fig. 3) depends only on the distance from the origin. In particular, the stability boundary $k=k_\mathrm{stab}$ is a circumference (black solid curve in Fig. 3), similar to the set of points corresponding to the maximum growth rate (dashed curve).

Fig. 3

Growth rate coefficients of three-dimensional disturbances for $Q = 50$, $E_a=50$, and $f=1.2$.

Further considerations are similar to those for a plane channel. In this case, the plot of the growth rate coefficients on the transverse wave numbers $k_y$ and $k_z$ is a surface. On this surface, we choose points forming a grid of admissible values of the wave numbers. The point lying on this surface higher than other points corresponds to the disturbance with the maximum growth rate coefficient. In Fig. 3, this disturbance is located somewhere near the dashed curve.

Using this procedure, we obtain the maximum growth rate coefficient as a function of the channel sizes (Fig. 4).

Fig. 4

Maximum growth rate coefficient of three-dimensional disturbances versus the channel sizes ($Q=50$, $E_a=50$, and $f=1.2$).

Fig. 5

Map of detonation regimes in rectangular channels for $Q = 50$, $E_a=50$, and $f=1.2$.

At small values of $L_y$ and $L_z$, there is a domain with no unstable disturbances (except for, possibly, one-dimensional disturbances). This fact again testifies to the existence of some minimum size of the channel necessary for the formation of the multifront structure.

As the channel width and height are increased, the maximum growth rate coefficient changes in a complicated and even intricate manner. For channels with large sizes in both directions, it approaches the maximum value for a plane DW propagating in an infinite space $\alpha_{\max} = \alpha _r (k_{\max} )$.

For prescribed values of $L_{y}$ and $L_{z}$, it is possible to find the numbers $N$ and $M$ determining the number of the most unstable harmonic. The entire plane ($L_{y},L_{z}$) is divided into domains corresponding to different values of $N$ and $M$(Fig. 5, the values of $N$ and $M$ for some domains are given). Recalling the hypothesis about the dominating character of the most unstable mode again, we can treat this figure as a map of detonation regimes determining the number of detonation cells and their size in both directions in the case of DW propagation in a rectangular channel.

DISCUSSION

We demonstrated that a simple hypothesis, which implies that the most unstable mode of the linear theory continues to dominate even in the developed nonlinear regime, makes it possible to correlate the data of the linear theory of DW stability with the characteristics of the multifront DW structure. In particular, one can predict the number and sizes of detonation cells formed in the case of DW propagation in a plane or rectangular channel of prescribed sizes and reproduce the typical features of changes in the multifront structure induced by changes in the channel sizes. Thus, it becomes possible to design a map of detonation regimes in rectangular channels.

It should be noted that a large number of theoretical models for predicting the detonation cell sizes have been proposed during the long story of investigations in this field; a detailed analysis of these models can be found in [17]. One of the most popular models is that developed by Barthel [18], which is based on considering the propagation of small perturbations (acoustic waves) emanating from the leading front in a nonuniform flow behind the DW front. The consideration is performed within the framework of geometrical acoustics, which is valid for short-wave acoustic perturbations. Comparisons with experimental data for hydrogen–oxygen mixtures diluted with argon [18] showed that the cell sizes predicted by the model are approximately twice greater than the experimentally observed values. In later studies [19, 20], however, the predictions of this model were found to be in good agreement with the results of numerical simulations of heterogeneous detonation in a mixture of aluminum particles with oxygen. This contradiction can be probably explained by the noticeable differences of gas detonation from heterogeneous detonation: in the latter case, the processes are less “vigorous" and closer to the linear regime.

We cannot fail to notice that the model developed in the present paper has many common features with the model of [18]. Propagation of small disturbances is considered in both models: on the basis of geometrical acoustics in one case and within the framework of the exact linear wave theory in the other case. However, the condition used to determine the cell size is essentially different. In the acoustic approach [18], the transverse cell size is taken as the distance along the front from the initial point to the point where the signal returning to the wave front within the shortest time arrives. In our case, the transverse cell size is understood as the wave length of the disturbance with the maximum growth rate coefficient.

The validity of the model can be quantified through comparisons of its predictions either with experimental data or with numerical simulation results. Comparisons with experiments are difficult because the simple model with one irreversible chemical reaction used to study DW stability does not describe any real chemically reacting mixture. The situation could have been improved by performing a stability study for a more realistic model. However, such studies have not been performed yet. Gorshkov et al. [21] formulated a system of stability equations for a detailed model of chemical kinetics, but no computations for any known chemical mechanism were reported (apparently, because these equations were too complicated).

Another method for verification of the formulated assumption is to perform direct numerical simulations of DW propagation in channels of various sizes on the basis of the full nonlinear Euler equations, determine the sizes of detonation cells being formed, and compare them with model predictions. Such a study is planned for the future.

The considered hypothesis is indirectly supported by the results of Taylor et al. [22], who modeled DW propagation in a channel at $Q=0.4$, $E_a=50$, and $f=1.2$. Four modes turned out to be unstable under these conditions, and mode 2 has the greatest growth rate coefficient. The growth of disturbances in the linear regime, the nonlinear stage of their development, and finally the formation of a developed cellular structure were reproduced. The amplitude of mode 2 was found to be noticeably greater than the amplitudes of other modes of disturbances even at the end of the computation.

The authors would like to express their gratitude to A. V. Trotsyuk (Lavrentyev Institute of Hydrodynamics of the Siberian Branch of the Russian Academy of Sciences) for the interest in this study and to D. V. Davidenko (ONERA, France) who turned our attention to Ref. [22].

This work was supported by the Russian Foundation for Basic Research (Grant No. 18-33-00740).