Shock wave interaction with cellular materials

The equations governing the head-on collision of a planar shock wave with a cellular material and a numerical scheme for solving the set of the governing equations were outlined. In addition, the condition for the transmitted compression waves to transform into a shock wave, inside the cellular material was introduced. It was proved analytically that a cellular material cannot be used as a means of reducing the pressure load acting on the end-wall of the shock tube. In subsequent papers, the interaction of planar shock waves with specific cellular materials, e.g., foams and honeycombs will be described in detail.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.     

Dynamics of stress wave propagation in a chain of photoelastic discs impacted by a planar shock wave; Part I, experimental investigation

The propagation of stress waves through a chain of discs has been studied experimentally. Optically transparent 20-mm diameter discs, made of epoxy, were loaded dynamically by head-on collision with an incident planar shock wave. The loading was done in a vertical shock tube. The head-on collision between the punch-plate, placed on top of the chain of discs, and the incident shock wave resulted in a head-on reflected shock wave inducing behind it a fairly uniform step-wise pressure pulse having duration of about 6 ms. The recorded fringe patterns of the stress field, in the discs-chain, show that the input pressure pulse was broken into several oscillating cycles. The back and forth bouncing of stress waves gave rise to two different modes of the contact stress oscillations, which continued until the overall stress reaches equilibrium with the input conditions. The registered propagation velocity of the stress wave was significantly lower than the appropriate speed of sound in the material from which the discs were made.      

Numerical simulation of the head-on reflection of a regular reflection

The head-on collision and subsequent reflection of a Regular Reflection (RR) from the end-wall of a shock tube has been investigated both experimentally and numerically for two different incident shock wave Mach numbers and two different reflecting wedge angles. The agreement between the double-exposure holographic interferograms and the numerical simulations which were obtained using a GRP based numerical code, was found to be excellent in the RR region and very good behind the head-on reflected RR. The overall good agreement between the computed and experimental constant-density contours (isopycnics) constitutes a validation of the computational method, including the oblique-wall boundary condition.     

Shock wave attenuation by grids and orifice plates

The interaction of weak shock waves with porous barriers of different geometries and porosities is examined. Installing a barrier inside the shock tube test section will cause the development of the following wave pattern upon a head-on collision between the incident shock wave and the barrier: a reflected shock from the barrier and a transmitted shock propagating towards the shock tube end wall. Once the transmitted shock wave reaches the end wall it is reflected back towards the barrier. This is the beginning of multiple reflections between the barrier and the end wall. This full cycle of shock reflections/interactions resulting from the incident shock wave collision with the barrier can be studied in a single shock tube test. A one-dimensional (1D), inviscid flow model was proposed for simulating the flow resulting from the initial collision of the incident shock wave with the barrier. Fairly good agreement is found between experimental findings and simulations based on a 1D flow model. Based on obtained numerical and experimental findings an optimal design procedure for shock wave attenuator is suggested. The suggested attenuator may ensure the safety of the shelter’s ventilation systems.     

Head-on collision of normal shock waves with rigid porous materials

The head-on collision of a planar shock wave with a rigid porous material has been investigated experimentally in a 75 mm × 75 mm shock tube. The experimental study indicated that unlike the reflection from a flexible porous material (e.g., polyurethane foam) where the transmitted compression waves do not converge to a sharp shock wave, in the case of a rigid porous material (e.g., alumina) the transmitted compression waves do converge to a sharp shock wave, which decays as it propagates along the porous material. In addition to this major difference, many other differences were observed. They are outlined in the following sections.Based on these observations a suggestion modifying the phenomenology of the reflection/interaction process in the case a porous material with large permeability is proposed.     

