Traveling Shock Wave Research Articles

Modelling of dust lifting using the Lagrangian approach

The subject of this paper is dust lifting behind shock waves, a process that is important for the formation of explosive dust clouds in air. While Eulerian–Eulerian has been the standard numerical technique for such simulations, the Eulerian–Lagrangian technique has been used in this paper, making it possible to take into account more physical phenomena, such as particle–particle and particle–wall collisions. The results of the simulations are shown mainly graphically, as snapshots of particle positions at given times after the passing of the shock wave. The results show that the collisions, and the coefficient of restitution assumed for them, is important in determining the mobility and lifting of dust behind shock waves. The results also show that the idea of a horizontally travelling shock wave is an oversimplification: the strong pressure gradient at the surface results in a series of reflected waves generated at the surface and travelling into the gas phase.

Linear interaction analysis applied to a mixture of two perfect gases

The linear interaction analysis is applied to mixtures of two perfect gases. Differences of molar mass and constant volume specific heat between both gases are taken into account as two new wave families. Transfer functions across the shock wave from these waves to acoustic, shear or entropy waves are given. The approach is validated against numerical simulations of the interaction of concentration spots with a shock wave. The case of turbulence creation by a shock wave traveling in an inhomogeneous mixture is considered.

Collective Bow Shock Ahead of a Transverse System of Spheres in a Supersonic Flow Behind a Moving Shock Wave

The results of investigating the dynamics and physical conditions of formation of a col- lective bow shock ahead of a system of spheres with the line of centers normal to the supersonic flow behind a traveling shock wave are presented. Two types of shock-wave patterns that necessarily precede the formation of the collective shock wave and correspond to regular and Mach interaction of the bow shocks were detected experimentally. On the basis of a local gasdynamic-discontinuity interference the- ory, quantitative criteria of the existence of these regimes and of the formation of a common shock wave are determined. These criteria are confirmed in a series of experiments for the transitional regimes.

Dynamics of Pattern Formation and Modulational Instabilities in the Quintic Complex Ginzburg–Landau Equation*

We study analytically modulational instability in an extended media, taking the one-dimensional quintic complex Ginzburg–Landau equation model as an example. We show that the quintic term can affect the conditions for modulational instability (MI), and the propagating regime. Numerical simulations demonstrate the validity of analytical predictions. The model presents a rich variety of patterns propagating into the system. The main feature of the travelling shock waves is the propagation of a secondary modulated wave at the same time with a stable plane wave.

Survival of a proto-atmosphere through the stage of giant impacts: the mechanical aspects

When a giant impact occurs, atmosphere loss may occur due to global ground motion excited by a strong shock wave traveling in the planetary interior. Here, the relations between the ground motion and the amount of the lost atmosphere are systematically investigated through calculations of a spherically one-dimensional atmospheric motion for various initial atmospheric conditions. The fraction of the lost atmosphere to the total mass of the atmosphere is found to be controlled only by the ground velocity and, insensitive to the initial atmospheric conditions. Unlike the previous studies (Ahrens, 1990, Origin of the Earth, H.E. Newson, J.H. Jones (Eds.), pp. 211–227; Ahrens, 1993, Annu. Rev. Earth Planet. Sci. 21, 525–555; Chen and Ahrens, 1997, Phys. Earth Planet. Inter. 100, 21–26); the estimated loss fraction for the giant impact is only 20%. Significant escape occurs only when the ground velocity is close to the escape velocity. Thus, most of the atmosphere should survive the giant impact. The cause of the difference from previous estimates is discussed from energetic and dynamic points of view. Moreover, if our estimates are applied to the atmosphere of the impactor planet, a significant fraction of it is carried to the target planet. Survival of the proto-atmosphere has very important effects on the origin and evolution of the terrestrial planets' volatile budget.

