Singularities and spectral asymptotics of a random nonlinear wave in a nondispersive system

In this paper, we study the propagation of high-intensity acoustic noise in free space and in waveguide systems. A mathematical model generalizing the Burgers equation is used. It describes the nonlinear wave evolution inside tubes of variable cross-section, as well as in ray tubes, if the geometric approximation for heterogeneous media is used. The generalized equation transforms to the common Burgers equation with a dissipative parameter, known as the “Reynolds–Goldberg number”. In our model, this number depends on the distance travelled by the wave. With a zero “viscous” dissipative term, the model reduces to the Riemann (or Hopf) equation. Its solution presents the field by an implicit function. The spectral form of this solution makes it possible to derive explicit expressions for both dynamic and statistical characteristics of intense waves. The use of a spectral approach allowed us to describe the high-intensity noise in media with zero and finite viscosity. Applicability conditions of these solutions are defined. Since the phase matching is fulfilled for any triplet of interacting spectral components, there is an avalanche-like increase in the number of harmonics and the formation of shocks. The relationship between these discontinuities and other singularities and the high-frequency asymptotic of intense noise is studied. The possibility is shown to enhance nonlinear effects in waveguide systems during the evolution of noise.     

The reflected pressure field in the interaction of weak shock waves with a compressible foam

This paper deals with the waves that are reflected from slabs of porous compressible foam attached to a rigid wall when impacted by a weak shock wave. The interest is in establishing possible attenuation of the pressure field after a shock or blast wave has struck the surface. Foam densities from 14 to 38 kg/m³ were tested over a range of shock wave Mach numbers less than 1.4. It is shown that the initial reflected shock wave strength is accurately predicted by the pseudo-gas model of Gelfand et al. (1983), with a pressure ratio of approximately 80% of the value for reflection off a rigid wall. Evidence is presented of gas entering the foam during the early stages of the process. A second wave emerges from the foam at a later stage and is separated from the first by a region of constant velocity and pressure. This second wave is not a shock wave but a compression front of significant thickness, which emerges from the foam earlier than predicted by the pseudo-gas model. Analysis of the origin of this wave points to much more complex flows within the foam than previously assumed, particularly in an apparent decrease in average wave front speed as the foam is compressed. It is shown that the pressure ratio across both these waves taken together is slightly higher than that for reflection off a rigid wall. In some cases this compression wave train is followed by a weak expansion wave.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1990.     

Reconstruction of multiply-scattered arrivals from the cross-correlation of waves excited by random noise sources in a heterogeneous dissipative medium

Imagining a medium composed of an arbitrary distribution of point-like heterogeneities, we study the reconstruction of scattered waves in Green's function derived from the cross-correlation function of waves excited by random noise sources of which the distribution is stationary and homogeneous. We show that the reconstruction process is intimately related to generalized forms of the optical theorem. The role of absorption in the formulation of the theorem is discussed. The reconstruction of multiply-scattered arrivals from the cross-correlation of two random wavefields is demonstrated to all orders of scattering for the simple case of two point scatterers, through application of the optical theorem for a single scatterer. In the case of N point scatterers, the cross-correlation of two Green's functions is expressed in the form of Feynman-like diagrams. The wavepaths that contribute to the reconstruction of an arbitrary multiply-scattered arrival of Green's function are identified. Repeated application of the generalized optical theorem, formulated as a diagrammatic rule, demonstrates the destructive interference between all spurious multiply-scattered arrivals.     

Evolution of shock waves and the primary vortex loop discharged from a square cross-sectional tube

In this paper, a numerical and experimental investigation of the evolution of a transmitting shock wave and its associated primary vortex loop, which are discharged from the open end of a square cross-sectional tube, is described. The experiments were conducted in the square tube connected to a diaphragmless shock tube and the flowfield was visualized from the axial direction with diffusive holographic interferometry. The numerical simulations were carried out by solving the three-dimensional Euler equations with a dispersion-controlled scheme. The numerical results were displayed in the form of interferograms to compare them with experimental interferograms. Good agreement between the numerical and experimental results was obtained. More detailed numerical calculations were carried out, from which the three-dimensional transition of the shock wave configuration from an initial planar to a spherical shape and the development of the primary vortex loop from a square shaped to a three-dimensional structure were clearly observed and interpreted. Received 29 January 1998 / Accepted 22 May 1998     

Deformation waves in thermoelastic media and the concept of internal variables

Summary Nonlinear thermoelastic continua with Fourier's type of heat conduction illustrate complex systems composed of internal and reaction-diffusion subsystems. An evolution equation is derived for the observable variable (deformation) when temperature effects are viewed as internal processes. This approach, which pertains to the so-called instantaneous wave analysis, shows that thermal losses accompanying the deformation wave correspond to the low-frequency approximation.     