Frontal collision of internal solitary waves of first mode

The dynamics and energetics of a frontal collision of internal solitary waves (ISW) of first mode in a fluid with two homogeneous layers separated by a thin interfacial layer are studied numerically within the framework of the Navier–Stokes equations for stratified fluid. It was shown that the head-on collision of internal solitary waves of small and moderate amplitude results in a small phase shift and in the generation of dispersive wave train travelling behind the transmitted solitary wave. The phase shift grows as amplitudes of the interacting waves increase. The maximum run-up amplitude during the wave collision reaches a value larger than the sum of the amplitudes of the incident solitary waves. The excess of the maximum run-up amplitude over the sum of the amplitudes of the colliding waves grows with the increasing amplitude of interacting waves of small and moderate amplitudes whereas it decreases for colliding waves of large amplitude. Unlike the waves of small and moderate amplitudes collision of ISWs of large amplitude was accompanied by shear instability and the formation of Kelvin–Helmholtz (KH) vortices in the interface layer, however, subsequently waves again become stable. The loss of energy due to the KH instability does not exceed 5%–6%. An interaction of large amplitude ISW with even small amplitude ISW can trigger instability of larger wave and development of KH billows in larger wave. When smaller wave amplitude increases the wave interaction was accompanied by KH instability of both waves.     

Three-dimensional propagation of the transmitted shock wave in a square cross-sectional chamber

An investigation into the three-dimensional propagation of the transmitted shock wave in a square cross-section chamber was described in this paper, and the work was carried out numerically by solving the Euler equations with a dispersion-controlled scheme. Computational images were constructed from the density distribution of the transmitted shock wave discharging from the open end of the square shock tube and compared directly with holographic interferograms available for CFD validation. Two cases of the transmitted shock wave propagating at different Mach numbers in the same geometry were simulated. A special shock reflection system near the corner of the square cross-section chamber was observed, consisting of four shock waves: the transmitted shock wave, two reflection shock waves and a Mach stem. A contact surface may appear in the four-shock system when the transmitted shock wave becomes stronger. Both the secondary shock wave and the primary vortex loop are three-dimensional in the present case due to the non-uniform flow expansion behind the transmitted shock.PACS: 43.40.Nm     

Head-on collision between normal shock waves and a rubber-supported plate,a parametric study

The equations governing the flow resulting from a head-on collision between a normal shock wave and a rubber-supported plate are listed. The non-dimensional parameters that may affect the resulting flow are specified and their influence on the post-collision flow and waves is studied numerically. It is shown that changes in: the area-ratio between the gas and the rubber cross-sections, the incident shock wave Mach number and the mass ratio between the rubber and the plate it supports, all have significant effects on the post-collision gas and rubber responses. Changes in the rubber elasticity constant also affect the post-collision flow. The extent of the effect that changes in the above mentioned parameters have on the post-collision flow responses depends upon the loading mode used. Three different modes were studied; uni-axial stress loading, bi-axial stress loading and uni-axial strain loading.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.     

Head-on collision of normal shock waves in dusty gases

The head-on collision of normal shock waves in dusty gases has been investigated numerically, using the modified random-choice method. The results concerning the various flow field properties as well as the waves configuration were compared with those of a pure gas case.     

爆轰波与激波对撞爆轰状态演化机理研究

对当量比氢氧混合气体中爆轰波与激波的正面对撞过程进行了二维数值研究. 采用了二阶精度NND差分格式与改进的两阶段化学反应模型,并以数值x-t纹影图以及烟迹图记录了对撞过程. 数值研究表明,透射爆轰波受到膨胀影响首先会衰减,甚至产生局部解耦现象;然后由于三波点的碰撞又能再次重新耦合. 在爆轰波对撞过程中,由于燃烧不均匀性而产生的弱横波对爆轰胞格的形成起着重要作用.      

The interaction between shock waves and foam in a shock tube

An experimental study and a numerical simulation were conducted to investigate the mechanical and thermodynamic processes involved in the interaction between shock waves and low density foam. The experiment was done in a stainless shock tube (80 mm in inner diameter, 10 mm in wall thickness and 5 360 mm in length). The velocities of the incident and reflected compression waves in the foam were measured by using piezo-ceramic pressure sensors. The end-wall peak pressure behind the reflected wave in the foam was measured by using a crystal piezoelectric sensor. It is suggested that the high end-wall pressure may be caused by a rapid contact between the foam and the end-wall surface. Both open-cell and closed-cell foams with different length and density were tested. Through comparing the numerical and experimental end-wall pressure, the permeability coefficients α and β are quantitatively determined.     

