A shock dynamics theory based analytical solution of double Mach reflections

The shock-dynamics based governing equations of double Mach reflection (DMR) wave configurations were developed and solved. The analytical results were compared to experimental results from various sources. Very good to excellent agreement was evident.     

激波在不同三角楔面内传播特性的数值研究

激波在收缩管内的反射与聚焦会形成高温高压区,点燃可燃混合气并诱导爆轰,因此对爆轰发动机的点火具有重要意义。本文基于二维N-S方程,结合五阶WENO格式,对马赫数为6的正激波在三角形楔面内的反射与聚焦现象进行了数值研究。结果表明,楔面顶角的变化对激波的反射类型以及聚焦均有明显的影响:随着顶角的增加,激波的反射类型从马赫反射向过渡马赫反射和双马赫反射转变,且壁面上的前向射流更加明显;三波点第一次碰撞产生的高温高压区足够满足可燃混合气体的点火条件,且其温度与压力值随顶角的增加而增大;当激波在楔面上发生临界双马赫反射时,温度与压力达到最大;当顶角增加到一定值时,激波在楔面反射转变为常规反射,不会产生激波对碰,因而没有高温高压区。     

Reconsideration of pseudo-steady shock wave reflections and the transition criteria between them

The shock wave reflection phenomenon in pseudosteady flows was reconsidered by replacing the Law-Glass assumption by models accounting for the interaction of the shock wave reflection and the shock induced flow deflection processes. As a result, the analytical predictions of the location of the kink of a transitional-Mach reflection and the second triple point of a double-Mach reflection improved tremendously. It has also been proven that based on gas dynamic considerations a triple-Mach reflection wave configuration is physically impossible. In addition, the transition lines between the various reflection configurations were also found to better agree with the experimental results when they were calculated using the proposed models.     

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.     

Structure of an oblique detonation wave induced by a wedge

The structure of an oblique detonation wave (ODW) induced by a wedge is investigated via numerical simulations and Rankine–Hugoniot analysis. The two-dimensional Euler equations coupled with a two-step chemical reaction model are solved. In the numerical results, four configurations of the Chapman–Jouguet (CJ) ODW reflection (overall Mach reflection, Mach reflection, regular reflection, and non-reflection) are observed to take place sequentially as the inflow Mach number increases. According to the numerical and analytical results, the change of the CJ ODW reflection configuration results from the interaction among the ODW, the CJ ODW, and the centered expansion wave.     

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.     

Existence criteria for reflection configurations in shock–vortex interactions

The length scale criteria is widely accepted as an explanation for transition and hence existence of different shock wave reflection configurations in pseudo-steady flows. However, there has not been any attempt to validate this criteria using information obtained from a time-dependent numerical simulation. A high resolution time-dependent numerical simulation in pseudo-steady flow is carried out in the present work. Time-dependent numerical data is used to calculate flow features in a laboratory frame of reference to verify validity of the length scale criteria for existence of different shock wave reflection configurations in pseudo-steady flow. This analysis is then extended to the study of unsteady shock wave reflection configurations in shock–vortex interactions. It is shown that the existence of regular reflection (RR) and Mach reflection (MR) configurations in an unsteady flowfield resulting from shock–vortex interactions can also be explained locally based on limiting conditions similar to that prescribed by the length scale criteria for pseudo-steady flow.

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Experimental investigation of the asymptotically approached Mach reflection over the second surface in a double wedge reflection

The assumption that the Mach reflection which is formed over the second surface of a double wedge with angles _w ¹ and kw/2 approaches asymptotically the Mach reflection which would have been obtained by an identical incident shock wave over a single wedge with an angle _w = _w ² was verified experimentally. The verification of this assumption supports the shock polar analysis suggested by Ben-Dor et al. (1987) for the study of the reflection process of a planar shock wave over a double wedge. Measurements of the rate of approach to the asymptotic value are also provided.     

