On shock wave propagation in a branched channel with particles

The problem of wave propagation in a dust–air mixture inside a branched channel has not been studied widely in literature, even though this topic has many important applications especially in process safety (dust explosions). In this paper, a shock wave interaction with a cloud of solid particles, and the further behaviour of both gas and particulate phases were studied using numerical techniques. The geometry mimicked a real channel where bends or branches are common. Two numerical approaches were used: Eulerian–Eulerian and Eulerian–Lagrangian. Using Eulerian-Lagrangian simulation, it was possible to include the effects of particle–particle and particle–wall collisions in a realistic and direct manner. Results are mainly shown as snap-shots of particle positions during the simulations and statistics for the particle displacement. The results show that collisions significantly influence the process of particle cloud formation. PACS47.40.Nm, 02.60.Cb, 47.55.kf     

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.      

Numerical simulation for nonstationary Mach reflection of a shock wave: A kinetic-model approach

A numerical simulation was performed for the process of formation of single Mach reflection on a wedge by solving a BGK type kinetic equation for the reduced distribution function with a finite difference scheme. The calculations were carried out for a shock Mach number 2.75 and wedge angle 25° in a monatomic gas, which corresponds to the conditions of single Mach reflection in the classical von Neumann theory. The calculations were performed for both diffuse and specular reflection of molecules at the wall surface. It is concluded that the diffuse reflection of molecules at the wall surface or the existence of the viscous or thermal layer is an essential factor for a nonstationary process at the initial stage of Mach reflection. Furthermore, the numerical results for diffuse reflection are found to simulate the experimental results very well, such as a transient process from regular reflection to Mach reflection along with shock propagation.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1990.     

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     

Computation and experimental validation of N_{2}-CH_{4}-Ar mixture flows behind normal shock wave

Different vibration-dissociation-vibration coupling models have been used to compute the nonequilibrium N-CH-Ar mixture flow behind a normal shock wave. A three-temperature model was used and the diffuse nature of vibrational relaxation at high temperatures was accounted for. The numerical results obtained with the Treanor and Marrone model (preferential or non preferential) and the Park model of vibration-dissociation-vibration coupling are compared. These results show that the temperatures and the concentrations are mainly affected by the value of the characteristic temperature U in the preferential model of Marrone and Treanor. An assessment of the more realistic model was realized by comparing numerical results with shock tube experiments. The influence of argon addition on the nonequilibrium emission of CN behind the shock wave was also numerically studied and compared to experimental measurements. Received 1 September 1995 / Accepted 10 December 1996     

Dynamics of stress wave propagation in a chain of photoelastic discs impacted by a planar shock wave: Part II, numerical investigation

The propagation of stress waves through a chain of discs has been studied experimentally in Part I (Glam et al. [1]) and is completed here with numerical investigation using the standard package ABAQUS. A fair agreement is found between experimental findings and their simulations. Based on this agreement, parametric study of wave propagation through disc-chains was conducted. Specifically, effects associated with changes in the disc diameter, material density, stiffness/rigidity and the number of discs in the chain on the stressed chain have been studied. It was found that the propagation velocity of the evolved waves increases with improving contacts between the chain’s discs by exposing the chain to a static load before its dynamic loading. The wave- propagation velocity decreases with increase in the discs material density and it increases when its diameter increases. In case of a chain composed of small diameter discs and/or small material density, the transmitted stress wave is first strengthened and only at discs further down the chain it starts decaying. When checking the influence of the dynamic-loading duration it was found that long dynamic-load duration dissolves quickly into short pulses. It was also found that there is a ‘characteristic’ wave for a given chain. This wave propagates with minimal dispersion. Dynamic loads having shorter time duration than the ‘characteristic’ one experiences significant attenuation.     