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.     

Wave propagation in a non-ideal dusty gas

In this paper we have studied the behavior of wave motion as propagating wavelets and their culmination into shock waves in a non-ideal gas with dust particles. In the absence of non-ideal effect the gas satisfies an equation of state of Mie–Gruneisen type. An expansion wave resulting from the action of receding piston is considered and the solutions to this problem showing effects of dust particles and non-idealness are obtained. The propagation of weak waves is considered and the flow variables in the region bounded by the piston and the characteristic wave front are found out. The expansive action of a receding piston undergoing an abrupt change in velocity is discussed. Cases of central expansion fan and shock fronts are studied and the solutions up to first order in the physical plane are obtained. The effects of non-idealness and dust particles are discussed in each case.     

Shock wave propagation in a branched duct

The propagation of a planar shock wave in a 90° branched duct is studied experimentally and numerically. It is shown that the interaction of the transmitted shock wave with the branching segment results in a complex, two-dimensional unsteady flow. Multiple shock wave reflections from the duct's walls cause weakening of transmitted waves and, at late times, an approach to an equilibrium, one-dimensional flow. While at most places along the branched duct walls calculated pressures are lower than that existing behind the original incident shock wave, at the branching segment's right corner, where a head on-collision between the transmitted wave and the corner is experienced, pressures that are significantly higher than those existing behind the original incident shock wave are encountered. The numerically evaluated pressures can be accepted with confidence, due to the very good agreement found between experimental and numerical results with respect to the geometry of the complex wave pattern observed inside the branched duct. Received 15 July 1996 / Accepted 20 February 1997     

Head-on collision between two mKdV solitary waves in a two-layer fluid system

In this paper, based on the equations presented in [2], the head-on collision between two solitary waves described by the modified KdV equation (the mKdV equation, for short) is investigated by using the reductive perturbation method combined with the PLK method. These waves propagate at the interface of a two-fluid system, in which the density ratio of the two fluids equals the square of the depth ratio of the fluids. The second order perturbation solution is obtained. It is found that in the case of disregarding the nonuniform phase shift, the solitary waves preserve their original profiles after collision, which agrees with Fornberg and Whitham's numerical result of overtaking collision161 whereas after considering the nonuniform phase shift, the wave profiles may deform after collision.     

General solution for interaction of solitary waves including head-on collisions

A corrected version of the Boussinesq equation for long water waves is derived and its general solution for interaction of any number of solitary waves, including head-on collisions, is given. For two solitary waves in head-on collision (which includes the case of normal reflection) the results agree with the experiments known.     

Dust suspensions accelerated by shock waves

The motion of dust suspensions accelerated by shock waves has been experimentally investigated in a vertical shock tube, in which a completely developed plane shock wave of moderate strength propagates into a homogeneously distributed dust suspension with a planar interface. Trajectories of the accelerated interfaces as well as transmitted and reflected shock waves are recorded by using a shadowgraph system with a Cranz-Schardin camera. Two kinds of particle samples, i.e. porous lycopodium particles 30 μm in diameter and corn starch particles with a mean diameter of 10 μm, are employed. The effects of shock wave strength and particle loading ratio are also examined. Experimental data are compared with theoretical results, and the agreement is good. Received: 7 October 1998/Accepted: 1 June 1999     

激波诱导两相流中影响阻力系数的特性参数研究

基于双流体模型 ,利用Euler Lagrange组合方法 ,对激波诱导的气固两相流场进行了数值计算 ,系统研究了影响颗粒群阻力系数的几个重要特性参数。结果表明 :目前采用激波管技术研究非定常条件下颗粒群阻力系数时界定这些因素的影响程度是必要的。     

Interaction of a weak discontinuity wave with a characteristic shock in a dusty gas

In this paper, the evolution of a characteristic shock in a dusty gas is investigated and its interaction with a weak discontinuity wave is studied. The transport equation for the amplitude of the weak discontinuity wave, which is of Bernoulli type, is obtained. The amplitudes of the reflected and transmitted waves after interaction of the weak discontinuity with the characteristic shock are evaluated by using the results of the general theory of wave interaction.      

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.