Wave front analysis of weak nonlinear waves in a two-phase flow of gas and dust particles

The non-linear behavior of waves including the characteristic front, the expansion wave front and the shock front, in a mixture of gas and dust particles has been studied. Such waves are conceived of as produced by a piston moving with a small velocity as compared with the speed of sound. The trajectories of these waves and the particle paths in the physical plane are determined. The effect of solid particles and the adiabatic heat exponent on the wave propagation is also investigated.     

Spherical shock wave generated by a moving piston in mixture of a non-ideal gas and small solid particles under a gravitational field

Self-similar solutions are obtained for one-dimensional isothermal and adiabatic unsteady flows behind a strong spherical shock wave propagating in a dusty gas. The shock is assumed to be driven out by a moving piston and the dusty gas to be a mixture of a non-ideal (or perfect) gas and small solid particles, in which solid particles are continuously distributed. It is assumed that the equilibrium flow-conditions are maintained and variable energy input is continuously supplied by the piston. The medium is under the influence of the gravitational field due to a heavy nucleus at the origin (Roche model). The effects of an increase in the mass concentration of solid particles, the ratio of the density of the solid particles to the initial density of the gas, the gravitational parameter and the parameter of non-idealness of the gas in the mixture, are investigated. It is shown that due to presence of gravitational field the compressibility of the medium at any point in the flow-field behind the shock decreases and all other flow-variables and the shock strength increase. A comparison has also been made between the isothermal and adiabatic flows. It is investigated that the singularity in the density and compressibility distributions near the piston in the case of adiabatic flow are removed when the flow is isothermal.     

Structure of normal shock waves in a gas-particle mixture

Summary The structure of normal shock wave propagating in a gas particle mixture has been obtained by including the particle-volume effect.

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Zusammenfassung Unter der Berücksichtigung des Teilchenvolumeneffektes wird die Struktur einer senkrechten Stoßwelle untersucht, die sich in einem Gasteilchengemisch bewegt.

     

Well-posedness of a modified initial-boundary value problem on stability of shock waves in a viscous gas. Part I

The shock wave in a viscous gas which is treated as a strong discontinuity is unstable against small perturbations [A.M. Blokhin, On stability of shock waves in a compressible viscous gas, Matematiche LVII (I) (2002) 3-19]. We suggest such additional boundary conditions that a modified (with account to these conditions) linear initial-boundary value problem on stability of the shock wave does not admit Hadamard-type ill-posedness examples.     

Solution of the Boltzmann equation for unsteady flows with shock waves in narrow channels

Unsteady rarefied gas flows in narrow channels accompanied by shock wave formation and propagation were studied by solving the Boltzmann kinetic equation. The formation of a shock wave from an initial discontinuity of gas parameters, its propagation, damping, and reflection from the channel end face were analyzed. The Boltzmann equation was solved using finite differences. The collision integral was calculated on a fixed velocity grid by a conservative projection method. A detector of shock wave position was developed to keep track of the wave front. Parallel computations were implemented on a cluster of computers with the use of the MPI technology. Plots of shock wave damping and detailed flow fields are presented.     

Numerical study of unsteady rarefied diatomic gas flows in a plane microchannel

The numerical solution of a kinetic equation for a diatomic gas (nitrogen) is used to study two-dimensional unsteady gas flows in a plane microchannel caused by discontinuous in the initial distributions of macroscopic gas parameters. The plane discontinuity fronts are perpendicular to the walls of the channel. The arising flows are model ones for gas flows in a shock tube and a microchannel. The reflection of an incident shock wave from a flat end face is studied. It is found that the gas piles up at the cold wall, which slows down the shock wave detachment. The numerical results are in qualitative agreement with experimental data.     

Similarity solution for a spherical shock wave in a non‐ideal gas under the influence of gravitational field and monochromatic radiation with increasing energy

The propagation of a spherical shock wave in a non‐ideal gas with or without gravitational effects is investigated under the action of monochromatic radiation. Similarity solutions are obtained for adiabatic flow between the shock and the piston. The numerical solutions are obtained using the Runge‐Kutta method of the fourth order. The density of the gas is assumed to be constant. The total energy of the shock wave is non‐constant and varies with time. The effects of change in values of non‐idealness parameter, gravitational parameter, shock Mach number, radiation parameter, and adiabatic exponent of the gas on shock strength and flow variables are worked out in detail. It is investigated that the presence of gravitational field increases the compressibility of the medium, due to which it is compressed and, therefore, the distance between the inner contact surface and the shock surface is reduced. A comparison is also made between the solutions in the cases of the gravitating and the non‐gravitating media. It is manifested that the gravitational parameter and the radiation parameter have in general opposite behaviour on the flow variables and the shock strength.     

