Experimental and numerical studies of shock wave focusing in water

The focusing of plane shock waves and spherical blast waves in water by parabolic and ellipsoidal reflectors is studied experimentally and numerically. The numerical computations are performed by a second order method based on an extension of Godunov's method, which is called piecewise-linear method. Tests of this method and a comparison with pressure measurements show a strong dependence of the maximum pressure attained in the focal region on the mesh size. Outside this focal region, however, good agreement between the pressure measurements and the result of the numerical computation is found. Furthermore it is shown by numerical computation, that non-linear effects in the focusing process may be partly avoided by an optimized reflector contour which depends on the strength of the incident shock wave. This modification provides that the maximum pressure is attained at the geometric focus and not at a location between reflector surface and geometrical focus.     

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.     

Shock focusing flow field simulated by a high-resolution numerical algorithm

Shock-focusing concave reflector is a very simple and effective tool to obtain a high-pressure pulse wave near the physical focal point. In the past, many optical images were obtained through experimental studies. However, measurement of field variables is not easy because the phenomenon is of short duration and the magnitude of shock waves is varied from pulse to pulse due to poor reproducibility. Using a wave propagation algorithm and the Cartesian embedded boundary method, we have successfully obtained numerical schlieren images that resemble the experimental results. By the numerical results, various field variables, such as pressure, density and vorticity, become available for the better understanding and design of shock focusing devices.     

Numerical investigation of axisymmetric shock wave focusing over paraboloidal reflectors

The problem of a plane shock wave incident to a paraboloidal reflector is numerically investigated. The numerical solver used is developed by an improved, implicit, upwind total variation diminishing scheme in a finite-volume approach. The real-gas effect is taken into account if high temperature occurs. The solver is validated on four test problems. The complicated flow fields of axisymmetric shock wave focusing for different-depth reflectors at various incident shock Mach numbers are studied. An interesting result of a maximum pressure happening at the reflector center is found. This is due to the occurrence of an implosion phenomenon. A maximum temperature might occur at the reflector center or at other locations, depending on the incident shock Mach number and the reflector depth. Moreover, vortical flows induced by shock wave focusing and their formation mechanism are explored. It was found that the vortices near the reflector are caused by a ring-shaped shock/slipline interaction. Owing to the slipline on the symmetry axis, a jet flow is induced, resulting in the formation of vortices near the symmetry axis. Received 13 January 1998 / Accepted 10 November 1998     

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。     

Anti-plane dynamic hole–crack interaction in a functionally graded piezoelectric media

The anti-plane dynamic problem of a functionally graded piezoelectric plane containing a hole–crack system is treated by a non-hypersingular traction-based boundary integral equation method. The material parameters vary exponentially in the same manner in an arbitrary direction. The system is loaded by an incident SH-type wave, and impermeable boundary conditions are assumed. Using a frequency-dependent fundamental solution of the wave equation, the boundary value problem is transformed into a system of integro-differential equations along the boundary of the hole and on the crack line. Its numerical solution yields the dynamic stress intensity factors and stress concentration factors. A parametric study reveals their dependence on the hole–crack scenario and its geometry, characteristics of the dynamic load and magnitude and direction of material inhomogeneity.     

Experimental and computational studies focusing processes of shock waves reflected from parabolic reflectors

This paper describes experimental and numerical studies of the focusing process of shock waves reflected from various shapes of a parabolic reflector. The effect of incident shock strength on the focusing process was also investigated. Experiments were carried out in a conventional shock tube and a test gas was air for incident shock Mach numbers ranging from 1.1 to 2.0. In the experiments, the process of shock focusing was visualized by schlieren method. Numerical simulations were conducted for incident shock Mach numbers up to 3.0 by solving the two-dimensional unsteady Euler equations. The numerical results were compared with experiment for various parabolic reflector shapes and for various incident shock Mach numbers. Based on the experimental and computational results, the pattern of shock focusing and shock focusing mechanism are discussed.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.     

激波聚焦诱导点火和爆轰的数值研究

以二维轴对称多组分Euler方程为基础，采用非正交结构化网格和改进的波传播算法，模拟了激波在抛物形反射壁面聚焦反射诱导点火和爆轰的过程，描述了其流场形态。讨论了预混气组成、入射激波强度及反射壁面形状对点火和爆轰的影响。结果表明，激波在抛物形反射壁面顶点处聚焦反射可形成局部高温高压区域，该区域在一定条件下可点燃预混气甚至形成爆轰，其中低稀释剂浓度的预混气、较大的入射激波Mach数和较深的反射壁面有利于可燃预混气形成爆轰。     

