Self-similar solution of cylindrical shock wave propagation in a rotational axisymmetric mixture of a non-ideal gas and small solid particles

Similarity solutions are obtained for one- dimensional isothermal and adiabatic unsteady flow behind a strong cylindrical shock wave propagating in a rotational axisymmetric dusty gas, which has a variable azimuthal fluid velocity together with a variable axial fluid velocity. The shock is assumed to be driven out by a moving piston and the dusty gas to be a mixture of non-ideal (or perfect) gas and small solid particles, in which solid particles are continuously distributed. It is assumed that the equilibrium flow-condition is maintained and variable energy input is continuously supplied by the piston. The shock Mach number is not infinite, but has a finite value. The azimuthal and axial component of the fluid velocity in the ambient medium are assumed to be vary and obey power laws, and the density of the ambient medium is taken to be constant. In order to obtain the similarity solutions the angular velocity of the ambient medium is assumed to be decreasing as the distance from the axis increases. Effects of the variation of the parameter of non-idealness of the gas in the mixture, the mass concentration of solid particles and the ratio of the density of solid particles to the initial density of the gas are investigated.     

Self-similar solutions for unsteady flow behind an exponential shock in an axisymmetric rotating dusty gas

Similarity solutions are obtained for one-dimensional unsteady isothermal and adiabatic flows behind a strong exponential cylindrical shock wave propagating in a rotational axisymmetric dusty gas, which has variable azimuthal and axial fluid velocities. The shock wave is driven by a piston moving with time according to an exponential law. Similarity solutions exist only when the surrounding medium is of constant density. The azimuthal and axial components of the fluid velocity in the ambient medium are assumed to obey exponential laws. The dusty gas is assumed to be a mixture of small solid particles and a perfect gas. To obtain some essential features of the shock propagation, small solid particles are considered as a pseudo-fluid; it is assumed that the equilibrium flow conditions are maintained in the flow field, and that the viscous stresses and heat conduction in the mixture are negligible. Solutions are obtained for the cases when the flow between the shock and the piston is either isothermal or adiabatic, by taking into account the components of the vorticity vector. It is found that the assumption of zero temperature gradient results in a profound change in the density distribution as compared to that for the adiabatic case. The effects of the variation of the mass concentration of solid particles in the mixture \(K_p\) , and the ratio of the density of solid particles to the initial density of the gas \(G_a\) are investigated. A comparison between the solutions for the isothermal and adiabatic cases is also made.     

Similarity solutions for the flow behind an exponential shock in a non-ideal gas

Similarity solutions for the flow of a non-ideal gas behind a strong exponential shock driven out by a piston (cylindrical or spherical) moving with time according to an exponential law are obtained. Similarity solutions exist only when the surrounding medium is of constant density. Solutions are obtained, in both the cases, when the flow between the shock and the piston is isothermal or adiabatic. It is found that the assumption of zero temperature gradient brings a profound change in the density distribution as compare to that of the adiabatic case. Effects of the non-idealness of the gas on the flow-field between the shock and the piston are investigated. The variations of density-ratio across the shock and the location of the piston with the parameter of non-idealness of the gas are also obtained.     

A self-similar solution of exponential shock waves in non-ideal magnetogasdynamics

A self similar method is used to analyze numerically the one-dimensional, unsteady flow of a strong cylindrical shock wave driven by a piston moving with time according to an exponential law in a plasma of constant density. The plasma is assumed to be a non-ideal gas with infinite electrical conductivity permeated by an axial magnetic field. Numerical solutions in the region between the shock and the piston are presented for the cases of adiabatic and isothermal flow. The general behaviour of density, velocity, and pressure profiles remains unaffected due to presence of magnetic field in non-ideal gas. However, there is a decrease in values of density, velocity and pressure in case of magnetogasdynamics as compared to non-magnetic case. It may be noted that the effect of magnetic field on the flow pattern is more significant in case of isothermal flow as compared to adiabatic flow. The effect of non-idealness, specific heat exponent and magnetic field strength on the variation of shock strength across the shock front is also investigated.     

