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.      

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.      

11th international Mach reflection symposium

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.     

Conical Mach reflection of moving shock waves

Conical Mach reflections differ from those of the equivalent plane, two-dimensional Mach reflection because in axisymmetry, the disturbances generated at the reflecting surface are modified by their more rapidly increasing or decreasing area as they move towards or away from the centerline. Equations for conical Mach reflection cases have now been developed using a simplified ray-shock theory formulation based on the initial assumption that the stem is straight and normal to the wall. These are in a form that applies generally. Their simple structure provides an easy conceptual understanding of self-similarity and non-self-similarity as well as a clear mathematical approach for the development of the curved triple-point locus of the latter by integration. They provide a quick and direct solution in all cases and can easily incorporate the Mach stem curvature by progressively calculating the new ray direction. A range of cases has been considered and results are presented for converging and diverging, self-similar and non-self-similar cases.     

Double Mach reflection of strong shock waves

The Mach reflection of shock waves in those cases in which the gas ideality condition is satisfied with high accuracy is well-known. The effects associated with the excitation of the internal degrees of freedom for the molecules lead to a qualitative change in the reflection pattern. The present study is an extension of [1, 2], devoted to the study of the Mach reflection of shock waves from a wedge under conditions in which the physical and chemical transformations in the gas heated by the shock wave play a significant role.     

Numerical simulation of Mach reflection in steady flows

The structure obtained when two shocks intersect is known to be highly sensitive to various parameters. In the so-called dual solution domain, both regular and Mach reflection patterns are possible, resulting in hysteresis. The phenomenon is important in inlets because of the substantial difference in entropy rise associated with the two manifestations, and the possibility of unstart with Mach reflection. The effect of various numerical and physical parameters on hysteresis are investigated with two-dimensional simulations. The effect of spanwise relief on a three-dimensional situation is also elucidated. It is confirmed that Mach-stem heights determined from inviscid computations are captured relatively accurately by comparison with experimental data and earlier Euler solutions reported in the literature. Near bifurcation points, however, the solution is highly sensitive to the scheme, and the van Leer and Roe schemes can yield converged solutions with different reflection configurations. Viscous terms and downstream conditions are observed to have relatively minor impact on the solution. The three-dimensional simulations reveal that beyond the spanwise limit of the compression surface, the overall shock-structure remains similar in form but the strengths of various shocks are rapidly muted by the expansion from the side surface. Additionally, the flow downstream of the shock that once formed the Mach reflection rapidly becomes supersonic. The Mach-stem height on the symmetry plane and its variation with spanwise position shows reasonable agreement with the experimental data of other investigators.     

Numerical simulation of Mach reflection of cellular detonations

The Mach reflection of cellular detonation waves on a wedge is investigated numerically in an attempt to elucidate the effect of cellular instabilities on Mach reflection, the dependence of self-similarity on the thickness of a detonation wave, and the initial development of the Mach stem near the wedge apex. A two-step chain-branching reaction model is used to give a thermally neutral induction zone followed by a chemical reaction zone for the detonation wave. A sufficiently large distance of travel of the Mach stem is computed to observe the asymptotic behavior in the far field. Depending on the scale at which the Mach reflection process occurs, it is found that the Mach reflection of a cellular detonation behaves essentially in the same way as a planar ZND detonation wave. The cellular instabilities, however, cause the triple-point trajectory to fluctuate. The fluctuations are due to interactions of the triple point of the Mach stem with the transverse waves of cellular instabilities. In the vicinity of the wedge apex, the Mach reflection is found to be self-similar and corresponds to that of a shock wave of the same strength, since the Mach stem is highly overdriven initially. In the far field, the triple-point trajectory approaches a straight line, indicating that the Mach reflection becomes self-similar asymptotically. The distance of the approach to self-similarity is found to decrease rapidly with decreasing thickness of the detonation front.     

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.     

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     

Shear-layer instability in the Mach reflection of shock waves

The Kelvin–Helmholtz instability (KHI) is an instability that takes the form of repeating wave-like structures which forms on a shear layer where two adjacent fluids are moving at a relative velocity to one another. Such a shear layer forms in the Mach reflection of shock waves. This work focuses on experimentally visualising the presence of the KHI in Mach reflection as well as its evolution. Experimentation was performed at shock Mach numbers of 1.34, 1.46 and 1.61. Plane test pieces and parabolic profiled pieces followed by a plane section having wedge angles of 30 \(^\circ \) and 38 \(^\circ \) were tested. Flow field visualisation was performed with a schlieren optical system. The KHI was best visualised with the camera-side knife edge perpendicular to the shear layer (i.e. the axis of sensitivity along the length of the shear layer). The structure and growth of the instability were readily identified. The KHI forms more readily with increasing Mach number and wedge angle. Second-order Euler, and Navier–Stokes numerical simulations of the flow field were also conducted. It was found that the Euler and laminar Navier–Stokes solvers achieved very similar results, both producing the KHI, but at a much less developed state than the experimental cases. The k \(-\epsilon \) solver, however, did not produce the instability.     

