Classification of pseudo-steady shock wave reflection types

Classification of various types of the reflections of a shock wave over a straight wedge is proposed. The idea about entire reflection phenomenon as a result of interaction of two processes—the shock wave reflection process and the flow deflection process—serves as a basis for the classification. To recognize the types of reflection, changes in the shapes of the reflected wave, Mach stem, and contact surface (slipstream) are taken into account. The boundaries and domains of existence for various types of reflection configuration are reported. New terms for some types of reflection are proposed. The domain of irregular non-Mach reflection is analyzed carefully. It is shown that the von Neumann reflection pattern can result from not only the weak shock reflection but also the strong shock reflection over thin wedges. Shadowgraph images of different types of irregular reflection that illustrate the suggested classification are presented. Emphasis is placed on near-wall behavior of the contact discontinuity in the Mach configuration.     

Shock wave diffraction on multi-facetted and curved walls

The two-dimensional diffraction of a shock wave over a wall made up of a series of plane and/or curved sections is considered. The analysis is based on the theory presented by, for the interaction of an originally plane shock wave with a corner. A method is presented by which the shock profile may be determined for a wall of any shape and for any incident Mach number, in regions where the characteristics form a simple wave. Comparisons are made between experimental measurements and theoretical predictions for convex walls consisting of a number of facets, and for circular arcs, for a range of incident shock wave Mach numbers. It is shown that the theory gives a satisfactory prediction of the wave shape, which improves as the Mach number increases. Modifications in the flow field behind the shock, compared to that for a simple corner made up of two plane walls is discussed, particularly relating to flow separation. For circular arc concave walls a inverse Mach reflection results experimentally, leading to regular reflection, for which the theory is of no use. PACS 47.40.Nm     

Shock-tube experiments on the stability of regular reflection in the dual-solution domain

Regular reflection (RR) and Mach reflection (MR) are theoretically both possible in the dual-solution domain of oblique shock reflection. The physical difference between the two types of reflection is the pressure behind the reflected shock wave: that of MR is lower than that of RR for strong shock reflection. The magnitude relation of these pressures is inverted for weak shock reflection. In the present paper, we performed two kinds of experiment, depending on whether the oblique shock reflection is weak or strong. For strong shock reflection, we decreased the pressure behind the reflected wave of RR using a convex double-wedge. For weak shock reflection, we increased this pressure using a concave double-wedge. Thus, we investigated the stability of RR against pressure disturbances. The results indicate that RR in a shock tube is stable, in the dual-solution domain, for both weak and strong incident shocks.     

Guderley reflection for higher Mach numbers in a standard shock tube

An experimental study shows that the Guderley reflection (GR) of shock waves can be produced in a standard shock tube. A new technique was utilised which comprises triple point of a developed weak Mach reflection undergoing a number of reflections off the ceiling and floor of the shock tube before arriving at the test section. Both simple perturbation sources and diverging ramps were used to generate a transverse wave in the tube which then becomes the weak reflected wave of the reflection pattern. Tests were conducted for three ramp angles (10°, 15°, and 20°) and two perturbation sources for a range of Mach numbers (1.10–1.40) and two shock tube expansion chamber lengths (2.0 and 4.0 m). It was found that the length of the Mach stem of the reflection pattern is the overall vertical distance traveled by the triple point. Images with equivalent Mach stem lengths in the order of 2.0 m were produced. All tests showed evidence of the fourth wave of the GR, namely the expansion wave behind the reflected shock wave. A shocklet terminating the expansion wave was also identified in a few cases mainly for incident wave Mach numbers of approximately 1.20.     

A revised three-shock solution for the Mach reflection of weak shocks (1.1<M i<1.5)

It is well known that the classical three-shock theory of von Neumann (1943) does not adequately describe the configuration of the shocks close to the triple-point of a Mach reflection of an incident shock with a Mach number less than about 1.5. The assumptions on which the three-shock theory is based have been examined and several of them are shown to be invalid. The assumption that may be of most significance is that the normal components of the flows behind the reflected and the Mach stem shocks are parallel. Dropping this assumption removes an essential equation in the three-shock solution. An alternative assumption, based on experimental observation, is that there is an approximate linear relationship between the pressure behind the reflected shock and the triple-point trajectory angle. This assumption permits a revised three-shock solution which gives results that are in agreement with experimental observations of reflections of incident shocks with Mach numbers between 1.1 and 1.5.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.     

Oblique reflection of shock waves from rigid porous materials

A study to determine the general gas dynamic behaviour associated with the impact of a shock wave on a porous wedge has been undertaken. A number of interesting features are noted. The pattern of wave reflection is shown to be significantly affected by the inflow of gas into the wedge. This has the effect of reducing the triple point trajectory angle for cases of Mach reflection and for strongly reducing the reflection angle in regular reflection. The permeability of the wedge has a significant effect on the strength of the reflected wave and in some cases this wave can be attenuated to the extent that it is almost eradicated. Pressure measurements taken under the wedge are characterized by oscillations which are of similar shape, for a given wedge, over a range of shock wave Mach numbers. It is shown that the wave transmitted into the wedge is attenuated to varying degrees depending on the material properties, and that for weak incident waves the mean propagation velocity can be less than the sound speed in the pore fluid. Photographs taken using a specially constructed wedge which allows the transmitted wave to be visualised, show that the transmitted wave is nearly plane.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.     

