Sound and weak shock wave propagation in gas-liquid foams

Propagation of sound and weak shock waves in gas-liquid foams is investigated theoretically and experimentally. An original physical model is developed to describe the evolution of small perturbations in a foam of polyhedral structure. The model developed takes into account both peculiarities of interface heat transfer in foam and liquid motion through the system of Plateau-Gibbs borders which results in the appearance of an additional hydrodynamic dissipative force. The Rayleigh equation analog, which takes into account the latter phenomenon, is obtained. Structure and dynamics of weak shock waves are investigated. A vertical shock tube was constructed and used to measure the parameters of weak shock wave propagation in gas-liquid foams of polyhedral structure. Spectral analysis of the data obtained shows that there are weak dispersion and strong dissipation of the initial signal. Comparison of the evolution of experimental and theoretical profiles permits to conclude that the suggested model allows to describe the peculiarities of acoustical perturbations in gas-liquid foam more precisely than it follows from the standard models. Received 15 September 1993 / Accepted 27 December 1993     

Attenuation of shock waves propagating in polyurethane foams

Shock wave attenuation in polyurethane foams is investigated experimentally and numerically. This study is a part of research project regarding shock propagation in polyurethane foams with high-porosities = 0.951 ~ 0.977 and low densities of ρc = 27.6 ~55.8 kg/m³. Sixty Millimeter long cylindrical foams with various cell numbers and foam insertion condition were installed in a horizontal shock tube of 50 mm i.d. and 5.4 mm in length. Results of pressure measurements in air/foam combination are compared with CFD simulation solving the one-dimensional Euler equations. In the case of a foam B fixed on shock tube wall, pressures at the shock tube end wall increases relatively slowly comparing to non-fixed foam, free to move and a foam A fixed on shock tube wall. This implies that elastic inertia hardly contributes to pressure build up. Pressures behind a foam C fixed on shock tube wall decrease indicating that shock wave is degenerated into compression wave. Dimensionless impulse and attenuation factor decrease as the initial cell number increases. The momentum loss varies depending on cell structure and cell number.     

Passage of shock waves through an interface of two-phase gas-liquid media

Among the problems connected with the motion of shock waves in two-phase media consisting of a mixture of a liquid and bubbles of gas, investigations of the passage of shock waves through an interface inside of a two-phase system, or at its surface are of special interest. Inside a two-phase system, interfaces between two two-phase media with different volumetric concentrations of the gas Β₁, Β₂ are possible. One of the values of Β, for example, Β₁, can revert to zero. There is then a passage of the wave from a two-phase system into an incompressible liquid, or vice versa. The investigation of both of the above cases, as well as of the transition two-phase medium (Β₁)—two-phase medium (Β₂) with Β₁≠Β₂ is not only of scientific, but also of practical importance. As is well known [1], the density of a two-phase mixture ρ with a small volumetric concentration of gas is calculated using the relationship ρ=(1?β)ρ_l+αρ₁.     

泡沫材料对冲击波的衰减特性

对冲击波与开式、闭式泡沫作用及其在空气中的传播特性开展实验研究,探讨不同结构的泡沫材料对冲击波衰减的力学特征。通过定量分析泡沫材料对冲击波的超压峰值、正冲量的损失,分析冲击波入射、反射、透射的正冲量。实验结果表明, 泡沫材料对冲击波的衰减体现在对冲击波的反射衰减等方面,其中开式泡沫对冲击波的衰减效果比闭式泡沫稍好,且它们衰减冲击波的具体力学过程也不尽相同。

     

Theory of regular and mach reflection of shock waves in a two-phase gas-liquid medium

