Mach Stem Research Articles

Analysis of double-Mach-reflection wave configurations with convexly curved Mach stems

The wave configuration of a double Mach reflection (DMR) with a convexly curved Mach stem and the resulted flow fields are analyzed. An analytical model describing the formation of a $DMR$ with a curved Mach stem and predicting its wave configuration is proposed. The transition criterion from a $DMR$ with a straight Mach stem to a $DMR$ with a curved Mach stem is also suggested. Predictions based on the analytical model are compared to experimental results. The agreement is found to be good to excellent.

An analytical study of Mach reflection in nonequilibrium steady flows

In the present work, we study the influence of both thermal and chemical relaxation phenomena on steady Mach reflections. To characterize the dependency of the scales of the phenomenon upon the relaxation we develop a quasi-one-dimensional model for a nonequilibrium diatomic gas coupled with the three shock theory. The results show that Mach reflections are strongly affected by vibrational relaxation and to a lesser extent by chemical excitation. The Mach stem height and its location, and the extent of the subsonic pocket are found to depend both on the geometrical setup and on the relaxation length, and do not scale linearly with the geometrical size as found in the ideal case. In addition, the relaxation is found to cause a forward displacement of the Mach stem thus counteracting Mach reflection instabilities. The nonlinearity has also been confirmed by a perturbation analysis of the equations for small Damköhler numbers.

Experimental investigation of the influence of downstream flow conditions on Mach stem height

An experimental work on the influence of downstream flow expansion process on Mach stem height has been carried out in steady hypersonic flows. The results showed clearly for the first time that the Mach stem height does not depend on the extent of expansion fan. These results contradict recent analytical findings from which a strong influence of downstream flow conditions on Mach stem height may be expected.

Shock waves in nonuniform gas

Shock waves propagating through a stratified gas are investigated numerically and analytically. A shock wave is produced by a piston that begins to move from rest abruptly at some constant velocity in a two-dimensional horizontal duct. Initially the gas in the duct has a temperature or density distribution only along the vertical axis at a constant pressure. The initial density distribution, which is assumed to change monotonically, has a zero spatial gradient at the upper and lower walls and then has a single inflection point. It is confirmed that at least three types of shock patterns can be realized asymptotically. The first is a single curved shock, the second is a shock with a Mach stem (Mach reflection), and the last is a shock with a reflected branch (regular reflection). The first is substantially steady but the latter two are essentially unsteady. The time evolution of the induced flow field is investigated in detail. Based on this information, an analytical solution for the substantially steady curved shock is obtained in the coordinate system fixed on the shock. The shock profile as well as the induced flow field is investigated in detail with this solution. It is shown that the analytical results can predict quite well the numerical results. Finally, the flow instability of this shock-induced flow is investigated, because the induced flow has a nonuniform horizontal velocity distribution along the vertical axis at a constant pressure.

Unsteady-State Simulation of Model Ram Accelerator in Expansion Tube

Steady- and unsteady-state numerical simulations have been carried out to investigate the ram accelerator flowfield that had been studied experimentally using an expansion tube facility at Stanford University. Navier-Stokes equations for chemically reactive flows were used for the modeling with a detailed hydrogen-air combustion mechanism. The governing equations were analyzed using a fully implicit and time-accurate total variation diminishing scheme. As a result, steady-state simulation reveals that the near-wall combustion regions are induced by aerodynamic heating in the separated flow region. This result agrees well with experiments in the case of the 2H 2 + O 2 + 17N 2 mixture but fails to reproduce the centerline combustion in the case of the 2H 2 + O 2 + 12N 2 mixture. To investigate the reason for this disagreement in the flow establishment process, unsteady-state simulations have been carried out, and the results show the detailed process of flow stabilization. The centerline combustion is revealed to be an intermediate process during flow stabilization. It is induced behind a Mach stem formed by the intersection of strong oblique shock waves at an early stage of the flow stabilization process

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.

On the role of turbulence in detonation induced by Mach stem reflection

A series of experiments conducted by Chan has shown that while some shock waves may not be strong enough to induce detonation when they collide with an obstacle the resulting Mach stem will induce detonation if it collides with a subsequent obstruction. A series of numerical simulations, however, failed to demonstrate the expected results if either the Euler or laminar Navier-Stokes equations are solved. On the other hand, calculations using the Favre averaged Navier-Stokes equations with a k-\(\epsilon\)-F turbulence model are able to reproduce the experimental results, indicating that turbulent effects may play an important role in the ignition process. A detailed examination of the results shows that turbulence causes the formation of activated kernels in a similar process to that observed in deflagration-detonation transition. The simulations in this paper have been undertaken using a modern high resolution hydrocode and a reduced kinetics mechanism for hydrogen combustion. The paper describes the reduced mechanism, the solution methods employed in the hydrocode and discusses the results of the simulations and their implications.