Discrete element simulation of shock wave propagation in polycrystalline copper

The implementation of the characteristic of compressive plasticity into the Discrete Element Code, DM2, while maintaining its quasi-molecular scheme, is described. The code is used to simulate the shock compression of polycrystalline copper at 3.35 and 11.0 GPa. The model polycrystal has a normal distribution of grain sizes, with mean diameter 14 μm, and three distinct grain orientations are permitted with respect to the shock direction; 〈1 0 0〉, 〈1 1 0〉, and 〈1 1 1〉. Particle velocity dispersion (PVD) is present in the shock-induced flow, attaining its maximum magnitude at the plastic wave rise. PVD normalised to the average particle velocity of and are yielded for the 3.35 and 11.0 GPa shocks, respectively, and are of the same order as those seen in the experiment. Non-planar elastic and plastic wave fronts are present, the distribution in shock front position increasing with propagation distance. The rate of increase of the spread in shock front positions is found to be significantly smaller than that seen in probabilistic calculations on nickel polycrystals, and this difference is attributed, in the main, to grain interaction. Reflections at free surfaces yield a region of tension near to the target free surface. Due to the dispersive nature of the shock particle velocity and the non-planarity of the shock front, the tensile pressure is distributed. This may have implications for the spall strength, which are discussed. Simulations reveal a transient shear stress distribution behind the shock front. Such a distribution agrees with that put forward by Lipkin and Asay to explain the quasi-elastic reloading phenomenon. Simulation of reloading shocks show that the shear stress distribution can give rise to quasi-elastic reloading on the grain scale.     

A holographic interferometric study of shock wave focusing in a circular reflector

A holographic interferometric study was made of the focusing of reflected shock waves from a circular reflector. A diaphragmless shock tube was used for incident shock Mach numbers ranging from 1.03 to 1.74. Hence, the process of reflected shock wave focusing was quantitatively observed. It is found that a converging shock wave along the curved wall undergoes an unsteady evolution of mach reflection and its focusing is, therefore, subject to the evolution of the process of shock wave reflections. The collision of triple points terminates the focusing process at the geometrical focus. In order to interprete quantitatively these interferograms, a numerical simulation using an Eulerian solver combined with adaptive unstructured grids was carried out. It is found numerically that the highest density appears immediately after the triple point collision. This implies that the final stage of focusing is mainly determined by the interaction between shock waves and vortices. The interaction of finite strength shock waves, hence, prevents a curved shock wave from creating the infinite increase of density or pressure at a focal point which is otherwise predicted by the linear acoustic theory.     

Gas combination for shock wave enhancement

It was recently demonstrated that shock wave enhancement could be achieved when a shock propagates in a constant cross-section duct through pairs of air–helium layers having a continually decreasing width (Igra and Igra in Shock Waves 16(3):199–207). A parametric study was conducted aimed at finding a two-layered, light–heavy gas arrangement that yields maximal shock enhancement; the heavy and the light gases used were air and helium, respectively. Effects associated with changes in following parameters were investigated: the number of alternating heavy/light gas layers, the applied reduction ratio between successive layers thickness, and the initial shock wave Mach number.      

Numerical simulation of the reflection of a planar shock wave over a double wedge

A numerical code based on the upwind TVD scheme for simulating the various reflection processes of a planar shock wave over a concave or convex double wedge has been developed. The numerical results were compared with actual experiments and excellent agreement was obtained. The excellent agreement serves also as a validation of the shock-capturing performance of the numerical scheme.     

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     

Numerical simulation of shock propagation in a polydisperse bubbly liquid

The effect of distributed bubble nuclei sizes on shock propagation in a bubbly liquid is numerically investigated. An ensemble-averaged technique is employed to derive the statistically averaged conservation laws for polydisperse bubbly flows. A finite-volume method is developed to solve the continuum bubbly flow equations coupled to a single-bubble-dynamic equation that incorporates the effects of heat transfer, liquid viscosity and compressibility. The one-dimensional shock computations reveal that the distribution of equilibrium bubble sizes leads to an apparent damping of the averaged shock dynamics due to phase cancellations in oscillations of the different-sized bubbles. If the distribution is sufficiently broad, the phase cancellation effect can dominate over the single-bubble-dynamic dissipation and the averaged shock profile is smoothed out.     