An equilibrium model of a two-phase gas-liquid medium, with allowance for the proportion, density, and compressibility of the components, and with a difference from [1] in that the adiabatic velocity of sound is introduced, has been used in order to study the regular and Mach (elementary theory) reflection of a shock wave of moderate intensity from a solid wall throughout the whole range of gas proportions. A complicated nonmonotonic variation has been found for the pressure on the wall behind the reflected wave, the angle of reflection, and the angle of departure of the triple point as functions of the gas proportion, the angle of incidence, and the intensity of the incident wave. In particular, it is shown that oblique reflection for moderate and low gas contents leads to the formation of a stronger reflected shock wave than does normal reflection. The effect of the gas proportion on the position of the boundary between the regions of regular and Mach reflection has already been studied in [2]. The results are described of serial calculations of the parameters of reflection for an air-water mixture, and these results agree fairly well for normal reflection with the known experimental data [3] for low and moderate gas contents. In the limiting case, the results agree with the known results for single-phase media [4, 5].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 188–190, March–April, 1985.     

Head-on interaction of planar shock waves with polyurethane foams — a semi-empirical model

Summary A semi-empirical model for calculating the velocity of the compaction wave which is transmitted into a polyurethane foam struck head-on by a planar shock wave is developed on the basis of some experimental observations. The compaction wave induced velocity and the induced density are determined. Predictions of the model are compared to experimental results and found to be satisfactory.

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Direkte Einwirkung ebener Stoßwellen auf Polyurethanschaumstoff — halbempirisches Modell

Übersicht Ein semi-empirisches Modell zur Berechnung der Verdichtungswellengeschwindigkeit bei planarer Stoßeinwirkung auf Polyurethanschaumstoff wird aufgrund der Analyse einiger experimenteller Beobachtungen aufgestellt. Das Modell ermöglicht die Berechnung der infolge der Stoßverdichtung induzierten Wellengeschwindigkeit und Dichtenänderung. Die Vorhersagen des Modells werden mit Versuchsergebnissen verglichen und können als ausreichend eingestuft werden.

     

Formation of shock waves in flow of charge carriers in semiconductors

A hydrodynamic model of the physics of semiconductors is studied numerically. It is shown that the solution of the problem of an (n⁺-n-n⁺) ballistic diode has a shock wave. This problem is solved using an iterative method. An economical conservative semi-implicit difference scheme is developed for search of a numerical solution. Siberian State Geodetic Academy, Novosibirsk 630108. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 5, pp. 3–10, September–October, 1999.     

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.     

Formation and propagation of cylindrical shock waves in a pulsed induction discharge

An experimental investigation was made into the formation and propagation of cylindrical collapsing shock waves in a pulsed induction gas discharge. The shock fronts were made visible and their propagation velocity measured by means of the schlieren method in conjunction with high-speed cine photography in the photoscanning regime and frame-by-frame detection, which made it possible, using the experimental data, to analyze the formation and propagation of the shock waves. A physical picture of the individual stages in the development of the discharge is given on the basis of the experimental results.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 129–133, January–February, 1981.We are thankful to Yu. P. Raizer for helpful discussions of the work.     

Critical phenomena and waves in gas-liquid systems

The results of the hydraulic studies of gas-liquid media, wave processes in two-phase media and critical phenomena are described. Some methodological foundations to describe these media and methods to obtain the basic similarity criteria for the hydraulics and gas-dynamics of bubble suspensions are discussed. A detailed consideration is given for the phase transition processes on interfaces and the interface stability. A relation has been revealed between the wave and critical phenomena in two-phase systems.Nomenclature a thermal diffusivity - Ar Archimedes number - B gas constant - C heat capacity - C _p heat capacity at constant pressure - C _v heat capacity at constant volume - c ₀ acoustic velocity in the mixture - c _l acoustic velocity in the liquid - C _f flow resistance coefficient - G mass rate of flow - g gravitational acceleration - L latent heat of evaporation - l initial perturbation width - M Mach number - Nu Nusselt number - P pressure - Pr Prandtl number - R bubble radius - (3P ₀/R ₀ ² _f )^–1 bubble resonance frequency square - T temperature - U medium motion velocity - W heavy phase velocity - W light phase velocity - We Weber number - heat release coefficient - dispersion coefficient - void fraction - adiabatic index - film thickness - dimensionless film thickness - kinematic viscosity coefficient - dynamical viscosity coefficient - dissipation coefficient in the mixture - dispersion parameter - _f liquid phase density - light phase density - heat conductivity - surface tension - frequency, ₀ ² =3P ₀/ _f R ₀ ²     