The discovery of the Mach reflection effect and its demonstration in an auditorium

This paper examines the historical background leading to the discovery of the Mach reflection effect and applies original documents from Mach's residue which are kept in the archives of the Ernst-Mach-Institut in Freiburg. Two experimental setups for the generation and demonstration of the Mach reflection effect, incorporating an overhead projector, are described: (a) Mach's historic mechanical shock wave reflection and interaction experiments with soot covered glass plates, performed in 1875. The Mach triple points sharply erase the soot which results in a residual picture of funnel-shaped V-formations. The head-on collision of two shock waves is marked as a narrow line of piled-up soot. (b) CalTech's hydraulic jump reflection experiments in a shallow ripple tank, performed during World War II. Regular reflection and its transition into a Mach reflection wave. Using a slightly inclined tank and providing a shoreline in the middle of the tank, Mach stem propagation slows down to zero when hitting the shore line and, therefore, can be observed live without the use of a slow motion technique.Visiting graduate student on an IAESTE grantThe International Mach Reflection Symposia have been held in Victoria, BC, Canada (1981); Sydney, Australia (1982); Freiburg, FRG and Melbourne, Australia (1983); Tokyo and Sendai, Japan (1984); Menlo Park, CA, USA (1985); Beer Sheva, Israel (1986); Albuquerque, USA (1987) and Toronto, Canada (1988)This article was processed using Springer-Verlag T_EX Shock Waves macro package 1990.     

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.     

Shock wave structure in a mixture of gas,vapour and droplets

The structure of the relaxation zone behind a shock wave of moderate strength in a mixture of gas, vapour and droplets is analysed. A model is presented for shock induced evaporation, which is based on wet-bulb equilibrium and on the absence of relative motion between droplets and gas. Experimental and numerical data on heterogeneous condensation induced by an unsteady rarefaction wave and on re-evaporation due to shock wave passage are reported for a mixture of water vapour, nitrogen gas and condensation nuclei. Pressure, temperature, saturation ratio and droplet size are experimentally obtained and are very well predicted by a numerical simulation based on the non-linear quasisteady wet-bulb model for phase transition, as well for the expansion wave as for the shock wave. During expansion, droplet number density decays much faster than predicted, which is not yet satisfactorily explained. Shock induced droplet evaporation is studied for post-shock saturation ratios ranging from 5×10^–3 to 0.2, corresponding to shock Mach numbers of 1.2 to 1.9. The evaporation times are well predicted by the theoretical model. No evidence is found for droplet break-up for Weber numbers up to 13, and droplet radii of the order of 1m.On leave at Institute of Fluid Science, Shock Wave Research Center, Tohoku University, Sendai 980, JapanThis 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 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     

Gas combination for shock wave enhancement

It was recently demonstrated that shock wave enhancement could be achieved when a shock propagates in a constant cross-section duct through pairs of air–helium layers having a continually decreasing width (Igra and Igra in Shock Waves 16(3):199–207). A parametric study was conducted aimed at finding a two-layered, light–heavy gas arrangement that yields maximal shock enhancement; the heavy and the light gases used were air and helium, respectively. Effects associated with changes in following parameters were investigated: the number of alternating heavy/light gas layers, the applied reduction ratio between successive layers thickness, and the initial shock wave Mach number.      

Re-entry body drag: shock tunnel experiments and computational fluid dynamics calculations compared