Mach reflection of weak shock waves from a rigid wall

On the basis of experimental observations and theoretical analysis of flow structure in the neighborhood of the triple point, it is shown that one should reject the condition for equality of the angle of deflection of flows passing through the Mach front and the two other fronts and replace it with some supplementary condition. The system of consistency equations in the indicated region is closed by an equation which is obtained under the assumption of the extremality of the deflection angle of a flow passing through the incident and reflected fronts. Calculations of the pressure drops behind the shock fronts agree with experimental data in this case.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 26–33, September–October, 1973.The authors thank S. A. Khristianovich for consideration of the work and advice.     

Numerical simulation for nonstationary Mach reflection of a shock wave: A kinetic-model approach

A numerical simulation was performed for the process of formation of single Mach reflection on a wedge by solving a BGK type kinetic equation for the reduced distribution function with a finite difference scheme. The calculations were carried out for a shock Mach number 2.75 and wedge angle 25° in a monatomic gas, which corresponds to the conditions of single Mach reflection in the classical von Neumann theory. The calculations were performed for both diffuse and specular reflection of molecules at the wall surface. It is concluded that the diffuse reflection of molecules at the wall surface or the existence of the viscous or thermal layer is an essential factor for a nonstationary process at the initial stage of Mach reflection. Furthermore, the numerical results for diffuse reflection are found to simulate the experimental results very well, such as a transient process from regular reflection to Mach reflection along with shock propagation.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1990.     

Analytical solution of a flow field for weak Mach reflection over a plane surface

In this paper, we present an analytical solution for the flow field of single weak Mach reflection caused by an advancing plane shock wave over a simple wedge surface. We developed an improvement of Lighthill's linearized theory for the correction due to nonlinearity of the flow field through a singular perturbation. The expressions obtained, including the one for the triple point path, compared favorably with existing experimental, computational, and theoretical results. PACS 01.50.Kw; 47.15.Pn Communicated by K. Takayama     

The Mach reflection of shock waves in converging,cylindrical channels

Shock waves impinging on axisymmetric converging passages of moderate wall angle undergo cyclical Mach reflection processes which are generally not self-similar during progression through the contraction. This paper presents the results of shock tube experiments in air where shocks of moderate strength are incident on cones having a range of wall angles of 10° to 30°. Mach reflection is maintained as the dominant reflection process throughout this range but the number of cycles decreases as wall angle increases. A ray-shock approach, used to calculate the triple-point trajectories gives good agreement with the experimental locations of their intersections with the centreline and wall.     

High temperature effects in moving shock reflection with protruding Mach stem

The influence of high temperature effects on the protrusion of Mach stem in strong shock reflection over a wedge was numerically investigated.A two-dimensional inviscid solver applies finite volume method and unstructured quadrilateral grids were employed to simulate the flow.Theoretical analysis was also conducted to understand the phenomenon.Both numerical and theoretical results indicate a wall-jet penetrating forward is responsible for the occurrence of Mach stem protrusion.The protrusion degree seems to depend on the thermal energy buffer capacity of the testing gas.Approaches to increase the energy buffer capacity,such as vibrational relaxation,molecular dissociation,and increase of frozen heat capacity,all tend to escalate the protrusion effect.     

Effect of boundary layer on Mach reflection over a wedge surface

Abstract. In this paper, we consider the phenomenon of unsteady Mach reflection generated by a plane shock wave advancing over a straight wedge surface, with particular attention to the deviation of the flow field from the self-similar nature. We examine the observed change in angle between incident and reflected shocks, which is in contrast to the fact that the angle should remain constant with time in a self-similar flow. The effect of the boundary layer behind the advancing shock wave over the surface of the wedge is considered to cause this, and boundary layer theory is utilized to estimate the thickness of the layer. It is found that the thickness increases as to the time t compared with t by the overall expansion in the self-similar flow. Assuming that the thicker boundary layer is effectively equivalent to a change in wedge angle, the effect of the boundary layer on the flow field should be less in later stages with larger t values in accordance with the observation above. Received 6 March 2000 / Accepted 23 April 2001     

A simple physical theory of weak Mach reflection over plane surfaces

It is shown that simple physical principles coupled with the inviscid shock jump relations can be applied to the problem of weak Mach reflection to the extent that the triple point path can be predicted from the incident shock Mach number , gas specific heat ratio and the inclination angle of the reflecting surface to the shock normal. Comparison with the Euler code data and with experiments show close agreement for conditions both far and close to transition and that the general shape of the reflected and Mach stem shocks follow simple curves except in the neighbourhood of the triple point. The conflict at the triple point in matching the flow deflection angles and pressures across the contact discontinuity remains. It is shown however that the simple model presented here gives a close match to the cfd and experimental overall shock and contact surface shapes although it cannot predict these or the flow properties in any detail. Received 10 May 1999 / Accepted 17 December 1999