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     

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.     

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.     

A numerical study of oblique shock wave reflections over wedges in an ideal quantum gas

The various oblique shock wave reflection patterns generated by a moving incident shock on a planar wedge using an ideal quantum gas model are numerically studied using a novel high resolution quantum kinetic flux splitting scheme. With different incident shock Mach numbers and wedge angles as flow parameters, four different types of reflection patterns, namely, the regular reflection, simple Mach reflection, complex Mach reflection and the double Mach reflection as in the classical gas can be classified and observed. Both Bose–Einstein and Fermi–Dirac gases are considered.      

Three-dimensional effects on regular reflection in steady supersonic flows

The reflection of shock waves between two symmetrical wedges is investigated for the case of three-dimensional flows. Oblique shadowgraphs at various optical angles of yaw and pitch were used to examine the nature of fully three-dimensional flows, with wedge aspect ratios as low as 0.25 being considered. These images were used to construct surface models of the overall flow field for various reflection patterns and aspect ratios, which provides a visual indication of the flow field shape. For a Mach number of 3.1, and suitable wedge angles, the flow field with regular reflection on the tunnel centreline and Mach reflection further out is examined. The point of transition from regular reflection to the peripheral Mach surfaces is identified for various wedge angles and aspect ratios. It is shown that the transition points move outwards from the central plane as the aspect ratio decreases. This shows that three-dimensional flows favor regular reflection, because of the increasing curvature of the incident shock as the wedge becomes narrower, causing a decrease in the local angle of incidence. The height of the Mach stem is shown to be highly dependent on the geometry of the test wedge models. The Mach stem height decreases with aspect ratio due to the three-dimensional relieving effect, where the increase in lateral flow relieves the pressure over the surfaces of the wedges. Experimental evidence of the existence of the strong oblique shock solution in steady flows is presented.Received: 7 July 2003, Revised: 20 October 2003, Accepted: 6 November 2003, Published online: 10 February 2004PACS: 47.40.Nm Correspondence to: B.W. Skews     

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.     

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.     

Simulation of the Mach reflection in supersonic flows by the CE/SE method

This study employs the Space-Time Conservation Element and Solution Element (CE/SE) method to determine the influence of downstream flow conditions on Mach stem height. The results indicate that the Mach stem height depends on the incident shock wave angle and the distance between the trailing edge and the symmetry plane. Furthermore, it is shown that the downstream length ratio and the trailing edge angle do not affect the Mach stem height nor the Mach reflection (MR) configuration, and the Space-Time Conservation Element and Solution Element method is able to simulate the MR as well as many other numerical schemes. Communicated by K. Takayama PACS 47.40.Nm     

环形激波绕射, 反射和聚焦的数值模拟研究

应用频散可控耗散格式对环形激波在圆柱形激波管内绕射、反射和聚焦的问题进行了数值模拟研究，研究结果表明环形激波形成强烈聚焦的关键因素是环形激波在圆柱形管道中向对称轴运动时，绕射激波就不断加速而不作通常情况下的衰减；不同马赫数的环形激波绕射也产生不同马赫数及形状的准柱形激波，导致聚焦效果和位置的差异；另外，环形激波聚焦于一个点而圆柱形激波聚焦于一条线，两者有本质不同。     

On flows behind shock waves reflected from a solid wall

The flow fields associated with the interaction of a normal shock wave with a plane wall kept at a constant temperature were studied based on kinetic theory which can describe appropriately the shock structure and its reflection process. With the use of a difference scheme, the time developments of the distributions of the fluid dynamic quantities (velocity, temperature, pressure and number density of the gas) were obtained numerically from the BGK model of the Boltzmann equation subject to the condition of diffusive-reflection at the wall for several cases of incident Mach number:M ₁=1.2, 1.5, 2.0, 3.0, 4.0, 5.0 and 6.0. The reflection process of the shocks is shown explicitly together with the resulting formation of the flow fields as time goes on. The nonzero uniform velocity toward the wall occurring between the viscous boundary layer and the reflected shock wave is found to be fairly large, the magnitude of which is of the order of several percent of the velocity induced behind the incident shock, decreasing as the incident Mach number increases. It is also seen that a region of positive velocity (away from the wall) within the viscous boundary layer manifests itself in the immediate vicinity of the wall, which is distinct for larger incident Mach numbers. Some of the calculated density profiles are compared with available experimental data and also with numerical results based on the Navier-Stokes equations. The agreement between the three results is fairly good except in the region close to the wall, where the difference in the conditions of these studies and the inappropriateness of the Navier-Stokes equations manifest themselves greatly in the gas behavior.This article was processed using Springer-Verlag TEX Shock Waves macro package 1990.     