Numerical Analysis of Shock Wave Reflection Transition in Steady Flows

Different aspects of the transition between regular and Mach reflections of strong shock waves in steady flows are numerically studied. Two approaches-kinetic (the direct simulation Monte Carlo method) and continuum (Euler equations)-are used to investigate the hysteresis phenomenon in the flow about two symmetrical wedges in two- and three-dimensional statements. The dependence of the final shock wave configuration on initial conditions, the transition from regular to Mach reflection by means of flow perturbations, and three-dimensional effects are examined. The three-dimensionality of the flow is shown to increase the angles of transition from regular to Mach reflection and back and to decrease the Mach stem height

Triple-shock entropy theorem and its consequences

For a convex equation of state, a general theorem on shock waves is proved: a sequence of two shocks has a lower entropy than a single shock to the same final pressure. We call this the triple-shock entropy theorem. This theorem has important consequences for shock interactions. In one dimension the interaction of two shock waves of the opposite family always results in two outgoing shock waves. In two dimensions the intersection of three shocks, such as a Mach configuration, must have a contact. Moreover, the state behind the Mach stem has a higher entropy than the state behind the reflected shock. For the transition between a regular and Mach reflection, this suggests that the von Neumann (mechanical equilibrium) criterion would be preferred based on thermodynamic stability, i.e. maximum entropy subject to the system constraint that the total specific enthalpy is fixed. However, to explain the observed hysteresis of the transition we propose an analogy with phase transitions in which locally stable wave patterns (regular or Mach reflection) play the role of meta-stable thermodynamic states. The hysteresis effect would occur only when the transition threshold exceeds the background fluctuations. The transition threshold is affected by flow gradients in the neighbourhood of the shock intersection point and the background fluctuations are due to acoustic noise. Consequently, the occurrence of hysteresis is sensitive to the experimental design, and only under special circumstances is hysteresis observed.

Approximate analytical calculation of the mach configuration of steady shock waves in a plane constricting channel

An approximate analytical model for calculation of the parameters of a steady gas flow inside a plane constricting channel formed by two symmetrically positioned wedges is suggested. A Mach configuration of shock waves (triple point) is formed in the channel when the wedge angles are larger than some critical value. The flow calculation in a constricting channel reduces to the solution of the iterative problem for a system of nonlinear algebraic equations. The configurations of shock waves, the slipstream, and the sonic line are described by the proposed model of a gas flow. A comparison of the results obtained using this model allows a fairly accurate calculation of the Mach stem and the length of the subsonic-flow region.

A Numerical Study of a Two-Dimensional H2-O2-Ar Detonation Using a Detailed Chemical Reaction Model

Two-dimensional computations of the propagation of a detonation in a low-pressure, argon-diluted mixture of hydrogen and oxygen were performed using a detailed chemical reaction mechanism. Cellular structure developed after an initial perturbation was applied to a one-dimensional solution placed on a two-dimensional grid. The energy-release pattern in a detonation cell showed that, in addition to the primary release of energy behind the Mach stem, there is a secondary energy release that starts about two-thirds of the way through the cell. Reignition, which occurs as transverse waves collide, results in an explosion that spreads over a region and releases a considerable amount of energy. Resolution tests showed convergence of the detonation mode (number of triple points or transverse waves) reached at the end of the computations, as well as global and local energy release. The computations were performed on massively parallel Connection Machines for which new approaches were developed to maximize the speed and efficiency of integrations.

Mach Reflection Wave Configuration in Two-Dimensional Supersonic Jets of Overexpanded Nozzles

The Mach reflection wave configurations and the flowfields associated with two-dimensional supersonic free jets of overexpanded nozzles are analysed. Based on the two- and the three-shock theories along with the classical gasdynamics theory, an analytical model for calculating the height of Mach stem and the cell size of the jet are proposed

Gradients at a curved shock in reacting flow

The inviscid equations of motion for the flow at the downstream side of a curved shock are solved for the shock–normal derivatives. Combining them with the shock–parallel derivatives yields gradients and substantial derivatives. In general these consist of two terms, one proportional to the rate of removal of specific enthalpy by the reaction, and one proportional to the shock curvature. Results about the streamline curvature show that, for sufficiently fast exothermic reaction, no Crocco point exists. This leads to a stability argument for sinusoidally perturbed normal shocks that relates to the formation of the structure of a detonation wave. Application to the deflection–pressure map of a streamline emerging from a triple shock point leads to the conclusion that, for non–reacting flow, the curvature of the Mach stem and reflected shock must be zero at the triple point, if the incident shock is straight. The direction and magnitude of the gradient at the shock of any flow quantity may be written down using the results. The sonic line slope in reacting flow serves as an example. Extension of the results – derived in the first place for plane flow – to three dimensions is straightforward.

Reflection of Underwater Shock Wave due to Underwater Explosion of High Explosives. 2nd Report. Curvatures of the vNR and the Transition Criteria from MR to vNR.