Cavitation phenomena in extracorporeal microexplosion lithotripsy

An experimental investigation was made of cavitation phenomena induced by underwater shock wave focusing applied to the extracorporeal microexplosion lithotripsy (microexplosion ESWL). Firstly an underwater microexplosion generated by detonation of a 10 mg silver azide pellet was studied and secondly underwater shock focusing and its induced cavitation phenomena were investgated. Underwater shock wave was focused by using a semi-ellipsoidal reflector in which a shock wave generated at the first focal point of the reflector was reflected and focused at the second focal point. It is found that an explosion product gas bubble did not produce any distinct rebound shocks. Meantime cavitation appeared after shock focusing at the second focal point where expansion waves originated at the exit of the reflector were simultaneously collected. A shock/bubble interaction is found to contribute not only to urinary tract stone disintegration but also tissue damage. The cavitation effect associated with the microexplosion ESWL was weaker in comparison with a spark discharge ESWL. The microexplosion ESWL is an effective method which can minimize the number of shock exposures hence decreasing tissue damage by conducting precise positioning of urinary tract stones.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.     

Numerical simulation of shock wave focusing over parabolic reflectors

The problem of a plane shock wave that propagates in an air media and then is reflected from a parabolic concave reflector and focuses at some region is considered. The shock focusing can greatly magnify the pressure and the temperature. The purpose of this study is to numerically simulate the shock focusing process of the reflection of shock waves from the parabolic reflectors with different depths and to analyze their associated flow fields in detail. The present solver developed is to solve the Euler equations using an improved, implicit, upwind Total Variation Diminishing scheme in a finite-volume approach. The effects of reflectors with different depths and of the incident shock Mach numbers on shock focusing are investigated. The real-gas effect is taken into account through a proper correction of the specific heat ratio of air, when high temperature occurs due to shock focusing.     

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     

Deformation of rock in the explosion of a cylindrical charge in a coaxial cavity filled with liquid

A study is made of the effect of a water sheath around a cylindrical explosive charge on the parameters of shock waves in rocks. A coupled problem for the detonation-product-water-rock system is examined within the framework of continuum mechanics. A binomial isentropic equation is used for the detonation products, water is modeled by a Tate-type equation, and the rock is modeled by an elastoplastic medium which undergoes dilatation. A numerical solution is obtained by a finite-difference scheme. The dependence of the wave processes in the rock and the dimensions of the crushing zone on the radius of the water sheath is studied with the use of sandstone and siltstone as examples. Institute of Hydromechanics, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 3, pp. 83–88, March, 1999.     

Formation of blast waves as a result of a low pressure domain decomposition

In this article the formation and propagation of blast wave as a result of a focusing of shock wave in a domain with low pressure and density are examined in the frame of an ideal gas model. We consider the decomposition of a pressure-density discontinuity on the boundary of the spherical (or cylindrical) domain which is filled by a gas whose pressure and density are lower than the pressure and density of gas that filling the external space. At initial moments of this decomposition the rarefaction wave propagates in the external space and the converging shock wave is formed in the domain of low pressure and density. The intensity of the converging shock wave gradually increases, and the wave transitions to the self-similar regime. After implosion, a diverging shock wave is formed which propagates through the disturbed gas. The values of wave intensity, wave impulse and other parameters for some magnitudes of initial parameters have been determined by means of numerical calculations. Received 10 August 1997 / Accepted 13 July 1998     

Generalized and exact solutions for oblique shock waves of real gases with application to real air

Calculation of the oblique shock wave of real gases is a difficult and time consuming problem because it involves numerical solution of a set of 10 equations, two of which (i.e., the equation of state and enthalpy function)—if available—are of a very complicated algebraic form. The present work presents a generalized method for calculating oblique shock waves of real gases, based on the Redlich-Kwong equation of state. Also described is an exact method applicable when the exact equation of state and enthalpy function of a real gas are available. Application of the generalized and the exact methods in the case of real air showed that the former is very accurate and at least twenty times faster than the latter. An additional contribution of the study is the derivation of real gas oblique shock wave equations, which are of the same algebraic form as the well known ideal gas normal shock wave relations.     

Perturbation Method in the Problem of Compression-Shear Shock Load for a Nonlinear Elastic Half-Space

On the example of a one-dimensional nonstationary problem of oblique impact on the boundary of a nonlinear elastic isotropic half-space, the question of the manifestation of nonlinear deformation effects via basic evolution equations is studied. Much attention is given to the behavior of the solution behind the leading edge of a quasi-transverse shock wave. For particular cases of boundary conditions, it is shown that the onset region of the evolution equation of a quasi-transverse wave is preceded by a series of preliminary transitions to the intermediate internal problems of the small parameter method determined by the type of preliminary bulk deformation. This deformation consistently affects the distortion of the characteristic coordinates and the leading edge of the quasitransverse process. As a consequence, the transition to the evolution equation of quasi-transverse waves occurs with simultaneous change of all independent variables of the boundary value problem.     

