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.     

Focusing of a shock wave in a rarefied gas A numerical study

The process of focusing of a shock wave in a rarefied noble gas is investigated by a numerical solution of the corresponding two dimensional initial–boundary value problem for the Boltzmann equation. The numerical method is based on the splitting algorithm in which the collision integral is computed by a Monte Carlo quadrature, and the free flow equation is solved by a finite volume method. We analyse the development of the shock wave which reflects from a suitably shaped reflector, and we study influence of various factors, involved in the mathematical model of the problem, on the process of focusing. In particular, we investigate the pressure amplification factor and its dependence on the strength of the shock and on the accommodation coefficient appearing in the Maxwell boundary condition modelling the gas-surface interaction. Moreover, we study the dependence of the shock focusing phenomenon on the shape of the reflector, and on the Mach number of the incoming shock. Received 25 May 1998 / Accepted 4 January 2000     

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     

A numerical study of shock wave reflections on low density foam

A continuum mixture theory is used to describe shock wave reflections on low density open-cell polyurethane foam. Numerical simulations are compared to the shock tube experiments of Skews (1991) and detailed wave fields are shown of a shock wave interacting with a layer of foam adjacent to a rigid wall boundary. These comparisons demonstrate that a continuum mixture theory describes well the shock interactions with low density foam.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.     

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     

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     

A computational study of shock speeds in high-performance shock tubes

This paper describes U2DE, a finite-volume code that numerically solves the Euler equations. The code was used to perform multi-dimensional simulations of the gradual opening of a primary diaphragm in a shock tube. From the simulations, the speed of the developing shock wave was recorded and compared with other estimates. The ability of U2DE to compute shock speed was confirmed by comparing numerical results with the analytic solution for an ideal shock tube. For high initial pressure ratios across the diaphragm, previous experiments have shown that the measured shock speed can exceed the shock speed predicted by one-dimensional models. The shock speeds computed with the present multi-dimensional simulation were higher than those estimated by previous one-dimensional models and, thus, were closer to the experimental measurements. This indicates that multi-dimensional flow effects were partly responsible for the relatively high shock speeds measured in the experiments. Received 15 November 1996 / Accepted 3 February 1997     

Behavior of the surface of a bubbly liquid after detonation wave impact

In safety engineering, one position of interest inside heterogeneous systems of the type liquid–gas is the contact surface between these two phases. Under certain conditions, e.g. shock wave impact, phenomena can take place at this position that can have a significant influence on the explosion behavior of the system. In this work an investigation is presented about the existence of such phenomena on the surface of liquid cyclohexane with or without the existence of oxygen containing bubbles. The observations have been performed during the time before, as well as after, a detonation wave reflection on that surface. High-speed pressure and optical measurements have been applied. Apart from the experimental observations, also a theoretical analysis and discussion is presented in this contribution, which contains the comparison between calculated and experimental values.     

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.     

A non-equilibrium multiscale simulation of shock wave propagation

A novel non-equilibrium multiscale dynamics (NEMSD) is proposed to simulate non-equilibrium thermal–mechanical processes. The model couples coarse-grain thermodynamics with a fine scale molecular dynamics. A Distributed Nośe-Hoover Thermostat Network is used, which regulates the temperature in each coarse scale Voronoi cell according to the finite element (FE) nodal temperature. The atoms in each element-cell, namely Voronoi cell-ensemble, are assumed to be in a local equilibrium state within one coarse scale time step. The change of FE nodal temperature provides a source of random forces, which drive the system out of equilibrium. The proposed NEMSD can successfully simulate shock wave propagation in a cubic lattice.     

The use of a slow sound speed fluorocarbon liquid for shock wave research

Fluorocarbon liquids have very low sound speeds in comparison with water. The measurement of shock waves in water is complicated by its relatively high sound speed. This paper presents a fluorocarbon liquid with a sound speed of 655 m/s for use in liquid shock experiments. Experimental and numerical results of shock wave reflection from various parabolas and wedges are given. Experiments were performed in a vertical liquid shock tube. The properties including an equation of state for the liquid are given. Numerical simulations using this equation of state are performed using a finite element program. It is shown that the investigation of non-linearities in water will require shock tubes that can withstand high pressures. Due to the high B/A parameter for this fluorocarbon liquid, it is demonstrated that non-linearities can be achieved and studied at much lower pressures. Received 1 July 1996 / Accepted 26 September 1996     

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.      

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.      

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.     

Supersonic liquid jets: Their generation and shock wave characteristics

The generation of high-speed liquid (water and diesel fuel) jets in the supersonic range using a vertical single-stage powder gun is described. The effect of projectile velocity and mass on the jet velocity is investigated experimentally. Jet exit velocities for a set of nozzle inner profiles (e.g. straight cone with different cone angles, exponential, hyperbolic etc.) are compared. The optimum condition to achieve the maximum jet velocity and hence better atomization and mixing is then determined. The visual images of supersonic diesel fuel jets (velocity about 2000 m/s) were obtained by the shadowgraph method. This provides better understanding of each stage of the generation of the jets and makes the study of their characteristics and the potential for auto-ignition possible. In the experiments, a pressure relief section has been used to minimize the compressed air wave ahead of the projectile. To clarify the processes inside the section, additional experiments have been performed with the use of the shadowgraph method, showing the projectile travelling inside and leaving the pressure relief section at a velocity of about 1100 m/s. Received 23 January 2001 / Accepted 2 July 2001     

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     

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     

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     

Wave pattern characteristics of a two-phase nozzle flow by shock propagation

In this paper, the wave pattern characteristics of shock-induced two-phase nozzle flows with the quiescent or moving dusty gas ahead of the incident-shock front has been investigated by using high-resolution numerical method. As compared with the corresponding results in single-phase nozzle flows of the pure gas, obvious differences between these two kinds of flows can be obtained. Received 14 June 1996 / Accepted 19 October 1996     

A study of shock wave interaction with a rotating cylinder

Shock wave reflection over a rotating circular cylinder is numerically and experimentally investigated. It is shown that the transition from the regular reflection to the Mach reflection is promoted on the cylinder surface which rotates in the same direction of the incident shock motion, whereas it is retarded on the surface that rotates to the reverse direction. Numerical calculations solving the Navier-Stokes equations using extremely fine grids also reveal that the reflected shock transition from RR MR is either advanced or retarded depending on whether or not the surface motion favors the incident shock wave. The interpretation of viscous effects on the reflected shock transition is given by the dimensional analysis and from the viewpoint of signal propagation.Received: 24 April 2002, Accepted: 16 August 2002, Published online: 25 March 2003