Over forty years of continuous research at UTIAS on nonstationary flows and shock waves

Analytical and experimental research on non-stationary shock waves, rarefaction waves and contact surfaces has been conducted continuously at UTIAS since its inception in 1948. Some unique facilities were used to study the properties of planar, cylindrical and spherical shock waves and their interactions. Investigations were also performed on shock-wave structure and boundary layers in ionizing argon, water-vapour condensation in rarefaction waves, magnetogasdynamic flows, and the regions of regular and various types of Mach reflections of oblique shock waves. Explosively-driven implosions have been employed as drivers for projectile launchers and shock tubes, and as a means of producing industrial-type diamonds from graphite, and fusion plasmas in deuterium. The effects of sonic-boom on humans, animals and structures have also formed an important part of the investigations. More recently, interest has focussed on shock waves in dusty gases, the viscous and vibrational structure of weak spherical blast waves in air, and oblique shock-wave reflections. In all of these studies instrumentation and computational methods have played a very important role. A brief survey of this work is given herein and in more detail in the relevant references.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1990.     

Experimental studies of ignition and transition to detonation induced by the reflection and diffraction of shock waves

This paper presents results from a program of experimental studies of ignition induced by the interaction of an initially planar shock wave with an obstacle in its path. With the aid of pressure measurements, spark schlieren photography and smoked foil techniques it is shown how, given favourable initial conditions, the two-dimensional multiple shock reflection and diffraction can promote ignition and transition to detonation in reactive gaseous mixtures. Comparison of the results with those of a non-reactive gas distinguishes the gas dynamic and chemical processes involved, and experimentally determined detonation cell sizes are compared with values predicted using chemical kinetic rate data. The systems investigated were argon, air, propane-air, propane-oxygen-argon and ethylene-oxygen-argon. Received: 3 December 1998 / Accepted: 27 October 1999     

Local and nonlocal conserved vectors of the system of two-dimensional generalized inviscid Burgers equations

The concept of non-linear self-adjointness for the construction of conservation laws has attracted a lot of interest in recent years. The most noteworthy aspect of it is the likelihood of explicitly constructing the conservation laws for any arbitrary systems of differential equations, in particular for those for which Noether׳s theorem is not applicable. In this study, we shall use both Noether׳s theorem and the non-linear self-adjoint method to construct local and nonlocal conserved vectors of the system of two-dimensional Burgers equations under consideration. The first integrals obtained not only give more credence to obtained results due to their generality with respect to any arbitrary functions of the velocity components but are also independent, nontrivial and infinitely many.     

A three-dimensional numerical study on instability of sinusoidal flame induced by multiple shock waves

The instabilities of a three-dimensional sinusoidally premixed flame induced by an incident shock wave with Mach = 1.7 and its reshock waves were studied by using the Navier–Stokes(NS) equations with a single-step chemical reaction and a high resolution, 9th-order weighted essentially non-oscillatory scheme. The computational results were validated by the grid independence test and the experimental results in the literature. The computational results show that after the passage of incident shock wave the flame interface develops in symmetric structure accompanied by large-scale transverse vortex structures. After the interactions by successive reshock waves, the flame interface is gradually destabilized and broken up, and the large-scale vortex structures are gradually transformed into small-scale vortex structures. The small-scale vortices tend to be isotropic later.The results also reveal that the evolution of the flame interface is affected by both mixing process and chemical reaction. In order to identify the relationship between the mixing and the chemical reaction, a dimensionless parameter, η, that is defined as the ratio of mixing time scale to chemical reaction time scale, is introduced. It is found that at each interaction stage the effect of chemical reaction is enhanced with time.The enhanced effect of chemical reaction at the interaction stage by incident shock wave is greater than that at the interaction stages by reshock waves. The result suggests that the parameter η can reasonably character the features of flame interface development induced by the multiple shock waves.     