Dynamic hohlraum driven inertial fusion capsules

A dynamic hohlraum is formed when an imploding annular cylindrical Z-pinch driven plasma collides with an internal low density convertor. This collision generates an inward traveling shock wave that emits x rays, which are trapped by the optically thick Z-pinch plasma and can be used to drive an inertial fusion capsule embedded in the convertor. This scheme has the potential to efficiently drive high yield capsules due to the close coupling between the intense radiation generation and the capsule. In prior dynamic hohlraum experiments [J. E. Bailey et al., Phys. Rev Lett. 89, 095004 (2002)] the convertor shock wave has been imaged with gated x-ray pinhole cameras. The shock emission was observed to be very circular and to be quite narrow in the radial direction. This implies that there is minimal Rayleigh–Taylor imprinting on the shock wave. Thus, the dominant source of radiation asymmetry is not random and in principle could be significantly decreased by proper design. Due to the closed geometry of the dynamic hohlraum, the most convenient way to diagnose the radiation symmetry is to image the x rays from the core of an imploded capsule. However, the core temperatures in the prior experiments were not high enough to obtain images. Using numerical simulations we have redesigned the dynamic hohlraum to obtain higher capsule core temperatures. This has enabled us to obtain x-ray pinhole images and Ar K-shell spectra from the imploded cores of 1.7–2.0 mm diameter CH-wall capsules filled with either D2 or CD4 and doped with a small amount of Ar. These capsules absorbed approximately 20 kJ of x-ray energy from the radiation drive, which peaked at a temperature of about 200 eV. Core temperatures of approximately 1 keV were inferred from the Ar spectrum. Our present understanding of the physics of dynamic hohlraums is presented along with our plans to improve this system.

Chemotactic Cellular Migration: Smooth and Discontinuous Travelling Wave Solutions

A simple model of chemotactic cell migration gives rise to travelling wave solutions. By varying the cellular growth rate and chemoattractant production rate, travelling waves with both smooth and discontinuous fronts are found using phase plane analysis. The phase plane exhibits a curve of singularities whose position relative to the equilibrium points in the phase plane determines the nature of the heteroclinic orbits, where they exist. Smooth solutions have trajectories connecting the steady states lying to one side of the singular curve. Travelling shock waves arise by connecting trajectories passing through a special point in the singular curve and recrossing the singular curve, by way of a discontinuity. Hyperbolic partial differential equation theory gives the necessary shock condition. Conditions on the parameter values determine when the solutions are smooth travelling waves versus discontinuous travelling wave solutions. These conditions provide bounds on the travel- ling wave speeds, corresponding to bounds on the chemotactic velocity or bounds on cellular growth rate. This analysis gives rise to the possibility of representing sharp fronts to waves of invading cells through a simple chemotactic term, without introducing a nonlinear diffusion term. This is more appropriate when cell populations are sufficiently dense.

Interaction of Decaying Freestream Turbulence with a Moving Shock Wave: Pressure Field