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.      

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。     

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 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     

Stability and analogy of shock wave reflection in steady flow

Numerical simulations have been performed to investigate the stability of shock wave reflection in supersonic steady flow. Wall deflection control has been applied just downstream of the reflection point in the regular reflection configuration. The results provide the magnitude of the disturbance required to cause transition from one configuration to the other throughout the range of incident shock angle. An argument focusing on the subsonic region generated behind the Mach stem in the Mach reflection configuration explains the mechanism of the transition. Numerical results show that both regular and Mach reflections are possible in the dual-solution domain, and also indicate the presence of the hysteresis effect. The transition processes and the stability of the possible states are shown to be described consistently by an analogy based on the potential energy of a particle on a surface. The necessity of more sophisticated experimental investigations is emphasized to verify the argument about the stability of shock reflections and proposed analogy. Received 17 March 1997 / Accepted 26 February 1998     

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.     

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.     

An experimental and theoretical study of converging polygonal shock waves

An experimental investigation was carried out to explore the possibility of producing converging polygonal shocks in an essentially two-dimensional cavity. Previous calculations by Apazidis and Lesser (1996) suggested that such configurations could be produced by reflecting a cylindrical outgoing shock from a smoothly altered circular boundary, the alteration having n-gonal symmetry. In the experiments the outgoing shock was produced by a spark discharge which yielded shocks in the Mach number range from 1.1 to 1.7 at a radius just prior to the reflection. Polygonal shocks were observed as predicted by using a modified form of geometrical shock dynamics, derived in the above paper. In addition, the modified theory was used to calculate the results of an experiment carried out by Sturtevant and Kulkarny (1976). The results of the numerical calculations were found to be in substantial agreement with both experiments, suggesting that the modifications in geometrical shock dynamics for non-uniform flow ahead of an advancing shock are useful in the case of shock focusing. The experiment also showed that the polygonal shapes were stable in the examined range of shock Mach numbers, a result that may be of importance for a number of practical situations in which shock focusing is present. Received 9 October 2001 / Accepted 7 January 2002 – Published online 11 June 2002     

Diffuse holographic interferometric observation of shock wave reflection from a skewed wedge

The pattern of shock wave reflection over a wedge is, in general, either a regular reflection or a Mach reflection, depending on wedge angles, shock wave Mach numbers, and specific heat ratios of gases. However, regular and Mach reflections can coexist, in particular, over a three-dimensional wedge surface, whose inclination angles locally vary normal to the direction of shock propagation. This paper reports a result of diffuse double exposure holographic interferometric observations of shock wave reflections over a skewed wedge surface placed in a 100 × 180 mm shock tube. The wedge consists of a straight generating line whose local inclination angle varies continuously from 30° to 60°. Painting its surface with fluorescent spray paint and irradiating its surface with a collimated object beam at a time interval of a few microseconds, we succeeded in visualizing three-dimensional shock reflection over the skewed wedge surface. Experiments were performed at shock Mach numbers, 1.55, 2.02, and 2.53 in air. From reconstructed holographic images, we estimated critical transition angles at these shock wave Mach numbers and found that these were very close to those over straight wedges. This is attributable to the flow three-dimensionality.      

Least energy as a criterion for transition between regular and Mach reflection

A new criterion is suggested to define the point of transition between regular and Mach reflection. The suggested criterion is based on the natural tendency of a physical system to minimize its energy. The increases of the specific energy behind the reflected shock of a regular reflection and behind the Mach stem of a Mach reflection are calculated. It is hypothesized that the type of reflection that will occur is that which produces the smaller change of specific energy. The transition angles predicted using this criterion show better agreement with experimental results than those predicted using the detachment criterion for incident shock waves with Mach numbers between 1.1 and 2.0.This article was processed using Springer-Verlag TEX Shock Waves macro package