Reflection of a spherical blast wave from a planar surface

Numerous authors have carried out rather extensive studies in the last twenty to thirty years of the problem of the interaction of shock and blast waves with obstacles in their paths. Owing to the complexity of the problem, they assumed certain limiting cases for the shock wave interactions in which the parameters behind the shock wave were usually taken to be constants. The first wave diffraction studies involving variable parameters behind the front were presented in [1, 2], wherein a development of the theory of “short waves” (blast waves at a substantial distance from the center of an explosion) and their reflection from a planar surface was given. The theory of short waves assumes that the jump in pressure at the wave front and the region over which the parameters vary are small. The problem concerning reflection of a blast wave from a surface was also considered in [3, 4], wherein a solution in the region behind the reflected wave was obtained at initial times. The initial stage in the reflection of a blast wave from a planar, cylindrical, or spherical surface (the one-dimensional case) was studied in [5]. In this paper we investigate the interaction of a spherical blast wave, resulting from a point explosion, with a planar surface; we consider both regular and non-regular reflection stages. In solving this problem we use S. L. Godunov's finite-difference method. We obtain numerical solutions for various values of the shock strength at the instant of its encounter with the surface. We present the pressure fields in the flow regions, the pressure distribution over the surface at various instants of time, and the trajectories of the triple point. The parameter values at the front of the reflected wave are compared with results obtained from the theory of regular reflection of shock waves.     

Regular reflection of a magnetohydrodynamic shock wave from a conducting wall

In two-dimensional supersonic gasdynamics, one of the classical steady-state problems, which include shock waves and other discontinuities, is the problem concerning the oblique reflection of a shock wave from a plane wall. It is well known [1–3] that two types of reflection are possible: regular and Mach. The problem concerning the regular reflection of a magnetohydrodynamic shock wave from an infinitely conducting plane wall is considered here within the scope of ideal magnetohydrodynamics [4]. It is supposed that the magnetic field, normal to the wall, is not equal to zero. The solution of the problem is constructed for incident waves of different types (fast and slow). It is found that, depending on the initial data, the solution can have a qualitatively different nature. In contrast from gasdynamics, the incident wave is reflected in the form of two waves, which can be centered rarefaction waves. A similar problem for the special case of the magnetic field parallel to the flow was considered earlier in [5, 6]. The normal component of the magnetic field at the wall was equated to zero, the solution was constructed only for the case of incidence of a fast shock wave, and the flow pattern is similar in form to that of gasdynamics. The solution of the problem concerning the reflection of a shock wave constructed in this paper is necessary for the interpretation of experiments in shock tubes [7–10].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 102–109, May–June, 1977.The author thanks A. A. Barmin, A. G. Kulikovskii, and G. A. Lyubimov for useful discussion of the results obtained.     

Supersonic flow of combustible gas mixture past sphere with account for ignition delay time

Assume an axisymmetric blunt body or a symmetric profile is located in a uniform supersonic combustible gas mixture stream with the parameters M₁, p₁, and T₁. A detached shock is formed ahead of the body and the mixture passing through the, shock is subjected to compression and heating. Various flow regimes behind the shock wave may be realized, depending on the freestream conditions. For low velocities, temperatures, or pressures in the free stream, the mixture heating may not be sufficient for its ignition, and the usual adiabatic flow about the body will take place. In the other limiting case the temperature behind the adiabatic shock and the degree of gas compression in the shock are so great that the mixture ignites instantaneously and burns directly behind the shock wave in an infinitesimally thin zone, i. e., a detonation wave is formed. The intermediate case corresponds to the regime in which the width of the reaction zone is comparable with the characteristic linear dimension of the problem, for example, the radius of curvature of the body at the stagnation point.The problem of supersonic flow of a combustible mixture past a body with the formation of a detonation front has been solved in [1, 2]. The initial mixture and the combustion products were considered perfect gases with various values of the adiabatic exponent .These studies investigated the effect of the magnitude of the reaction thermal effect and flow velocity on the flow pattern and the distribution of the gasdynamic functions behind the detonation wave.In particular, the calculations showed that the strong detonation wave which is formed ahead of the sphere gradually transforms into a Chapman-Jouguet wave at a finite distance from the axis of symmetry. For planar flow in the case of flow about a circular cylinder it is shown that the Chapman-Jouguet regime is established only asymptotically, i. e., at infinity.This result corresponds to the conclusions of [3, 4], in which a theoretical analysis is given of the asymptotic behavior of unsteady flows with planar, spherical, and cylindrical detonation waves.Available experimental data show that in many cases the detonation wave does not degenerate into a Chapman-Jouguet wave as it decays, bur rather at some distance from the body it splits into an adiabatic shock wave and a slow combustion front.The position of the bifurcation point cannot be determined within the framework of the zero thickness detonation front theory [1], and for the determination of the location of this point we must consider the structure of the combustion zone in the detonation wave. Such a study was made with very simple assumptions in [5].The present paper presents a numerical solution of the problem of combustible mixture flow about a sphere with a very simple model for the structure of the combustion zone, in which the entire flow behind the bow shock wave consists of two regions of adiabatic flow-an induction region and a region of equilibrium flow of products of combustion separated by the combustion front in which the mixture burns instantaneously. The solution is presented only for subsonic and transonic flow regions.     

