Reflection of shock waves from a solid boundary in a mixture of condensed materials. 1. Equilibrium approximation

The process of reflection of shock waves (SW) from a solid wall in a two-component mixture of condensed materials is studied within the framework of mechanics of heterogeneous media. The velocity of a reflected SW and the values of the parameters behind its front are analytically determined as functions of the velocity of the incident wave and the initial parameters of the mixture. It is shown that the absolute value of the velocity of the reflected SW can be greater than the velocity of the incident SW in mixtures with a small content of the light component and at low velocities of the incident shock wave. The nonmonotonic character of the dependence of pressure in the final equilibrium state behind the incident SW on the initial volume concentration of particles is demonstrated. The velocity of the incident SW is estimated for the case where a similar effect is also observed behind a reflected SW. It is established that, for weak shock waves, the dependence of the amplification factor of the reflected SW on the initial volume concentration of the light component is nonmonotonic and has a local maximum. It is noted that, as the velocity of the incident SW increases, the effect of compacting of the mixture (increase in concentration of the heavy component) behind the reflected SW becomes much less pronounced than in a passing SW. Institute of Theoretical and Applied Mechanics, Siberian Division, Russian Academy of Sciences. Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 5, pp. 73–78, September–October, 1999.     

Interaction of shock waves with a combined discontinuity in two-phase media. 2. Nonequilibrium approximation

Interaction of a shock wave and a motionless combined discontinuity separating two twocomponent mixtures with different initial volume concentrations is studied on the basis of numerical simulation of unsteady processes. The calculations were performed using a modified method of coarse particles and a highaccuracy TVD difference scheme adapted to calculation of twophase flows. Flow parameters determined by analytical dependences coincide with those obtained by numerical simulation at large times of the process. Upon interaction of the shock wave and the combined discontinuity, the type of the transient or reflected shock wave may coincide with or differ from the type of the incident shock wave. The possibility of existence of a pressure difference at the combined discontinuity boundary, which was earlier predicted analytically, is confirmed.     

Shock waves in a mixture of materials. Hydrodynamic approximation

We formulate equations of motion and of hydrodynamics for the calculation of shock adiabats of a mixture of condensed materials under very high pressures, the assumption being made that strength properties can be neglected. Use is made of the general principles for constructing models of interacting continuous media [1–3] systematically presented in [4]. In our calculations we involve the difference of the pressures in the component materials of the mixture. In this regard we invoke the following conditions: the condition of proportional (with respect to mass) shock increase of energy; the consistency condition requiring equality of particle velocities and shock velocity in the individual particles; and, finally, also the condition of proportionality of the pressures in the individual phases. We present numerical calculations for mixtures of tungsten and paraffin and also for mixtures of aluminum and epoxy resin. Our calculations agree with experimental data and also with calculations made upon specifying the equality of the phase pressures [5–8]; they are also in agreement with calculations made in accordance with the additive rule (see [9–11]).Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 113–126, July–August, 1976.     

On flows behind shock waves reflected from a solid wall

The flow fields associated with the interaction of a normal shock wave with a plane wall kept at a constant temperature were studied based on kinetic theory which can describe appropriately the shock structure and its reflection process. With the use of a difference scheme, the time developments of the distributions of the fluid dynamic quantities (velocity, temperature, pressure and number density of the gas) were obtained numerically from the BGK model of the Boltzmann equation subject to the condition of diffusive-reflection at the wall for several cases of incident Mach number:M ₁=1.2, 1.5, 2.0, 3.0, 4.0, 5.0 and 6.0. The reflection process of the shocks is shown explicitly together with the resulting formation of the flow fields as time goes on. The nonzero uniform velocity toward the wall occurring between the viscous boundary layer and the reflected shock wave is found to be fairly large, the magnitude of which is of the order of several percent of the velocity induced behind the incident shock, decreasing as the incident Mach number increases. It is also seen that a region of positive velocity (away from the wall) within the viscous boundary layer manifests itself in the immediate vicinity of the wall, which is distinct for larger incident Mach numbers. Some of the calculated density profiles are compared with available experimental data and also with numerical results based on the Navier-Stokes equations. The agreement between the three results is fairly good except in the region close to the wall, where the difference in the conditions of these studies and the inappropriateness of the Navier-Stokes equations manifest themselves greatly in the gas behavior.This article was processed using Springer-Verlag TEX Shock Waves macro package 1990.     

On one-dimensional shock waves in composite materials modelled as interpenetrating solid continua

A one dimensional version of a theory of composite materials modelled as interpenetrating solid continua is used to study the propagation of shock waves in composites with two identifiable constituents. It is found that two distinct types of shock waves may propagate except when one of the constituents is a chopped fiber. The speeds at which the shock waves propagate are determined as are the differential equations which govern the evolutionary behaviour of the amplitudes of the waves. The implications of these results are studied in detail in a number of particular situations. Finally, the special results which hold when the amplitudes of the shock waves are infinitesimal are also presented.     

