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.     

Investigation of certain techniques for stabilizing detonationwaves in a supersonic flow

The salient features of detonation wave propagation in a supersonic flow of a stoichiometric hydrogen-air mixture in plane channels of both constant and variable cross-section are numerically investigated for the purpose of determining the conditions ensuring stabilized detonation. The propagation of a detonation wave formed in a variable-cross-section channel is studied. For different inflow Mach numbers the geometric parameters of a channel providing the detonation combustion stabilization are determined. An investigation of detonation wave stabilization in a supersonic flow of a combustible gas mixture in a plane channel with parallel walls using additional weak discharges is continued. The effects of the flow Mach number, the additional discharge energy, and the discharge location on detonation wave stabilization are studied.     

Fundamentals of rotating detonations

A rotating detonation propagating at nearly Chapman–Jouguet velocity is numerically stabilized on a two-dimensional simple chemistry flow model. Under purely axial injection of a combustible mixture from the head end of a toroidal section of coaxial cylinders, the rotating detonation is proven to give no average angular momentum at any cross section, giving an axial flow. The detonation wavelet connected with an oblique shock wave ensuing to the downstream has a feature of unconfined detonation, causing a deficit in its propagation velocity. Due to Kelvin–Helmholtz instability existing on the interface of an injected combustible, unburnt gas pockets are formed to enter the junction between the detonation and oblique shock waves, generating strong explosions propagating to both directions. Calculated specific impulse is as high as 4,700 s.      

Flow of a gas behind a Chapman-Jouguet detonation wave in the vicinity of a line of discontinuity of a surface wave curvature

This paper discusses questions of constructing a solution of the gasdynamic equations near a line of curvature discontinuity at the surface of a detonation wave, propagating under Chapman—Jouguet conditions. It describes the construction of the solution in two cases: in a flow arising with the initiation of a detonation along a half-plane in a quiescent homogeneous combustible gas and in a flow arising with the initiation of a detonation along a half-line under these same conditions.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 120–126, January–February, 1978.     

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.     

Head-on Collision of a Detonation with a Planar Shock Wave

The phenomenon that occurs when a Chapman–Jouguet (CJ) detonation collides with a shock wave is discussed. Assuming a one-dimensional steady wave configuration analogous to a planar shock–shock frontal interaction, analytical solutions of the Rankine–Hugoniot relationships for the transmitted detonation and the transmitted shock are obtained by matching the pressure and particle velocity at the contact surface. The analytical results indicate that there exist three possible regions of solutions, i.e. the transmitted detonation can have either strong, weak or CJ solution, depending on the incident detonation and shock strengths. On the other hand, if we impose the transmitted detonation to have a CJ solution followed by a rarefaction fan, the boundary conditions are also satisfied at the contact surface. The existence of these multiple solutions is verified by an experimental investigation. It is found that the experimental results agree well with those predicted by the second wave interaction model and that the transmitted detonation is a CJ detonation. Unsteady numerical simulations of the reactive Euler equations with both simple one-step Arrhenius kinetic and chain-branching kinetic models are also carried out to look at the transient phenomena and at the influence of a finite reaction thickness of a detonation wave on the problem of head-on collision with a shock. From all the computational results, a relaxation process consisting of a quasi-steady period and an overshoot for the transmitted detonation subsequent to the head-on collisions can be observed, followed by the asymptotic decay to a CJ detonation as predicted theoretically. For unstable pulsating detonations, it is found that, due to the increase in the thermodynamic state of the reactive mixture caused by the shock, the transmitted pulsating detonation can become more stable with smaller amplitude and period oscillation. These observations are in good agreement with experimental evidence obtained from smoked foils where there is a significant decrease in the detonation cell size after a region of relaxation when the detonation collides head-on with a shock wave.     

Stability of gas mixture flow with a stabilized detonation wave in a plane channel with a constriction

The stability of a flow with a stabilized detonation wave is studied within the framework of a detailed kinetic mechanism of the chemical interaction. The flow is due to the initiation of detonation combustion of a stoichiometric hydrogen-air mixture that enters into a plane channel with a constriction at a supersonic velocity greater than that of the self-sustained detonation propagation. The flow under consideration is numerically investigated using the software package developed by the authors. It is established that the flow formed in the channel, whose geometric parameters ensure the detonation stabilization in the case of the inflow Mach number M₀ = 5.2, is stable against strong disturbances of a certain type. The effect of an increase in the inflow Mach number and the dustiness of the combustible gas mixture entering into the channel on the stabilization of detonation combustion in the flow is investigated.     