Normal interaction of shock waves in the presence of vibrational relaxation

The effect of nonequilibrium physicochemical processes on the flow resulting from the normal collision and reflection of shock waves is studied by the example of nonequilibrium excitation of molecular oscillations in nitrogen. It is shown that the thermal effect of vibrational relaxation is small and the problem can be linearized around a known solution [1]. A similar approach to the solution of the problem of flow around a wedge and certain one-dimensional non-steady-state problems was used earlier in [2–4]. The solution of these problems was constructed in an angular domain, bounded by the shock wave and a solid wall (or the contact surface) and was reduced to a well-known functional equation [6]. The solution of this problem, because of the presence of two angular domains divided by a tangential discontinuity, reduces to a functional equation of more general form than in [6]. The results are obtained in finite form. In the special case of shocks of equal intensity, the normal reflection parameters are obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 90–96, July–August, 1976.     

Gas content factor in the interaction of shock waves of different intensity in a two-phase gas-liquid medium

The transition from regular to Mach interaction is investigated in connection with the interaction of two plane weak or moderate shock waves of different intensity in a two-phase gas-liquid medium over the entire range of gas contents. A nonmonotonic dependence of the transition limit and the flow parameters on the gas content is detected. The investigation extends the results of [1] corresponding to the reflection of a shock wave from a wall. At intermediate gas contents in the case of opposing shock waves, analogous to the normal reflection of a shock wave from a solid wall, the results are in agreement with [2]. In the case of weak shock waves non-linear asymptotic expansions [3] are employed. In the extreme cases of single-phase media the results coincide with the findings of [3, 4]. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 172–174, November–December, 1986.     

Effect of reflection type on detonation initiation at shock-wave focusing

Abstract. From practical and theoretical standpoints, the initiation of combustion in gaseous media due to the shock waves focusing process at various reflectors is a subject of much current interest. The complex gas flowfield coupled with chemical kinetics provides a wide spectrum of possible regimes of combustion, such as fast flames, deflagration, detonation etc. Shock wave reflection at concave surfaces or wedges causes converging of the flow and produces local zones with extremely high pressures and temperatures. The present work deals with the initiation of detonation due to shock waves focusing at parabolic and wedge reflectors. Particular attention has been given to the determination of the critical values of the incident shock wave (ISW) Mach number, parameters of the combustible mixture, and geometrical sizes of reflector at which different combustion regimes could be generated. Received 30 August 1999 / Accepted 23 February 2000     