Enhancement of shock waves

The flow field developed behind a shock wave propagating inside a constant cross-section conduit is solved numerically for two different cases. First, when the density of the ambient gas into which the shock propagates has a logarithmic change with distance. In the second, and the more practical case, the ambient gas is composed of pairs of air–helium layers having a continually decreasing width. It is shown that in both cases meaningful pressure amplification can be reached behind the transmitted shock wave. It is especially so in the second case. By proper choice of the number of air–helium layers and their width reduction ratio, pressure amplification as high as 7.5 can be obtained.      

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.     

Revisiting shock waves in metals

We cast Wallace's theory of thermoplastic flow in conservative form. We point out the difference between our formulation, which accounts for contact with an external energy reservoir, and previous formulations of thermoplastic flow. The theory is exploited to show that the experiments of Johnson and Barker on 6062-T6 Al can be interpreted as a weak shock wave that splits into an infinite sequence of “infinitesimal”, shocks, caused by increasing plasticity, leading to the observed smooth temporal velocity profile (a dispersed wave). We predict that overdriven shock waves in metals will split as well. We also re-examine the need for invoking a heat dissipation mechanism for overdriven shocks. It is briefly pointed out that our approach of casting the theory of thermoplastic flow in divergence form can be generalized easily to account for heat release in energetic solids. Received 25 March 1996 / Accepted 20 August 1996     

Propagation of modulated nonlinear waves in a relaxing gas-liquid mixture

The propagation of nonstationary weak shock waves in a chemically active medium is essentially dispersive and dissipative. The equations for short-wavelength waves for such media were obtained and investigated in [1–4]. It is of interest to study quasimonochromatic waves with slowly varying amplitude and phase. A general method for obtaining the equations for modulated oscillations in nonlinear dispersive media without dissipation was proposed in [5–8]. In the present paper, for a dispersive, weakly nonlinear and weakly dissipative medium we derive in the three-dimensional formulation equations for waves of short wavelength and a Schrödinger equation, which describes slow modulations of the amplitude and phase of an arbitrary wave. The coefficients of the equations are particularized for the considered gas-liquid mixture. Solutions are obtained for narrow beams in a given defocusing medium as well as linear and nonlinear solutions in the neighborhood of a diffraction beam. A solution near a caustic for quasimonochromatic waves was found in [9].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 133–143, January–February, 1980.     

Propagation of compaction waves in metal foams exhibiting strain hardening

One-dimensional models for compaction of cellular materials exhibiting strain hardening are proposed for two different impact scenarios. The models reveal the characteristic features of deformation under the condition of decreasing velocity during the impact event. It was established that an unloading plastic wave of strong discontinuity propagates in the foam and it has a significant dynamic effect on the foam compaction and energy absorption. The proposed models are based on the actual experimentally derived stress strain curves. The compaction mechanism in three aluminium based foam materials, two of them with relatively low density – Alporas and Cymat with 9% and 9.3% relative density, respectively and a higher density Cymat foam with 21% relative density, is analysed. Numerical simulations were carried out to verify the proposed models.The predictions of the proposed models are compared with published analytical models of compaction of cellular materials which assume a predefined densification strain. It is shown that the approximation of a cellular material with significant strain hardening by the Rigid Perfectly-Plastic-Locking (RPPL) model can lead to an overestimation of the energy absorption capacity for the observed stroke due to the non-uniform strains along the compacted zone of the actual material in contrast to the predefined constant densification strain in the RPPL model. The assumption of a constant densification strain leads also to an overestimation of the maximum stress, which occurs under impact.