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.     

Effects of a Single-pulse Energy Deposition on Steady Shock Wave Reflection

The effects of energy deposition in the free stream on steady regular and Mach shock wave reflections are studied numerically. A short-duration laser pulse is focused upstream of the incident shock waves. It causes formation of the expanding blast wave and the residual hot-spot interacting in a complex way with the steady shock wave reflection. It was found that the laser energy addition in the free stream may force the transition from regular to Mach reflection in the dual solution domain. In contrast to previously reported numerical results, the transition from Mach to regular reflection has not been reproduced in our refined computations since the Mach reflection is restored after the flow perturbation.     

Numerical solution of the problem of wave diffraction by a wedge

The systematic development of the theory of shock reflection from a solid wall started in [1]. Regular reflection and a three-shock configuration originating in Mach reflection were considered there under the assumption of homogeneity of the domains between the discontinuities and, therefore, of rectilinearity of these latter. The difficulties of the theoretical study include the essential nonlinearity of the process as well as the instability of the tangential discontinuity originating during Mach reflection. Analytic solutions of the problem in a linear formulation are known for a small wedge angle or a weak wave (see [2–4], for example). The solution in a nonlinear formulation has been carried out numerically in [5, 6] for arbitrary wedge angles and wave intensities. Since the wave was nonstationary, the internal flow configuration is difficult to clarify by means of the constant pressure and density curves presented. A formulation of the problem for the complete system of gasdynamics equations in self-similar variables is given in [7] and a method of solution is proposed but no results are presented. Difficulties with the instability of the contact discontinuity are noted. The problem formulation in this paper is analogous to that proposed in [7]. However, a method of straight-through computation without extraction of the compression shocks in the flow field is selected to compute the discontinuous flows. The shocks and contact discontinuities in such a case are domains with abrupt changes in the gasdynamics parameters. The computations were carried out for a broad range of interaction angles and shock intensities. The results obtained are in good agreement with the analytical solutions and experimental results. Information about the additional rise in reflection pressure after the Mach foot has been obtained during the solution.     

A weak shock wave reflection over wedges

When a weak shock wave reflects from wedges its reflection pattern does not appear to be a simple Mach reflection. This reflection pattern is known to be von Neumann Mach reflection in which a Mach stem can not necessarily be straight. In this paper the local change of the Mach stem curvature was experimentally and numerically investigated. A distinct triple point, at which the curvature becomes infinite as appears in a simple Mach reflection, was not observed but the Mach stem curvature became a maximum between foot of the Mach stem and a point, P₁, at which an incident shock met with a reflected shock. Maximum curvature point P₂ and P₁ do not coincide for small wedge angles and tend to merge over a certain wedge angle. Experimental results agreed with numerical results. The trajectory angle of P₂ was found to be expressed well by Whitham's shock-shock angle.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.     

Irregular reflection of a strong shock wave from a thin wedge

The problem of irregular reflection of a strong shock wave from a rigid wall has been studied [1–3] mainly within the framework of the linear theory. It has been found that near the front of a shock wave there exist a region of large gradients of gasdynamic parameters in which the linear theory is no longer valid [4]. In the present paper we consider the nonlinear problem of Mach reflection when there is interaction between a shock wave of high intensity and a thin wedge. The solution of the problem is constructed on the assumption that the ratio of densities along the front of the impinging shock wave is small [5, 6].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 55–61, September–October, 1974.In conclusion, the authors wish to express their gratitude to A. A. Grib for his interest in the subject and to L. A. Rumyantsev for his help in carrying out the calculations.     

The reflection of a shock wave over a cone

An investigation was made of the reflection of planar shock waves from cones. 86 cones, the half apex angle of which varied from 10° to 52° at every 0.5°, were installed in a 60 mm×150 mm diaphragmless shock tube equipped with holographic interferometry. The diaphragmless shock tube had a high degree of reproducibility with which the scatter of shock wave Mach number was within ±0.25% for shock wave Mach number ranging from 1.16 to approximately 2.0. The reflection of shock waves over cones was visualized using double exposure holographic interferometry. Whitham's geometrical shock wave dynamics was used to analyse the motion of Mach stems over cones. It is found that for relatively smaller apex angles of cones trajectory angles of resulting irregular reflections coincide with the so-called glancing incidence angles and their Mach stems appear to be continuously curved from its intersection point with the incident shock wave, which shows the chractericstic of von Neumann reflection. The domain of the existence of the von Neumann reflection was analytically obtained and was found to be broadened much more widely than that of two-dimensional reflections of shock waves over wedges.     