Supersonic flow of a combustible gas mixture past a sphere

In recent years considerable interest has developed in the problems of steady-state supersonic flow of a mixture of gases about bodies with the formation of detonation waves and slow combustion fronts. This is due in particular to the problem of fuel combustion in a supersonic air stream.In [1] the problem of supersonic flow past a wedge with a detonation wave attached to the wedge apex is solved. This solution is based on using the equation of the detonation polar obtained in [2]-the analog of the shock polar for the case of an exothermic discontinuity. In [3] a solution is given of the problem of cone flow with an attached detonation wave, and [4] presents solutions of the problems of supersonic flow past the wedge and cone with the formation of attached adiabatic shocks with subsequent combustion of the mixture in slow combustion fronts. In the two latter studies two different solutions were also found for the problem of flow past a point ignition source, one solution with gas combustion in the detonation wave, the other with gas combustion in the slow combustion front following the adiabatic shock. These solutions describe two different asymptotic pictures of flow of a combustible gas mixture past bodies.In an experimental study of the motion of a sphere in a combustible gas mixture [5] it was found that the detonation wave formed ahead of the sphere splits at some distance from the body into an ordinary (adiabatic) shock and a slow combustion front. Arguments are presented in [6] which make it possible to explain this phenomenon and in certain cases to predict its occurrence.The present paper presents examples of the calculation of flow of a combustible gas mixture past a sphere with a detonation wave in the case when the wave does not split. In addition, the flow near the point at which the detonation wave splits is analyzed for the case when splitting occurs where the gas velocity behind the wave is greater than the speed of sound. This analysis shows that in the given case the flow calculation may be carried out without any particular difficulties. On the other hand, the calculation of the flow for the case when the point of splitting is located in the subsonic portion of the flow behind the wave (or in the region of influence of the subsonic portion of the flow) presents difficulties. This flow case is similar to the problem of the supersonic jet of finite width impacting on an obstacle.     

Structure of detonation waves in gas suspensions of fuel containing the oxidant

The structure of detonation waves in air suspensions of unitary fuels (fuels containing an oxidant such as gunpowder and high explosives) is investigated. In such systems, complete combustion of the particles is possible at a high mass concentration of the fuel. As a result, the structure of detonation differs from that in gas-drop [1–3] and gas [4, 5] mixtures. The shock adiabats characteristic for air suspensions [6, 7] are used to investigate the field of integral curves which describe the structure of detonation waves in disperse media. Calculated distributions of the parameters which characterize the gas and particles in the detonation front are given. The influence of the rate of combustion of the particles and the intensity of interphase friction on the structure of the detonation is investigated. Results of the calculation of the structure of relaxation shock waves in gas suspensions of the solid fuel of rockets are given in [8]. Unsteady problems of convective combustion and the transition of combustion of air suspensions into detonation are analyzed in [9, 10].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 47–53, September–October, 1981.     

Supersonic motion of bodies in a gas with shock waves

The authors consider the problem of supersonic unsteady flow of an inviscid stream containing shock waves round blunt shaped bodies. Various approaches are possible for solving this problem. The parameters in the shock layer on the axis of symmetry have been determined in [1, 2] by using one-dimensional theory. The authors of [3, 4] studied shock wave diffraction on a moving end plane and wedge, respectively, by the through calculation method. This method for studying flow around a wedge with attached shock was also used in [5]. But that study, unlike [4], used self-similar variables, and so was able to obtain a clearer picture of the interaction. The present study gives results of research into the diffraction of a plane shock wave on a body in supersonic motion with the separation of a bow shock. The solution to the problem was based on the grid characteristic method [6], which has been used successfully to solve steady and unsteady problems [7–10]. However a modification of the method was developed in order to improve the calculation of flows with internal discontinuities; this consisted of adopting the velocity of sound and entropy in place of enthalpy and pressure as the unknown thermodynamic parameters. Numerical calculations have shown how effective this procedure is in solving the present problem. The results are given for flow round bodies with spherical and flat (end plane) ends for various different values of the velocities of the bodies and the shock waves intersected by them. The collision and overtaking interactions are considered, and there is a comparison with the experimental data.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 141–147, September–October, 1984.     