Numerical solutions of the Boltzmann equation: comparison of different algorithms

In the paper we compare different algorithms for numerical solutions of the Boltzmann equation. For this comparison we have taken the standard problem of the shock wave structure in a mono-atomic rarefied gas. Different parameters characterizing the shock structure have been calculated by a Monte Carlo simulation (DSMC), a second order time-relaxed Monte Carlo method (TRMC2), a fully deterministic discrete velocity method (DV), a discrete velocity method with Monte Carlo calculations of collision integral (DVMC) and a molecular dynamics method (MD). Results of these calculations have been compared with the shock wave structure obtained in experiments in a shock tube. The results of the comparison are not conclusive. We have observed general agreement between numerical and experimental results but there is no single numerical method which fits best to the experimental measurements.     

Focusing of weak shock waves on a target in a parabolic chamber

Theoretical study of a weak shock focusing process in a confined chamber filled with liquid is presented. The chamber has a form of a thin cylinder with a parabolic cross-section, planar bottom and an arbitrary, although slowly varying, upper bounding surface. Analytical, numerical and experimental studies of weak shock wave focusing have been previously performed in the elliptic and ellipsoidal cases with a shock wave generated at one of the foci by means of an electric discharge or a microexplosion. In the present case a planar shock, perpendicular to the axis of the parabolic cross-section, sent in the inner of the chamber will converge at the focus after the reflection off the chamber wall, thus offering a different technical realization of the shock generation. The problem is solved within the frame of the geometrical acoustics approximation and a relation between the form of the upper bounding surface of the chamber and the pressure distribution behind the converging wavefront is obtained. It is shown that a desired pressure distribution may be obtained by an appropriate choice of the upper bounding surface.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.     

水下爆炸作用下基于声学的爆源子结构输入方法

无限域吸收边界和爆炸波输入是水下爆炸载荷数值模拟的两个关键问题.本文借鉴基于内部子结构的地震波动输入方法和爆源子结构多尺度分析方法,考虑水下爆炸载荷与地震载荷同属于波动问题,提出一种水下爆炸作用下爆源子结构的爆炸波输入方法,该方法首先通过对爆源区域进行水下爆炸波自由场的波场分解,并利用地震波的内部子结构输入方法,将该自由波场运动转换为等效爆炸波输入载荷,从而实现了水下爆炸问题中冲击波的输入;本文采用圆形爆源子结构区域,并采用AUTODYN软件中一维自由场模型计算水下爆炸作用下该区域的自由波场压力时程.进一步,基于连分式近似方法提出了一种模拟无限域水体辐射效应的高精度时域人工边界条件,该吸收边界可设置于离结构和爆源子结构较近处.本文提出的分析方法通过圆形爆源子结构进行水下爆炸载荷的转化,并采用高精度吸收边界大大减少计算区域,既保证了计算精度,又降低了单元数量,具有较高的计算效率和较强的实用性.通过数值算例分析,验证了本文模型与方法的准确性,模拟了水下爆炸作用下的冲击波和气泡脉动阶段测点的压力时程曲线,并研究了圆形结构对水下近场爆炸波散射效应的影响规律.     

The distribution of particles in a shock-induced boundary layer of a dusty gas over a solid surface

The laminar boundary layer behind a constant-speed shock wave moving through a dusty gas along a solid surface is studied. The Saffman lift force acting on a spherical particle in a gas boundary layer is taken into account. A method for calculating the density profile of dispersed phase near the wall is proposed and some numerical results are given. It is shown that behind the shock wave, there exists a curved thin layer where the density of particles is many times higher than the original one. This dust collection effect may be of essential importance to the problem of dust explosion in industry.     

可燃气体中激波聚焦的点火特性

数值模拟了二维平面激波从抛物面上反射在可燃气体中聚焦的过程，研究了形 成爆轰波的点火特性. 对理想化学当量比氢气/空气混合气体，在初始压强20kPa的条件下， 马赫数2.6-2.8的激波聚焦能产生两个点火区：第1个点火区是反射激波会聚引起的，第 2个点火区是由入射激波在抛物面上发生马赫反射引起的. 这种条件下流场中会出现爆燃转 爆轰，起爆点分别分布在管道壁面、抛物反射面和第2点火区附近. 起爆机理分别为激波管 道壁面反射、点火诱导激波的抛物面反射和点火诱导的激波与第2点火区产生的爆燃波的相 互作用. 不同的点火和起爆过程导致了不同的流场波系结构，同时影响了爆轰波传播的波动 力学过程.