激波诱导含灰气体边界层中的粒子分布

本文研究当激波沿着一个固体表面等速地穿越含灰气体运动时所诱导的层流边界层特性。考虑了作用在气体边界层中球形粒子的 Saffman 升力,建议了一种计算近壁区中弥散相密度剖面的方法,并给出了数值计算结果。本文结果表明:在激波后方存在着一个弯曲的薄层区域,其中的粒子密度可以比其波前原始值增加许多倍。这种粒子聚集效应对于工业中粉尘爆炸等实际问题具有重要意义。     

The distribution of particles in a shock-induced boundary layer of a dusty gas over a solid surface

The laminar boundary layer behind a constant-speed shock wave moving through a dusty gas along a solid surface is studied. The Saffman lift force acting on a spherical particle in a gas boundary layer is taken into account. A method for calculating the density profile of dispersed phase near the wall is proposed and some numerical results are given. It is shown that behind the shock wave, there exists a curved thin layer where the density of particles is many times higher than the original one. This dust collection effect may be of essential importance to the problem of dust explosion in industry.     

Self-similar flow of a mixture of a non-ideal gas and small solid particles behind a shock wave in presence of heat conduction, radiation heat flux and a gravitational field

Similarity solutions are obtained for one-dimensional unsteady flow of a dusty gas behind a spherical shock wave with heat conduction and radiation heat flux under a gravitational field of heavy nucleus at the centre (Roche model). The dusty gas is assumed to be a mixture of small solid particles and a non-ideal gas. The equilibrium flow conditions are assumed to be maintained, and the heat conduction is expressed in terms of Fourier’s law and the radiation is considered to be of the diffusion type for an optically thick grey gas model. The thermal conductivity K and the absorption coefficient α _R are assumed to vary with temperature and density. In order to obtain similarity solutions the density of the undisturbed medium is assumed to be constant. The effects of an increase in the value of the parameter of non-idealness of the gas in the mixture $\bar{b}$ , the mass concentration of the solid particles in the mixture K _p , the ratio of the density of the solid particles to the initial density of the gas G ₁ and the variation of the heat transfer parameters Γ _R and Γ _c are obtained.     

Flow behind magnetogasdynamic exponential shock wave in self-gravitating gas

The propagation of a cylindrical (or spherical) shock wave driven out by a piston moving with time according to an exponential law, in a self-gravitating ideal gas with azimuthal magnetic field is investigated. The initial magnetic field is assumed to be varying according to an exponential law. Solutions are obtained for both the cases of isothermal and adiabatic flows. The effects of variation of ambient magnetic field, gravitational parameter and adiabatic exponent are worked out in detail. It is manifested that the increase in strength of ambient magnetic field has decaying effect on the shock wave however increase in the value of gravitational parameter has reverse effect on the shock strength. The compressibility of the medium is increased in the presence of gravitational field. Also, a comparison between the solutions obtained in the case of isothermal and adiabatic flows is done. Density, pressure, velocity and magnetic field increases, however mass decreases as we move inward from the shock front towards the piston.     

Ideal gas flow behind a finite-amplitude shock wave

The equations of one-dimensional (with a plane of symmetry) adiabatic motion of an ideal gas are transformed to a form convenient for studying flows between a moving piston and a shock wave of variable intensity. The solution is found for the equations of a motion containing a shock wave which propagates through a quiescent gas with variable initial density and constant pressure. This solution contains four arbitrary constants and, in a particular case, gives an example of adiabatic shockless compression by a piston of a gas initially at rest.     

Boundary layer flows behind constant speed shock waves moving into a dusty gas

Laminar boundary layer flows behind constant speed shock waves moving into a dusty gas are analyzed numerically. The basic equations of two-phase flows are derived in shock fixed coordinates and solved by an implicit finite-difference method for the side wall boundary layer in a dusty gas shock tube. The development of the boundary layer and resulting velocity and temperature profiles, respectively, for the gas and particles are given from the shock front to far downstream. The effects of diaphragm pressure ratio, mass loading ratio of particles and particle size upon the flow properties are discussed in detail.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1990.     

Shock wave acceleration in a gravitational field. A special case

A one-dimensional problem of shock wave acceleration in a uniform gravitational field is exactly solved. In front of the shock wave, the medium state is initially in equilibrium and its density decreases according to a power law. The shock wave is generated using a piston moving freely in the gravitational field. The adiabatic index is assumed to be equal to 3. The obtained solution is represented in terms of elementary functions.     