Application of unsteady mixing equations to some aerodynamic problems

Tangential discontinuities [1] are introduced in solving several transient and steady-state problems of gas dynamics. These discontinuities are unstable [2] as a result of the effects of viscosity and thermal conductivity. Therefore it is advisable to replace the tangential discontinuity by a mixing region and account for its interaction with the inviscid flows, establishing on the boundaries of this region the conditions of vanishing friction stress and equality of the velocity and temperature components to the corresponding velocity and temperature components of the inviscid flows. This formulation improves the accuracy of the solution of such problems by posing them as problems with irregular reflection and intersection of shock waves [1].The consideration of the interaction of unsteady turbulent mixing regions with the inviscid flow also permits the formulation of several problems in which the effects of viscosity lead to complete rearrangement of the flow pattern (the lambda-configuration) with the interaction of the reflected shock wave with the boundary layer in the shock tube [3,4], the formation of zones of developed separation ahead of obstacles, etc.).In this connection, §1 presents an analysis of the self-similar solutions of the unsteady turbulent mixing equations (a corresponding analysis of the laminar mixing equations which coincide with the boundary layer equations is presented in [1]). It is shown that these self-similar solutions describe, along with the several problems noted above, the problems of the formation of steady jets and mixing zones in the base wake.As an example, §2 presents, within the framework of the proposed schematization, an approximate solution of the problem of the interaction of a shock wave reflected from a semi-infinite wall with the boundary layer on a horizontal plate behind the incident shock wave. The results obtained are applied to the analysis of reflection in a shock tube. Computational results are presented which are in qualitative agreement with experiment [3, 4].     

正向爆轰驱动高焓激波风洞的数值模拟

对充满氢氧可燃气体、带扩容腔的正向爆轰驱动的激波风洞进行了数值模拟。计算采用了欧拉方程,频散可控耗散差分格式（DCD）和改进的二阶段化学反应模型。在扩容腔附近采用二维轴对称计算模型,而在驱动段和被驱动段的直管道部分则采用一维计算模型。本文分析了爆轰波在管道中的传播、反射和绕射过程。计算结果表明扩容腔的尺寸对爆轰波的传播、反射、汇聚等起着决定性的作用;带扩容腔的正向爆轰驱动的激波风洞能够得到平稳的持续时间较长的气流,提高了实验的精确度和可重复性。     

Experimental investigation of three-dimensional supersonic flow of a gas in the interference region of intersecting surfaces

A contemporary high-speed aircraft represents a complex three-dimensional configuration, where supersonic gas flow is accompanied by numerous local flow interaction zones, in particular, near the intersection of different surfaces. Such a flow is characterized by three-dimensional systems of shock and expansion waves, and close to the surfaces one finds interaction of boundary layers and, above all, interaction of shock waves with the boundary layer. In general, the angular configurations are formed by intersection or contact of nonplanar surfaces with swept-back or blunted leading edges. This makes it practically impossible to obtain a rigorous theoretiical solution to the problem of gas flow over these surfaces, and presents considerable difficulty in an experimental investigation. It is therefore of interest to study the physical features of gas flow in corner configurations of very simple form [1–3]. The present paper examines the results of an experimental investigation of typical features of symmetric and asymmetric interaction of compressive, expansive, and mixed flows in the interference region of planar surfaces intersecting at an angle of less than 180?.     

Asymptotic theory of the flow around an obstacle by a sonic flow

The first investigation of the problem of the flow around an obstacle by a gas flow whose velocity is equal to the speed of sound at infinity was carried out in [1, 2], where it is shown in particular that the principal term of the appropriate asymptotic expansion is a self-similar solution of Tricomi's equation, to which the problem reduces in the first approximation upon a hodographic investigation. The requirement that the stream function be analytic as a function of the hodographic variables on the limiting characteristic was an important condition determining the selection of the self-similarity exponent n (xy^–n is an invariant of the self-similar solution). The analytic nature of the velocity field everywhere in the flow above the shock waves, which arise from necessity upon flow around an obstacle, follows from this condition. The latter was found in [3], where one of the branches of the solution obtained in [1] was used in the region behind the shock waves. The principal and subsequent terms of the asymptotic expansion describing a sonic flow far from an obstacle were discussed in [4], where the author restricted himself to Tricomi's equation. Each term of the series constructed in [4] contains an arbitrary coefficient (we will call it a shape parameter) which is not determined within the framework of a local investigation, and consideration of the problem of flow around a given obstacle as a whole is necessary in order to determine these shape parameters. It follows from the results of [4] that the problem of higher approximations to the solution of [1] coincides with the problem, of constructing a flow in the neighborhood of the center of a Laval nozzle with an analytic velocity distribution along the longitudinal axis (a Meyer-type flow). Along with the Meyer-type flow in the vicinity of the nozzle center, which corresponds to a self-similarity exponent n=2, two other types of flow are asymptotically possible with n=3 and 11, given in [5]. The appropriate solutions are written out in algebraic functions in [6]. The results of [5] show that the condition that the velocity vector be analytic on the limiting characteristic in the flow plane is broader than the condition that the stream function be analytic as a function of the hodographic variables, which is employed in [1, 2, 4]. Therefore, the necessity has arisen of reconsidering the problem of higher approximations for the obstacle solution of F. I. Frankl'. It has proved possible for the region in front of the shock waves to use a series which is more general than in [4], which implies the inclusion of an additional set of shape parameters. The solution is given in the hodograph plane in the form of the sum of two terms; the series discussed in [4] corresponds to the first one, and the series generated by the self-similar solution with n=3 or with n=11 corresponds to the second one.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 99–107, May–June, 1979.The authors thank S. V. Fal'kovich for a useful discussion.     