A numerical study of oblique shock wave reflections over wedges in an ideal quantum gas

The various oblique shock wave reflection patterns generated by a moving incident shock on a planar wedge using an ideal quantum gas model are numerically studied using a novel high resolution quantum kinetic flux splitting scheme. With different incident shock Mach numbers and wedge angles as flow parameters, four different types of reflection patterns, namely, the regular reflection, simple Mach reflection, complex Mach reflection and the double Mach reflection as in the classical gas can be classified and observed. Both Bose–Einstein and Fermi–Dirac gases are considered.      

Oblique shock wave with sweep

An attached planar, oblique shock with sweep is investigated for the inviscid flow of a perfect gas. The ratio of specific heats, freestream Mach number, and wedge angle in the plane of the freestream velocity are prescribed, with the sweep angle as a free parameter. Explicit relations are provided for jump and detachment conditions. A number of trends, some non-intuitive, are discussed, e.g., the downstream Mach number may increase with sweep.     

Simulation of blast wave propagation over a cylinder

A numerical study of the interaction of plane blast waves with a cylinder is presented. Computations are carried out for various blast-wave durations and comparisons are obtained with the corresponding results of planar shock-wave. Both inviscid and viscous results based on the solution of the Euler and Navier-Stokes equations are presented. The equations are solved by an adaptive-grid method and a second-order Godunov scheme. The shock wave diffraction over the cylinder is investigated by means of various contour plots, as well as, pressure and skin-friction histories. The study reveals that the blast-wave duration significantly influences the unsteady flow over the cylinder. The differences between the viscous and inviscid results are also discussed. Received 2 March 1996 / Accepted 28 February 1997     

Reflection of a ring shock wave from a rigid wall

A new type of an implosion has been observed experimentally and simulated in a numerical calculation: the formation of a quasi-spherical converging shock wave after the reflection of a ring shock wave from a solid wall. The conversion of the ring shock wave into the quasi-spherical converging shock wave intensifies the local implosion properties.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.     

Experimental and numerical studies of underwater shock wave attenuation

The attenuation of an underwater shock wave by a thin porous layer is studied both experimentally and numerically. The shock waves are generated by exploding 10 mg silver azide pellets and the pressures at different distances from the explosion center are measured. Measurements are also carried out with a gauze layer placed between the explosion source and the pressure gauge. The results with and without the gauze layer are compared evaluating the shock wave attenuation. Numerical simulations of the phenomenon are also carried out for a simple wave attenuation model. The results are compared with the experimental data. Despite the simple mathematical model of wave attenuation, the agreement between the experimental and numerical results is reasonable.Received: 22 October 2002, Accepted: 17 June 2003, Published online: 5 August 2003PACS: 47.11.+j, 47.40.Nm, 47.55.Mh     

Oblique shock wave with sweep II

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.      

The formation of a secondary shock wave behind a shock wave diffracting at a convex corner

This paper deals with the formation of a secondary shock wave behind the shock wave diffracting at a two-dimensional convex corner for incident shock Mach numbers ranging from 1.03 to 1.74 in air. Experiments were carried out using a 60 mm 150 mm shock tube equipped with holographic interferometry. The threshold incident shock wave Mach number () at which a secondary shock wave appeared was found to be = 1.32 at an 81° corner and = 1.33 at a 120° corner. These secondary shock waves are formed due to the existence of a locally supersonic flow behind the diffracting shock wave. Behind the diffracting shock wave, the subsonic flow is accelerated and eventually becomes locally supersonic. A simple unsteady flow analysis revealed that for gases with specific heats ratio the threshold shock wave Mach number was = 1.346. When the value of is less than this, the vortex is formed at the corner without any discontinuous waves accompanying above the slip line. The viscosity was found to be less effective on the threshold of the secondary shock wave, although it attenuated the pressure jump at the secondary shock wave. This is well understood by the consideration of the effect of the wall friction in one-dimensional duct flows. In order to interpret the experimental results a numerical simulation using a shock adaptive unstructured grid Eulerian solver was also carried out. Received 1 May 1996 / Accepted 12 September 1996