The unsteadiness of shock waves propagating through gas-particle mixtures

A shock wave which is incident onto a gas-particle mixture or initiated within such a mixture needs a certain distance to reach a constant velocity. This effect is due to the inertia and the heat capacity of the particles. In general the shock wave is decelerated and the frozen pressure jump is decaying. A vertical shock tube was used in order to produce a plane shock wave incident onto a homogeneous gas-particle mixture. In addition to measurements of the shock velocity and the pressure history along the total low pressure section, the particle velocity was measured within the relaxation zone far downstream of the diaphragm using a laser-Doppler-velocimeter. Thus a drag law describing the particle acceleration within the relaxation zone was derived from the measurements. To compare the experiments with theoretical results, calculations were performed by the random-choice method.     

Numerical investigations of shock waves in gas-particle mixtures

The propagation of shock waves in gas-particle mixtures in one- and two-dimensional geometries is numerically investigated. Two schemes for approximating conservation laws for particles, which are collectively treated as a continuum medium, are compared and discussed. Different models of the drag coefficient and Nusselt number, directly affecting the interaction between the gas and particle phases, are used for obtaining shock profiles, and the results are compared. The oblique shock reflections at a solid wedge in a gas-particle mixture are simulated. The results demonstrate that the reflection pattern changes as the shock propagates along the wedge, revealing strong non-selfsimilarity of the phenomenon.Received: 22 May 2003, Accepted: 28 August 2003, Published online: 12 November 2003 Correspondence to:T. Saito     

Numerical study of the lift force influence on two-phase shock tube boundary layer characteristics

The importance of the lift force acting on the dispersed phase in the boundary layer of a laminar gas-particle dilute mixture flow generated by a shock wave is investigated numerically. The particle phase is supposed to form a continuum and is described by an Eulerian approach. The ability of the Eulerian model to simulate particle flows and the importance of the two-way coupling are proven by comparison with experimental data as well as with the numerical results from schemes based on a Lagrangian approach. The models used for the lift force are discussed through comparisons between numerical and experimental results found in the literature. Some results about the formation of a dust cloud are numerically reproduced and show the major role of the lift force. Simulations of two-dimensional two-phase shock tube flows are also performed including the lift force effects. Although the wave propagation is weakly influenced by the lift force, the force modifies substantially the dynamics of the flow near the wall. Received 17 February 2000 / Accepted 30 November 2000     

The flow structure of dilute gas-particle suspensions behind a shock wave moving along a flat surface

The asymptotic and numerical investigations of shock-induced boundary layers in gas-particle mixtures are presented.The Saffman lift force acting on a particle in a shear flow istaken into account.It is shown that particle migration across the boundary layer leads tointersections of particle trajectories.The corresponding modification of dusty gas model isproposed in this paper.The equations of two-phase sidewall boundary layer behind a shock wave moving at aconstant speed are obtained by using the method of matched asymptotic expansions.Themethod of the calculation of particle phase parameters in Lagrangian coordinates isdescribed in detail.Some numerical results for the case of small particle concentration aregiven.     

Numerical study of shock diffraction in dusty gases

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Reflection of shock wave from a compression corner in a particle-laden gas region

We investigated in this paper the progression of a shock-wave reflected from a compression corner in a particle-laden gas medium using a TVD class numerical technique and a MacCormack scheme. For a gas-only flow, the numerical results agreed well with the existing experimental data, suggesting that the gas phase is correctively solved. The effect of particle size and mass fraction ratio is investigated for a dilute gas-particle flow. It has been shown that the shock-wave diffraction and the flow configuration after the shock can become remarkably different from the gas-only flow depending on the particle parameters. Relaxation phenomenon due to the momentum drag and the heat exchange between the gas and the particle phases is explained.Graduate Student of Korea Advanced Institute of Science and TechnologyThis article was processed using Springer-Verlag T_EX Shock Waves macro package 1990.     

The effect of an unsteady drag force on the structure of a non-equilibrium region behind a shock wave in a gas-particle mixture

For numerical analysis of shock wave propagation in gas-particle mixtures, drag coefficients of a sphere in steady flows are generally used. However, it is shown both experimentally and numerically that a shock loaded solid sphere experiences unsteady drag forces. The paper describes a model of unsteady drag force and its effect on the structure of the non-equilibrium region behind a shock front traveling in a dusty gas. The results are compared with those obtained by using a steady drag coefficient and are discussed. It is demonstrated that the large drag force at the early stage of the interaction between shock-wave induced flow and a solid particle affects the flow structure that is obtained with a steady drag force.      