气相爆轰波正反射激波加速研究

实验采用压力传感器测量了指定点压力时间曲线。数值模拟基于二维反应欧拉方程和基元反应模型,采用二阶附加半隐的龙格-库塔法和5阶WENO格式分别离散时间和空间导数项,获得了指定点数值压力时间曲线。理论分析基于爆轰理论和激波动力学,分析了气相爆轰波反射过程所涉及的复杂波系演变并获得了反射激波速度。结果表明：本文数值模拟和理论计算定性上重复并解释了实验现象。气相爆轰波在右壁面反射后,右行稀疏波加速反射激波。其加速原因是：尽管激波波前声速减小,但激波马赫数增大,波前气流速度减小。在低初压下,可能还由于爆轰波后未反应或部分反应气体的作用,导致反射激波加速幅度比高初压下大。     

Self-similar magnetogasdynamic flows accompanied by detonation and combustion waves

Magnetogasdynamic (MGD) flows with detonation waves and combustion fronts have attracted more and more attention in recent years. Intensive heat supply assures such a significant increase in the temperature and pressure behind the heat liberation fronts that the gaseous combustion products become conductive so that the flow map in the electric and magnetic fields can vary substantially as compared with ordinary gasdynamics. In the case of finite gas conductivity, when the magnetic Reynolds numbers R_m are low, the asymptotic laws of detonation wave propagation which either go over into the Chapman-Jouguet (CJ) mode (in a number of cases at a finite distance from the initiation source) or remain overcompressed, have been studied [1]. Stationary flow modes behind detonation waves have been investigated in [2] and the problem of the detonation wave originating at the closed end of the tube emerging in the stationary mode in crossed homogeneous magnetic and electric fields has been examined. Results are presented in this paper of an investigation of one-dimensional self-similar flows caused by piston motion in a hot gas mixture in which a detonation wave or combustion front is propagated. The motion is realized in external electric and magnetic fields which exert a substantial effect on the flow of the conductive combustion products. Domains of application of the governing parameters in which the various flow modes are realized are found by using a qualitative and numerical analysis. The results obtained are used to solve problems about the hypersonic gas flow around a thin wedge in an axial magnetic field.     

气相爆轰波在障碍物上Mach反射的实验验证

本文公布了气相爆轰波沿收缩管道传播时发生Mach反射的实验证据。在爆轰波通过的管道中安装不同楔角的楔块,形成管道的收缩。爆轰波在通过楔块时会发生Mach反射。利用烟熏玻璃片记录到了爆轰波Mach反射时形成的三波点迹线及其两侧胞格尺寸和密度的变化。据我们掌握的资料,这是首次用胞格结构变化的记录证实,气相爆轰波与无化学反应的空气中的冲击波一样,在一定的入射条件下会发生Mach反射。这一实验结果可使我们更深入了解爆轰波的本质,也为数值模拟气相爆轰波在障碍物上Mach反射现象提供了可对比的依据。     

Distinctive features of galloping detonation in a supersonic combustible-mixture flow under an inert gas layer

