Numerical simulation of Mach reflections

In this paper, the “FLIC” difference method with triangular mesh is adopted to numerically simulate the regular and Mach reflections that occur when a shock wave pass around a wedge. The compuational result is compared with the shock tube experimental results of G. Ben-Dor and I. I. Glass. The comparison shows that the position, shape of shock wave and height of Mach stem all show a good agreement. Consequently, the “FLIC” difference method with triangular mesh is quite satisfactory in numerical simulation of the regular and Mach reflections.     

Interaction of two Mach reflections over concave double wedges-analytical model

A simplified analytical model for solving the wave configuration resulting when two triple points join together over a concave double wedge is developed. The model equations are solved analytically. Predictions of the model are compared to experimental results and fairly good agreement is obtained. It is believed that the present model can be used to better understand complex wave interaction phenomena. Received 2 May 1995 / Accepted 18 December 1997     

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.     

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     

Analysis of the wave configuration resulting from the termination of an inverse Mach reflection

An analytical model for solving the wave configuration which is formed when an inverse Mach reflection terminates after its triple point collides with the reflecting surface has been developed. The predictions of the model were compared with available experimental results and good agreement was obtained.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1990.     

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     

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.     

Analytical prediction of the wave configuration size in steady flow Mach reflections

Analitycal model for predicting the size of the Mach reflection wave configuration in steady flows has been improved (Azevedo 1989; Azevedo and Liu (1993)). Predictions based on the modified analytical model were compared to available experimental results. The agreement was found to be better than that obtained by Azevedo (1989) and Azevedo and Liu (1993). The reason for the better agreement is due to the fact that, unlike Azevedo's original model, downstream effects were not neglected in the modified model which was developed in the course of this study. Received 12 May 1996 / Accepted 15 January 1997     

DSMC approach to nonstationary Mach reflection of strong incoming shock waves using a smoothing technique

The direct simulation Monte Carlo (DSMC) method is applied to simulation of nonstationary Mach reflection of strong shock waves. Normally the DSMC method is very time consuming in solving unsteady flow field problems especially for high Mach numbers, because of the necessity of iterative calculations to eliminate the inherent statistical fluctuation caused by a finite sample size. A central weighted smoothing technique is introduced to process the DSMC results, so that the iteration time can be significantly reduced. In spite of some relaxations of the shock wave structure, the smoothing technique is verified to be useful to estima te the flow fields qualitatively and even quantitatively by using a relatively small sample size. The comparison between the present approach and the kineticmodel approach (Xu et al. 1991a, 1991b) on the application to unsteady rarefied flow fields was also carried out.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.     

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     

A numerical study of shock wave reflections on low density foam

A continuum mixture theory is used to describe shock wave reflections on low density open-cell polyurethane foam. Numerical simulations are compared to the shock tube experiments of Skews (1991) and detailed wave fields are shown of a shock wave interacting with a layer of foam adjacent to a rigid wall boundary. These comparisons demonstrate that a continuum mixture theory describes well the shock interactions with low density foam.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.     

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     

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.      

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     

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.     

准定常强激波反射马赫杆突出变形准则的探讨

分析了滑移线延长线与楔面交点附近的流动特点,研究了准定常强激波反射中马赫杆的变形,建立了马赫杆突出变形消失的过渡准则,并用激波极曲线方法进行了求解。对马赫杆的变形过程进行了描述,研究了比热比、马赫数的变化对马赫杆突出变形消失条件的影响,并对突出变形消失区域与终点双马赫反射区域进行了比较。结果表明:马赫杆突出变形的消失是低比热比介质中出现的一种激波反射现象。马赫杆突出变形的消失,导致出现一种新的无射流、直马赫杆的双马赫反射结构。     

Mach reflection of a plane shock wave passing a mountain of 45° inclination

The transition from regular reflection (RR) to Mach reflection (MR) as a plane shock wave diffracts around a triangular mountain of 45° inclination is analysed in this paper, both by optical measurement in a shock tube and by numerical simulation the numerical method developed by Li Yingfan^[1] is of the FLIC type with triangular mesh. The dependence of the critical transition point Lk ofRR→MR on shock Mach numberM _i is analyzed and the variations of the incidence angle ω _i of the impinging shock and the reflection angle ω _r with the distanceL ^* are investigated. Our experimental and numerical results agree well with the theoretical results of Iton and Italya.     

Investigation of shock waves interference and associated hysteresis effect at variable-Mach-number upstream flow

Experimental investigations and numerical simulations have been performed to study the transition between steady regular and Mach reflections induced by flow Mach number variation. The experiments have been carried out in the supersonic wind tunnel SIGMA 4B, at the Institut Aérotechnique (IAT) in St Cyr L'Ecole, France. Symmetric and asymmetric arrangements of wedges have been tested. No significant hysteresis phenomenon has been detected experimentally. However, this phenomenon has been revealed by numerical computations obtained by solving the Navier-Stokes equations.Received: 9 October 2001, Accepted: 11 October 2002, Published online: 21 February 2003