Detonation and deflagration initiation at the focusing of shock waves in combustible gaseous mixture

Abstract. Detonation and deflagration initiation under focusing conditions in a lean hydrogen-air mixture was experimentally investigated. The experiments were carried out in a shock tube equipped with the laser schlieren system and pressure transducers. Two-dimensional wedges (53° and 90°), semi-cylinder and parabola, were used as the focusing elements. The peculiarities of mild and strong ignition inside the reflector cavity were visualized. A hydrogen-nitrogen mixture was taken for comparison between reactive and inert mixture. It was found that mild ignition inside the reflector cavity can lead to detonation initiation outside the cavity. Schlieren pictures of the process were obtained and the dependence of the distance of detonation initiation on Mach number of the incident shock wave was established. Received 30 August 1999 / Accepted 23 February 2000     

On the influence of elevated surface temperatures on hypersonic shock wave/boundary layer interaction at a heated ramp model

Although important flow parameters as Mach number, Reynolds number and total enthalpy can be reproduced in most hypersonic experiments quite well, due to different surface temperature effects in wind tunnel and flight, scaling as well as specific flow properties of shock wave/boundary layer interactions are different. This especially holds for short-duration facilities like, e.g. shock tunnels where due to short running times the models remain more or less at ambient temperature. To overcome this shortcoming, an experimental study has been conducted using a preheatable ramp model with 15° ramp angle. This allowed us to adjust the surfaces to an arbitrary temperature just before the experiment started. Pressure and heat flux measurements clearly showed the effect of varying surface and free stream temperatures. These results are supported by schlieren pictures and infrared measurements. The comparison of the measurements with theoretical and numerical results shows a good agreement. Separation bubble scaling laws proposed by Katzer and Davis have been applied and partially confirmed using the local conditions of the boundary layer at separation.     

Pseudo-schlieren CT measurement of three-dimensional flow phenomena on shock waves and vortices discharged from open ends

1A pseudo-schlieren technique is applied to the interferometric computed tomography (CT) measurement of three-dimensional (3-D) shock waves discharged from a square open end and a pair of circular open ends in a shock tube experiment. The experiment is performed for incident shock Mach numbers of 2.0 and 2.2 in nitrogen gas under supersonic post shock flow conditions at the open end. The 3-D density-gradient distributions are evaluated from the CT data of the 3-D density distributions, and are depicted in gray-scale CT images of the gradient magnitude and in pseudo-color CT images of the gradient component. The resultant pseudo-schlieren CT images clearly illustrate the 3-D flow features of shock waves, contact surfaces, and the other sharp density fronts. Their image characteristics and meaning in gas dynamics are discussed in comparison with the pseudo-color images of the density. We demonstrate that the pseudo-schlieren CT technique is a useful tool for studying 3-D problems in shock dynamics. Communicated by K. Takayama PACS 47.32.cd; 47.40.Nm; 47.80.jk     

Numerical investigation of the propagation of shock waves in rigid porous materials: flow field behavior and parametric study

A numerical parametric study of the flow field which develops when a planar shock wave impinge on a rigid porous material is presented. This study complements an earlier study (Levy et al. 1996a) where the values of some dominating parameters were estimated and the dependence of the resulting flow field on these values was not checked. Received 22 April 1996 / Accepted 5 January 1997     

Particle morphology and packing effects on the shock loading of powders

A physically based model for the shock Hugoniot of a powdered material is described which allows separate identification of the cold and thermal contributions to pressure and specific internal energy. Special features of this model are provision for the effects of porosity on the stress state and an empirically determined cold loading contribution to pressure. The model was tested against published Hugoniot data for iron and gave excellent agreement for shock pressures ranging from low to high values.This shock Hugoniot was used to explore the shocked state of 4 samples of iron powder derived from commercially available material. The purpose of this study was to investigate the effect of powder particle characteristics and initial starting densities on the shocked state.The powder samples investigated had a range of morphologies and sizes. Powders with either a large shape factor or high internal friction, as determined in shear cell experiments, exhibited a higher stiffness in the cold loading curve. In the shocked state, this translated into a higher cold component of pressure and energy than found in the other powders.The effect of initial powder density was studied by applying the Hugoniot model to two impact initiated shock loadings, one for a stainless steel flyer impacting at 0.5 km/s and one at the higher velocity of 2.0 km/s. Both were applied to iron powder targets preloaded to a range of initial densities. For a given impact event, the proportion of shock energy in the thermal mode was found to decrease with increasing initial density. This decrease was more pronounced at higher shock strengths. As a result of the decreasing component of thermal energy with higher initial density, there was a reduction in the continuum temperature behind the shock. However, the corresponding increase in the component of cold energy with the falling relative contribution from the thermal energy lead to increasing density behind the shock suggesting that there is a trade off in terms of temperature and density achievable with a given impact event.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.     