We have previously reported our studies on the phenomenon of von Neumann reflection of underwater shock waves by experimental and numerical techniques. In this paper, as a further supplement to the previous report, we mainly made our efforts to investigate the change trend of the Mach stem curvature in von Neumann reflection of underwater shock wave. We presented a theoretical analysis model on describing the change of Mach stem curvature on the basis of Whitham's Ray-shock theory. It has been shown that our analytical results agree quite well both experimental and numerical results. Finally, we also presented the theoretical prediction on the domains and transition boundaries between regular reflection, Mach reflection and von Neumann reflection, of underwater shock wave by the above theory, as well as the comparison with those from the experiment and numerical calculation.

Numerical analysis of shock wave reflection transition in steady flows

Different aspects of the transition between regular and Mach reflections of strong shock waves in steady flows are numerically studied. Two approaches-kinetic (the direct simulation Monte Carlo method) and continuum (Euler equations)-are used to investigate the hysteresis phenomenon in the flow about two symmetrical wedges in two- and three-dimensional statements. The dependence of the final shock wave configuration on initial conditions, the transition from regular to Mach reflection by means of flow perturbations, and three-dimensional effects are examined. The three-dimensionality of the flow is shown to increase the angles of transition from regular to Mach reflection and back and to decrease the Mach stem height

A parametric study of Mach reflection in steady flows

The flow fields associated with Mach reflection wave configurations in steady flows are analysed, and an analytical model for predicting the wave configurations is proposed. It is found that provided the flow field is free of far-field downstream influences, the Mach stem heights are solely determined by the set-up geometry for given incoming-flow Mach numbers. It is shown that the point at which the Mach stem height equals zero is exactly at the von Neumann transition. For some parameters, the flow becomes choked before the Mach stem height approaches zero. It is suggested that the existence of a Mach reflection not only depends on the strength and the orientation of the incident shock wave, as prevails in von Neumann's three-shock theory, but also on the set-up geometry to which the Mach reflection wave configuration is attached. The parameter domain, beyond which the flow gets choked and hence a Mach reflection cannot be established, is calculated. Predictions based on the present model are found to agree well both with experimental and numerical results.

Three-dimensional shock wave reflection over a corner of two intersecting wedges

This paper presents an experimental and numerical investigation of three-dimensional shock wave reflections over a corner of two wedges intersecting perpendicularly in a shock tube. Experiments were conducted in a \(100\,{\rm mm} \times 180\,{\rm mm}\) diaphragmless shock tube equipped with double-exposure diffuse holographic interferometry in which the time interval between the first and second exposure was set to be \(1 \,{\rm \mu s}\). This arrangement clearly visualized complex configurations of three-dimensional shock wave reflections. A numerical study was also carried out for interpreting these holographic interferometric observations by using the Weighted Average Flux (WAF) method to solve the three-dimensional unsteady compressible Euler equations. It was found that along the line of the intersection of these two wedges, two Mach stems intersected each other resulting in the formation of a Mach stem which leaned forward.

A difference scheme with anti-diffusion terms to improve the computation of contact discontinuities and the study of weak discontinuities in compressible flow

The investigation of Mach reflection formed after the impingement of a weak plane shock wave on a wedge with shock Mach number M s near 1, is still an open problem[12]. It's difficult for shock tube experiments with interferometer to detect contact discontinuities if it is too weak; also difficult to catch with due accuracy the transition condition between Mach reflection and regular reflection. The interest to this phenomenon is continuing, especially for weak shocks, because there was systematic discrepancy between simplified three shock theory of von Neumann [8] and shock tube results [15] which was named by G. Birkhoff as “von Neumann Paradox on three shock theory” [18].In 1972, K.O.Friedrichs called for more computational efforts on this problem. Recently it is known that for weak impinging shocks it's still difficult to get contact discontinuities and curved Mach stem with satisfactory accuracy. Recent numerical computation sometimes even fails to show reflected shock wave[6]. These explain why von Neumann paradox of the three shock theory in case of weak discontinuities is still a problem of interesting [9,12,14]. In this paper, on one hand, we investigate the numerical methods for Euler's equation for compressible inviscid flow, aiming at improving the computation of contact discontinuities, on the other hand, a methodology is suggested to correctly plot flow data from the massive information in storage. On this basis, all the reflected shock wave , contact discontinuities and the curved Mach stem are determined. We get Mach reflection under the condition when over-simplified shock theory predicts no such configuration[5].

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.

Numerical investigation of shock wave reflections in steady flows

To better understand the shock wave reflection phenomenon in steady flows in general, and the transition between regular reflection and Mach reflection in particular, a two-dimensional flowfleld model and simulation over a wedge in supersonic flows was investigated. The numerical simulation enabled one to determine the height of the Mach stem in a Mach reflection wave configuration. The determination of the height was pointed out as one of the yet unsolved problems in steady shock wave reflections. The numerical results were compared both with steady flow experiments performed at nominal freestream flow Mach numbers ranging from 2.8 to 5 and with analytical results. Based on the presented numerical investigation, a new geometric parameter that controls the transition between regular reflection and Mach reflection wave configurations was extracted for the flow Mach number range 2.8-7.