Supersonic low-concentration dusty-gas flow past a plane cylinder in the presence of an oblique shock wave interacting with the bow shock

The motion of an inertial dispersed admixture near a plane cylinder immersed in a steady-state hypersonic dusty flow in the presence of an oblique shock wave interacting with the bow shock is considered. It is assumed that the free-stream particle mass concentration is small and the particles do not affect the carrier flow. The III and IV shock wave interaction regimes are considered. The gas flow parameters in the shock layer are calculated from the numerical solution of the full Navier-Stokes equations for the perfect gas. A TVD second-order finite-difference scheme constructed on the basis of a finite volume method is used. For calculating the dispersed-phase parameters, including the concentration, the full Lagrangian method is used. On a wide range of variation of the particle inertia parameters, the patterns of the particle trajectories, velocity, concentration, and temperature in the shock layer are studied. The possibility of aerodynamic focusing of the particles behind the shock wave intersection point and the formation of narrow beams with a high particle concentration is revealed. These beams impinge on the cylinder surface and result in a sharp increase in the local heat fluxes. The maximal possible increase in the heat fluxes caused by the particles colliding with the cylinder surface is estimated for the flows with and without the incident oblique shock wave.     

Flow of a dispersed phase in the Laval nozzle and in the test section of a two-phase hypersonic shock tunnel

An unsteady gas-particle flow in a hypersonic shock tunnel is studied numerically. The study is performed in the period from the instant when the diaphragm between the high-pressure and low-pressure chambers is opened until the end of the transition to a quasi-steady flow in the test section. The dispersed phase concentration is extremely low, and the collisions between the particles and their effect on the carrier gas flow are ignored. The particle size is varied. The time evolution of the particle concentration in the test section is obtained. Patterns of the quasi-steady flow of the dispersed phase in the throat of the Laval nozzle and the flow around a model (sphere) are presented. Particle concentration and particle velocity lag profiles at the test-section entrance are obtained. The particle-phase flow structure and the time needed for it to reach a quasi-steady regime are found to depend substantially on the particle size. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 5, pp. 102–113, September–October, 2008.     

Hypersonic flow past a supersonic two-phase source

The interaction of a uniform hypersonic gas flow with a supersonic two-phase gas-particle source is considered. In the symmetry-axis neighborhood between the bow and termination shock waves, an approximate analytical solution for the carrier-phase parameters is found. On the basis of parametric numerical calculations, the behavior of the particle trajectories and the concentration distribution in the shock layers are studied for both continuum and free-molecule flow regimes around the particles. The appearance of regions with multiple intersections of the particle trajectories and the formation of "layer structures" in the particle concentration distributions (particle accumulation regions near the envelopes of the particle trajectories) are indicated. The dependence of the number of the high concentration layers on the governing parameters is studied. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 134–147, May–June, 1998. The work received financial support from the Russian Foundation for Basic Research (project No. 96-01-00313) and the National Foundation for Natural Sciences of China (joint RFBR-NFNS grant No.96-01-00017c).     

A multiphase shock tube for shock wave interactions with dense particle fields

Currently there is a substantial lack of data for interactions of shock waves with particle fields having volume fractions residing between the dilute and granular regimes. To close this gap, a novel multiphase shock tube has been constructed to drive a planar shock wave into a dense gas–solid field of particles. A nearly spatially isotropic field of particles is generated in the test section by a gravity-fed method that results in a spanwise curtain of spherical 100-micron particles having a volume fraction of about 20%. Interactions with incident shock Mach numbers of 1.66, 1.92, and 2.02 are reported. High-speed schlieren imaging simultaneous with high-frequency wall pressure measurements are used to reveal the complex wave structure associated with the interaction. Following incident shock impingement, transmitted and reflected shocks are observed, which lead to differences in particle drag across the streamwise dimension of the curtain. Shortly thereafter, the particle field begins to propagate downstream and spread. For all three Mach numbers tested, the energy and momentum fluxes in the induced flow far downstream are reduced about 30–40% by the presence of the particle field.     