Strong shock waves propagating through a bubbly mixture

Relatively strong shock waves propagating through a mixture of an aqueous glycerol solution and small helium or argon bubbles are investigated experimentally. Shock strength and void fraction are varied in the experiments. Results regarding shock speed and other aspects are presented and discussed in connection with previous studies. Good agreement with isothermal theory is found.     

Distribution of spherical shock waves in a two-phase fuel mixture

A study has been made of the distribution of spherical shock waves in a two-phase fuel mixture. It is shown that interaction with the liquid fuel droplets increases the wave intensity. Conditions for spherical shock wave amplification in a two-phase fuel mixture are marked out.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 112–117, September–October, 1973.The authors wish to thank V. V. Adushkin for a useful discussion of the results obtained in this work.     

Structure,propagation, and reflection of shock waves in a mixture of solids (the hydrodynamic approximation)

Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, No. 4, pp. 10–18, July–August, 1992.     

Regular reflection of a shock wave from a solid wall in a stream of fuel mixture

The two-dimensional stationary problem of regular reflection of a shock wave from a plane solid wall in a fuel gas mixture is examined in the case when the mixture is ignited at the intersection of the incident wave with the wall and a flame front is formed behind the reflected shock wave. The shock waves and the flame front are considered plane surfaces of discontinuity. The fuel mixture and the reaction products are considered perfect, inviscid, and non-heat-conducting gases.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 161–163, July–August, 1978.     

Oblique reflection of shock waves from rigid porous materials

A study to determine the general gas dynamic behaviour associated with the impact of a shock wave on a porous wedge has been undertaken. A number of interesting features are noted. The pattern of wave reflection is shown to be significantly affected by the inflow of gas into the wedge. This has the effect of reducing the triple point trajectory angle for cases of Mach reflection and for strongly reducing the reflection angle in regular reflection. The permeability of the wedge has a significant effect on the strength of the reflected wave and in some cases this wave can be attenuated to the extent that it is almost eradicated. Pressure measurements taken under the wedge are characterized by oscillations which are of similar shape, for a given wedge, over a range of shock wave Mach numbers. It is shown that the wave transmitted into the wedge is attenuated to varying degrees depending on the material properties, and that for weak incident waves the mean propagation velocity can be less than the sound speed in the pore fluid. Photographs taken using a specially constructed wedge which allows the transmitted wave to be visualised, show that the transmitted wave is nearly plane.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1.0 and the AMS fonts, developed by the American Mathematical Society.     

Reflection and transmission of plane waves from fluid-piezothermoelastic solid interface

The reflection and transmission of plane waves from a fluid-piezothermoelastic solid interface are studied. The expressions for amplitude ratios and energy ratios corresponding to reflected waves and transmitted waves are derived analytically. The piezo-thermoelastic solid half-space is assumed to have 6mm type symmetry and assumed to be loaded with water. The effects of angle of the incidence, the frequency, the specific heat, the relaxation time, and the thermal conductivity on the reflected and transmitted energy ratios are studied numerically for a particular model of cadmium selenide (CdSe) and water. Some special cases are also studied.     

The interaction between shock waves and solid spheres arrays in a shock tube

When a shock wave interacts with a group of solid spheres, non-linear aerodynamic behaviors come into effect. The complicated wave reflections such as the Mach reflection occur in the wave propagation process. The wave interactions with vortices behind each sphere‘s wake cause fluctuation in the pressure profiles of shock waves. This paper reports an experimental study for the aerodynamic processes involved in the interaction between shock waves and solid spheres. A schlieren photography was applied to visualize the various shock waves passing through solid spheres. Pressure measurements were performed along different downstream positions. The experiments were conducted in both rectangular and circular shock tubes. The data with respect to the effect of the sphere array, size, interval distance, incident Mach number, etc., on the shock wave attenuation were obtained.     

Numerical simulations of shock wave propagation in condensed multiphase materials

We propose to find out numerical solutions of a travelling shock wave in condensed mixtures by using a direct numerical simulation. Condensed multiphase materials under shock wave conditions are mechanically characterized by a unique pressure and a unique velocity. In this study, the mixture is considered as a collection of grains separated by interface between each material: this problem of interfaces is solved by a diffuse interface method. The results will be compared to existing one-dimensional numerical models, analytical solutions and also experimental data. The volume fraction (or the phase temperature) is not measured in experiments and it is then important to verify the behaviour of a phase quantity through various methods. A non-monotonous evolution of the volume fraction is obtained with analytical solution as well as numerical simulation.      

Use of a potential approximation to calculate flows with shock waves

Errors arising in the calculation of ideal gas flows with shock waves of moderate intensity in a potential approximation are analyzed. Recommendations for decreasing them are given in the framework of the potential approximation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 121–129, March–April, 1986.The authors are grateful to V. A. Vostretsova for her help in the work.