A physical model of the structure and attenuation of shock waves in metals

In this paper, a physical model of the structure and attenuation of shock waves in metals is presented. In order to establish the constitutive equations of materials under high velocity deformation and to study the structure of transition zone of shock wave, two independent approaches are involved. Firstly, the specific internal energy is decomposed into the elastic compression energy and elastic deformation energy, and the later is represented by an expansion to third-order terms in elastic strain and entropy, including the coupling effect of heat and mechanical energy. Secondly, a plastic relaxation function describing the behaviour of plastic flow under high temperature and high pressure is suggested from the viewpoint of dislocation dynamics. In addition, a group of ordinary differential equations has been built to determine the thermo-mechanical state variables in the transition zone of a steady shock wave and the thickness of the high pressure shock wave, and an analytical solution of the equations can be found provided that the entropy change across the shock is assumed to be negligible and Hugoniot compression modulus is used instead of the isentropic compression modulus. A quite approximate method for solving the attenuation of shock wave front has been proposed for the flat-plate symmetric impact problem.     

Theoretical considerations on shock reflections and their implications on the evaluation of air intake performance

Many theoretical studies have shown the existence of a hysteresis effect in the solution of oblique shock reflections. In fact, a wide domain of free-stream Mach number and shock angle values exists where regular reflection and Mach reflection are both possible solutions for the same flow conditions. Part of this domain overlaps the typical operating conditions of supersonic air intakes, and therefore it is of practical interest to obtain a deeper understanding of the theoretical problem. Indeed, although both solutions are theoretically possible, they yield very different flowfields and consequently large discrepancies in the evaluation of the air intake performance. Numerical solutions for steady configurations have been carried out and compared with the flow evolution obtained for time-dependent cases. The results have confirmed numerically the existence of the multiple solution domain where hysteresis takes place in time-dependent simulations. The analysis of the physical and numerical problems encountered has provided indications for a correct simulation in practical applications. Received 10 August 1999 / Accepted 6 October 2000     

An investigation into the effects of electrostatic and squeeze-film non-linearities on the shock spectrum of microstructures

We present an investigation into the effects of some of the common microelectromechanical systems (MEMS) non-linearities on their shock response and shock spectrum. As a case study, a capacitive accelerometer is selected to investigate theoretically and experimentally the effect of non-linearities due to squeeze film damping (SQFD) and electrostatic actuation. For the theoretical investigation, a non-linear single-degree-of-freedom model is used to simulate the response of the device. It is shown that, in the case of light damping, the electrostatic forces soften the microstructure and raise its deflection significantly. Dynamic pull-in instability is predicted near the dynamic range zone of the shock spectrum. On the other hand, SQFD is found to highly suppress the deflection of the microstructure in the dynamic range, while it is of less effect in the quasi-static range. Experimentally, the capacitive accelerometer is powered with a DC load and then subjected to acceleration pulses generated by a shaker. Tests are conducted while the accelerometer is operated in air, where the squeeze film effect is significant, and while placed inside a vacuum chamber. Simulation results are compared to experimental data showing excellent agreement.     

椭球罩作用下的水下爆炸冲击波反射聚焦模型

基于冲击波传播、非线性反射和聚焦理论,建立水下爆炸冲击波在椭球罩作用下的反射聚焦模型。讨论自由传播、壁面反射和定向聚焦阶段的冲击波特性和压力计算方法,利用波前和波法线的近似方程构建压力场的数值计算域,进而模拟聚焦过程,并与现有实验结果进行对比,结果表明:所建模型可为正聚焦压力提供满足工程精度的预测,并能描述水下冲击波及产生的拉伸波聚焦过程中的一些细节;椭球罩能有效地聚焦水下冲击波,在动力学焦点附近获得有效增益区,在近轴方向上明显削弱冲击波压力衰减;理想条件下的动力学焦点一般位于几何焦点之前,但实际的反射罩变形和背向位移将使其发生后迁,甚至能越过几何焦点。     

On the interaction of water waves and seabed by the porous medium model

The interaction of water waves and seabed is studied by using Yamamoto's model, which takes into account the deformation of soil skeletal frame, compressibility of pore fluid flow as well as the Coulumb friction. When analyzing the propagation of three kinds of stress waves in seabed, a simplified dispersion relation and a specific damping formula are derived. The problem of seabed stability is further treated analytically based on the Mohr-Coulomb theory. The theory is finally applied to the coastal problems in the Lian-Yun Harbour and compared with observations and measurements in soil-wave tank with satisfactory results. The project supported by the National Science Foundation of China