The structure of particle-laden,underexpanded free jets

Underexpanded, supersonic gas-particle jets were experimentally studied using the shadowgraph technique in order to examine the influence of the dispersed particles on the shape of the free jet and the structure of the imbedded shock waves. The particle mass loading at the nozzle exit was varied between zero and one, and two sizes of particles (i.e. spherical glass beads) with mean number diameters of 26 and 45 m were used. It was found that the Mach-disc moves upstream towards the orifice with increasing particle loading. The laser light sheet technique was also used to visualize the particle concentration distribution within the particle jet and the spreading rate of the particle jet. Furthermore, the particle velocity along the jet centerline was measured with a modified laser-Doppler anemometer. These measurements revealed that the particles move considerably slower than the gas flow at the nozzle exit. This is mainly the result of the particle inertia, whereby the particles are not accelerated to sonic speed in the converging part of the nozzle.In order to further explore the particle behavior in the free jet, numerical studies were performed by a combined Eulerian/Lagrangian approach for the gas and particle phases, including full coupling between the two phases. The numerical results showed that the application of different particle velocities at the nozzle exit as the inlet conditions, which were below the sonic speed of the gas phase has a significant influence on the free jet shape and the configuration of the shock waves. These results demonstrate that the assumption of equilibrium flow (i.e. zero slip between the phases) at the nozzle exit which has been applied in most of the previous numerical studies is not justified in most cases. Furthermore, the numerical calculations of the free jet shape and the particle velocity along the jet axis were compared with the measurements. Although correlations for rarefaction and compressibility effects in the drag coefficient were taken into account, the particle velocity along the center line was considerably overpredicted.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.     

Interaction of a shock wave with a loose dusty bulk layer

The interaction of a planar shock wave with a loose dusty bulk layer has been investigated both experimentally and numerically. Experiments were conducted in a shock tube. The incident shock wave velocity and particle diameters were measured with the use of pressure transducers and a Malvern particle sizer, respectively. The flow fields, induced by shock waves, of both gas and granular phase were visualized by means of shadowgraphs and pulsed X-ray radiography with trace particles added. In addition, a two-phase model for granular flow presented by Gidaspow is introduced and is extended to describe such a complex phenomenon. Based on the kinetic theory, such a two-phase model has the advantage of being able to clarify many physical concepts, like particulate viscosity, granular conductivity and solid pressure, and deduce the correlative constitutive equations of the solid phase. The AUSM scheme was employed for the numerical calculation. The flow field behind the shock wave was displayed numerically and agrees well with our corresponding experimental results.      

Smooth particle hydrodynamics method for modeling cavitation-induced fracture of a fluid under shock-wave loading

It is demonstrated that the method of smoothed particle hydrodynamics can be used to study the flow structure in a cavitating medium with a high concentration of the gas phase and to describe the process of inversion of the two-phase state of this medium: transition from a cavitating fluid to a system consisting of a gas and particles. A numerical analysis of the dynamics of the state of a hemispherical droplet under shock-wave loading shows that focusing of the shock wave reflected from the free surface of the droplet leads to the formation of a dense, but rapidly expanding cavitation cluster at the droplet center. By the time t = 500 µs, the bubbles at the cluster center not only coalesce and form a foam-type structure, but also transform to a gas-particle system, thus, forming an almost free rapidly expanding zone. The mechanism of this process defined previously as an internal “cavitation explosion” of the droplet is validated by means of mathematical modeling of the problem by the smoothed particle hydrodynamics method. The deformation of the cavitating droplet is finalized by its decomposition into individual fragments and particles.     

Investigations of gas and particle dynamics in first generation needle-free drug delivery devices

Abstract. Transdermal powdered drug delivery involves the propulsion of solid drug particles into the skin by means of high-speed gas-particle flow. The fluid dynamics of this technology have been investigated in devices consisting of a convergent-divergent nozzle located downstream of a bursting membrane, which serves both to initiate gas flow (functioning as the diaphragm of a shock tube) and to retain the drug particles before actuation. Pressure surveys of flow in devices with contoured nozzles of relatively low exit-to-throat area ratio and a conical nozzle of higher area ratio have indicated a starting process of approximately 200 s typical duration, followed by a quasi-steady supersonic flow. The velocity of drug particles exiting the contoured nozzles was measured at up to 1050 m/s, indicating that particle acceleration took place primarily in the quasi-steady flow. In the conical nozzle, which had larger exit area ratio, the quasi-steady nozzle flow was found to be overexpanded, resulting in a shock system within the nozzle. Particles were typically delivered by these nozzles at 400 m/s, suggesting that the starting process and the quasi-steady shock processed flow are both responsible for acceleration of the particle payload. The larger exit area of the conical nozzle tested enables drug delivery over a larger target disc, which may be advantageous. Received 12 March 2000 / Accepted 8 June 2000     