Theunsteady interaction of a moving shock wavewith nearly homogeneousand isotropic decaying compressible turbulence has been studied experimentally in a large-scale shock-tube facility by using rectangular grids of various sizes. The interaction has been investigated by measuring velocity, total pressure, and temperature inside thee owe eldandstaticpressureatthewalloftheshocktubewithinstrumentationoftemporalandspatialresolution. Attenuation of theshock wavestrength has been found to take placeas a result oftheinteraction with theturbulent e owe eld, which depends on the mesh size of the grid or its Reynolds number, the e ow Mach number, and the initial Mach number of the shock wave. Finer grids produce turbulence, which attenuates the shock wave more than coarse grids at the same Mach number. Amplie cation of pressure e uctuations after the interaction has been observed in all experiments, depending on grid size (initial turbulencelevel )and shock strength. Furthermore, this amplie cation is not the same through the whole range of wave numbers resolved. I. Introduction T RAVELINGshockwavesorexpansionwavescanbegenerated inside turbines or compressors of turbine engines as a result of unsteady e ow phenomena. These moving waves can interact with rotating or stationary blades, as well as with the e ow, and may result in phenomena that can affect the operation and performance of the engine. One of these phenomena is the interaction of passing shock waves with turbulence. Of particular interest to those studying turbomachinery is the interaction of traveling shock waves with freestream turbulence, which can be considered as nearly isotropic and nearly homogeneous. In general, interactions of shock waves with turbulent e ows are of great practical importance in external or internal aerodynamics of engineering applications because they considerably modify the e uid e eld by vorticity and entropy production and transport. PastworkthathasbeenrecentlyreviewedbyAndreopoulosetal. 1 hasshownthattheinteractionbetweentheshockwaveandturbulent e ow is mutual and that the coupling between them is very strong. Complex linear and nonlinear mechanisms are involved that can cause considerable changes in the structure of turbulence and its statistical properties and alter the dynamics of the shock wave motion. Amplie cation of velocity e uctuations and substantial changes in length scales are the most important outcomes of interactions of shock waves with turbulence. This indicates that such interactions may greatly affect mixing. The use of shock waves, for instance, has been proposed by Budzinski et al. 2 as means to enhance mixing

Actaptive grid technique based on the variational principle and its weight functions

Two computational cases that have analytic solutions are employed for studying the adaptive grid technique based on the variational principle. The results show that for the computational case of traveling shock waves the weight function, with the 2nd-order derivation terms taken into consideration, can more effectively reduce the error than one with gradient terms. For the case of cyclonic frontogenesis, weight functions only related to the gradient are unable to enhance the computational accuracy while ones with the wind field and frontogenesis function taken into consideration can more reasonably arrange the grid. Compared with analytic solutions, the adaptive grid technique suggested in this paper can improve computational accuracy and it displays the prominent advantage of saving memory.

Bimodal nanoparticle size distributions produced by laser ablation of microparticles in aerosols

Silver nanoparticles were produced by laser ablation of a continuously flowing aerosol of microparticles in nitrogen at varying laser fluences. Transmission electron micrographs were analyzed to determine the effect of laser fluence on the nanoparticle size distribution. These distributions exhibited bimodality with a large number of particles in a mode at small sizes (3–6-nm) and a second, less populated mode at larger sizes (11–16-nm). Both modes shifted to larger sizes with increasing laser fluence, with the small size mode shifting by 35% and the larger size mode by 25% over a fluence range of 0.3–4.2-J/cm2. Size histograms for each mode were found to be well represented by log-normal distributions. The distribution of mass displayed a striking shift from the large to the small size mode with increasing laser fluence. These results are discussed in terms of a model of nanoparticle formation from two distinct laser–solid interactions. Initially, laser vaporization of material from the surface leads to condensation of nanoparticles in the ambient gas. Material evaporation occurs until the plasma breakdown threshold of the microparticles is reached, generating a shock wave that propagates through the remaining material. Rapid condensation of the vapor in the low-pressure region occurs behind the traveling shock wave. Measurement of particle size distributions versus gas pressure in the ablation region, as well as, versus microparticle feedstock size confirmed the assignment of the larger size mode to surface-vaporization and the smaller size mode to shock-formed nanoparticles.

Protective properties of hot‐air screens

An analytical and numerical study of protective properties of hot‐air screens is performed for a plane wave with a triangular profile. The reduction of the shock‐wave pressure behind the screen is only observed at temperatures higher a certain critical value; at lower temperatures, the pressure is higher than that in a shock wave traveling in an isothermal gas. With allowance for real properties of air, an analytical relation between the critical temperature of the hot screen and the incident‐wave intensity is obtained.

A note on the travelling wave solution to the perturbed Burgers' equation

In this work, utilizing the hyperbolic tangent method, a travelling wave solution is presented for the perturbed Burgers' equation, which has not been studied before in the existing literature. It is observed that the perturbed Burgers' equation still assumes a travelling shock wave type of solution with decaying amplitude in time.

