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.     

Mach reflection in detonations propagating through a gas with a concentration gradient

In accident scenarios where detonations can occur a concentration gradient constitutes a more realistic initial condition than a perfectly homogeneous mixture. In this paper, the influence of a concentration gradient on detonation front shape, detonation instabilities and pressure distribution is studied. First, a simple method to determine the front shape from a given fuel distribution is presented. It is based on Huygens’ principle and includes a correction to satisfy the boundary conditions on the enclosing walls. Next, the presented highly resolved Euler computations demonstrate the influence of a concentration gradient on detonation instabilities. In configurations with a strong concentration gradient, Mach reflection occurs and leads to an asymmetric pressure load on the enclosing geometry. In this case, the impulse on the wall is higher than in configurations with homogeneous fuel distribution, although the fuel content is much lower.     

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.     

不稳定性对爆轰波楔面马赫反射的影响规律研究

采用一种两步化学反应模型对胞格爆轰波的楔面马赫反射过程进行了数值研究,从而澄清和解释胞格不稳定性对马赫反射发展模式和自相似性的影响。考虑到反应欧拉方程源项的刚性问题,本文采用附加RungeKutta方法耦合非刚性对流项和刚性反应源项,对流项的离散采用五阶精度的WENO格式。计算结果表明,对于稳定胞格爆轰波而言,其马赫反射过程本质上与ZND爆轰波的马赫反射是一致的,整体上不存在自相似性,胞格不稳定性只是造成了三波点轨迹线局部小振幅的波动。在楔面顶点附近,由于马赫杆是强过驱的,爆轰波的马赫反射过程是自相似的。在远场,爆轰波马赫反射的三波点轨迹线渐近的趋向于一条直线,说明重新获得了自相似性。对于不稳定的爆轰波,由于自身的不稳定性可以与马赫反射的强度相匹配,定义其三波点的轨迹是困难的,进行自相似性分析没有意义。     

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.     

Numerical simulation of boundary-layer transition at Mach two

The linear and early nonlinear stages of boundary-layer transition at free-stream Mach numberM ==2.0 are investigated by direct numerical simulation of the compressible Navier-Stokes equations. Results from simulations with a large computational box and small-amplitude random initial conditions are compared with linear stability theory. The growth rates of oblique waves are reproduced correctly. Two-dimensional waves show a growth that is modulated in time, indicating the presence of an extra unstable mode which moves supersonically relative to the free stream. Further simulations are conducted to investigate the nonlinear development of two- and three-dimensional disturbances The transition due to oblique disturbance waves is the most likely cause of transition at this Mach number, and is found to lead to the development of strong streamwise vortices.     

Numerical simulation of the Vasilev reflection

We present a high resolution numerical solution of the Vasilev reflection configuration within the framework of depth averaged two-dimensional inviscid shallow water flow. The study provides the details of the steady flow field and shock wave pattern close to the triple point which confirm the four-wave theory. The shape of the reflected shock in the region upstream of the supercritical patch is also investigated.      

Numerical simulation of the head-on reflection of a regular reflection

The head-on collision and subsequent reflection of a Regular Reflection (RR) from the end-wall of a shock tube has been investigated both experimentally and numerically for two different incident shock wave Mach numbers and two different reflecting wedge angles. The agreement between the double-exposure holographic interferograms and the numerical simulations which were obtained using a GRP based numerical code, was found to be excellent in the RR region and very good behind the head-on reflected RR. The overall good agreement between the computed and experimental constant-density contours (isopycnics) constitutes a validation of the computational method, including the oblique-wall boundary condition.     

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.      