Experiments on Guderley Mach reflection

Experiments have been conducted in a large shock tube to examine the four-wave shock reflection pattern, now known as Guderley reflection (GR). The fourth wave, an expansion, is clearly identified, as is the supersonic patch behind the reflected wave. A shocklet terminating the supersonic patch behind the reflected wave is identified, which forms a second triple point further down the Mach stem. Evidence is presented showing the presence of more than one expansion wave and more than one shocklet, thus indicating the existence of more than one supersonic patch. In order to distinguish between cases with a single patch without the shocklet as originally proposed by Guderley and found in some computations, and the indications of a multi-patch geometry found here, and also in other computations, this latter case is designated Guderley Mach reflection (GMR). Multi-exposure images of the shock propagation superimposed on a single image frame enable estimates to be made of the strength of the major waves, and it is shown that the reflected wave is very weak.      

Numerical investigations of transition between regular and Mach reflections caused by free-stream disturbances

Numerical simulations have been performed to study the influence of the free-stream disturbances on the alternation of the steady shock wave reflection configurations in the dual solution domain. Different types of disturbances have been considered. The analysis of interaction between disturbances and the incident shock wave can be substantially simplified for the localized density disturbances. It is shown that such disturbances can indeed cause the transition from regular reflection to Mach reflection and back, so that within a certain range of angles of incidence the shock wave reflection configuration can be considered as a bi-stable system. The threshold amplitude of the localized density disturbance, able to induce the transition, has been estimated theoretically. The results of numerical computations convince of higher stability of the Mach reflection in the dual solution domain compared to the regular reflection, which is in accordance with available experimental data. Received 10 May 2001 / Accepted 15 November 2001 Published online 8 July 2002     

Regular versus Mach reflection for converging polygonal shocks

The onset of Mach reflection or regular reflection at the vertices of a converging polygonal shock wave was investigated experimentally in a horizontal annular shock tube. The converging shock waves were visualized by schlieren optics. Two different types of polygonal shock convergence patterns were observed. We compared the behavior during the focusing process for triangular and square-shaped shocks. It is shown that once a triangular shaped shock is formed, the corners in the converging shock will undergo regular reflection and consequently the shape will remain unaltered during the focusing process. A square-shaped shock suffers Mach reflections at the corners and hence a reconfiguring process takes place; the converging shock wave alternates between a square and an octagon formation during the focusing process.      

爆轰波在突扩通道中传播的数值模拟研究

建立了描述甲烷 空气混合物爆轰波传播的单步化学反应爆轰模型 ,通过数值模拟研究了在二维突扩通道中爆轰波的强度变化和各种波行为。结果表明 :爆轰波在进入突扩通道初始阶段的衍射使爆轰波局部向爆燃转变 ;爆炸波在壁面发生马赫反射形成的高温高压区域将直接诱导自持爆轰波的重新形成。     

Transition between regular and Mach reflection of shock waves: new numerical and experimental results

New numerical and experimental results on the transition between regular and Mach reflections of steady shock waves are presented. The influence of flow three-dimensionality on transition between steady regular and Mach reflection has been studied in detail both numerically and experimentally. Characteristic features of 3D shock wave configuration, such as peripheral Mach reflection, non-monotonous Mach stem variation in transverse direction, the existence of combined Mach-regular-peripheral Mach shock wave configuration, have been found in the numerical simulations. The application of laser sheet imaging technique in streamwise direction allowed us to confirm all the details of shock wave configuration in the experiments. Close agreement of the numerical and experimental data on Mach stem heights is shown. Received 23 November 2000 / Accepted 25 April 2001     

金属铅在斜冲击波对碰加载下动态行为的数值模拟

高强度冲击加载作用下金属材料的动态物理行为是当前冲击波领域基础研究和工程应用最为关注的焦点。采用光滑粒子法(SPH)开展不同位置起爆诱发的斜冲击波对碰加载金属铅的二维数值模拟研究,得到了金属铅内入射斜冲击波的角度和强度,并利用极曲线方法理论上导出发生马赫反射时的临界入射角和入射马赫数关系。根据计算结果可知,金属铅内入射斜波对碰后将发生马赫反射。随着起爆位置与金属铅表面距离的增加,不仅金属铅内入射冲击波强度和入射角增加,而且形成的马赫杆宽度也在增加。由自由面速度剖面给出了马赫杆宽度及张角,结果与理论预测的结果吻合较好。     

Application of streak camera photography for the study of shock wave reflections over a double wedge

The streak camera technique was employed to investigate the reflection process of a planar shock wave over a double wedge. It is shown that this technique is superior to others in precisely determining whether a Mach reflection is direct, stationary or inverse. Furthermore, it provides an excellent means of exactly determining the location of the regular to Mach reflection transition over a double wedge.     

Investigation of the hysteresis phenomena in steady shock reflection using kinetic and continuum methods

The problem of transition of planar shock waves over straight wedges in steady flows from regular to Mach reflection and back was numerically studied by the DSMC method for solving the Boltzmann equation and finite difference method with FCT algorithm for solving the Euler equations. It is shown that the transition from regular to Mach reflection takes place in accordance with detachment criterion while the opposite transition occurs at smaller angles. The hysteresis effect was observed at increasing and decreasing shock wave angle. Received September 1, 1995 / Accepted November 20, 1995