Towards evolutionary game models of financial markets

Evolutionary game models analyse strategic interaction over time; equilibrium emerges (or fails to emerge) as players/traders adjust their actions in response to the payoffs they earn. This paper sketches some early and some recent evolutionary game models that contain ideas useful in modelling financial markets. It spotlights recent work on adaptive landscapes. In an extended example, the distribution of player/trader behaviour obeys a variant of Burgers' partial differential equation, and solutions involve travelling shock waves. It is conjectured that financial market crashes might insightfully be modelled in a similar fashion.

SOHO/Energetic and Relativistic Nucleon and Electron Experiment Measurements of Energetic H, He, O, and Fe Fluxes during the 1997 November 6 Solar Event

A brilliant solar X-ray and Hα flare and a coronal mass ejection (CME) on 1997 November 6 were associated with high particle fluxes in interplanetary space at energies above MeV. The CME had an exceptionally high leading edge velocity (1560 km s-1) as observed by the Large Angle and Spectrometric Coronagraph (LASCO) on board the Solar and Heliospheric Observatory (SOHO). The Energetic and Relativistic Nucleon and Electron experiment (ERNE), also on SOHO, measured high H, He, O, and Fe fluxes in several energy channels from 3 to 200 MeV nucleon-1. The oxygen energy spectrum in energy range 3-200 MeV nucleon-1 was of a broken-power-law form with a break around 50 MeV nucleon-1. In addition, ERNE observed abrupt changes in the intensity, elemental composition, and anisotropy of high-energy particles, which may indicate two energetic particle sources during several hours after the solar flare eruption. The observational results lead us to conclude that, most of the time, the O and Fe nuclei were injected by the interplanetary shock associated with the coronal mass ejection emitted around 12 UT on November 6. However, during two time periods the injection source might have been different or complementary. The first period was in the very beginning of the event, 13:20-13:40 UT, when the particle streaming showed very strong anisotropy with the maximum intensity from the direction of the Sun. The second was between 17 UT November 6 and 5 UT November 7, when particle fluxes were dominated by a particle population with a different elemental composition and a different spectral shape of O as compared with the particle population prevailing in the beginning and during the long decay phase of the event. We propose that the source of these particles was associated with a coronal shock wave traveling in the low solar atmosphere.

Shock propagation in the outer heliosphere: 3. Pickup ions, MHD, cosmic rays, and energetic particles

It has been shown previously that to study the propagation charcteristics of shock waves accurately in the outer helioshpere, the heliospheric model should contain pickup ions (PIs). The inclusion of a magnetic field was also shown to affect shock wave propagation but was found to be far less important than the effect of the PIs. To further this study, we consider here the effect of galactic cosmic rays and energetic particles on interplanetary traveling shock waves in the outer heliosphere. The energetic particles are taken to be protons and are assumed to be produced by injection and acceleration at the shock fronts. The galactic cosmic rays are found to have almost no effect on the shock waves. Depending on their injection efficiency and diffusion coefficient, the energetic protons, on the other hand, can significantly modify the shock structure.

Modeling the after-effects of a heliospheric traveling shock wave through the Earth's radiation belts

It is known that interplanetary traveling shock waves can strongly disturb the Earth's magnetosphere (e.g., Blake et al., 1992; Reeves, 1996). In recent years, published hydromagnetic simulations have illustrated how such shock waves cause large distortions in the geomagnetic field, substantial transient induction electric fields and a compression of the magnetospheric topology (e.g., Li et al., 1993, 1996; Hudson et al., 1996, 1997). Particle observations in the magnetosphere following the March 1991 event show that pre-existing particle fluxes can be strongly and suddenly accelerated on a time scale of a small fraction of a trapped particle drift time scale around the Earth. Such short time scales appear incompatible with bodily injection phenomena from the exterior of the magnetosphere. Modeling has been carried out starting with the suddenly accelerated in-situ particle fluxes and following the subsequent relaxation process.