星系螺旋结构的气体激波理论

本文用星际气体自引力星系激波来解释星系的螺旋结构、恒星的扰动引力场并非必要条件.我们首先证明,即使扰动引力场为零,也可以存在局部的星系激波解.这种解要求|ω_η0|>α,而且只要气体的密度反差比较大,就只能用激波解来解释螺旋结构.用叠代的方法求出了星际气体的自引力激波宏图.对一种特定的扰动引力场模拟气体自引力,可以在速度平面上定性分析激波解的特性.初始原星系盘中的物质分布不均匀性,通过缠卷过程、不稳定性增长和波动叠加.可以发展成星系激波宏图.这样,对星系激波的起源,演化和维持给出一个完整的图象.利用这个图象,可以解释星系螺旋结构的大量观测结果和分类特性.     

Numerical Simulation of Detonation Excitation at Shock Wave Interaction with Dust Cloud in Channel

The study of detonation ability of reactive particle gas mixtures is necessary to prevent industrial explosions in industries where dispersed powders are used. The present paper focuses on numerical simulation of the shock wave interaction with semiinfinite aluminum dust cloud, which is situated inside a plane channel. The cloud fills entirely or partly the channel cross‐section and has initially a rectangular shape. The scenarios of detonation initiation in the cloud are determined depending on the incident shock wave amplitude values. The processes of transformation and spreading of finite width clouds under weak incident shock wave action (when the particles do not ignite) are investigated. The types of an oblique shock wave reflection from the plane of symmetry in the cloud are analyzed. The processes of particle ignition and detonation structure formation at strong incident shock wave action are investigated. Nonstationary periodic fuctuations take place in the detonation flow due to transversal wave effect. Nevertheless the detonation structure established propagates in quasistationary regime. If the incident shock wave is attenuated with a rarefaction wave then the detonation formation fails at clouds of insufficient width.     

On the acceleration of shock waves and concentration of energy

By a series of simple examples related to exact solutions of problems in gas dynamics and magnetohydrodynamics, possible mechanisms of acceleration of shock waves and concentration of energy are elucidated. The acceleration of a shock wave is investigated in the problem of motion of a plane piston at a constant velocity in the case when the initial density of the medium drops in the presence of constant counterpressure. It is shown that in this situation a “blow-up” regime is induced by a shock wave going to infinity in finite time even for limited work of the piston. A simple spherically symmetric solution with a converging shock wave is constructed and shown to lead to the concentration of energy. A general method for solving one-dimensional non-self-similar problems related to matching the equilibrium state to a motion with homogeneous deformation on a shock wave is discussed; this method leads to a solution in quadratures.     

Vanishing pressure limits of Riemann solutions to the isentropic relativistic Euler system for Chaplygin gas

The Riemann solutions to the isentropic relativistic Euler system for Chaplygin gas with a small parameter are considered. Unlike the polytropic or barotropic gas cases, we find that firstly, as the parameter decreases to a certain critical number, the two-shock solution converges to a delta shock wave solution of the same system. Moreover, as the parameter goes to zero, that is, the pressure vanishes, the solution is nothing but the delta shock wave solution to the zero-pressure relativistic Euler system. Meanwhile, the two-rarefaction wave solution tends to the vacuum solution to the zero-pressure relativistic system, and the solution containing one rarefaction wave and one shock wave tends to the contact discontinuity solution to the zero-pressure relativistic system as pressure vanishes.     

Detailed simulation of the pulsating detonation wave in the shock-attached frame

The paper is devoted to the numerical investigation of the stability of propagation of pulsating gas detonation waves. For various values of the mixture activation energy, detailed propagation patterns of the stable, weakly unstable, irregular, and strongly unstable detonation are obtained. The mathematical model is based on the Euler system of equations and the one-stage model of chemical reaction kinetics. The distinctive feature of the paper is the use of a specially developed computational algorithm of the second approximation order for simulating detonation wave in the shock-attached frame. In distinction from shock capturing schemes, the statement used in the paper is free of computational artifacts caused by the numerical smearing of the leading wave front. The key point of the computational algorithm is the solution of the equation for the evolution of the leading wave velocity using the second-order grid-characteristic method. The regimes of the pulsating detonation wave propagation thus obtained qualitatively match the computational data obtained in other studies and their numerical quality is superior when compared with known analytical solutions due to the use of a highly accurate computational algorithm.     