Decay of shock waves in a dusty-gas shock tube

A numerical study is performed for the unsteady nonequilibrium flow of a gas-particle mixture in a shock tube, where a semi-empirical formula for a single particle is assumed to calculate the drag and heat transfer rate of the particle cloud. To simulate actual flows of the mixture in which the size of the particles is distributed over a finite range, the motion of the particles is analyzed by dividing them into several groups according to their different diameters. It is shown that the particles of diameter larger than the average value cause a significant delay in the relaxation of the gas-particle flow. Good agreement is obtained between the numerical and the experimental results of the decrease in the shock propagation velocity, except for strong shock waves transmitted into dusty gas with a high loading ratio.     

Compressible flow of a two-phase fluid between finite vessels—I: Ideal carrier gas

The transfer of a multiphase fluid from a high-pressure vessel to one initially at lower pressure is investigated. The fluid is composed of two phases which do not undergo any change. The phases consist of an ideal gas, and solid particles (or liquid droplets) having constant density. The mixture is assumed to be stagnant and always perfectly mixed as well as at thermal equilibrium in each constant volume vessel. The fluid also remains homogeneous and at equilibrium while flowing between vessels. The transport properties of the mixture are taken to be zero. One important finding is that the expanding mixture or pseudo-fluid behaves similarly to a polytropic Abel-Noble gas. The mixture thermodynamic properties, the end state in each vessel at pressure equilibrium, the critical parameters and time-dependent results are given for the adiabatic and isothermal limiting cases. The results include both initially sonic and initially subsonic transfers. No mathematical restriction is placed on the particle concentration, although some limiting results are given for small particle volume fraction. The mass transferred at adiabatic pressure equilibrium can be significantly less than that when thermal equilibrium is also reached. Furthermore, the adiabatic pressure equilibrium level may not be the same as that obtained at thermal equilibrium, even when all initial temperatures are the same. Finally, it is shown that the transfer times can be very slow compared to those of a pure gas due to the large reduction possible in the mixture sound speed.     

The quasi-stationary structure of radiating shock waves. I. The one-temperature fluid

We calculate the quasi-stationary structure of a radiating shock wave propagating through a spherically symmetric shell of cold gas by solving the time-dependent equations of radiation hydrodynamics on an implicit adaptive grid. We show that this code successfully resolves the shock wave in both the subcritical and supercritical cases and, for the first time, we have reproduced all the expected features – including the optically thin temperature spike at a supercritical shock front – without invoking analytic jump conditions at the discontinuity. We solve the full moment equations for the radiation flux and energy density, but the shock wave structure can also be reproduced if the radiation flux is assumed to be proportional to the gradient of the energy density (the diffusion approximation), as long as the radiation energy density is determined by the appropriate radiative transfer moment equation. We find that Zel'dovich and Raizer's (1967) analytic solution for the shock wave structure accurately describes a subcritical shock but it underestimates the gas temperature, pressure, and the radiation flux in the gas ahead of a supercritical shock. We argue that this discrepancy is a consequence of neglecting terms which are second order in the minimum inverse shock compression ratio [, where is the adiabatic index] and the inaccurate treatment of radiative transfer near the discontinuity. In addition, we verify that the maximum temperature of the gas immediately behind the shock is given by , where is the gas temperature far behind the shock. Received 21 September 1998/ Accepted 2 February 1999     

Interaction of a shock wave with a loose dusty bulk layer

The interaction of a planar shock wave with a loose dusty bulk layer has been investigated both experimentally and numerically. Experiments were conducted in a shock tube. The incident shock wave velocity and particle diameters were measured with the use of pressure transducers and a Malvern particle sizer, respectively. The flow fields, induced by shock waves, of both gas and granular phase were visualized by means of shadowgraphs and pulsed X-ray radiography with trace particles added. In addition, a two-phase model for granular flow presented by Gidaspow is introduced and is extended to describe such a complex phenomenon. Based on the kinetic theory, such a two-phase model has the advantage of being able to clarify many physical concepts, like particulate viscosity, granular conductivity and solid pressure, and deduce the correlative constitutive equations of the solid phase. The AUSM scheme was employed for the numerical calculation. The flow field behind the shock wave was displayed numerically and agrees well with our corresponding experimental results.      