The best approach for conducting the research necessary for developing hypersonic flight vehicles is a close coupling between experiments that employ rapid measurement techniques and computational fluid dynamics (CFD) that appropriately accounts for the freestream nonuniformities, as well as for hypervelocity flow phenomena. This approach has been employed here, where stress wave force measurements and CFD calculations have been combined in an investigation of the axial drag on a generic re-entry body. Experiments were performed in argon and nitrogen, with test flows ranging in total enthalpy between 3 MJ/kg and 12 MJ/kg and Mach numbers varying from 6 to 13. The associated measured drag forces ranged from 300 to 360 N. For Mach 12 argon flows, the CFD overpredicted the drag by 8%, while for two hypervelocity nitrogen flows the CFD overpredicted the drag by at most 5%. Considering uncertainties in the force measurements and the CFD boundary conditions, the agreement is good, and the work highlights both the ability of the force measurement technique to respond to rapid changes in flow conditions and the importance of carefully accounting for flow gradients in the CFD boundary conditions. This paper was based on work that was presented at the 3rd International Symposium on Interdisciplinary Shock Wave Research, Canberra, Australia, March 1–3, 2006.     

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.     

Numerical confirmation of the hysteresis phenomenon in the regular to the Mach reflection transition in steady flows

Numerical calculations based on the Navier-Stokes equations are carried out to investigate the reflection of shock waves over straight reflecting surfaces in steady flows. The results for a flow Mach number of M₀=4.96 confirm the recent experimental findings of Chpoun et al. (1995) concerning the transition from regular to Mach reflection. Numerical calculations as well as experimental results show a hysteresis phenomenon during this transition and the regular reflection is found to be stable in the dual-solution domain in which theoretically both regular and Mach reflection wave configurations are possible.     

Effect of yaw on the starting point of curvature for reflected diffracted shock wave (subsonic and sonic cases)

Lighthill (Proc. R. Soc. A 198, 454–470, 1949) considered the diffraction of a normal shock wave passing over a small bend. The bend being small Lighthill was able to linearize the flow equations and solved the problem through several mathematical techniques. Following Lighthill (Proc. R. Soc. A 198, 454–470, 1949), Srivastava and Chopra (J. Fluid Mech. 40, 821–831, 1970) extended the work to the diffraction of oblique shock waves. Srivastava (AIAAJ 33, 2230–2231, 1995) considered the problem of starting point of curvature and extended the work to yawed wedges (Srivastava in Proceedings of the 14th International Mach reflection symposium Sun Marina Hotel, Yonezawa, Japan, 1–5 October 2000, pp. 225–249, 2002). Srivastava (Shock waves 13, 323–326, 2003) considered the problem for starting point of curvature when the relative outflow behind reflected shock before diffraction has been subsonic and sonic. The present work is an extension of the work published in Srivastava (Shock waves 13, 323–326, 2003) when the wedge has been yawed through an angle. The results have been obtained for two angles χ = 60° and χ = 40° (χ is the angle of yaw).      

Stability of converging cylindrical shock waves

An experimental and numerical study was made of converging cylindrical shock waves. The goal of the present study was to clarify the movement and instability of the converging cylindrical shock waves. Experiments were conducted in an annular shock tube of 230 mm o.d. and 210 mm i.d. connected to a cylindrical test section of 210 mm diameter. Double exposure holographic interferometry was used to visualize the converging cylindrical shock waves. Incident shock Mach numbers ranged between 1.1 and 2.0 in air. A numerical simulation was conducted using the TVD finite difference scheme. It was found in the experiments that although the initial shock wave configuration looked cylindrical, it was gradually deformed with propagation towards the center and finally showed mode-four instability. This is attributable to the existence of initial disturbances which were introduced by the struts which supported the inner tube of the annular shock tube. This trend was significant for stronger shock waves indicating that at the last stage of shock wave convergence the initial perturbations of the converging cylindrical shock wave were amplified to form the triple point of Mach reflection. The numerical results correctly predicted the experimental trend.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1990.     

Reflection of an Oblique Shock Wave in a Reacting Gas with a Finite Relaxation–Zone Length

Reflection of an oblique shock wave in a reacting gas with a finite length of the chemical–reaction zone is studied. Shock polars for an arbitrary heat release behind the oblique shock wave are constructed. Transition criteria from regular to Mach reflection and back are obtained. It is shown that transition criteria are significantly changed if the reaction–zone length is taken into account.