Formation of particle jetting in a cylindrical shock tube

A dense, solid particle flow is numerically studied at a mesoscale level for a cylindrical shock tube problem. The shock tube consists of a central high pressure gas driver section and an annular solid powder bed with air in void regions as a driven section with its far end adjacent to ambient air. Simulations are conducted to explore the fundamental phenomena, causing clustering of particles and formation of coherent particle jet structures in such a dense solid flow. The influence of a range of parameters is investigated, including driver pressure, particle morphology, particle distribution and powder bed configuration. The results indicate that the physical mechanism responsible for this phenomenon is twofold: the driver gas jet flow induced by the shock wave as it passes through the initial gaps between the particles in the innermost layer of the powder bed, and the chaining of solid particles by inelastic collision. The particle jet forming time is determined as the time when the motion of the outermost particle layer of the powder bed is first detected. The maximum number of particle jets is bounded by the total number of particles in the innermost layer of the powder bed. The number of particle jets is mainly a function of the number of particles in the innermost layer and the mass ratio of the powder bed to the gas in the driver section, or the ratio of powder bed mass (in dimensionless form) to the pressure ratio between the driver and driven sections.     

Interaction of a shock wave with a two-phase interface

The acceleration by an incident shock of a planar interface between a gas and a particle-gas mixture has been investigated experimentally and numerically. The experiments were conducted in a newly developed vertical shock tube in which the planar interface of the particle-gas mixture was generated and its particle concentration history was measured. Polydisperse corn starch particles with a mean diameter of 10m were used. We recorded the motion of the interface, as well as of the incident and reflected shock by using a 4 channel spark shadowgraph. The experimental conditions were Mach numberM _s=5.15 and initial pressurep ₁=50kPa for various particle concentrations in nitrogen. The reflected shock appears with a delay after the incident shock enters the particle-gas mixture. Numerical methods were employed to solve the two-phase governing equations. Experiments and numerical solutions are in good agreement.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.     

DRAG FORCE IN DENSE GAS—PARTICLE TWO—PHASE FLOW

Numerical simulations of flow over a stationary particle in a dense gas-particle two-phase flow have been carried out for small Reynolds numbers (less than 100). In order to study the influence of the particles interaction on the drag force, three particle arrangements have been tested: a single particle, two particles placed in the flow direction and many particles located regularly in the flow field. The Navier-Stokes equations are discretized in the three-dimensional space using finite volume method. For the first and second cases, the numerical results agree reasonably well with the data in literature. For the third case, i.e., the multiparticle case, the influence of the particle volume fraction and Reynolds numbers on the drag force has been investigated. The results show that the computational values of the drag ratio agree approximately with the published results at higher Reynolds numbers (from 34.2 to 68.4), but there is a large difference between them at small Reynolds numbers. The project supported by the Special Funds for Major State Basis Research Projects in China (G19990222).     

Measurement and simulation of the two-phase velocity correlation in sudden-expansion gas-particle flows

In this paper the present authors measured the gas-particle two-phase velocity correlation in sudden expansion gas-particle flows with a phase Doppler particle anemometer(PDPA) and simulated the system behavior by using both a Reynolds-averaged Navier-Stokes(RANS) model and a large-eddy simulation(LES).The results of the measurements yield the axial and radial time-averaged velocities as well as the fluctuation velocities of gas and three particle-size groups(30 μ m,50 μ m,and 95 μ m) and the gas-particle velocity correlation for 30 μ m and 50 μ m particles.From the measurements,theoretical analysis,and simulation,it is found that the two-phase velocity correlation of sudden-expansion flows,like that of jet flows,is less than the gas and particle Reynolds stresses.What distinguishes the two-phase velocity correlations of sudden-expansion flow from those of jet and channel flows is the absence of a clear relationship between the two-phase velocity correlation and particle size in sudden-expansion flows.The measurements,theoretical analysis,and numerical simulation all lead to the above-stated conclusions.Quantitatively,the results of the LES are better than those of the RANS model.     

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