Stability condition for strong shock waves in the problem of flow around an infinite plane wedge

We consider the thermodynamical equilibrium state flow of an inviscid non-heat-conducting gas flowing around a plane infinite wedge, and study the stationary solution to this problem–the so-called strong shock wave; the flow behind the shock front is subsonic.We find the solution to the linear analog of the original mixed problem, prove that the solution trace on the shock wave is the superposition of the direct and reflected waves, and (the main point) justify the Lyapunov asymptotical stability of the strong shock wave provided that the uniform Lopatinsky condition is fulfilled. The initial data have a compact support, and the solvability conditions occur.     

Asymptotic behavior of solutions for the 1-D isentropic Navier–Stokes–Korteweg equations with free boundary

In this paper, we consider the problem with a gas–gas free boundary for the one dimensional isentropic compressible Navier–Stokes–Korteweg system. For shock wave, asymptotic profile of the problem is shown to be a shifted viscous shock profile, which is suitably away from the boundary, and prove that if the initial data around the shifted viscous shock profile and its strength are sufficiently small, then the problem has a unique global strong solution, which tends to the shifted viscous shock profile as time goes to infinity. Also, we show the asymptotic stability toward rarefaction wave without the smallness on the strength if the initial data around the rarefaction wave are sufficiently small.     

Dispositif d'éclatement commandé des diaphragmes associé à un tube à ondes de choc pour générateur magnétohydrodynamique

Summary In the experimental device, the MHD generator receives the current from a gas in the shock tube. This gas, argon, heated beforehand to 1200°C and seeded with alkaline metal (potassium), is brought to the pseudoplasma state under the influence of a shock wave. This wave is obtained by means of a membrane and a driving gas (helium). The use of a pulsed magnetic field at the level of the MHD generator makes a time delay necessary between the beginning of the discharge of the condensers, the source of energy which has to create the pulsed magnetic field, and the rupture of the membrane producing the shock wave (the rise time of the magnetic field is longer than the travelling time of the shock wave from the membrane to the entrance of the generator).A projectile shot along the axis of the tube and behind the membrane which is suitably scratched and near rupture under the influence of the helium pressure, will insure its rupture without disturbing the flow in the tube. The passage of this projectile is detected behind the membrane by a photo-electric cell. The signal obtained triggers the discharge of the condensers in such a way that the arrival of the shock wave in the generator coincides with the beginning of the peak of the magnetic field.The projectile is propelled by a compressed gas pistol operating under varible pressure which makes it possible to adjust the time delay to within ±0.2 millisecond.     

Nonlinear geometric optics for contact discontinuities and shock waves in one-dimensional Euler system for gas dynamics

The aim of this paper is to study the rigorous theory of nonlinear geometric optics for a contact discontinuity and a shock wave to the Euler system for one-dimensional gas dynamics. For the problem of a contact discontinuity and a shock wave perturbed by a small amplitude, high frequency oscillatory wave train, under suitable stability assumptions, we obtain that the perturbed problem has still a shock wave and a contact discontinuity, and we give their asymptotic expansions.     

绝热流Chaplygin气体的Riemann问题

本文研究了绝热流Chaplygin气体动力学方程组,利用特征分析方法,在得到所有基本波的基础上,构造出Riemann问题的所有解.Riemann解由前向疏散波(激波)、后向疏散波(激波)、接触间断以及δ波构成.     

Study of the stability in the problem on flowing around a wedge. The case of strong wave

We consider the flow of an inviscid nonheatconducting gas in the thermodynamical equilibrium state around a plane infinite wedge and study the stationary solution to this problem, the so-called strong shock wave; the flow behind the shock front is subsonic.We find a solution to a mixed problem for a linear analog of the initial problem, prove that the solution trace on the shock wave is the superposition of direct and reflected waves, and, the main point, justify the Lyapunov asymptotical stability of the strong shock wave provided that the angle at the wedge vertex is small, the uniform Lopatinsky condition is fulfilled, the initial data have a compact support, and the solvability conditions take place if needed (their number depends on the class in which the generalized solution is found).     

Computation of a shock wave structure in monatomic gas with accuracy control

The structure of a shock wave in a monatomic one-component gas was computed by solving the Boltzmann kinetic equation with accuracy controlled with respect to computational parameters. The hard-sphere molecular model and molecules with the Lennard-Jones potential were considered. The computations were performed in a wide range of Mach numbers with the accuracy no less than 3% for the shock front width and 1% for local values of density and temperature. The shock wave structure was studied in terms of macroscopic gas characteristics and in terms of the molecular velocity distribution function.