Traveling Shock Waves Arising in a Model of Malignant Invasion

Members of a family of traveling wave solutions for a simple two-variable model of malignant invasion driven by haptotaxis are shown to have shocks (which satisfy an entropy condition) and to be computationally stable. By seeking traveling wave solutions, a PDE model, equivalent to three first order PDEs defined on the real space and time axes, may be reduced to an ordinary differential equation system, which can be studied in a two-dimensional phase plane. This phase plane is shown to contain a singular barrier and a "hole" in this barrier which admits two singular phase trajectories. Thus orbits are able to smoothly cross this barrier along either of the two singular trajectories. Once this has occurred, the barrier seems to prevent any heteroclinic steady-state connections. However, we will show that shocks admitted by the PDEs allow the orbit to jump back over the singular barrier. This process leads to a previously unnoticed family of solutions for malignant invasion which have slower wavespeeds than the family of smooth traveling wave solutions (extending over the whole real axis) admitted by the model. The existence of a minimum-speed wave with semi-infinite support is demonstrated. The practical stability of these new shock solutions is confirmed via numerical solutions of the model. The minimum speed solution is found to evolve from specific nonsmooth initial data, whereas smooth initial data extending over the whole real axis evolves to faster waves.

Detonation type waves in phase (chemical) transformation processes in condensed matter

Fast self sustained waves (autowaves) associated with chemical or phase transformations are observed in many situations in condensed matter. They are governed neither by diffusion of matter or heat (as in combustion processes) nor by a travelling shock wave (as in gaseous detonation). Instead, they result from a coupling between phase transformation and the stress field, and may be classified as gasless detonation autowaves in solids. We propose a simple model to describe these regimes. The model rests on the classical equations of elastic deformations in a 1-dimensional solid bar, with the extra assumption that the phase (chemical) transformation induces a change of the sound velocity. The transformations are assumed to occur through a chain branched mechanism, which starts when the mechanical stress exceeds a given threshold. Our investigation shows that supersonic autowaves exist in this model. In the absence of diffusion (dissipation factor, losses), a continuum of travelling wave solutions is found. In the presence of diffusion, a steady state supersonic wave solution is found, along with a slower wave controlled by diffusion.

A PHASE-PLANE DESCRIPTION OF NONLINEAR TRAVELING WAVES IN BUBBLY LIQUIDS

One-dimensional traveling wave solutions to the fully nonlinear continuity and Euler equations in a bubbly liquid are considered. The elimination of velocity from the two equations leaves a single nonlinear algebraic relation between the pressure and density profiles in the mixture. On assuming the bubbles to have identical size and taking the volume fraction of bubbles in the medium to be small, an equation of state which relates the mixture pressure to the density and its first two material time-derivatives is derived. When this equation of state is linearized and combined with the laws of conservation of mass and momentum, a nonlinear, second-order, ordinary differential equation is obtained for the density as a function of the single traveling wave coordinate. A phase-plane analysis of this equation reveals the existence of two fixed points, one of which is a saddle and the other a node. A single trajectory connects the two fixed points and corresponds to a traveling shock wave solution when the Mach number of the wave, defined as the ratio of traveling wave speed to the low-frequency speed of sound in the bubbly liquid, exceeds unity. The analysis provides a qualitative explanation of the oscillations behind shocks seen in experiments on bubbly liquids.

Intersonic crack growth in bimaterial interfaces : an investigation of crack face contact

Steady-state intersonic interfacial crack growth accounting for crack face contact is analyzed. The results clearly predict the essential features of experimental observations based on two different but complementary optical techniques and high speed photography. The solution features a large scale contact zone and two distinct traveling shock waves, one emanating from the crack tip and the other from the end of the contact zone. In addition, the solution predicts a non-zero energy dissipation rate due to crack face contact.