Detonation Initiation by Shock Reflection from an Orifice Plate

Results from an experimental investigation of the interaction of a “non-ideal” shock wave and a single obstacle are reported. The shock wave is produced ahead of an accelerated flame in a 14 cm inner-diameter tube partially filled with orifice plates. The shock wave interacts with a single larger blockage orifice plate placed 15–45 cm after the last orifice plate in the flame acceleration section of the tube. Experiments were performed with stoichiometric ethylene–oxygen mixtures with varying amounts of nitrogen dilution at atmospheric pressure and temperature. The critical nitrogen dilution was found for detonation initiation. It is shown that detonation initiation occurs if the chemical induction time based on the reflected shock state is shorter than the time required for an acoustic wave to traverse the orifice plate upstream surface, from the inner to the outer diameter. The similarity between the present results and those obtained from previous investigators looking at detonation initiation by ideal shock reflection produced in a shock tube indicates that the phenomenon is not sensitive to the detailed structure of the shock front but only on the average shock strength.This paper is based on work that was presented at the 20th International Colloquium on the Dynamics of Explosions and Reactive Systems, Montreal, Canada, July 31–August 5, 2005.     

The criterion of the existence or inexistence of transverse shock wave at wedge supported oblique detonation wave

A simplified theoretic method and numerical simulations were carried out to investigate the characterization of propagation of transverse shock wave at wedge supported oblique detonation wave.After solution validation,a criterion which is associated with the ratio Φ (u 2 /u CJ) of existence or inexistence of the transverse shock wave at the region of the primary triple was deduced systematically by 38 cases.It is observed that for abrupt oblique shock wave (OSW)/oblique detonation wave (ODW) transition,a transverse shock wave is generated at the region of the primary triple when Φ < 1,however,such a transverse shock wave does not take place for the smooth OSW/ODW transition when Φ > 1.The parameter Φ can be expressed as the Mach number behind the ODW front for stable CJ detonation.When 0.9 < Φ < 1.0,the reflected shock wave can pass across the contact discontinuity and interact with transverse waves which are originating from the ODW front.When 0.8 < Φ < 0.9,the reflected shock wave can not pass across the contact discontinuity and only reflects at the contact discontinuity.The condition (0.8 < Φ < 0.9) agrees well with the ratio (D ave /D CJ) in the critical detonation.     

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.     

The shock–vortex interaction patterns affected by vortex flow regime and vortex models

We have used a third-order essentially non-oscillatory method to obtain numerical shadowgraphs for investigation of shock–vortex interaction patterns. To search different interaction patterns, we have tested two vortex models (the composite vortex model and the Taylor vortex model) and as many as 47 parametric data sets. By shock–vortex interaction, the impinging shock is deformed to a S-shape with leading and lagging parts of the shock. The vortex flow is locally accelerated by the leading shock and locally decelerated by the lagging shock, having a severely elongated vortex core with two vertices. When the leading shock escapes the vortex, implosion effect creates a high pressure in the vertex area where the flow had been most expanded. This compressed region spreads in time with two frontal waves, an induced expansion wave and an induced compression wave. They are subsonic waves when the shock–vortex interaction is weak but become supersonic waves for strong interactions. Under a intermediate interaction, however, an induced shock wave is first developed where flow speed is supersonic but is dissipated where the incoming flow is subsonic. We have identified three different interaction patterns that depend on the vortex flow regime characterized by the shock–vortex interaction.      

Detonation interaction with a diffuse interface and subsequent chemical reaction

We have investigated the interaction of a detonation with an interface separating a combustible from an oxidizing mixture. The ethylene-oxygen combustible mixture had a fuel-rich composition to promote secondary combustion with the oxidizer in the turbulent mixing zone that resulted from the interaction. Diffuse interfaces were created by the formation of a gravity current using a sliding valve that initially separated the test gas and combustible mixture. Opening the valve allowed a gravity current to develop before the detonation was initiated. By varying the delay between opening the valve and initiating the detonation it was possible to achieve a wide range of interface conditions. The interface orientation and thickness with respect to the detonation wave have a profound effect on the outcome of the interaction. Diffuse interfaces result in curved detonation waves with a transmitted shock and following turbulent mixing zone. The impulse was measured to quantify the degree of secondary combustion, which accounted for 1–5% of the total impulse. A model was developed that estimated the volume expansion of a fluid element due to combustion in the turbulent mixing zone and predicted the resulting impulse increment.      

Explosive dispersal of solid particles

Abstract. The rapid dispersal of inert solid particles due to the detonation of a heterogeneous explosive, consisting of a packed bed of steel beads saturated with a liquid explosive, has been investigated experimentally and numerically. Detonation of the spherical charge generates a blast wave followed by a complex supersonic gas-solid flow in which, in some cases, the beads catch up to and penetrate the leading shock front. The interplay between the particle dynamics and the blast wave propagation was investigated experimentally as a function of the particle size (100–925 m) and charge diameter (8.9–21.2 cm) with flash X-ray radiography and blast wave instrumentation. The flow topology during the dispersal process ranges from a dense granular flow to a dilute gas-solid flow. Difficulties in the modeling of the high-speed gas-solid flow are discussed, and a heuristic model for the equation of state for the solid flow is developed. This model is incorporated into the Eulerian two-phase fluid model of Baer and Nunziato (1986) and simulations are carried out. The results of this investigation indicate that the crossing of the particles through the shock front strongly depends on the charge geometry, the charge size and the material density of the particles. Moreover, there exists a particle size limit below which the particles cannot penetrate the shock for the range of charge sizes considered. Above this limit, the distance required for the particles to overtake the shock is not very sensitive to the particle size but remains sensitive to the particle material density. Overall, excellent agreement was observed between the experimental and computational results. Received 16 August 1999 / Accepted 26 June 2000     