Mathematical simulation of Mach reflection of shock waves in media with different adiabatic indices

Some peculiarities of the processes of regular and Mach reflection at constant adiabatic index =c_p/c_v=1.4 were investigated theoretically in [1]. It was demonstrated that increase in incident-wave intensity above some value leads to the appearance of an internal compression discontinuity (Fig. 1) and a break in the reflected wave (at point h), both of which had been observed previously only in experiment [2–4]. In the present study the method described in [1] is used to study the influence of adiabatic index on these peculiarities of the Mach reflection process which lead to a significant increase in pressure (to a maximum value P_m) on the surface wedge in the vicinity of point i. Pressure and density curves along the wedge surface are presented. It is found that increase in leads to the same qualitative changes in the pressure and density curves on the surface as are observed upon increase in semiaperture angle of the wedge or upon decrease in Mach number M_f of the shock-wave front incident on the wedge ab. These similarities in the shock-wave reflection process were first noted in [5] for weak shock waves in which the internal compression discontinuity does not appear.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 8, pp. 90–94, May–June, 1976.The author thanks O. S. Ryzhov for the many valuable remarks offered in his evaluation of the study.     

爆轰波在楔面上反射数值分析

应用基元反应模型和频散可控耗散格式(DCD)对氢氧爆轰波在楔面反射进行了数值模拟,计算中氢氧混合物的化学反应采用了8种组分20个反应方程式,在处理化学反应引起的刚性问题时采用了时间算子分裂的方法,模拟了爆轰波在楔面反射由马赫反射向规则反射转变的过程,得到了反射转变临界角,同时考虑了初始压力和组分的影响,并和实验及理论分析结果进行了比较,结果是令人满意的。     

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.     

Mach reflection parameters for plexiglas cylinders

The collapsing air gap method has been used to investigate the development and limiting parameters of the Mach reflection of a conical convergent shock in plexiglas cylinders arranged along the axis of a detonating explosive charge. The diameters of the cylindrical specimens varied from 15 to 100 mm. It is shown that on the stationary interval of development of the triple-shock configuration, where the velocity of the head wave is equal to the detonation velocity, there is a linear relationship between the diameter of the head wave, its radius of curvature, and the diameter of the cylinder.     

Analytical study of idealized two-dimensional cellular detonations

In this study, the idealized two-dimensional detonation cells were decomposed into the primary units referred to as sub-cells. Based on the theory of oblique shock waves, an analytical formula was derived to describe the relation between the Mach number ratio through triple-shock collision and the geometric properties of the cell. By applying a modified blast wave theory, an analytical model was developed to predict the propagation of detonation waves along the cell. The calculated results show that detonation wave is, first, strengthened at the beginning of the cell after triple-shock collision, and then decays till reaching the cell end. The analytical results were compared with experimental data and previous numerical results; the agreement between them appears to be good, in general. Received 13 February 2001 / Accepted 2 August 2001     

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 a pulsating flow arising over an axisymmetric spiked blunt body at Mach 2.21 and Mach 6.00

The present paper reports a numerical simulation of the supersonic/hypersonic unsteady flow over a spiked blunt body. Axisymmetric compressible Navier-Stokes equations are solved using a high-resolution unfactored implicit upwind Roe's scheme and a time-accurate pseudo-time method is employed for advancing in time. Unsteady flows arising at Mach 2.21 and Mach 6.00 around a spiked cylinder are simulated and the computational results are compared with measurements. The simulated results are used to increase understanding of the mechanisms of the flow. Received 28 September 1999 / Accepted 26 July 2000     

Three-dimensional numerical simulation of detonations in coaxial tubes

Three-dimensional numerical simulation of detonations in both a circular tube and a coaxial tube are simulated to reveal characteristics of single spinning and two-headed detonations. The numerical results show a feature of a single spinning detonation which was discovered in 1926. Transverse detonations are observed in both tubes, however, the single spinning mode maintains the complex Mach reflection whereas the two-headed mode develops periodically from the single Mach reflection to the complex one. The calculated cell aspect ratio for the two-headed mode changes from 1.09 to 1.34 as the radius of axial insert increases from r ₁/R = 0.1 to 0.9. The calculated cell aspect ratio for r ₁/R = 0.1 is close to the experimental results without an axial insert. The formation of an unreacted gas pocket behind the detonation front was not observed in the single spinning mode; however, the two-headed mode has unreacted gas pocket behind the front near the axial insert.      

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.