Lifting of Disperse Particles from a Cavity Behind the Front of an Unsteady Shock Wave with a Triangular Velocity Profile

The gas flow in plane shock waves slipping along an impermeable surface with a rectangular cavity where solid disperse particles are suspended is considered numerically. The motion of the gas and particles (gas suspension) is modeled by equations of mechanics of multiphase media. Some laws of the behavior of the dusty cloud in the cavity are established for the case of wave interaction with the cavity.     

Supersonic low-concentration dusty-gas flow past a plane cylinder in the presence of an oblique shock wave interacting with the bow shock

The motion of an inertial dispersed admixture near a plane cylinder immersed in a steady-state hypersonic dusty flow in the presence of an oblique shock wave interacting with the bow shock is considered. It is assumed that the free-stream particle mass concentration is small and the particles do not affect the carrier flow. The III and IV shock wave interaction regimes are considered. The gas flow parameters in the shock layer are calculated from the numerical solution of the full Navier-Stokes equations for the perfect gas. A TVD second-order finite-difference scheme constructed on the basis of a finite volume method is used. For calculating the dispersed-phase parameters, including the concentration, the full Lagrangian method is used. On a wide range of variation of the particle inertia parameters, the patterns of the particle trajectories, velocity, concentration, and temperature in the shock layer are studied. The possibility of aerodynamic focusing of the particles behind the shock wave intersection point and the formation of narrow beams with a high particle concentration is revealed. These beams impinge on the cylinder surface and result in a sharp increase in the local heat fluxes. The maximal possible increase in the heat fluxes caused by the particles colliding with the cylinder surface is estimated for the flows with and without the incident oblique shock wave.     

Propagation of shock and detonation waves in dust-laden gases

This article examines the flows of a two-phase mixture of a gas with solid particles arising as a result of the propagation of shock waves or detonation waves through a homogeneous medium at rest. It is assumed that the basic assumptions of the mechanics of mutually penetrating continua hold [1], whereby it is possible to describe the flow of each phase of the mixture within the framework of the mechanics of a continuous medium. We assume that the solid phase consists of identical, incompressible, and nondeformable particles of spherical shape. It is assumed that the temperature inside the particles is homogeneous. Collisions between particles and their Brownian motion are ignored. It is assumed that the carrier phase is an ideal gas (the viscosity is only allowed for in the interaction forces between phases). The contribution of the volume of the particles is not considered. On the basis of these assumptions, the following problems are considered: the propagation of a detonation wave in a mixture of a detonating gas and chemically inert particles and the motion of a dust-gas mixture in a shock tube in the presence of combustion of the particles.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6. pp. 93–99, November–December, 1984.     

Experimental study of planar shock wave interactions with dense packed sand wall

A dense packed sand wall is impacted by a planar shock wave in a horizontal shock tube to study the shock-sand wall interaction. The incident shock Mach number ranges from 2.18 to 2.38. A novel device for actively rupturing diaphragm is designed for the driver section of the shock tube. An apparatus for loading particles is machined by the electrical discharge cutting technique to create a dense packed particle wall. High-speed schlieren imaging system and synchronized pressure measurement system are used together to capture the wave structures and particle cloud velocity. The dynamic evolution model from dense packed particles to dense gas–solid cloud at the initial driving stage is established. The blockage and permeation effects of the sand wall work together and influence each other. The high pressure gas behind the incident shock wave blocked by the sand wall pushes the upstream front of the wall forward like a piston. Meanwhile, the high speed gas permeating through the sand wall drags the sands of the most downstream layer forward. The incident shock strength, initial sand wall thickness and particle diameter are varied respectively to investigate the shock attenuation and the wall acceleration. Increasing the sands diameter or mixing in small diameter sands can significantly attenuate the incident shock. The smaller particles or the particles in thinner wall can be dispersed into a larger range in the process of transform from dense packed particles to dense gas–solid cloud. Moreover, the stronger incident shock can disperse the particles into a larger region.