Calculation of hydrogen-air combustion behind detached shock wave for supersonic flow past a sphere

Many of the published theoretical studies of quasi-one-dimensional flows with combustion have been devoted to combustion in a nozzle, wake, or streamtube behind a normal shock wave [1–6].Recently, considerable interest has developed in the study of two-dimensional problems, specifically, the effective combustion of fuel in a supersonic air stream.In connection with experimental studies of the motion of bodies in combustible gas mixtures using ballistic facilities [7–9], the requirement has arisen for computer calculations of two-dimensional supersonic gas flow past bodies in the presence of combustion.In preceding studies [10–12] the present author has solved the steady-state problem under very simple assumptions concerning the structure of the combustion zone in a detonation wave.In the present paper we obtain a numerical solution of the problem of supersonic hydrogen-air flow past a sphere with account for the nonequilibrium nature of eight chemical reactions. The computations encompass only the subsonic and transonic flow regions.The author thanks G. G. Chernyi for valuable comments during discussion of the article.     

超音速来流中爆轰波衍射和二次起爆过程研究

预爆管技术被广泛地应用在爆轰波发动机的起爆过程中，但是在超音速来流中基于预爆管技术起始爆轰波的研究并未被广泛地开展。基于此，本文中数值研究了横向超音速来流对半自由空间内爆轰波的衍射和自发二次起爆、及管道内的衍射和壁面反射二次起爆两种现象的影响。数值模拟的控制方程为二维欧拉方程，空间上使用五阶WENO格式进行数值离散，采用带有诱导步的两步链分支化学反应模型。所模拟的爆轰波具有规则的胞格结构，对应于用惰性气体高度稀释过的可爆混合物中形成的爆轰波。结果表明:在半自由空间内，在本文所模拟的几何尺寸下，爆轰波并未成功发生二次起爆现象，但是爆轰波的自持传播距离随着横向超音速来流强度的增强而增加。在核心的三角形流动区域外，波面诱导产生了更多的横波结构；在管道内，横向的超音速来流在逆流侧对出口气流产生了压缩作用，能有效提高波面压力，因此反射后的激波压力也比较高。在同样的几何尺寸下，爆轰波在静止和超音速(Ma=2.0)气流中分别出现了二次起爆失败和成功两种现象，这是由于在超音速来流中化学反应面的褶皱诱导产生了横波结构，横波与管壁以及其他横波之间的碰撞提高了前导激波的强度，并最终促进了爆轰波在超声速流主管道内的成功起始。     

气相爆轰胞格结构和马赫反射数值模拟

采用简化二阶段化学反应模型和改进的高精度时空守恒方法 (CE/SE)对二维可燃气体DDT过程和爆轰波在楔面上的马赫反射进行了数值模拟 ,计算和实验结果的比较令人满意。     

激波诱导火焰失稳与爆轰的条件研究

采用九阶WENO和十阶中心差分格式数值求解激波与火焰作用过程,考察了激波强度、火焰尺寸对激波与球形火焰作用过程的影响。结果表明,增大激波强度或火焰尺寸均可在流场中引发爆轰,但激波强度的影响更大,并且其引发的爆轰可使火焰迅速膨胀,放热率提高,从而影响燃烧特性;此外,爆轰波传播过程中会迅速消耗可燃预混气,合并原有的反射激波,并在流场中形成局部高压区,极大地改变流场结构。     

可燃介质中飞行圆球诱导斜爆轰的流场结构

基于带化学反应的二维轴对称Euler方程,利用带有Superbee限制函数的波传播算法,对氢/氧/ 氮预混气中飞行圆球诱导斜爆轰进行了数值模拟。结果表明,在达姆科勒数Da略大于临界情形时,圆球诱导 的驻定爆轰是一个由强过驱斜爆轰、弱过驱斜爆轰、反应激波和惰性激波组合而成的复合结构。波后的圆球 绕流流场内存在2个亚音速区和1个超音速区。在圆球背风表面,还形成了第2道斜激波。     

Interaction of shear flows of an ideal incompressible fluid in a channel

The problem of the decay of an arbitrary discontinuity for the equations describing plane-parallel shear flows of an ideal fluid in a narrow channel is considered. The class of particular solutions corresponding to fluid flows with piecewise constant vorticity is studied. In this class, the existence of self-similar solutions describing all possible unsteady wave configurations resulting from the nonlinear interaction of the specified shear flows is established. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 6, pp. 34–47, November–December, 2006.