The problem of detonation initiation in a supersonic flow of a stoichiometric propane-air mixture occupying partially or completely the cross-section of a plane channel is considered. The initiation in the flow is produced by a step or a wall completely cutting off the flow. The study is conducted within the framework of one-stage combustion kinetics. A numerical method based on the Godunov scheme is employed. The critical conditions for detonation formation are determined in terms of the oncoming flow velocity. A previously unknown mechanism of detonation propagation is found; it is related with the presence of the combustible mixture in the wall layer under an inert gas layer. It is due to the formation of a complicated wave structure of the flow characterized by the penetration of a shock wave formed in the inert gas layer into a combustible mixture layer ahead of the detonation wave with the result that the latter layer is heated and ignited. The process as a whole is periodic in nature, as distinct from the conventional cellular detonation in a homogeneous fluid. Many problems arise in connection with the use of detonation in engines and other power plants. The most important among them are detonation excitation and stabilization in combustion chambers. The detonation initiation within a layer under conditions of unbounded space and a fluid at rest was experimentally investigated in [1]. In the case of a combustion chamber bounded in the transverse direction, some new effects accompanying the detonation might be expected.     

Relaxation of overdriven detonation waves with finite reaction rate

A numerical study is made of the interaction of a detonation wave having finite reaction velocity with a rarefaction wave of different intensity which approaches it from the rear, for the Zeldovich-Neumann-Doring (ZND) model with a single irreversible reaction A B. It is found that, for a fixed value of the parameter characterizing the initial supercompression (depending on the activation energy and the heating value of the mixture), the considered interaction leads either to a gradual relaxation of the detonation wave and its transition to the Chapman-Jouguet (CJ) regime, or to the development of undamped oscillations.Interest in the problems of detonation and supersonic combustion has increased in recent years. This is associated with the appearance and development of new experimental and theoretical techniques; it is also associated with the further development of air-breathing reaction engines, and other practical requirements. The present state of detonation theory is reflected in the survey [1].It has been established [2] that the detonation wave in gases nearly always has a complex nonuniform structure. Transverse disturbances are observed under a wide range of conditions and differ both in amplitude and wavelength. At the same time, behind the detonation leading front there is a region of uncompletely burned gas corresponding to the effective ignition induction period [3]. In spinning detonation the induction period is significantly longer than the heat release period and transverse detonation waves traveling in the induction zone of the head wave appear [3, 4]. Such a secondary detonation wave is free of transverse disturbances. The same is true of the detonation waves observed in the wake behind a body moving at high speed in a combustible medium [5] or in a gas which has been preheated by a shock wave [6].Although it is possible, under favorable conditions, to study in detail the system of discontinuities accompanying detonation, information on the extensive zones in which heat release takes place is scarce, the mechanism of detonation wave autonomy (in particular, the role of the rarefaction zone behind the wave) is not entirely clear, and the fact that, in spite of the complex structure, an autonomous detonation propagates with the CJ velocity calculated on the basis of one-dimensional theory has not yet been explained.In studying the nonlinear phenomena associated with the finite reaction rate it is quite acceptable to investigate only the simple one-dimensional detonation model, with which it is convenient to restrict ourselves to a single effective chemical reaction. This model is particularly reasonable since, in certain cases, the real detonation is virtually one-dimensional.The question of the stability of the one-dimensional detonation wave to disturbances of its structure has been examined by several authors [7–13]. The use of computers makes possible the direct computation of flows with heat release and the study of their properties. This method has been used in [11–13] to study the stability problem for a detonation wave with respect to finite disturbances.In the present paper we present a numerical study of the interaction of a detonation wave having finite chemical reaction rate with a rarefaction wave of different intensity approaching it from the rear for the ZND model with a single irreversible reaction A B. It is found that for a fixed value of the parameter characterizing the difference between detonation and the CJ waves, depending on the activation energy E and the mixture heating value Q_m, the interaction in question leads either to a gradual relaxation of the detonation wave and its transition to the CJ regime (this relaxation may be accompanied by decaying oscillations) or to the appearance of undamped oscillations (the unstable regime). The parameters E and Q_m affect the wave stability differently: with increase of Q_m, the wave is stabilized; with increase of E, it is destabilized. The boundary between the stable and unstable detonation wave propagation regimes is found. This boundary has a weak dependence on the rarefaction wave intensity. Estimates and calculated examples show that the amplitude of the unstable wave oscillations is finite and that the average detonation propagation velocity is close to the CJ velocity computed for the given heating value Qm.The author wishes to thank G. G. Chernyi for his guidance and L. A. Chudov for advice on computational questions.